Rf Ch7 Fet Mixer 2008

(e) FET Mixers have conversion gain (not loss) Pozar (RF Ch 7)

R A Pucel D Masse and R Bera Performance of GaAs MESFET Mixers at X Band IEEE TramMicrowave Theory and Techniques vol MTT-24 pp 351-360 June 1976 Single-Ended FET Mixer

There are several FET parameters that offer nonlinearities used for mixing The strongest is the transconductance gm when the FET is operated in a

common source configuration with a negative gate bias (Vgs )

When used as an amplifier the gate bias voltage is near zero or positive so the transconductance is near its maximum value and operates as a linear device

When the gate bias Vgs is near the pinch-off region where the transconductance approaches zero a small variation of gate voltage causing a large change in transconductance

leading to a nonlinear response

2008 copyH-R Chuang EE NCKU

7-2

Thus the LO voltage can be applied to the gate of the FET to pump the transconductance to switch the FET between high amp low transconductance states =gt provide mixing as the switching model

(see the Diode Large-Signal Model for Mixer)

RF chokes are used to bias the gate at a negative voltage near pinch-off and to provide a positive bias for the drain of the FET

A bypass capacitor at the drain provides a return path for the LO signal and a LPF provides the final IF output signal

Based on the standard unilateral equivalent circuit for a FET

⎩⎨⎧

ω=ω=

tVtvtVtv

LOLOLo

RFRFRFcos)(cos)(

Zg = Rg + jXg Thevenin source impedance for the RF input port ZL = RL + jXL Thevenin source impedance at the IF output port

LO port has a real generator impedance of ZO

=gt since we are not concerned with maximum power transfer for the LO signal

The same as for the large-signal analysis of the diode mixer the LO pumped FET transconductance is espressed as a Fourier series of harmonics of LO signal

suminfin

=

ω+=1

0 cos2)(n

LOn nggtg

not having an explicit formula for the transconductance must rely on measurements for values of ng in the switching model the desired down-conversion is due to (n = 1) only need 1g coefficient amp the typical measured value in the range of 10 mS


7-3

Conversion gain of the FET mixer can be found as ()

2

22

224

4

RF

IFD

L

Lg

gRF

LLIF

D

availRF

availIFc V

V

Z

RR

RV

ZRV

PPG ===

minus

minus (see ch3 p3-26 conjugate matching formula)

IFDV IF drain voltage

Zg amp ZL chosen for maximum power transfer at the RF and IF ports

The RF signal across the gate-to-source capacitance is given as

)(1)]1()[( gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

VCjZRCj

VV+ω+

=ω++ω

= ()

tVtv RFRF

cRFc ω= cos)( amp

[ ] [ ]+ωω+ω=

ωω+=

ω+=

sum

suminfin

=

infin

=

)cos()cos(2)cos(

)cos()](cos2[)()(

)cos(2)(

10

10

10

ttVgtVg

tVtnggtvtg

tnggtg

LORFRF

cRFRF

c

RFRF

cn

LOnRFcm

nLOn

From

The down-converted IF signal can be extracted from the second term by using

the usual trigonometric identity

)cos(]|)()[( 1 tVgtvtg IFRF

cRFcm IF ω=ω=ω ()

Then the IF component of the drain voltage (in phasor form) is (by using () )

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+ω+minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus=minus=

Ld

Ld

gigsRF

RF

Ld

LdRFcLd

RFc

IFD

ZRZR

ZRCjVg

ZRZRVgZRVgV

)(1

)(

1

11


7-4

The conversion gain GC (before conjugate matching) is then

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+ω+minus==

Ld

Ld

gigsRFRFIF

DRF

IFD

L

Lgc ZR

ZRZRCj

VgVVV

Z

RRG

)(14

1

2

2 amp from

we have

])[(])()[(2

)(4|

22212

21

2

)(11

2

LLd

L

Cggi

g

gsRF

d

RF

ZRZR

ZRCjV

L

Lg

matchednotc

XRRR

XRR

RCRg

V

g

Z

RRG

gsRF

LdLd

gigsRFRF

++minus++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

⎟⎠⎞⎜

⎝⎛minus

=

ω

++ω+

By conjugately matching the RF amp IF ports

( 01 ==ω== LdLgsRFgig XRRCXRR )

igsRF

d

d

d

i

i

gsRF

dc RC

RgR

RR

RCRgG 2

21

2222

21

42

])0()2[(])0()2[(2

ω=

++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

Practical mixer circuit generally use matching circuits to transform the FET impedance to 50 ohm for the RF LO amp IF ports


7-5

Diode Large-Signal Model for Mixer (Pozar RF P233)

)1()( minus= αVs eIVI

tVtVVVtv oorrLoRF ω+ω=+= coscos)( For a diode small-signal approximation

( ) ( )( ) ( )

( )

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ω+ω+ωminusω+ω+ω++=

ω+ωω+ω=

ω+ω=primerArr

+prime++=+

prime

prime

prime

tVVtVVtVtVVV

tVttVVtV

tVVGv

GvvGItvVI

ororororooroG

ooororrG

oorrG

d

ddoo

rrd

rd

d

cos2cos22cos2cos

coscoscos2cos

coscos2

2)(

22224

22222

22

2

2

signal IF desired

- Usually LO power (typical 5~10 dBm) will violate the small signal approximation of the diode

- Large signal model is needed for fully nonlinear analysis


7-6


7-7


7-8


7-9


7-10


7-11


7-12

EXAMPLE 72 MIXER CONVERSION GAIN A single-ended FET mixer is to be designed A single-ended FET mixer is to be designed for a wireless local area network receiver operating at

24 GHz The parameters of the FET are Rd = 300 R = 100 Cgs =03 pF and g = 10 mS Calculate the maximum possible conversion gain

Solution This is a straightforward application of the formula for conversion gain given in (754) Note that this value does not include losses due to the necessary impedance matching networks FIGURE 712 A dual-gate FET mixer

tgtggtg LOLO ω+ω+= 2cos2cos2)( 210

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22

min minusminusminus+

=minus+

=minus

minus+minusminus+=minus

LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13

Mixer Conversion Noise 1dB 3rd Order Type Gain Figure Compression Intercept Diode -5 dB 5-7dB -6 to ndash1 dBm 5 dBm FET 6 dB 7-8dB 5 to 6 dBm 20 dBm

1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=

+ωω+ω= ttVgtVgtvtg LORFRF

cRFRF

cRFcm coscos2cos)()( 10

tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14

Using the small-signal approximation of (720) gives the total diode current as The first term in (724) is the DC bias current which will be blocked from the IF by the DC blocking

capacitors The second term is a replication of the RF and LO signals which will be filtered out by the low-pass IF filter This leaves the third term can be rewritten using trigonometric identities as

This result is seen to contain several new signal components only one of which produces the desired

IF difference product The two DC terms again will be blocked by the blocking capacitors and the 2ampgtRF 2ampgtLo^ and amp)RF + (ULO terms will be blocked by the low-pass filter This leaves the IF output current as

where (UIF = O)RF a LO is the IF frequency The spectrum of the down- converting single-ended

mixer is thus identical to that of the idealized mixer shown in Figure 71b Large-Signal Model While the small-signal analysis of a mixer demonstrates the key process of frequency conversion

it is not accurate enough to provide a realistic result for conversion loss This is primarily because the power supplied to the mixer LO port is usually large enough to violate the small-signal approximation Here we consider a fully nonlinear analysis of a resistive diode mixer [3]-[4] with the goal of deriving an expression for the conversion loss defined in (710) The term resistive in this context means that reactances associated with the diode junction and package are ignored to simplify the analysis Our results should be useful in understanding the nonlinear operation and losses of the diode mixer but for actual design purposes modem computer-aided design (CAD) software is preferred [5]Such software can model the diode nonlinearity as well as the effects of diode reactances and impedance matching networks

We again assume a diode I-V characteristic as given by (718) with a relatively low-level RF input voltage given by (722) and a much larger LO pump signal given by (723) A DC bias current may also be present but will not directly enter into our analysis As we have seen from the small-signal mixer analysis these two AC input signals generate a multitude of harmonics and other frequency products

amp)RF RF input signal (low power) ampIIF = ampIRF LO IF output signal (low power) (DIM = (ULO tîF image signal (low power) (ULO LO input signal (high power)

( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

gtLO harmonics of LO (high power) racfLO MIF harmonic sidebands of LO (low power)

is leaves the IF output current as In a typical mixer harmonics of the LO and the harmonic sidebands are terminated reactively

and therefore do not lead to much power loss This leaves three signal frequencies of most importance ampIRF agtw and n)iw To evaluate conversion loss we will find the available power of the RF input signal the power of the IF output signal and the power lost in the image signal The image signal is important because it is relatively close in frequency to the RF signal and thus sees essentially the same load We will see that approximately half the input power gets converted to the image frequency Note that the image term at frequency ampgtIM = WLO - RF = ^LO - lt^RF was not explicitly shown in the small-signal expansion of (724) since this product is generated by the u3 term of (720)

Under the assumption that the RF input voltage is small we can write the AC diode current as a Taylor series expansion about the LO voltage as

This Taylor series is similar in form to (720) as used for the small-signal analysis but with the

important difference that the expansion point here is about the LO voltage where as(720) was expanded about the DC bias point The first term in (726) is due only to the LO input and does not enter into the calculation of conversion loss The second term is a function of the RF and LO input voltages and will provide a good approximation for the three products at frequencies ww w^ and ampgtIM with a large LO pump signal The coefficient of the second term has dimensions of conductance so we can use (718) to write the differential conductance as

Then for small input voltages v(t) we can write the resulting diode current as We see from (727) that g(t) is a real number (consistent with our description as a resistive mixer)

and is a periodic function of the LO frequency Thus g(t) can be expressed as a Fourier cosine series in terms of harmonics of wô-

with Fourier coefficients given by where In(x) is the modified Bessel function of order n defined in Appendix B Now let the AC

diode current consist of three components at the frequencies amp)RF (DIF and where pp IF and [M are the amplitudes of the RF IF and image signals to be determined If the

RF voltage of (722) is applied to the diode through a source resistance Rg and the IF and image ports are terminated in load resistances ip and Rg respectively then the voltage

FIGURE 74 Equivalent circuit for the large-signal model of the resistive diode mixeracross the

diode can be written as (732) The equivalent circuit consists of a three-port network with one port for each of the frequency

components at amp)RF agtip and agtm[ as shown in Figure 74 We assume the terminations for the RF and image ports are identical because aiRp is very close to ugtwi while the termination for the IF port may be different

Using the first three terms of the Fourier series of (729) for the diode differential conductance gives


7-16

(733) Multiplying the voltage of (732) by the conductance in (733) and matching like frequency

terms with the current of (731) gives a system of three equations for the unknown port currents (734) where VRF is the source voltage and the gn s are defined in (730) Note that multiplication of

(732) by (733) creates several frequencies in addition to oipp (UIF and (UIM but we assume these frequencies to be reactively terminated so that they do not lead to significant power dissipation

The easiest way to find the available power from the IF port is to first find the Norton equivalent source for the IF port As shown in Figure 75 this consists of a current source equal to sc the short-circuit current at the IF port and Gip the conductance seen looking into the IF port This conductance can be found as GIF = sc Voc where Voc is the open-circuit voltage of the IF port The short-circuit IF port current can be found by setting R^p == 0 in (734) and solving for ip After some straightforward algebra we obtain

FIGURE75 Norton equivalent circuit for the IF port of the large-signal model of the resistive diode mixer

The open-circuit IF port voltage is found by setting ip = 0 and solving (734) for Vy Then the Norton conductance of the IF port is The available output power at the IF port is and the available input power from the RF source is So from (710) the conversion loss is (not in dB) Note that the conversion loss does not depend on the IF port termination ip because of the use of

available powers It does depend on Rg the RF and image port terminations so it is possible to minimize the conversion loss by properly selecting Rg If we let x = I a = go + g2and b = 2g^go then (739) can be rewritten as

(740) Differentiating with respect to x and setting the result to zero gives the optimum value of x as for which the minimum value of conversion loss is (741) We can evaluate this result by approximating values for go gi and g^ For an LO input power of

10 mW VLO is about 0707 V rms and a = 128 mV so aV^n the argument of the modified Bessel functions for gn given in (730) is approximately 25 Thus the modified Bessel functions can be approximated asymptotically using the large-argument formula given in Appendix B and the gns simplified as

and the minimum conversion loss of (742) reduces to L( = 2 or 3 dB This means that half the


7-17

RF input power is converted to IF power and half is converted to power at the image frequency In principle this result could be improved by terminating the image port with a reactive load but it is usually difficult in practice to separate the image termination from the RF termination Also this result is highly idealized in that it assumes no power loss at higher harmonic frequencies and it ignores diode reactances

This same model can be used to derive the SSB noise temperature for the resistive mixer as (745) where n is the diode ideality factor and T is the physical temperature of the diode [3] Switching Model The large-signal model suggests that the diode mixer can be viewed as a switch As the LO

voltage cycles between positive and negative values of cosecant the diode becomes conducting or nonconducting respectively Thus the diode conductance (the ratio of diode current to diode voltage) switches between large values and zero at the same rate as the LO voltage Figure 76 shows a typical diode conductance waveform where T = ITCJWÔ is the period of the LO waveform

The conductance waveform of Figure 76 can be calculated directly from the diode V-I characteristic of (718) or from the Fourier series representation of (729) But since a conductance greater than a few Siemens is essentially a short circuit we can approximate the diode conductance as the square wave shown in Figure 77 This square wave has a Fourier transform given by

(746) which is similar in form to the Fourier series of (729) An equivalent circuit of the diode mixer then consists of the RF input voltage applied across a

load resistor in series with an ideal switch as shown in Figure 78 The time-varying FIGURE 76 Conductance waveform of a mixer diode pumped with a large-signal FIGU運77 LO voltage waveform and idealized square-wave diode conductance waveform for the switching model of a diode mixer switch conductance is given by (746) The diode current can be found by multiplying the RF input

voltage of (722) by the conductance of (746) Filtering all but the lowest-frequency component for the n = 1 term of the summation gives the

desired IF output as The switching model is useful for mixers of any type including the FET mixer discussed in the

following section Note that the switching model of a mixer can be considered as a linear but time-varying circuit

FIGURE 78 Equivalent circuit for the switching model of the diode mixer 73 - FET MIXERS Mixers can also be implemented by using the nonlinear properties of transistors FETs in

particular offer low noise characteristics and easy integration with other circuitry such as switches and low-noise amplifiers Transistor mixers can provide conversion gain but their noise figure is generally not as good as can be obtained with diode mixers PET mixers also offer


7-18

higher dynamic range The following table compares the characteristics of typical diode and FET mixers

Because a FET mixer has conversion gain but usually worse noise figure the proper comparison with a diode mixer should include the cascade effect of adjacent stages

In this section we will analyze the single-ended FET mixer and derive an expression for its conversion gain We will also discuss a few other popular FET mixer configurations

Single-Ended FET Mixer There are several FET parameters that offer nonlinearities that can be used for mixing but the

strongest is the transconductance ^ when the FET is operated in a common source configuration with a negative gate bias Figure 79 shows the variation of transconductance with gate bias for a typical FET When used as an amplifier the gate bias voltage is near zero or positive so the transconductance is near its maximum value and the transistor operates as a linear device When the gate bias is near the pinch-off region where the transconductance approaches zero a small variation of gate voltage can cause a large change in transconductance leading to a nonlinear response Thus the LO voltage can be applied to the gate of the FET to pump the transconductance to switch the FET between high and low transconductance states and provide mixing in much the same manner as the switching model discussed in the previous section

The circuit for a single-ended FET mixer is shown in Figure 710 A diplexing coupler is used to combine the RF and LO signals at the gate of the FET An impedance matching net-work is also usually required between the inputs and the FET which typically presents a verylow input impedance RF chokes are used to bias the gate at a negative voltage near pinch-off and to provide a positive bias for the drain of the FET A bypass capacitor at the drain pro-vides a return path for the LO signal and a low-pass filter provides the final IF output signal

FIGURE 79 Variation of FET transconductance versus gate-to-source voltage

FIGURE 710 Circuit for a single-ended FET mixer

Our analysis of the mixer of Figure 710 follows the original work described in reference [6] The simplified equivalent circuit is shown in Figure 711 and is based on the standard unilateral equivalent circuit for a FET The RF and LO input voltages are given in (722) and (723) Let Zg = Rg + jXg be the Thevenin source impedance for the RF input port and ZL = RL + JX^ be the Thevenin source impedance at the IF output port These impedances are complex to allow us to conjugately match the input and output ports for maximum power transfer The LO port has a real generator impedance of Zo since we are not concerned with maximum power transfer for the LO signal

As we did for the large-signal analysis of the diode mixer we express the LO pumped FET transconductance as a Fourier series in terms of harmonics of the LO signal

Because we do not have an explicit formula for the transconductance we cannot calculate directly

the Fourier coefficients of (748) but must rely on measurements for these values As in the case of the switching model the desired down-conversion result is due to the n = 1 term of the Fourier series so we only need the g coefficient Measurements typically give a value in the range of 10 mS for g

The conversion gain of the FET mixer can be found as where V^f is the IF drain voltage and the impedances Zg and Z^ are chosen for maximum power

transfer at the RF and IF ports The RF frequency component of the phasor voltage FIGURE 711 Equivalent circuit for the FET mixer for Figure 710


7-19

across the gate-to-source capacitance is given in terms of the voltage divider between Z RSindCgs

Multiplying the transconductance of (748) by ^(r) = V^COSCURF gives terms oftr form The down-converted IF frequency component can be extracted from the second ten of (751) using the usual trigonometric identity Then the IF component of the drain voltage is in phasor form where (750) has been used Using this result in (749) gives the conversion gain (before conjugate

matching) as We must now conjugately match the RF and IF ports Thus we let Rg = RiX^ = loiRpCg RL = Rd

and XL = 0 which reduces the above result to The quantities gt R^ R and Cg are all parameters of the FET Practical mixer circuit generally use

matching circuits to transform the FET impedance to 50 ^ for the RF LC and IF ports EXAMPLE 72 MIXER CONVERSION GAIN A single-ended FET mixer is to be designed A single-ended FET mixer is to be designed for a wireless local area network receiver operating at


Solution This is a straightforward application of the formula for conversion gain given in (754) Note that this value does not include losses due to the necessary impedance matching networks FIGURE 712 A dual-gate FET mixer Other FET Mixers There are several variations of mixer circuits that can be implemented using FET Figure 712 shows a single-ended mixer using a dual-gate FET where the RF and LO inputs

are applied to separate gates of the PET This provides a high degree of RF-LO isolation generally an inferior noise figure relative to the transconductance mixer of Figure 710

Another configuration is shown in Figure 713 using two FETs in a differential amplifier configuration The balun (balanced-to-unbalanced) networks on the LO and IP ports provide a transition between a two-wire line that is balanced with respect to ground and a single line that is unbalanced relative to ground Baluns may be implemented with center-tapped transformers or with 180 hybrid junctions

The differential mixer operates as an altenating switch with the LO turning the top two FETs on and off on alternate cycles of the LO These PETs are biased slightly above pinch-off so each PET will be conducting for slightly more than half of each LO cycle Thus one of the upper FETs is always conducting and the lower FET will remain in saturation The RF and LO ports should each be impedance matched The IF output circuit must provide a return path to ground for the LO signal

An extension of the differential FET mixer is the Gilbert cel1 mixer show in Figure 714 This mixer us differential FET mixer stages to form a double balanced mixer This circuit


7-20

achieves high RF-LO isolation and a high dynamic range It also cancels all even-order intermodulation products This circuit is very popular for wireless integrated circuits

FIGURE 713 A differential FET mixer

FIGURE 714 A Gilbert cell mixer 74 OTHER MIXER CIRCUITS The single-ended diode and FET mixers discussed above provide frequency conversion but often

have poor RF input matching and RF-LO isolation This reduces the performance of wireless systems but fortunately it is possible to improve these characteristics by combining two or more single-ended mixers with hybrid junctions

Balanced Mixers RF input matching and RF-LO isolation can be improved through the use of a balanced mixer

which consists of two single-ended mixers combined with a hybrid junction Figure 715 shows the basic configuration with either a 90hybrid (Figure 715a) or a 180hybrid (Figure 715b) As we will see a balanced mixer using a 90hybrid junction will ideally lead to a perfect input match at the RF port over a wide frequency range while the use of a 180hybrid will ideally lead to perfect RF-LO isolation over wide frequency range In addition both mixers will reject all even-order intermodulation products

Microwave quadrature or ring hybrids [1] can be used to implement balanced mixers but at lower frequencies a center-tapped transformer can be used As shown in Figure 716 the secondary of the transformer provides outputs with a 180phase shift to the two mixer diodes The LO signal is applied to the center tap of the secondary

The double-balanced mixer of Figure 717 uses two hybrid junctions or transformers and provides good isolation between all three ports as well as rejection of all even harmonics of the RP and LO signals This leads to very good conversion loss but less than ideal input matching at the RF port The double-balanced mixer also provides a higher third-order intercept point than either a single-ended mixer or a balanced mixer

FIGURE 715 Balanced mixer circuits (a) Using a 90hybrid (b) Using a 180 hybrid

FIGURE 716 Balanced mixer using a hybrid transformer FIGURE 717 Double-balanced mixer circuit The following table summarizes the characteristics of several types of mixers Small-Signal Analysis of the Balanced Mixer We can analyze the performance of a balanced mixer using the small-signal approachthat was

used in Section 72 Here we will concentrate on the balanced mixer with a 90 hybrid shown in Figure 715a and leave the 180hybrid case as a problem

As usual let the RF and LO voltages be defined as The scattering matrix for the 90hybrid junction is [1] where the ports are numbered as shown in Figure 715a Then the total RF and LO voltages applied

to the two diodes can be written as Using only the quadratic term from the small-signal diode approximation of (720) gives the diode

currents as


7-21

where the negative sign on (2 accounts for the reversed diode polarity and K is a constantfor the quadratic term of the diode response Adding these two currents at the input to the low-pass filter gives

where the usual trigonometric identities have been used and ampIIF = WRF (ULO is the IF

frequency Note that the DC components of the diode currents cancel upon combining After low-pass filtering the IF output is

as desired We can also calculate the input match at the RF port and the coupling between the RF and LO

ports If we assume the diodes are matched and each exhibits a voltage reflection coefficient F at the RF frequency then the phasor expression for the reflected RF voltages at the diodes will be

and (761a)

These reflected voltages appear at ports 2 and 3 of the hybrid respectively and combine to form the following outputs at the RF and LO ports

Thus we see that the phase characteristics of the 90hybrid lead to perfect cancellation of reflections

at the RF port The isolation between the RF and LO ports however is dependent on the matching of the diodes which may be difficult to maintain over a reasonable frequency range

Image Reject Mixer We have already discussed the fact that two distinct RF input signals at frequencies ampIRF ==

ampgtLO ÎF will down-convert to the same IF frequency when mixed with iMLo These two frequencies are the upper and lower sidebands of a double-sideband signal The desired response can be arbitrarily selected as either the LSB (ampJLO ^w) or the USB ((ULO + 展F) assuming a positive IF frequency The image reject mixer shown in Figure 718 can be used to isolate these two responses into separate output signals The same circuit can also be used for up-conversion in which case it is usually called a single-sideband modulator In this case the IF input signal is delivered to either the LSB or the USB port of the IF hybrid and the associated single sideband signal is produced at the RF port of the mixer

We can analyze the image reject mixer using the small-signal approximation Let the RF input signal be expressed as

FIGURE 718 Circuit for an image reject mixer

where Vu and VL represent the amplitudes of the upper and lower sidebands respectively Using the S-matrix given in (757) for the 90hybrid gives the RF voltages at the diodes as

After mixing with the LO signal given in (756) and low-pass filtering the IF inputs to the IF hybrid

are where K is the mixer constant for the squared term of the diode response The phasor representation of the IF signals of (765) is Combining these voltages in the IF hybrid gives the following outputs which we see are the separate sidebands of the downconverted input signal of (763) These outputs can be expressed in time-domain form as


7-22

which clearly shows the presence of a 90phase shift between the two sidebands Aso note that the

image rejection mixer does not incur any additional losses beyond the usual conversion losses of the single rOection mixer

A practical difficulty with image rejection mixers is in fabricating a good hybrid at the relatively low IF frequency Losses and hence noise figure are also usu3ly greater than for a simpler mixer

REFERENCES [1] D M Pozar Microwave Engineering 2nd edition Wiley New York 1998 [2] S Y Yngvesson Microwave Semiconductor Devices Kluwer Academic Publishers 1991 [3] K Chang Handbook of Microwave and Optical Components vol 2 Chapter 2 Mixers and

Detectors by E L Kollberg Wiley InterScience New York 1990 [4] C T Torrey and C A Whitmer Crystal Rectifiers MIT Radiation Laboratory Series vol 14

McGraw-Hill New York 1948 [5] S A Maas Microwave Mixers 2nd edition Artech House Dedham MA 1993 [6] R A Pucel D Masse and R Bera Performance of GaAs MESFET Mixers at X Band IEEE

TramMicrowave Theory and Techniques vol MTT-24 pp 351-360 June 1976


7-23


7-24

33 57GHz CMOS雙端平衡式混波器設計與製作

為了減少共模雜訊以及clock feed-through透過基板寄生耦合電容對電路的干擾且增

加LO-IFLO-RF隔離度抑制RF和LO訊號的偶次項諧波故選擇double-balanced mixer為主要架構(圖310)Double-balanced mixer 架構較single-balanced mixer架構複雜在於混波

器的輸入端需接上一個單端轉雙端的balun增加電路面積與功率消耗由於使用在外差

式系統中LO頻率(5465~5525GHz)遠高於IF(280MHz)的情形下LO洩漏至IF輸出端之

訊號可以輕易被中頻濾波器所衰減更提高了LO-IF隔離度

VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1

圖310 double-balanced mixer電路架構圖

在設計double-balanced mixer時首先決定轉導電晶體大小及偏壓M1M2為轉導電晶體

負責將RF電壓訊號轉為電流訊號因此其轉導大小為首要考量轉導要大偏壓電流勢必要

增大需要較大的功率消耗然而較大的直流偏壓卻可獲得較佳的雜訊表現所以首先考量

轉導電晶體M1M2在偏壓電流各為1mA的條件下參考[8]找出最佳雜訊表現之電晶體寬

度決定轉導電晶體M1M2大小為各為20μm018μm 本地振盪信號VLO+VLO-為大訊號輸入M3 M4M5M6閘極端形成開關電晶體負責

切換RF電流訊號達到混波目的開關電晶體大小決定了開關特性通常電晶體寬度越大開

關特性越佳但是其source端較大的雜散電容容易使RF電流訊號衰減因此需審慎的選擇

M3~M6的大小經過模擬微調後選擇各為40μm018μm已知訊號產生器輸出阻抗為50歐

姆為了達到最大的功率傳輸利用on chip電阻與電容再加上bondwire等效電感將開關電

晶體閘極輸出阻抗匹配至50歐姆由於本地振盪信號為雙端差模輸入故利用一rat race ring將原本單端本地振盪訊號分為雙端差動訊號

混波器負載部分利用RL電阻接成diode connected PMOS型式負載大小由電阻RL與

PMOS汲極與遠源級間的阻抗所構成M7M8電晶體寬度大小取決於負載壓降與負載阻抗

在相同偏壓電流情況下為了達到最小壓降可增大M7M8電晶體寬度此時電晶體較容易

操作在低飽和區易造成線性度因非線性負載而下降增加M7M8電晶體寬度的結果也增加

了電晶體本身之寄身效應導致整體負載阻抗因寄生效應而變小造成轉換增益下降取決

於線性度與轉換增益之考量下選擇M7M8電晶體寬度為160μm018μm考量量測時所

使用的量測儀器皆是50歐姆輸入阻抗之負載為了避免嚴重之負載效應在混波器核心負載


7-25

輸出端與50歐姆負載端間置入一緩衝放大器此緩衝放大器由一共源極電晶體所組成因此

對混波訊號也有放大之功能 Double-balanced mixer 之電路與電路佈局圖如圖 311 所示輸出為雙端輸出因

為 mixer 後級接單端中頻濾波器因此加上 21 的 balun 作為雙端轉單端電路[10]IF端 balun 使用 TOKO 616PT-1039insertion loss 為 3dBBalun 單端輸出為 480MHz 中頻訊號利用 off-chip 的晶片電感 56nH並聯電容 7pF 匹配至 50 歐姆

M1ampM2 gate width = 20um018um

pad equivalent circuit

bondwire equivalent circuit

2nH

0065pF 625O

065O

M3ampM4ampM5ampM6 gate width = 40um018um

M7ampM8 gate width = 160um018umM9ampM10 gate width = 15um018um

IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF

圖311 57GHz CMOS double-balanced mixer電路及晶片佈局圖


7-26

34 模擬與量測結果 Double-balanced mixer設計電路的RF輸入範圍為5725~5825GHz中頻輸出為480MHz頻寬

20MHzmixer核心部分直流偏壓為18V18mA緩衝放大器部分為各18V14mA在IF balun 插入損耗為3dB代入轉換損代入模擬結果轉換增益約1276dB input P1dB約-15dBmIIP3約-69dBmLO-RF的隔離度皆在42dB左右LO-IF的隔離度約100dB

量測上利用FR-4製作測試基板來量測晶片mixer核心部分直流偏壓量測為18V2mA緩

衝放大器部分為各18V1mARF輸入訊號端與本地振盪埠(LO)的輸入訊號由訊號產生器

經由rat race ring產生差動訊號饋入混波器量測特性結果為轉換增益約1106dBinput P1dB約-164dBmIIP3約-75dBm LO-RF的隔離度皆在19dB左右LO-IF的隔離度約

50dB模擬與量測結果比較如圖312表31所示晶片照片圖測試板佈局圖及測試板

照片如圖313所示 35 結果與討論

量測結果除了LO-RF隔離度外在輸入輸出匹配轉換增益P1dBOIP3與模擬大

致吻合而在LO-IF隔離度方面設計之時因為balun沒有model可以模擬因此利用一個

理想balun加上3dB的insertion loss代入模擬軟體由於量測使用之IF balun的高頻響應對LO有額外的衰減因此得到比模擬更佳的LO-IF隔離度LO-RF隔離度方面模擬量測差距較

大理想上single-balanced的架構LO-RF隔離度應該是相當大原因可能是因為製程偏移

使得M2及M3電晶體些微不對稱或是substrate coupling造成LO洩漏至RF端改進之處為

佈局方面可以再將M2及M3電晶體對稱性提高及將substrate coupling問題列入佈局考量

55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)

57GHz double-balanced mixermeasurementsimulation

300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)


(a) RF input return loss (b) IF output return loss

5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)

57GHz double-balanced mixerLO-RF isolation

5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)

57GHz double-balanced mixerLO-IF isolation

(c) LO-RF isolation (d) LO-IF isolation


7-27

-50 -45 -40 -35 -30 -25 -20 -15 -10RF Input Power (dBm)

6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)

57GHz double balanced mixermeasurementsimulation

470 475 480 485 490

IF Channel Frequency (MHz)

10

11

12

13

14

15

Noi

se F

igur

e(dB

)


(e)轉換增益及input P1dB (f)雜訊指數

479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7

(g) Two tone test OIP3模擬 (h) Two tone test OIP3量測

圖312 57GHz CMOS double-balanced mixer模擬量測結果

(a) (b) (c)

圖313 57GHz CMOS double-balanced mixer (a)晶片照片圖 (b)測試板照片圖 (c)測試板佈局圖


7-28

表31 57GHz CMOS double-balanced mixer模擬量測特性表

57GHz CMOS Double-Balanced Mixer (TSMC 018μm) RF Frequency Range 5725~5825GHz

IF Frequency 480MHz LO Frequency 5265~5325GHz

Vdd 18V Simulation Measurement

LO Power -3dBm -3dBm Core Each Buffer Current 1814mA 21mA

Conversion Gain 1276dB 1106dB RF Input Return Loss gt21dB gt18dB Output Return Loss 15dB280MHz 20dB280MHz

LO-RF Isolation(LO=-3dBm) 42dB 19dB LO-IF Isolation(LO=-3dBm) gt100dB gt50dB RF-IF Isolation(RF=-30dBm) gt200dB -

Noise Figure 124dB 128dB IIP3 -69dBmRF=-28dBm -75dBmRF=-28dB

Input P1dB -15dBm -164dBm Die size 0627 x 0649 mm2


7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H

DesiredChannel Image

imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

Band SelectFilter Response

BPF1

1LOω

Channel SelectFilter Response

BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A

Duplexer Image-RejectFilter

IFFilter

Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6

dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO

Figure 635 Calculation of noise figure in a cascade of stages

Duplexer LNA Image-RejecFilter

Mixer IF Filter

IF Amplifier

Stage Gain (dB) Voltage Power

-2 -2

15 15

-6 -6

15 5

-5

Cumulative Voltage Gain (dB)

Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)

Stage IP3 +100 dBm 12 dBm +100 dBm +5 dBm 1000 Vrms 700mVrms

Cumulative IP3 -106 dBm -126 dBm +11 dBm +5 dBm 221Vrms 700mVrms

Figure 636 Level diagranm corresponding to the cascade of Fig 635

B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB

Figure 632 Cascade of a mixer and an IF amplifier

YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band

Figure 617 Folding of RF and image noise into the IF band


7-32

Filter 1 Filter 2 Detector

Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st

(Tx ) ( Rx ) For a transceiver if transnitting and receiving use different frequencies (for example

celluLOr phone system) a duplexer is used to Separate T Rx xamp duplexer is a kind of BPF Hence a duplexer will also act as a BPF ( before RF stage ) for image rejection

( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω

V V VRF Lo= +α α1 2 darr darr depending on combiner Where to add DC bias lumped circuit form impedance matching is implemmted in the inductor-coie winding


7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port

f - fLO IFf + fLO IF

3f - fLO IF

f LO

2fLO3fLO

2f - fLO IF2f + fLO IF 3f + fLO IF

(e) Dual-Gate MESFET Mixer MESFET V V I mA G dBV V I V j k

DS DS

G S G S

= = == minus = minus minus

3 65 2 5 80 78 0 78 980 11 2

output impedance = Ω Ω

RF amp LO signal are combined in dual-gate FET structure MODEL RR-12 LO 1 to 2 GHz RF 1 to 2 GHz IF 0 to 064 GHz Characteristic Min Typ Max Test ConditionConcersion


7-34

Loss dB 5 6 Phase Deviation From 90o

Deg

plusmn4

Amplitude Unbalance dB

plusmn05

All measurements made

lsolation dB LO-RF LO-IF RF-IF

18 20 20

25 25 25

in 50 Ohm system RF=1-2GHz LO=1-2GHz

VSWR RF LO IF

15 15 15

20 20 20

IF=003GHz PLO = +7 dBm PRF =minus10 dBm

1dB Compression dBm

+5

3-rd order HP dBm

+15

ltConversion Loss of Mixergt In mixer design several frequencies (RFLOIF) and Their harmonics are involved Impedance matching design at three ports (RFLOIF) is complicated by the above

situation Undesired harmonic signal can be dissipated in resistive termination rarr increase mixer loss or blocked with reactive terminetion rarr frequency dependent An important figure of merit of the mixer Conversion loss Lc = 10 log avaialable RF input power

IF output power

typical Lc= 5 ~ 8 dB for passire mixersa active mixer can have a gain High local oscillator signal (pump) power can rgeduce the mixer conversion loss


7-35

minimum Lcusually sccurs for 0 ~ 10 dBm LO powers But (0 ~ 10 dBm) LO power will violate the small signal approximation of the mixer

diode precise design requires numerical ssolution of the nonlinear equation theat describes

the diode charactoristics

( )( ) ( ) ( )

let

ω ω ω ω ω ω ω

ω ω ω ω

r o i r o i i

RF r r r o r o r o i

IF

V V t V V t V V t

= + minus =

= rarr minus prime = prime

rarr

cos sin sin2 2

( )

( ) ( ) ( )let

prime = minus minus = minus

= prime prime rarr prime prime minus = minus prime

rarrω ω ω ω ω ω

ω ω ω ω

r o i r o i

RF r r r o r o r o iV V t V V t V V tcos sin sin2 2

f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭

all produce a same IF frequency and cause image interference

Fig 12 Measured characteristics of a 24 GHz bandpass filter (FDK 2450B)

( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus

= + minus + minus minus

⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90

= minus + minus minus

= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36

Mixer Performance Characteristics

bull Mixer Conversion Loss [ ]Lc =10 log available RF input powerIF output power dB

=gt stringly depend on LO power level

bull Mixer Noise Figure

混頻器除了射頻(RF)頻帶之雜訊混至中頻(IF)帶外同時有image頻帶及贅餘(spurious)頻帶之雜訊混入中頻

fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band

bull 單旁波帶(SSB single side band)雜訊指數與雙埠放大器之定義相同 bull 測量上多採用雙旁波帶(DSBdouble side band) 雜訊指數較方便iexcl

=gt 雙旁波帶雜訊指數為單旁波帶雜訊指數的兩倍

Mixer

LOSignal Generator

Broad bandNoise Source

Pre Amp

Noise Figure Meter

Mixer DSB noise figure measurement

混頻器為雙旁波帶雜訊指數測量

RF IF

LO

f

f

ffRF

fLO

f fRF LOminus f fRF LO+


7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38

APPENDIX 3 SOME APPLICATIONS OF MICROSTRIP CIRCUITS Fooks Microwave Engineering using Microstrip Circuits (Ch 12) This part introduces selected practical microstrip circuits or subsystems It brings together

some of the combination of microwave passive components amp active devices to produce functioning self-contained building blocks that in turn may be a part of a complete microwave systems

FIGURE 1116 Frequency conversion in a receiver and transmitter (a) Down-conversion in a heterodyne (b) Up-conversion in a transmitter f f fIF RF Lo= minus and a much higher frequency signal f fRF Lo+ (filtered out)

2-stage mixers rarr for vetter image frequency rejetion

Portable communication Receiver BLOck Diagram ltMixer in Rransmittergt ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω

conside only quadratic term of the diode which givers thedesired mixer product

混頻器的雜訊指數根據IEEE的定義單旁波帶(SSBsingle side band)雜訊指數與雙埠放大器之定義相同如(24)式但在實際測量上較為困難若以濾波器阻隔假像頻率(image

grequency)頻帶之雜訊則濾波器將造成輸入阻抗改變且增加溫度雜訊因此在測量上多採用雙旁波帶(DSBdouble side band)雜訊指數較方便一般的定義雙旁波帶雜訊指數為單旁波帶雜訊指數的兩倍圖25說明混頻器雜訊之造成混頻器除了射頻(RF)頻帶之雜訊混至中頻(IF)帶外同時也將假像(image)頻帶及贅餘(spurious)頻帶之雜訊混入中頻因此混頻器的雜訊指數將比放大器來得大圖26為測量混頻器雜訊指數之架構所得為雙旁波帶雜訊指數


7-39

fIF2

=gt channel bandwidth

IF smallrarr amp requiring higher Q of the IF filter


7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector

(c) Mixer (frequency conversion)

RF IF

LO

f

f

ffRF

fLO


For a single-ended mixer the noise terms will be ( ) ( )[ ] ( )tVVtVtVV nonno 212

21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42

Mixer Nonlinear Simulation by Libra

( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier

LOf Up-conversion(for transmitting)

DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter

Down-conversion (receiving)

bull The band-stop response of the BPF will determine the image-rejection ratio


7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω

LNAChannelSelectFilter

LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90

( )[ ] ttVVV onoLo ω+= cos

tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=

+ΓminusminusΓminusΓ=+minus=

+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ

port LO at appears signal but reflection no

port RF at appears signal but reflection no

rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=

+ΓminusminusΓminusΓ=


ΓminusΓ=minusΓ+Γ=+=

ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22

monolithic quad DMOS FET for mixer application


7-45

FET mixer has a gain (Diode mixer has no gain) TABLE 111 Frequencies and Relative Amplitudes of the Square-Law Output of a

Detected AM signal Frequency Relative Amplitude 0 1 22+ m

ωω

mm2 2

22m

m rarrdesired demodulated output

2ωo 1 22+ m 2ω ωo mplusmn m

( )2 ω ωo mplusmn m2 4

EXAMPLE 14 A diode in as axial-lead package has the following equivalent circuit parameters Cp =010 pF Lp=20nH Cj=015pF Rs=10Ω and I s=01μA Calculate and plot the

impedance of this diode from 4 to 14 GHz for a bias current Io=0 and Io=60μA Ignore the change in Cj with bias and assume α =125 mV

solution From (1127) the junction resistance for the two bias states is for ( )I Ro j I I

mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω

for ( ) ( )I A Ro j I ImV

Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω

Then the input impedance can be calculated from the equivalent circuit of Figure 1112 the result is plotted versus frequency on a 50Ω Smith chart in Figure 1113

Diode impedance is frequency dependent FIGURE 1113 Impedance of the diode of Example 114 for to bias states from 4 to

14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o

prime = primecos cosω ω


7-46

bull FM broadcasting IF = 107MHz bull Cellular phone IF = 45MHz IF = 455MHz

RF

trr ωcosMatchingnetwork

Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias

tv ori )cos( ωminusω

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF

2LOω1LOωA B C D E F G

H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter IF

Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel

Fig 6 Measured characteristics of a 24-GHz single-ended resistive FET mixer


7-47

Mixer Noise Figure 混頻器除了射頻(RF)頻帶之雜訊混至中頻(IF)帶外同時有image頻帶及

贅餘(spurious)頻帶之雜訊混入中頻iexcl

fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

bull 單旁波帶(SSB single side band)雜訊指數與雙埠放大器之定義相同

bull 測量上多採用雙旁波帶(DSBdouble side band) 雜訊指數較方便iexcl

=gt 單旁波帶雜訊指數為單旁波帶雜訊指數的兩倍

Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2

LP filtertVV rrRF ω= cos

tVV ooLo ω= cos

RFVΓ

LOVΓ

)2()2(1 Lor VjVv minus+=

1vΓ

2vΓ

tVKVi iorIF ωminus= sin2

)2()2(2 Lor VVjv +minus=

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓtVKVi iorIF ωminus= sin2



7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Single-ended mixer circuit

LO

IF

tVtVVVv oorrLoRF ω+ω=+= coscos For a diode small-signal approximation

( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53

(a) Single-ended Mixer

Pozar (RF Ch 7)

Mixer in a Transmitter

bull In a transmitter a mixer is used to mix with IF signal to up-convert the signal frequency for efficient radio-wave transmission from antenna

MixerBaseband filterfM

Antenna

Local oscillator

+ fMfLo

Poweramplifier

LOfUp-conversion(for transmitting)

t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff

ff Double sideband (DSB)= USB + LSB

=gt For a single sideband transmission (SSB)

( )USB f fLo IF+ or ( )LSB f fLo IFminus


7-54

We use a sideband filter or an image rejection mixer to remove a sideband signal ( also celled single sideband modulator )

direct-conversion transmitter

tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA

drawback leakage of PA output to LO

LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)

measurementsimultaioon

400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)

measuemsntsimulation

(c) (d)


7-55

(中正大學電機MS Thesis)


3f - fLO IF

f LO

2fLO3fLO


Spurious chart due to LO harmonics


7-56


7-57

image rejection filter The BPF band-stop response

determines the image-rejection ratio

RF IF

LOf f

fRF f fRF LOminus f fRF LO+


fLOf

DetectorIFamplifierLNA

fIF

Mixer

Local oscillator

IF filter

MLORF fff plusmn= Mf

)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t

Down-conversion (for receiving)

ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω

Image RejectFilter Response

DesiredBand

image

ωrω

DesiredBand

image

A 24 GHz bandpass filter =gtpassband = 100 MHz =gt insertion loss lt 1 dB


7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector Then the input to the two mixers through a 90o hybrid is

RF inputs rarr ⎪⎩

⎪⎨

⎧

minusωminusω+minusω+ω=

ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

ωminusωminusω+ωminus=


ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22

bull After mixing with an LO signal of cos ωot the IF outputs of the mixers are

IF outputs rarr ⎪⎩

⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV

bull Combining these two signals in the 90o hybrid at the IF output gives

component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=

ωminusminusω+ωminusω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus+ωminusminusminusω

+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59

bull Then the input to the two mixers through a 90o hybrid is


⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

bull After mixing with an LO signal of cos ωot amp lowpass filtered the IF outputs of the mixers are


⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV

bull Combining these two signals in the 90o IF hybrid (transformer) at the IF output gives

component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus+ω+minusminusω

+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1

bull After mixing with an LO signal of cos ωot amp lowpass filtered the IF outputs of the mixers are (K = the mixer constant for the squared term of the diode)

IF inputs (to IF hybrid)rarr ⎪⎩

⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22

Phasor representation ( )

( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90

IF hybrid(transformer)

o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cosRFVΓ

LOVΓ


1vΓ

2vΓ tVKVi iorIF ωminus= sin2



7-2

Thus the LO voltage can be applied to the gate of the FET to pump the transconductance to switch the FET between high amp low transconductance states =gt provide mixing as the switching model

(see the Diode Large-Signal Model for Mixer)

RF chokes are used to bias the gate at a negative voltage near pinch-off and to provide a positive bias for the drain of the FET

A bypass capacitor at the drain provides a return path for the LO signal and a LPF provides the final IF output signal

Based on the standard unilateral equivalent circuit for a FET

⎩⎨⎧

ω=ω=

tVtvtVtv

LOLOLo

RFRFRFcos)(cos)(

Zg = Rg + jXg Thevenin source impedance for the RF input port ZL = RL + jXL Thevenin source impedance at the IF output port

LO port has a real generator impedance of ZO

=gt since we are not concerned with maximum power transfer for the LO signal

The same as for the large-signal analysis of the diode mixer the LO pumped FET transconductance is espressed as a Fourier series of harmonics of LO signal

suminfin

=

ω+=1

0 cos2)(n

LOn nggtg

not having an explicit formula for the transconductance must rely on measurements for values of ng in the switching model the desired down-conversion is due to (n = 1) only need 1g coefficient amp the typical measured value in the range of 10 mS


7-3


2

22

224

4

RF

IFD

L

Lg

gRF

LLIF

D

availRF

availIFc V

V

Z

RR

RV

ZRV

PPG ===

minus





)(1)]1()[( gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

VCjZRCj

VV+ω+

=ω++ω

= ()

tVtv RFRF

cRFc ω= cos)( amp

[ ] [ ]+ωω+ω=

ωω+=

ω+=

sum

suminfin

=

infin

=

)cos()cos(2)cos(

)cos()](cos2[)()(

)cos(2)(

10

10

10

ttVgtVg

tVtnggtvtg

tnggtg

LORFRF

cRFRF

c

RFRF

cn

LOnRFcm

nLOn

From






( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+ω+minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus=minus=

Ld

Ld

gigsRF

RF

Ld

LdRFcLd

RFc

IFD

ZRZR

ZRCjVg

ZRZRVgZRVgV

)(1

)(

1

11


7-4


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+ω+minus==

Ld

Ld

gigsRFRFIF

DRF

IFD

L

Lgc ZR

ZRZRCj

VgVVV

Z

RRG

)(14

1

2

2 amp from

we have

])[(])()[(2

)(4|

22212

21

2

)(11

2

LLd

L

Cggi

g

gsRF

d

RF

ZRZR

ZRCjV

L

Lg

matchednotc

XRRR

XRR

RCRg

V

g

Z

RRG

gsRF

LdLd

gigsRFRF

++minus++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

⎟⎠⎞⎜

⎝⎛minus

=

ω

++ω+



igsRF

d

d

d

i

i

gsRF

dc RC

RgR

RR

RCRgG 2

21

2222

21

42

])0()2[(])0()2[(2

ω=

++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=



7-5




( ) ( )( ) ( )

( )

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


ω+ωω+ω=

ω+ω=primerArr

+prime++=+

prime

prime

prime

tVVtVVtVtVVV

tVttVVtV

tVVGv

GvvGItvVI

ororororooroG

ooororrG

oorrG

d

ddoo

rrd

rd

d

cos2cos22cos2cos

coscoscos2cos

coscos2

2)(

22224

22222

22

2

2

signal IF desired




7-6


7-7


7-8


7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-3


2

22

224

4

RF

IFD

L

Lg

gRF

LLIF

D

availRF

availIFc V

V

Z

RR

RV

ZRV

PPG ===

minus





)(1)]1()[( gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

VCjZRCj

VV+ω+

=ω++ω

= ()

tVtv RFRF

cRFc ω= cos)( amp

[ ] [ ]+ωω+ω=

ωω+=

ω+=

sum

suminfin

=

infin

=

)cos()cos(2)cos(

)cos()](cos2[)()(

)cos(2)(

10

10

10

ttVgtVg

tVtnggtvtg

tnggtg

LORFRF

cRFRF

c

RFRF

cn

LOnRFcm

nLOn

From






( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+ω+minus=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus=minus=

Ld

Ld

gigsRF

RF

Ld

LdRFcLd

RFc

IFD

ZRZR

ZRCjVg

ZRZRVgZRVgV

)(1

)(

1

11


7-4


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+ω+minus==

Ld

Ld

gigsRFRFIF

DRF

IFD

L

Lgc ZR

ZRZRCj

VgVVV

Z

RRG

)(14

1

2

2 amp from

we have

])[(])()[(2

)(4|

22212

21

2

)(11

2

LLd

L

Cggi

g

gsRF

d

RF

ZRZR

ZRCjV

L

Lg

matchednotc

XRRR

XRR

RCRg

V

g

Z

RRG

gsRF

LdLd

gigsRFRF

++minus++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

⎟⎠⎞⎜

⎝⎛minus

=

ω

++ω+



igsRF

d

d

d

i

i

gsRF

dc RC

RgR

RR

RCRgG 2

21

2222

21

42

])0()2[(])0()2[(2

ω=

++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=



7-5




( ) ( )( ) ( )

( )

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


ω+ωω+ω=

ω+ω=primerArr

+prime++=+

prime

prime

prime

tVVtVVtVtVVV

tVttVVtV

tVVGv

GvvGItvVI

ororororooroG

ooororrG

oorrG

d

ddoo

rrd

rd

d

cos2cos22cos2cos

coscoscos2cos

coscos2

2)(

22224

22222

22

2

2

signal IF desired




7-6


7-7


7-8


7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-4


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+ω+minus==

Ld

Ld

gigsRFRFIF

DRF

IFD

L

Lgc ZR

ZRZRCj

VgVVV

Z

RRG

)(14

1

2

2 amp from

we have

])[(])()[(2

)(4|

22212

21

2

)(11

2

LLd

L

Cggi

g

gsRF

d

RF

ZRZR

ZRCjV

L

Lg

matchednotc

XRRR

XRR

RCRg

V

g

Z

RRG

gsRF

LdLd

gigsRFRF

++minus++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

⎟⎠⎞⎜

⎝⎛minus

=

ω

++ω+



igsRF

d

d

d

i

i

gsRF

dc RC

RgR

RR

RCRgG 2

21

2222

21

42

])0()2[(])0()2[(2

ω=

++⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=



7-5




( ) ( )( ) ( )

( )

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


ω+ωω+ω=

ω+ω=primerArr

+prime++=+

prime

prime

prime

tVVtVVtVtVVV

tVttVVtV

tVVGv

GvvGItvVI

ororororooroG

ooororrG

oorrG

d

ddoo

rrd

rd

d

cos2cos22cos2cos

coscoscos2cos

coscos2

2)(

22224

22222

22

2

2

signal IF desired




7-6


7-7


7-8


7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-5




( ) ( )( ) ( )

( )

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


ω+ωω+ω=

ω+ω=primerArr

+prime++=+

prime

prime

prime

tVVtVVtVtVVV

tVttVVtV

tVVGv

GvvGItvVI

ororororooroG

ooororrG

oorrG

d

ddoo

rrd

rd

d

cos2cos22cos2cos

coscoscos2cos

coscos2

2)(

22224

22222

22

2

2

signal IF desired




7-6


7-7


7-8


7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-6


7-7


7-8


7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-7


7-8


7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-8


7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-9


7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-10


7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-11


7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-12





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

gIM

IFIF

gRFRF

IM

IF

RF

RIRI

RIV

ggggggggg

III

012

101

210

gg

RFRIFSC RgRg

VgIIIF 20

10 1 ++=minus= =

)1(2 002021

10

ggg

RFIIFOC

RggRggRgVgVV

IF +minusminus== =

gg

g

OC

SCIF RgRg

Rgg

VIG

20

21

0 12

++minus==

IF

SCavailIF G

IP

4

2=minus

g

RFavailRF R

VP

4

2=minus

[ ]g

ggggg

availIF

availRFc

Rg

RgRgRggRgRgPPL 2

1

2120020 2)1()1( minus++++

==minus

minus

bxbaxaxLc

))((2 minus++=

)( baaxopt minus=

[ ][ ] [ ]abab

bbaaa

baabbaababaaa

Lc 11112

)(2)(

)()(22


=minus+

=minus


LO

VsLOnsn V

eIVIIgLO

πααcongαα=

α

2)(


7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-13


1(

2

)200

21 cong+

=ggg

gab

)1(2

minus= ce LnTT

tnnn

tg LOn

ωπ

π+= sum

infin

=cos

2sin2

21)(

1

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡ωminusω+ω+ω

ππ

+ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ωω

ππ

+ω==

sum

suminfin

=

infin

=

tntnn

tV

ttnnn

ttvtgti

LORFRFRFn

RFRF

RFLOn

RFRF

)cos()cos(2

sin2cos21

coscos2

sin2cos21)()()(

1

1

suminfin

=ω+=

10 cos2)(

nLOn nggtg

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIF

D

availRF

availIFc V

V

Z

RR

RV

Z

RV

PPG ===

minus

minus

( ) )(1 gigsRF

RF

gsRFgigsRF

RFRFc ZRCj

V

CjZRCj

VV+ω+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ωminus+ω

=


cRFRF


tVgtvtg IFRF

cRFcm RF

ω=ω cos|)()( 1

⎟⎟⎠

⎞⎜⎜⎝

⎛++ω+

minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

minus=Ld

Ld

gigsRF

RF

Ld

LdRFc

IFD ZR

ZRZRCj

VgZR

ZRVgV)(1

11


7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-14










( )( )[ ]222

2

21

1

2|LLd

L

gsRFggi

g

gsRF

d

matchednotc

XRRR

CXRR

RCRgG

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

ωminus++

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω=

RiCRgG

gsRF

dc 22

21

4ω=


7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-15

















7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-16












7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-17













7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-18
















7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-19













7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-20














currents as


7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-21






and (761a)












7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-22









7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-23


7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-24







VLO-

RLRL

VDD

VLO-VLO+

M3 M4 M5M6

M7 M8

VIF+VIF-

VRF-M2VRF+ M1




















7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-25







2nH

0065pF 625O

065O



IF_OUT

11V

07V

LO+LO+LO -

MLI

N

IF -

MLIN

IF +

18V

18V

08V

M7

45K45K

M8

M6M5M4M3

M1 M2

M10 M9

IF +

IF -

56nH

7pF



7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-26














55 555 56 565 57 575 58 585 59 595 6

RF Frequency (GHz)

-35

-30

-25

-20

-15

-10

-5

0

S 11(

dB)


300 350 400 450 500 550 600 650 700

IF Frequency (MHz)

-35

-30

-25

-20

-15

-10

-5

0

S 22(

dB)



5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

15

17

19

21

23

25

LO-R

F is

olat

ion

(dB

)


5245 5265 5285 5305 5325 5345 5365

LO Frequency (MHz)

48

49

50

51

52

LO-IF

isol

atio

n(dB

)




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-27


6

8

10

12

14

16

Con

vers

ion

Gai

n (d

B)


470 475 480 485 490


10

11

12

13

14

15

Noi

se F

igur

e(dB

)



479 481477 483

-200

-150

-100

-50

-250

0

freq MHz

dBm

(OU

T)

m1

m7



(a) (b) (c)



7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-28











7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-29

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel

DesiredChannel


BPF1

1LOω


BPF3

2LOω


BPF4

IF Amp

1BPF

Band SelectFilter


BPF2


7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-30

A


IFFilter


dB12

dB5dB15

4

4

4

=

==

NF

AA

p

v

dB12dB15

2

2==

NFAv

dB21 =LdB63 =L

A BLNA

C D E F G

IF Amplifier

dB55 =LdB106 =NFLO



Mixer IF Filter

IF Amplifier


-2 -2

15 15

-6 -6

15 5

-5


Stage NF (dB) 2 2 6 12 5 10

Cumulative NF (dB)




B C D E F

-2 13 7 22 17

879 679 201 141 15 10


7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-31

LO

B

NF=10dB

IFA

DDV

1R Ω500dB5dB15

==

p

vAA

SSB NF =10 dB


YXsR

Loω

+inV

Spectrum at X

Spectrum at YLoω

IFω

ω

ω

ThermalNoise

SignalBand Image

Band



7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-32


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st



( )( )

V t V t

V t V tRF r r

Lo o o

=

=

⎧⎨⎪

⎩⎪

cos

cos

ω

ω



7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-33

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port

RF IF

LO

RF IF

LO

KTB Frequency

Am

plitu

de a

tm

ixer

RF

port


3f - fLO IF

f LO

2fLO3fLO



DS DS

G S G S


3 65 2 5 80 78 0 78 980 11 2




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-34


Deg

plusmn4


plusmn05



18 20 20

25 25 25


VSWR RF LO IF

15 15 15

20 20 20


1dB Compression dBm

+5

3-rd order HP dBm

+15



IF output power



7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-35




( )( ) ( ) ( )

let


ω ω ω ω

r o i r o i i


IF

V V t V V t V V t

= + minus =


rarr

cos sin sin2 2

( )

( ) ( ) ( )let




ω ω ω ω

r o i r o i


f f ff f f

fr LO IF

im LO IFIF

= minus

= +⎧⎨⎩

⎫⎬⎭



( ) ( )

( )[ ] ( )[ ]V t t

V t t

rA V

o iV

o i

rB V

o io V

o io

V t VV t V

U L

U L

rA

o iA

rB

o iB

= + + minus


⎧

⎨⎪

⎩⎪

rarrrarr

2 2

2 290 90

cos cos

cos cos

coscos

ω ω ω ω

ω ω ω ω

ωω

IF mixing

( )[ ]

( ) ( )

V V t V t V t V t

kV t

V kV t kV t

V kV t kV t

kU i L i U i

oL i

L i

iA

U i L i

iB

U io

L io

1 2180

2

90 90


= minus

= minus

= minus minus +

⎧⎨⎪

⎩⎪

sin sin sin sin

sin

sin sin

sin sin

ω ω ω ω

ω

ω ω

ω ω


7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-36






fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF

Loωω

ω

ThermalNoise

SignalBand Image

Band



Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter



RF IF

LO

f

f

ffRF

fLO



7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-37

KTB

f + fLO IF

f - fLO IF

f LO

fRF

Frequency

Imagenoise

DesiredSignal

Am

plitu

de a

tm

ixer

RF

port


7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-38






( )( ) ( ) ( ) ( )

( )

v t V t V V t

V t V V t

v t V t V V t

V t V V t

r ro

o n oo

r r o n o

r ro

o n oo

r r o n o

1

2

90 180

180 90

= minus + + minus

= + +

= minus + + minus

= minus + +

⎫

⎬⎪⎪

⎭⎪⎪

cos cos

sin cos

cos cos

cos sin

ω ω

ω ω

ω ω

ω ω





7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-39

fIF2




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-40

Pt

Gt Gr

Pr

R

fRF

t

RF DC

f

t

f

t

f

t

f

RF IF

LO

f

f

ffRF

fLO


fRF fm

ModulatedRF

Modulation

(a) Diode rectifier

(b) Diode detector


RF IF

LO

f

f

ffRF

fLO



21 =++


7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-41

f

A

f

B

f

C

f

D

f

E

f

F

f

G

f

H


imωrω rω

rω 1IFω

imω

ImageDesiredChannel

imω

Image

DesiredChannel

interfers

1IFω 2IFω

2IFω 2IFω


7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-42


( )

( ) ( )

( ) ddso

VsVdVV

jjdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

RRGIIeIdVdI

eIVI

oooo

oo

prime=α=+α=

α===rArr

==+α=α=rArr

minus=

α

minusα

α

2

222

1

)1()(

resistance junction

Since

( )

( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

prime=α=+α=

α===

==+α=α=

minus= α

minusα

α

resistance junction

Since

ddso

VsVdVV

j

jdsoV

sV

Vs

GGII

eIdVdGdVdVdIddVId

R

RGIIeIdVdI

eIVI oooo

oo

2

222

1

)1()(

Mixer

Basebandfilter

fMAntenna

Localoscillator

+ fMfLo

Poweramplifier


DetectorIFamplifier

LNA

+ fMfIF fIFfMfLo +

fMfLo-fo =

Mixer

Localoscillator

IF filter




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-43

bull

1BPF 2BPF 3BPF 4BPF

2LOω1LOω

A B C D E F G H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifierLNA

1stMixer

2ndMixer

ImageRejectFilter

t0cosω


LO

MixerRF IF

Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

o90


tVV rrRF ω= cos

1v

2v


7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-44

3-dBpowerdivider

0Z

rvRF input

Arv

Brv

RF

RF

Mixer B

Mixer A

LO LOinput

oω

hybrido90

hybrido90

1v

2v

LSB

USB

IFout

⎩⎨⎧

ω+ω+ω+ω

=tVtV

vioL

ioUr )(cos

)cos(

LO

IF

IF

( ) ( ) ( )( ) ( ) ( )

( )

( )⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=


+=ΓminusΓ=+=

ΓΓΓ

ΓΓ

ΓΓΓΓΓ



rLo

rLOLOLO

LOr

Lo

LorLorLoRF

LoRFRFRFRF

VV

VjVVV

VV

Vj

VVjjVjVVVj

VVVjVVVV

21

22

1121

21

21

21

21

21

2

222

( ) ( ) ( )

( ) ( )[ ]( ) ( )[ ]

( )

( )⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

Γminus=+=

Γminus=




ΓΓΓ

ΓΓΓ



rLo

rLOLOLO

LOr

LO

LOrLOr

LOrLOr

RFRFRF

VV

VjVVV

VV

Vj

VVjjVjV

VVjjVjV

VjVjVVVVV

21

21

21

21

21

21

2121

21 22



7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-45



ωω

mm2 2

22m







mVAo s

= = = = times+

0 22 5 101 2501

5 α μ Ω


Ao s= = = =

+ +60 4171 25

60 01μα μ

Ω



14 GHz (1127) ( )R Gj d I Io s

= =minus+

1 1α

( )V t 2

( )( )

( )14

222

2

12

2

V Gd m t to mV Gd m

V

mo

o



7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-46


RF


Combiner

LO

tv oo ωcos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias


Diode2 LP

filters

+ If output

Diode1 1i

2i

RF input

LO input3 dB hybrid

)18090( ooor

Fig

1019

1BPF 2BPF 3BPF 4BPF


H

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter

ChannelSelectFilter

IFAmplifier

1BPF 2BPF 3BPF 4BPF

A B C D E F GH

BandSelectFilter

ImagerejectFilter

ChannelSelectFilter


Amplifier

f

DesiredChannel

Image

imωrω

interfersinterfers

1LOω 2LOω

2IFω

DesiredChannel



7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-47



fIF

fLO

fRF

Mixer noise

2 fLO

- fIF

2 fLO

+ fIF




Mixer

LOSignal Generator


Pre Amp

Noise Figure Meter




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-48


7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-49

LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ

2vΓ



LP filter

+ If outputDiode 1

1i

2i

RF input

LO input

3 dB hybrid)90( o

Diode 2


tVV ooLo ω= cos

RFVΓ

LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-50


7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-51

RF

tvV rrRF ω= cos

Matchingnetwork

Combiner

tvV ooLO ω= cos

DCreturn

or

orωplusmnω

ωωLP filter

DCbias



LO

IF


( ) ( )

( ) ( )

( )( )[ ] tVVtVV

tVttVVtV

tVVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

=ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

coscoscos2cos

coscos2

)coscos2

24

22222

22

2

2

output IF

(note


7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-52


7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-53


Pozar (RF Ch 7)




Antenna

Local oscillator

+ fMfLo

Poweramplifier


t

( )( )⎪⎩

⎪⎨⎧

minus

+

LSB SidebandLower

USB SidebandUpper

IFLo

IFLo

ff





7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-54



tcωcos

BasebandQ

BasebandI

tcωsin

MatchingNetwork

Duplexer

PA


LO

I

Q

BPF

LOω ω

PA

56 565 57 575 58 585 59Frequency (GHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S11

(dB

)


400 420 440 460 480 500 520 540 560

Frequency (MHz)

-40

-35

-30

-25

-20

-15

-10

-5

0

S22

(dB

)


(a) (b)

-40 -35 -30 -25 -20 -15 -10 -5 0input power (dBm)

0

2

4

6

8

10

12

14

Con

vers

ion

Gai

n (d

B)


(c) (d)


7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-55



3f - fLO IF

f LO

2fLO3fLO




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-56


7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-57



RF IF

LOf f



fLOf


fIF

Mixer

Local oscillator

IF filter


)( IFLO

off

fminus=

MIF

MoLO

oRF

fffff

ff

plusmn=plusmnminus=

minust

t


ImageRejectFilter

LNA

tA LOωcos0

IFω2ωrω imω


DesiredBand

image

ωrω

DesiredBand

image



7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-58

bull

fRF

t

RF DC

f

t

f

t

f

t

ffRF fm

ModulatedRF

Modulation

(a) Diode rectifier



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧



ωminusω+ω+ω=


tt

ttV

tt

ttV

IFoV

IFoV

oIFo

VoIFo

VBr

IFoV

IFoV

oIFo

VoIFo

VAr

LU

LU

LU

LU

)cos()cos(

]180)cos[(]180)cos[(

)sin()sin(

]90)cos[(]90)cos[(

22

22

22

22



⎪⎨⎧

+ωminusminusω=

ωminusω=

)90sin()90sin(

sinsino

IFLo

IFUB

i

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB tkV

tVtVtVtV

tkVtkV

tkVtkVV

IFL

IFLo

IFUIFLIFUk

ooIFL

ooIFU

IFLIFU

ωminus=


⎥⎥⎦

⎤

⎢⎢⎣

⎡


+ωminusω=

sin2

]sin)180sin(sinsin[

)]9090sin()9090sin([

)sinsin(

2

21

1


Injectionfilter

~ ~

st1 mixer mixer2nd

RFamplifier

1 IFstages

st 2 IFstages

nd

2 localoscillator

nd1 localoscillator

st

1 IFamplifier

st


7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-59



⎪⎨

⎧


ωminusω+ω+ω=

]90)cos[(]90)cos[(

)cos()cos(

22

22o

IFoVo

IFoVB

r

IFoV

IFoVA

r

ttV

ttV

LU

LU



⎪⎨⎧

+ω+minusω=

ω+ω=

)]90cos()90cos([

]coscos[

221

221

oIFL

oIFU

Bi

IFLIFUA

i

tkVtkVV

tkVtkVV


component LSB

]

tV

tVtVtVtV

tkVtkV

tkVtkV

V

IFLk

IFLIFLo

IFUIFUk

ooIFL

ooIFU

IFLIFU

ω=

ω+ω+minusω+ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


+

ω+ω

=

=

cos

]coscos)180cos(cos[

)]9090cos()9090cos([

coscos[

2

04

221

221

21

1

component USB tV

tVtV

tVtV

V

IFUk

oIFL

oIFU

oIFL

oIFU

k

ω=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minusω+minusω

+

+ω+minusω

=

sin

)]90cos()90cos([

)]90cos()90cos([

2

221

221

22


7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-60


RF inputs rarr

[ ][ ]

[ ][ ]⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ωminusω+ω+ω=


ωminusω+ω+ω=


minus tVtV

ttVttVv

tVtV

ttVttVv

IFoLIFoU

oIFoL

oIFoU

BRF

IFoLIFoU

oIFoL

oIFoU

ARF

)cos()cos(

)180cos()180cos(

)sin()sin(

)90cos()90cos(

212

12

12

1



⎪⎨⎧

ω+=

ωminus=

minus tVVVv

tVVVv

IFLULOKB

IF

IFLULOKA

IF

cos][

sin][

22

22


( )⎪⎪⎩

⎪⎪⎨

⎧

+minus

=

minusminus

=

LULOB

IF

LULOA

IF

VVVKV

VVVjKV

22

22


222

222

2

1

ULOB

IFA

IF

LLOB

IFA

IF

VjKVVjVV

VKVVVjV

=minusminus=

=minusminus=

sin2

)(

cos2

)(

2

1

tVKVtv

tVKVtv

IFULO

IFLLO

ωminus

=

ω=

3-dBpowerdivider

0Z

rvRF input

ARFv

BRFv

RF

RFMixer B

Mixer A

LO LOinput

)cos( tVv

oLO

LOω

=

hybrido90


o90

1v

2v

LSB

USB

IFout

⎪⎩

⎪⎨

⎧

ωminusω+

ω+ω=+

tV

tVvv

IFoL

IFoU

RFIM)(cos

)cos()

LO

IF

IFLPF

LPF

AIFv

BIFv

IFω2rω imω

Desired Bandimage

rω

Desired Band

imω

image


7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ




7-61

( ) ( )

( ) ( )

( )[ ] tVVtVV

tVttVVtV

tVtVGv

tVtVVVvGvvGIVI

IForG

ororG

ooororrG

oorrG

d

oorrLoRFddo

dd

rd

d

ω=ωminusω=rArr

ω+ωω+ω=

ω+ω=primerArr

ω+ω=+=+prime++=

primeprime

prime

prime

coscos2

)coscoscos2cos(

coscos2

)coscos2

24

22222

22

2

2

output IF

(note

poorisolation

LP filter

+ IF outputDiode 1

1i

2i

RF

LO

3 dB hybrid)90( o

Diode 2



LOVΓ


1vΓ



Documents

Rf Ch7 Fet Mixer 2008