b3 Angle Modulation -comm theorem

SZE 3533SZE 3533COMMUNICATION PRINCIPLESCOMMUNICATION PRINCIPLES

Topic III – Angle ModulationTopic III – Angle Modulation

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3.1 Introduction3.1 Introduction

• Beside AM technique, there is another technique that used modulating signal to change frequency and phase of carrier signal.

• Both are known as Angle Modulation. • Also known as Exponent Modulation.• Introduced in 1931 (Edwin H. Armstrong).

• Generally sinusoidal signal expression:

• Therefore, we can change the amplitude and angle of the carrier signal in order to send information signal.

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)](cos[)( tAts

3.2 Basic Concept of Angle Modulation3.2 Basic Concept of Angle Modulation

Graph shown the characteristic of sinusoidal signal

The angle of sinusoidal signal :octt )(

Gradient for θ(t)=ωct+o is an angle frequency, ωc for sinusoidal signal.

For nonlinear process, θ(t)=θx(t), the gradient represents instantaneous angle frequency, ωi for sinusoidal signal.

)()(

)( tdt

tdti

t

i dt0

)()(

and the instantaneous angle value is given by integration of:

Therefore, we can calculate the instantaneous angle frequency, ωi at time t by calculating the gradient of graph θ(t) at time t i.e:

This can be seen at the time interval t (t1

and t2) both signal are the same.

t

• Therefore, it is shown that information signal, vm(t) can be transmitted with the amplitude of the carrier signal is held constant and the angle either the phase or frequency of the carrier is varied linearly with the information signal, vm(t).

• Let the carrier signal:

• And the instantaneous angle value:

)]([cos)( ttEtv cccc

)()( ttt ccc

tdt

tdt cc

ci

)()(

ttdtt cc

t

ic 0

)(

Pemodulatan Sudut

Ec c(t)

ct

c(t)

i(t)

c(t)

c(t)

Therefore, the instantaneous angle frequency and instantaneous angle value are given by:

Angle ModulationAngle Modulation

tdt

tdt cc

ci

)()(

3.3 Phase Modulation (PM)3.3 Phase Modulation (PM)

)()( ttt ccc

)()( tvktt mpcc

)]([)( tvktkosEtv mpccPM

dt

tdvk

dt

tdt m

pcc

i

)()()(

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PM implies that the phase deviation of the carrier, c is

proportional to the modulating signal, vm(t):

where kp is the phase deviation constant in radians/sec/volt

And the instantaneous angle frequency:

Therefore:

3.4 Frequency Modulation (FM)3.4 Frequency Modulation (FM)

t

mfccFM dttvktkosEtv0

t

mfc

t

mfcc dttvktdttvkt00

)( )()(

Pemodulatan Sudut

)()( tvkt mfci

tdt

tdt cc

ci

)()(

FM implies that the frequency deviation of the carrier,

is proportional to the modulating signal, vm(t): tc

where kf is the frequency deviation constant in radians per volt

Integrate:

Therefore FM signal :

3.5 Relationship between FM and PM3.5 Relationship between FM and PM

• We can generate FM signal by using PM modulator and vice versa.

• From the above block diagrams, it can be shown that the generation of FM and PM signals are mutually related.

DifferentiatorFM

Modulatorvm(t) vPM(t)

dt

d

PMModulator

vm(t) vFM(t)Integrator

dt

Generation of FM

Generation of PM

Pemodulatan Sudut

)]([)( tvktkosEtv mpccPM

t

mfccFM dttvktkosEtv0

• Demodulation process is used to get back the information signal.

• For FM demodulator in order to get back information signal from FM signal : PM modulator is used and the signal is pass through differentiator.

• In contrast for PM demodulator : FM demodulator is used and the signal is pass through the integrator.

• This shows the close relationship between FM and PM.

• Hence we can discuss only either one technique in angle modulation.

Differentiator

dt

dPM

Demodulatorvm(t)vFM(t)

FM Demodulator

FMDemodulator

vm(t)vPM(t) Integrator

dt

PM Demodulator

Pemodulatan Sudut

t

mfccFM dttvktkosEtv0

)]([)( tvktkosEtv mpccPM

3.6 Analysis of AM signal3.6 Analysis of AM signal

)cos()( tEtv mmm

]sin[cos

])cos([cos)(0

tEk

tE

dttEktEtv

mmm

fcc

t

mmfccFM

Pemodulatan Sudut

t

mfccFM dttvktEtv0

cos

Assuming that the modulating signal, vm(t):

Substitute in the equation:

Take:

mm f

f

mf Ek

]sin[cos)( ttEtv mccFM

Pemodulatan Sudut

]sin[cos)( tEk

tEtv mmm

fccFM

rad/s, as a maximum frequency deviation

• Define the modulation index as a ratio of maximum frequency deviation to modulating signal frequency:

• Hence equation FM yields:

• Trigonometric identities:

)](sin[sin)sin()]sin(cos[)cos()( ttEttEtv mccmccFM

genapn

mnm tnJJt )cos()(2)()]sin(cos[ 0

ganjiln

mnm tnJt )sin()(2)]sin(sin[

n = even

n = odd

)sin()sin()cos()cos()cos( BABABA

Pemodulatan Sudut

]sin[cos)( ttEtv mccFM

• Hence :

Expand using Fourier series yields:

Where cos[βsin(ωmt)] dan sin[βsin(ωmt)] is a trigonometric series called as Bessel Function(Fungsi Bessel).

• Using Bessel identities :

])cos())[cos((

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

)sin()sin()(2

)cos()cos()(2)cos()(

)sin()(2)sin(

])cos()(2)()[cos()(

0

0

0

tntnJE

tntnJEtJE

tntJE

tntJEtJE

tnJtE

tnJJtEtv

mcgenapn

mcnc

mcganjiln

mcnccc

ganjilnmcnc

genapnmcnccc

ganjilnmncc

genapnmnccFM

Pemodulatan Sudut

Substitute in vFM

oddn

evenn

J

JJ

n

nn

nn

n JJ 1

• Hence FM equation also known as WBFM:

Pemodulatan Sudut

])cos[(]))cos[((...

])4cos[(])4)cos[((

])3cos[(])3)cos[((

])2cos[(])2)cos[((

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

)cos()()(

4

3

2

1

0

tntnJE

ttJE

ttJE

ttJE

ttJE

tJEtv

mcmcnc

mcmcc

mcmcc

mcmcc

mcmcc

ccFM

Sideband 1

Sideband 2

Sideband 3

Sideband 4

Sideband n

Carrier band

])cos[()()( tnJEtv mcncFM

Expand the equation yields :

3.6.1 Frequency Spectrum of FM signal3.6.1 Frequency Spectrum of FM signal

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mc mc c

β = 0.25

)( 1rads

BW

mc 4 mc 4c

β = 2

)( 1rads

BW=2nfm=8fm

mc 8 mc 8c

β = 5

)( 1rads

BW=2nfm=16fm

The number of sidebands depend on value: (Rujuk Jadual Bessel ms 102)

Bessel Function for n=0 to n=4Bessel Function for n=0 to n=4

Bessel Function PlotBessel Function Plot

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Bessel Function TableBessel Function Table

• Frequency spectrum consists of carrier component at fc and also sideband at fc±nfm where n is an integer (n = 1,2,3,…)

• The number of sideband depends on index modulation value, β.

• Magnitude of carrier signal decreases as β increases. • Amplitude of the frequency spectrum depends on

value of Jn(β).

• The bandwidth of modulated signal increases when index modulation, β increases. BW > 2∆fm is expected.

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Summary of FM spectrum:

3.6.2 Carlson’s Rule3.6.2 Carlson’s Rule

• Even though FM signal has infinite number of sidebands but from the experiment conducted, it is shown that errors (herotan) due to the band limited signal of FM can be neglected if 98% of the power of the signal has been transmitted.

• Based on Bessel function, 98% of signal power has been transmitted if the number of the sidebands transmitted equal to 1+β.

• Therefore the BW needed for FM was :

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m

m

fffBW

212

3.6.3 Narrow Band FM (NBFM)3.6.3 Narrow Band FM (NBFM)

• For FM signal with the small index modulation i.e β < 0.2, is called Narrow Band FM (FM jalur sempit)

• For FM signal that we have studied previously also known as WBFM and the equation is given by :

• Let :

• Hence, the equation yields:

• NBFM with β = small , therefore;

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)sin()( tt m

)](sin[sin)sin()]sin(cos[)cos()( ttEttEtv mccmccFM

)]([sin)sin()](cos[)cos()( ttEttEtv ccccFM

1)sin()( tt m

Pemodulatan Sudut

])cos[(2

])cos[(2

)cos(

)sin()sin()cos(

)sin()()cos()(

tE

tE

tE

ttEtE

ttEtEtv

mcc

mcc

cc

cmccc

ccccFM

1)](cos[ t )()](sin[ tt and

tkosmE

tkosmE

tkosEtam mcc

mcc

ccFCDSB 22)(

• Therefore :

• Hence NBFM equation yields :

• Compared with amDSB-FC signal:

• It is shown from both equations for NBFM and amDSB-FC consist of one carrier component and two sidebands components. But LSB component for NBFM the phase shift is varies for 90° (quadrature).

1)sin()( tt m

3.7 Differences between FM and AM3.7 Differences between FM and AM

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• Frequency spectrum

• Phase diagram (Rajah pemfasa)

)(VAmplitud

)( 1radsc mc mc 0

cA

2cmA

2cmA

22mc AmA

Di mana

)(VAmplitud

)( 1radsc mc mc 0

cA

2

cA

2

cA

2

cA

2

cA

cA

m

c

m)(tvFM

)(t

2cmA

2cmA

cA

mcm)(tam FCDSB

AM FM

3.8 Power in FM signal3.8 Power in FM signal• Power signal depends on the amplitudes and not on the

frequencies. • The amplitude of the FM signal is constant and therefore the

power transmitted depends only on the amplitudes of the signal. It does not depends on the modulation index.

• For AM signal the power transmitted depends on the modulation index.

• It can be seen from the Bessel equation:

• In other word the total power of FM signal consists of the power in carrier component and all the power in the sidebands.

Pemodulatan Sudut

1

2

...2

0

3210

nJJ

JJJJJT

n

n

PP

PPPPPP

1

220

223

22

21

20 122...222

nnn JJJJJJJ

• FM equation is given by:

• And therefore the total power transmitted :

1

220

2

2223

222

221

220

2

2)(

2)(

2)(

2)(

2)(

)(

22

2...

2222

2

...2

...2

3210

3210

nn

c

nccccc

rmsJrmsJrmsJrmsJrmsJ

JJJJJFMT

JJR

E

R

JE

R

JE

R

JE

R

JE

R

JE

R

V

R

V

R

V

R

V

R

V

PPPPPP

n

n

Pemodulatan Sudut

])cos())[cos((...

])3cos()3)[cos((

])2cos()2)[cos((

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

3

2

10

tntnJ

ttJ

ttJ

ttJtJEtv

mcmcn

mcmc

mcmc

mcmcccFM

Ex. 1 :

A carrier with a peak value of 2000 V is frequency modulated with a message signal of 5 kHz. The modulation index obtained is 2. Calculate the average power in:

(i) Highest sideband (ii) Lowest sideband . Given R = 50 Ω .

Solution :

For β = 2 from Bessel table :

The highest sideband is : 58.01 JThe lowest sideband is : 01.05 J

01.02

58.02

5

1

J

J

=>

R

EP C 1

2

58.02

1

(i)

kW 5.1350

1

2

200058.02

50

1

2

200001.02

5

P

W4

(ii)

Ex. 2 :

(a) Determine the BW required to transmit FM signal when the modulating frequency, fm = 10 kHz and maximum frequency deviation is 20 kHz.

From Bessel table the components obtained is J0 , J1 , J2 , J3 , J4 and J5 That means J1 will be at 10 kHz, J2 at 20 kHz, J3 at 30 kHz etc.

Therefore BW = BFM = 2nfm = 2 x 5 x 10 = 100 kHz

210

20

mf

f

Amplitud

fc fc+fm fc+2fm

J0

J1

J5

fc-fm

f (kHz)

m

m

fffBW

212

Carson Rule

Solution :

(b) Repeat (a) with fm = 5 kHz

From Bessel table the highest component is J7

Therefore BW = 2 x 7 x 5 = 70 kHz

45

20

mf

f

m

m

fffBW

212

Carson Rule

Solution :

Ex. 3 :

A FM signal, 2000 cos (2π x 108 t + 2 Sin π x 104t) is transmitted using an antenna with the resistance of 50 Ω. Determine

(i) Carrier frequency (ii) Modulation index (iii) Information signal

(iv) Power transmitted (v) Bandwidth (vi) Power in highest and lowest sidebandsPenyelesaian :

]sin[cos)( ttEtv mccFM Bandingkan :

(i) fc = 108 Hz = 100 MHz

(ii) β = 2

(iii)fm = 104 / 2 = 5 kHz

i) (v) β = 2 => bilangan jalursisi 4

BW = BFM = 2nfm = 2 x 4 x 5 = 40 kHz

Carson - BW = 2(β + 1)fm = 2(2 + 1)5 = 30 kHz

i) (iv) Ec = 2000 V => Ec(rms) = 2000 / 2

Kuasa dipancarkan PT = V2 (rms)

/ R

= (2000 / 2)2 / 50

= 40 kW

(vi) Dari jadual J1 jalursisi amplitud tertinggi

Nilai puncak jalursisi = 0.58 x 2000

Kuasa P = (0.58 x 2000/2)2 / 50 Ω

= 13.27 kW untuk satu jalursisi

Dua jalursisi = 2 x 13.27 kW = 26.54 kW

Kedua-dua jalursisi berada pada

fc fm = 100 MHz 5 kHz

Kuasa jalursisi terkecil J4

P = (0.03 x 2000/2)2 / 50 Ω = 36 W

Contoh 3.1

Satu isyarat FM mempunyai persamaan berikut :

t

mfccFM dttvktEtv0

cos

di mana , , , tfEtv mmm 2sin kHz 10fkV 100cE

kHz 5dan V 1 , MHz 2.106 mmc fEf

(i) Kirakan sisihan frekuensi (frequecy deviation)

(ii) BW menggunakan aturan Carson

(iii)Kuasa yang dipancarkan

(iv)Jika indek pemodulatan adalah kecil, dapatkan persamaan isyarat NBFM

Penyelesaian :

tfdttvkEtfEv

R

EP

ffBWf

f

Ekf

cmfcccNBFM

cFM

mm

mf

2sin22cos (iv)

1Ranggapan dengan ;kW 512

100

2 (iii)

kHz 3051022 25

10 (ii)

kHz 10110 (i)

22

)sin()(sin)cos()( ttEtEtv cmcccNBFM

3.8.1 Isyarat FM dan PM dalam Domain Masa3.8.1 Isyarat FM dan PM dalam Domain Masa

Pemodulatan Sudut

FM

PM

3.9 Generation of FM signal3.9 Generation of FM signal

2 techniques – direct and indirect methods (kaedah langsung dan tidak angsung)

Require a system that able the frequency of the output signal to vary in accordance to an information signal amplitude.

3.9.1 Direct method/Kaedah langsung

1. Varactor diode

2. Reactance modulation/Pemodulatan Regangan

3. Voltage Controlled Oscillator/Pengayun terkawal voltan (VCO)

Output frequency is proportional to the input voltage.

Ex: VCO manufactured by Signetics, SE/NE 566 or HCT 4046

http://www.see.ed.ac.uk/~gjrp/EE3/Comms/Lecture10/sld003.htm

http://www.ycars.org/EFRA/Module%20B/directfm.htm

Varactor diode

L

C = kvm dimana k adalah pemalar dan vm adalah voltan ketika isyarat maklumat

1. Varactor diode

Varactor diode characteristic

CCL o

21

CCC oT T

oLC

f2

1 ;

Analisa matematik :

O

CLC

f2

1Bila vm=

0;

OO

O

CC

LC

f

12

1

2

1

12

1

OOC

C

LC

Varactor diode’s capacitance depends on the voltage across it.

Audio signals placed across the diode cause its capacitance to change, which in turn, causes the frequency of the oscillator to vary.

Using Binomial expansion :

OCO C

Cff

21

O

mC C

kvf

21

From the equation it can be seen that the FM signal can be obtained because the output frequency is dependant to the information signal amplitude, vm .

OO C

C

C

C

211

2

1

OC

Cif is small;

A reactance modulator is a circuit in which a transistor is made to act like a variable reactance.

The reactance modulator is placed across the LC circuit of the oscillator and as the modulator’s reactance varies in response to an applied audio signal, the oscillator frequency varies as well.

2. Reactance modulator

Frequency modulation using these techniques are not able to create a signal with large frequency deviation. It means it is not suitable for WBFM. To address this issue, the Crosby modulator was developed. The Crosby circuit incorporates an automatic frequency control (AFC).

The VCO’s output frequency is proportional to the voltage of the input signal.

If audio is applied to the input of a VCO, the output is an FM signal.

3. VCO

Direct method - Crosby circuit

AFC Circuit

To transmit and fed back an error control voltage to a modulator in order to control frequency oscillator at 5 MHz (to prevent drift of the carrier and frequency deviation). This method is called Automatic Frequency Control (Kawalan frekuensi automatic).

Crosby circuit – to generate WBFM

Let us look at an example. An FM station operates at 106.5 MHz with a maximum deviation of 75 KHz. The FM signal is generated by a reactance modulator that operates at 3.9444 MHz, with a maximum deviation of 2.7778 KHz. The resulting FM signal is fed through 3 frequency triplers, multiplying the carrier frequency and deviation 27 times. The final carrier frequency is 27*3.9444 = 106.5 MHz and the final deviation is 27*2.7778 = 75 KHz.

It is important to remember that frequency multiplication multiplies both the carrier frequency and the deviation.

http://www.see.ed.ac.uk/~gjrp/EE3/Comms/Lecture10/sld004.htm

3.9.2 Indirect method

Pemodulatan Sudut

~

vWBFM(t) Mixer Penapis Lulus Jalur

Local Oscillator cos(ωLOt)

vz(t)vy(t)

ωc1 Nωc1

PemodulatNBFM

Pekali Frekuensi, N

vm(t)

vNBFM(t)

Armstrong methodArmstrong method

First generate NBFM. Then multiplies NBFM frequency with multiplier N. This frequency multiplication multiplies both the carrier frequency and the deviation.

Then signal vy(t) is tuned at the frequency desired and is suitable to the ranges of LO frequency, fLO . BPF is then used to filter the desired

frequency components.

3.9.3 Generation of NBFM3.9.3 Generation of NBFM

• FM modulation : The amplitude of the modulated carrier is held constant and the time derivative of the phase of the carrier is varied linearly with the information signal.

• The instantaneous frequency of FM is given by:

• Hence

Pemodulatan Sudut

)()( tvkt mfci

)()( tvkt mfc tdt

tdt cc

ci

)()( where

~

∫dt

k fvm(t))(tc

X ∑

90°

vNBFM(t)

Eccos(ωct)Ecsin(ωct)

-

+Pemodulat Fasa

• The angle of the FM signal can be obtained by integrating the instantaneous frequency.

• vm(t) is a sinusoidal signal, hence:

Pemodulatan Sudut

)sin(

)sin(

)cos()(0

t

tEk

dttEkt

m

mm

mf

t

mmfc

ttdttt cc

t

ic 0

)()(

t

mfc dttvkt0

)()(Notes:

1)()(0

t

mfc dttvkt

Notes:

1)sin( tmFor NBFM

Therefore

t

mfc

t

mfcc

dttvkt

dttvkt

0

0

)(

)()(

• General equation for FM signal

Pemodulatan Sudut

)](sin[)(sin)](cos[)(cos

)]([cos)(

ttEttE

ttEtv

cccccc

cccFM

)(sin)()(cos)( tEttEtv cccccNBFM

• Therefore NBFM signal can be generated using phase modulator circuit as shown.

• To obtain WBFM signal, the output of the modulator circuit (NBFM) is fed into frequency multiplier circuit and the mixer circuit.

• The function of the frequency multiplier is to increase the frequency deviation or modulation index so that WBFM can be generated.

• Hence :

1)( tcFor NBFM therefore 1)](cos[ tc )()](sin[ tt cc and

Summary:

vWBFM(t)

~

Mixer Penapis Lulus Jalur

Penjana Tempatan cos(ωLOt)

vz(t)vy(t)

ωc1 Nωc1

PemodulatNBFM

Pekali Frekuensi, N

vm(t)

vNBFM(t)

3.9.4 Generation of WBFM3.9.4 Generation of WBFM

Analisa Matematik :

• The instantaneous value of the carrier frequency is increased by N times.

)()()( 1 ttt cci Let :

)(

)]([

)()(

2

12

tN

tN

tNt

cc

cc

Output of the frequency multiplier :

cc N 2

Notes

And :

Pemodulatan Sudut

)()()( 222 tNttdt

dt cc

cc N 2

Nota:)sin(

)sin()(

2

1

t

tNtN

m

mc

12 N

• It is proven that the modulation index was increased by N times following this equation.

)sin(

)sin(

)cos()(0

t

tEk

dttEkt

m

mm

mf

t

mmfc

• The output equation of the frequency multiplier :

• Pass the signal through the mixer, then WBFM signal is obtained :

• BPF is used to filter the WBFM signal desired either at ωc2+ ωLO

or at ωc2- ωLO .

• Hence the output equation :

)]([

)]([cos)(

2

2

tNtkosE

tEtv

ccc

cFM

Pemodulatan Sudut

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

)cos(2 x )]([cos)(

22

2

tNtEtNtE

ttNtEtv

cLOcccLOcc

LOcccFM

)]()[(

)]()[()(

2

2

tNtkosE

tNtkosEtv

cLOcc

cLOccWBFM

3.9.5 Comparison between FM and AM3.9.5 Comparison between FM and AM

• Advantages– SNR is much better than AM can be obtained, if the BW is greater

enough.

– SNR can be increased by increasing the transmitted power.

– Constant amplitudes made the non linear preamplifier to be used effectively.

• Disadvantages– BW is usually larger than AM.

– Circuitry is more complex.

Pemodulatan Sudut

3.10 Demodulation of FM signal3.10 Demodulation of FM signal

• Demodulation process is done in order to recover/get back the information signal transmitted.

• Basic concepts of demodulation circuit is to detect the frequency variation.

• Two techniques can be used:

Pemodulatan Sudut

Penyahmodulatan FM

Secara Tak Terus

Secara Terus

• Pembezalayan/Discriminator Phase Lock Loop(PLL)/Gelung Terkunci Fasa

3.10.1 Conversion circuit - FM to AM 3.10.1 Conversion circuit - FM to AM ((DiscriminatorDiscriminator) – K.Terus) – K.Terus

Pemodulatan Sudut

• This technique is required to convert FM signal to AM signal and then by using AM demodulation circuit is to get back the information signal.

• This technique is called pengesan kecerunan (slope detection) or discriminator.

• Block diagram of the detection circuit is as shown below:

t t t

y(t)

Pengesan Sampuldt

dvFM(t) y(t) tvFM

tvFMvFM(t)

Pemodulatan Sudut

))(cos()(0t

mfccFM dttvktEtv

)]([ tvkE mfcc

Mathematical analysis :

Differentiate; yields :

FM equation :

dttvkttvkEdt

tdvmfcmfcc

FM sin

• From the above equation it can be seen that the amplitude of the signal contains the information signal.

• The amplitude of the signal is an envelope of the signal and the equation is given by :

• For envelope detector to be used the frequency deviation, Δω required must be smaller than the carrier frequency, ωc or otherwise an envelope detector cannot be used.

Pemodulatan Sudut

cmf tvk )(

0][ ccE for all t

)()( tvkEty mfc

• In practice a limiter circuit (litar penghad amplitude) can be used.

• It is due to the FM signal received at the antenna was influenced by the noise and therefore the amplitudes of the signal were varied and not constant.

• Hence the output equation of the envelope detector :

• Therefore the envelope equation can be written as:

Pemodulatan Sudut

• For effective detection the constant amplitude of the FM signal is required. Therefore an amplitude limiter is used.

• Below is a block diagram of FM detection circuit with limiter circuits.

11

)(ovcos(θ) > 0

cos(θ) < 0 vi(θ)

vo(θ)

1

-1

Penghad BPF )](cos[)( tttE ccc )](cos[4

tt cc

Penghad Amplitud (Limiter)

PenghadAmplitud

Pengesan Sampuldt

dvFM(t) y(t)

Discriminator

• A limiter will limits the output to +1 or -1 depends on the positive or negative cycles of the FM signal and Ec(t) ≥ 0.

• Output of the limiter is a square wave signal as shown below.

• For FM signal the angle varied in accordance to the amplitude of the information signal.

Pemodulatan Sudut

])([)]([0t

mfcoo dttvktvtv

...)5cos(

5

1)3cos(

3

1)cos(

4)(

ov

vo(θ)

θ

2

2

32

5-1

1 Fourier series equation for square wave:

t

mfc dttvktt0

)()(

t

vo[θ(t)]

• Therefore the limiter output is a function of θ(t) and the equation can be written as :

• Output of limiter :

• Output of BPF :

Pemodulatan Sudut

])(cos[4

)(0t

mfco dttvktte

eo(t)

4

t

4

...])(55cos[5

1

])(33cos[3

1])(cos[

4

])([)]([

0

00

0

t

mfc

t

mfc

t

mfc

t

mfcoo

dttvkt

dttvktdttvkt

dttvktvtv

Analysis (continued) : Pengesan kecerunan/Slope detection

Bandpasslimiter

Pengesan Sampuldt

dvFM(t) y(t)

v2(t)v1(t)

Limiter output : ]cos[

4)(1 ttVtv cL

Differentiator output : ]sin[4

)(2 ttdt

tdVtv ccL

)](cos[)()( tttEtv ccFM t

mf dttvkt0

)()(where

FM signal :

Output of the envelope detector :

dt

tdVty cL

4

)(

Since dt

dc

dt

tdVty cL

4

)(;

tvkVVty mfLcL

44

)(

which indicates that the output consists of a dc voltage plus the ac voltage, which is proportional to the modulation on the FM signal.

Therefore :

dc ac

Slope detector circuit

The slope detector is essentially a tank circuit which is tuned to a frequency either slightly above or below the FM carrier frequency. It is not widely used

because of the characteristics of LC tuned circuit which is nonlinear especially for FM signal with large f .AM

t

Is addressed by using - Balanced Slope Detector (Pembezalayan terimbang) – Using two tuned circuit.

To create wider linear region for signal with large f – achieved by using two diodes and tuned at two different tuning frequency.

AM

Pemodulatan Sudut

Kaedah yang lebih popular adalah : Menggunakan peranti litar bersepadu (IC).

• Litar Round TravisLitar Round Travis• Pembezalayan Foster–Seely • Pengesan nisbah

D 1

D 2

f01

f02

C

C

R

R

V FM V o

I1

I2

3.10.2 Litar Round Travis3.10.2 Litar Round Travis

3.10.3 Litar Foster Seeley3.10.3 Litar Foster Seeley

Pemodulatan Sudut

D 1

D 2

C 2

C

C 3

C 4

R 1

R 2

V 12V o

I1

I2

C 1L

Ip

1

2

6

7

3

4

5

http://en.wikipedia.org/wiki/Detector_(radio)

The Foster-Seeley discriminator is a widely used FM detector. The detector consists of a special center-tapped transformer feeding two diodes in a full wave DC rectifier circuit. When the input transformer is tuned to the signal frequency, the output of the discriminator is zero when there is no deviation of the carrier; both halves of the center tapped transformer are balanced. As the FM signal swings in frequency above and below the carrier frequency, the balance between the two halves of the center-tapped secondary are destroyed and there is an output voltage proportional to the frequency deviation.

3.10.4 Litar Pengesan Nisbah3.10.4 Litar Pengesan Nisbah

Pemodulatan Sudut

L1 L2

D 1

D 2

C

C R

R 1

R 2

V 12

V o

C 1L

Ip

1

2

3

4

5

V DC

+

-

The ratio detector is a variant of the Foster-Seely discriminator, but, the diodes conduct in opposite directions. The output in this case is taken between the sum of the diode voltages and the center tap. The output across the diodes is connected to a large value capacitor, which eliminates AM noise in the ratio detector output. While unlike the Foster-Seely discriminator, the ratio detector will not respond to AM signals, however the output is only 50% of the output of a discriminator for the same input signal.

3.10.5 Phase-Locked Loop (PLL) – 3.10.5 Phase-Locked Loop (PLL) – Indirect MethodIndirect Method

• Above is a block diagram of FM detector using Phase-Locked Loop (PLL).

• The input is FM signal:

Pemodulatan Sudut

Penapis Lulus Rendah

Voltage-ControlledOscillator (VCO)

Xvin(t)ve(t)

vvco(t)

vo(t)

))(cos(

)](cos[)(

0

t

mfcc

cccin

dttvktE

ttEtv

)](sin[)( ttEtv ocovco

)()]()([ ttt eoin

)()](sin[ tt ee Then

1)( teIf

Phase-Locked LoopPhase-Locked Loop• VCO output:

• Multiplier in the circuit will function as a phase variation detector/pengesan perubahan fasa :

• LPF will pass all the lower frequency components and filtered all the higher frequency components:

)]()(sin[2

)]()(2sin[2

)](sin[)](cos[

)()()(

ttEE

tttEE

ttttEE

tvtvtv

oinoc

oincoc

ocincoc

vcoine

)(2

)(sin2

)]()(sin[2

)(

tEE

tEE

ttEE

tv

eoc

eoc

oinoc

o

)](sin[)( ttEtv ocovco t

ooo dttvkt0

)()(where

ee

avo

vo

Figure shows the plot of vo vs e . Using this plot we can explain the tracking mechanism.

• Frequency generated at the VCO output is proportional to the input voltage of the VCO.

• Therefore

• Output of the PLL is given by:

• Given:

• Hence:

)()( tvkt ooo

Pemodulatan Sudut

dttvkdtttt

oo

t

oo )()()(00

dt

td

ktv o

oo

)(1)(

1)()()( ttt oine )()( tt oin

)()()(1)(1

)( tkvtvk

k

dt

td

kdt

td

ktv mm

o

fin

o

o

oo

3.11 3.11 PREEMPHASIS/DEEMPHASIS PREEMPHASIS/DEEMPHASIS (PRATEGASAN (PRATEGASAN DAN NYAHTEGASAN)DAN NYAHTEGASAN)

In an FM system the higher frequencies contribute more to the noise than the lower frequencies. The situation become more complex due to the amplitude of the signal at higher frequencies are smaller than at the lower frequencies.

Because of this all FM systems adopt a system of preemphasis at the transmitter and deemphasis at the receiver.

Preemphasis – The higher frequencies are increased in amplitude before being used to modulate the carrier and therefore will be less affected to noise.

Deemphasis – is the mirror of pre-emphasis process.

http://en.wikipedia.org/wiki/FM_radio#Pre-emphasis_and_de-emphasis

The characteristics is as shown : Audio Input

Xc = 1/jC ; 1 = 1/R1C ; 2 = 1/R2C

Xc = 1/jC ; 1 = 1/RC ; 2 = 1/ RC

(a) Preemphasis

(b) Deemphasis

1

2

• Vo(VR2) = VinR2 / (R2+ZR1C)

• Vo(Vc) = VinXc / (R+XC)

High frequency caused the reactance of C to decrease and provides and easier path for high frequency to pass through.

Constant Amplitude

Amplitude gain up to 17 dB to maintain SNR

3dB occurs at 2120Hz predicted by RC time constant (RC=75s-US)

3.11 Penerima Radio FM3.11 Penerima Radio FM

Prategasan

FMPemodula

tanFM

fc

88 – 108 MHz

Penguat RF

Pencampur (Mixer)

Lebarjalur IF 200 kHz Penghad

Pengesan FM

NyahtegasanPenguat Audio

LO : fLO = fc + 10.7 MHz

Talaan sepunya (common tuning)

fIF = 10.7 MHz

APenghantar FM

Pemodulatan Sudut

3.11 Penyiaran Radio FM3.11 Penyiaran Radio FM

CH1

CH2

CH 3

CH 99

CH100

88MHz 108MHz20MHz Jalur Penyiaran Radio FM

BW=200kHz

fc1=88.1MHz fc2=88.3MHz

25k

Hz

Gu

ard

Ban

d

25k

Hz

Gu

ard

Ban

d

25k

Hz

Gu

ard

Ban

d

25k

Hz

Gu

ard

Ban

d

BW=200kHz

150kHz(Δf=±75kHz)

150kHz(Δf=±75kHz)

Channel 1 Channel 2Julat frekuensi isyarat maklumatfm = 50Hz – 15kHzSisihan frekuensi maksimumΔf = ±75kHzJulat indek pemodulatanβmin = (75kHz/15kHz) = 5βmax = (75kHz/50Hz) = 1500Lebar jalur bagi setiap channelBW = 200kHzBilangan channelN = 5(f-47.9)

3.12 FM STEREO3.12 FM STEREO(a) FM STEREO TRANSMITTER(a) FM STEREO TRANSMITTER

(b) SPECTRUM STEREO SIGNAL (b) SPECTRUM STEREO SIGNAL

Dua isyarat (0Hz –15 kHz) digunakan untuk memodulat pembawa

(c) FM STEREO RECEIVER(c) FM STEREO RECEIVER

FM MONO

2R

2L

Penerima pula dapat memisahkan isyarat ini manjadi isyarat ‘kanan’ dan ‘kiri’ dan seterusnya menguat dan mengeluarkan kedua–dua isyarat pada pembesar suara yang berasingan

http://www.see.ed.ac.uk/~gjrp/EE3/Comms/Lecture10/index.htm

3.13 Hingar di dalam FM3.13 Hingar di dalam FM

• Persamaan hingar boleh ditunjukkan dalam bentuk berikut:

• Ia juga boleh ditulis sebagai

• Di mana

• Keluaran penapis lulus jalur adalah

)sin()()()()( ttntkostntn cscc

Pemodulatan Sudut

∑PenghadAmplitud

xfm(t) so(t) DiscriminatorPenapis

Lulus JalurPenapis

Lulus Rendah

n(t)

)]([)()( ttkostrtn ncn

)()()( 22 tntntr scn

)(

)(tan)( 1

tn

tnt

c

sndan

)]([)(

)]([)()]([

)()()(

ttkostR

ttkostrttkosA

tntxtx

c

ncnccc

fm

• Persamaan tersebut boleh digambarkan dengan menggunakan gambarajah pemfasa di atas.

• Anggapkan Ac >> rn(t) , maka

Pemodulatan Sudut

)(trn

)()( tt c )()( tt cn

)(tR

cA

)]()([)(

)]()(sin[)(tan)()( 1

ttkostrA

tttrtt

cnnc

cnnc

)]()(sin[)(

tan)(

)]()(sin[)(

tan)()(

1

0

1

ttA

trdmk

ttA

trtt

cnc

nt

f

cnc

nc

3.13 Hingar di dalam FM3.13 Hingar di dalam FM

• Persamaan hingar boleh ditunjukkan dalam bentuk berikut:

• Setelah melalui penapis hingar yang wujud dikenali sebagai band-limited noise

• Di mana

)sin()()()()( ttntkostntn cscc

Pemodulatan Sudut

∑PenghadAmplitud

xfm(t) so(t) DiscriminatorPenapis

Lulus JalurPenapis

Lulus Rendah

n(t)

)()]([)( 22 tntnAtr scc

)(

)(tan)( 1

tnA

tnt

cc

sdan

)()( tntkosAn ccb

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

)sin()()()()()(

ttkostrttntkostnA

ttntkostntkosAtn

c

csccc

csccccb

Hingar Amplitud Hingar Fasa

• Oleh kerana isyarat FM dan hingar melalui penghad amplitud, maka hingar amplitud boleh diabaikan.

• Kita akan hanya menganalisa hingar fasa sahaja.

• Anggapkan Ac >> nc(t) dan Ac >> ns(t) , maka

• Jadi hingar pada keluaran adalah

• Jadi spektrum kuasa hingar

Pemodulatan Sudut

)(

)(tan)( 1

tnA

tnt

cc

s

c

s

c

s

A

tn

A

tnt

)()(tan)( 1

dt

tdn

Adt

tdtn s

co

)(1)()(

h(t)ns(t) no(t)dt

d

Ath

c

1)( j

AH

c

1)(

)()()()(2

22

sso nc

nn SA

SHS

• Pada penapis lulus rendah

• Maka

• Isyarat keluaran pengesan adalah

• Oleh yang demikian SNRo

2

2

2

2

2

2

2

2

22

)]()([

)()(

c

c

LPFcncnc

nc

n

A

A

SSA

SA

Sso

Pemodulatan Sudut

2

32

22

32)(

c

m

coo A

dA

tnNm

m

)()( tmkts fo maka kuasanya adalah )()( 222 tmktsS foo

3

222 )(3

m

fc

o

oo

tmkA

N

SSNR

• Jika isyarat masukan adalah

• maka

• Oleh yang demikian SNRo

( ) dan m m f mm t A kos t k A

Pemodulatan Sudut

2

2 ( )2mA

m t

2 2 2

3

2 2 2

3

2 2

3 ( )

3

2

3 di mana

2

c fo

m

c f m

m

f mc

m m m

A k m tSNR

A k A

k AA

Perbandingan Hingar AM dan FMPerbandingan Hingar AM dan FM

• Kuasa hingar keluaran pada IF penerima AM

• Di mana di dalam perhubungan AM kebanyakkan isyarat yang dipancarkan adalah didominasi oleh isyarat pembawa maka kuasa isyarat yang dipancarkan adalah

1

2

c m

c m

c

m

N d

Pemodulatan Sudut

2

2 ( )2c

c

AS c t

• Oleh yang demikian jika dibandingkan di antara SNRo(FM) dan SNRo(AM) maka

Pemodulatan Sudut

2 2

( )

22

( ) ( )

3

2

3 di mana = 2

co FM

m

co AM o AM

m

ASNR

ASNR SNR

2

( ) 2c c

o AMc m

S ASNR

N

• Dan untuk kes m = 100%