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ECE 461/561COMMUNICATIONS I
Dr. Mario Edgardo Magaña
EECS
Oregon State University
2
OBJECTIVE: The objective of a communication system is to efficiently transmit
an information-bearing signal (message) from one location to another through a
communication channel (transmission medium).
WISH LIST:
• High reliability
• Minimum signal power
• Minimum transmission bandwidth
• Low system complexity (low cost)
EFFICIENT TRANSMISSION: Can be achieved by processing the signal through
a technique called modulation.
MODULATION TYPES:
• Analog modulation
• Digital modulation
3
Amplitude modulation
In this type of modulation the amplitude of a sinusoidal carrier is varied
according to the desired message signal. Let m(t) be the message signal we
would like to transmit, ka be the amplitude sensitivity (modulation index), and
c(t) = Accos(2πfct) be the sinusoidal carrier signal, where Ac is the amplitude of the
carrier and fc is the carrier frequency.
The transmitted AM signal waveform is described by
Requirements:
1.
2. , where Wm is the message bandwidth
)2cos()(1)( tftmkAts cacAM
ttmka ,1)(
mc Wf
4
Taking the Fourier transform of the modulated waveform, we get
Let |M(f)| be described by
Magnitude spectrum of m(t)
)()(2
)()(2
)2cos()()2cos(
)()(
ccca
ccc
caccc
AMAM
ffMffMAk
ffffA
tftmkAtfAF
tsFfS
5
Then the magnitude spectrum of sAM(t) is
Magnitude spectrum of sAM(t)
Observations:
1.
2. Carrier is transmitted explicitly
mAM WB 2
6
Let , then
)2.0cos()( ttm
)2cos()2.0cos(1)( tftkAts cacAM
7
8
To conserve transmitted power, let us suppress the carrier, i.e., let the
transmitted waveform be described by
This is called double side-band suppressed carrier (DSB-SC) modulation.
).2cos()()( tftmAts ccDSB
9
In the frequency domain,
The modulated waveform now has the following spectral characteristics:
Magnitude spectrum of sDSB(t)
ccc
cc
DSB
ffMffMA
tftmAF
tsFfS
2
2cos)(
)()(
10
Observations:
1. (same as standard AM)
2. Transmitted power is less than standard AM
Example: Let a sinc pulse be transmitted using DSB-SC modulation.
mDSB WB 2
11
12
Receivers for AM and DSB
Receivers can be classified into coherent and non-coherent categories.
Definition: If a receiver requires knowledge of the carrier frequency and phase to extract the message signal, then it is called coherent.
Definition: If a receiver does not require knowledge of the phase (only rough knowledge of the carrier frequency) to extract the message signal, then it is called non-coherent.
Non-coherent demodulator (receiver) for standard AM
Peak envelope detector (demodulator)
)(ˆ tm)(tS AM R C
+
- -
+D
13
Observations:
• The net effect of the diode is the multiplication (mixing) of the signals applied
to its input. Therefore, its output will contain the original input frequencies, their
harmonics and their cross products.
• The network is a lowpass filter with a single pole (lossy integrator) that
removes most of the high frequencies. R and C have to be judiciously picked
so that = RC is neither too short (rectifier distortion ) nor too long (diagonal
clipping ). Rule of thumb is period of the carrier.
• A rule of thumb for choosing is based on the highest modulating signal
(message) frequency that can be demodulated by a peak envelope detector
without attenuation, that is,
where is the maximum modulating signal (message) frequency in Hz.
1,
2
1/1
(max)
2
am
a kf
k
(max)mf
14
Let the message signal be described by the sinusoidal tone .
Then, the following figures show the types of distortion that can occur when is
improperly chosen.
tftm m2cos)(
15Illustration of rectifier distortion and diagonal clipping
16
Coherent demodulator for DSB-SC
At the output of the mixer,
and
cos4cos)(2
ˆ
2cos)(2cosˆ
)(2cosˆ)(
tftmAA
tftmAtfA
tstfAtv
ccc
cccc
DSBcc
cos2
ˆ22
4
ˆ
)(
fMAA
ffMffMAA
tvFfV
cccc
cc
17
Let the message signal have the following magnitude spectrum
Then, if fc>Wm,
18
Suppose is such that
Then, if ,
fH lpf
mcm WfBW 2 cos2
ˆˆ fM
AAfM cc
19
Coherent Costas loop receiver for DSB-SC :
20
I-channel:
After downconverwsion,
At the output of the lowpass filter, with |H(0)| = 1,
Q-channel:
cos2cos)(2
coscos)()(
ttmA
tttmAtv
cc
cccI
)(cos2
)( tmA
tm cI
sin2sin)(2
)( ttmA
tv cc
Q
)(sin2
)( tmA
tm cQ
21
Feedback path:
At the output of the multiplier,
At the output of the lowpass filter,
The purpose of hf(t) is to smooth out fast time variations of me(t)
The output of the VCO is described by
2sin)(8
cossin)(4
)(
22
22
tmA
tmA
tm
c
ce
dthmtm feef )()()(
( ) cos ( )VCO cx t t t
22
Where c is the VCO’s reference frequency and is the
residual phase angle due to the tracking error. The constant kv is the frequency
sensitivity of the VCO in rad/s/volt
The instantaneous frequency in radians/sec of the VCO’s output is given by
Clearly, if (t) were small, then the instantaneous frequency would be close to
c and the output of the I-path would also be proportional to m(t).
Single-Sideband Modulation
Motivation: Bandwidth occupancy can be minimized by transmitting either the
lower or the upper sideband of the message signal.
,)()(0t
efv dmkt
),(
)(tmk
dt
ttdefvc
c
23
Consider the following transmitter
Suppose we want to transmit the upper sideband, then the following choice of
bandpass filter yields the correct result
24
At the output of the bandpass filter,
Observation: A practical bandpass filter will cause severe distortion.
Suppose the original message signal contains a gap in its magnitude spectrum around the origin, i.e.,
25
Then, SSB demodulation can be obtained by
Synthesis and analysis of SSB:
Let the spectrum of a DSB-SC signal be given by
26
Let this DSB-SC signal be the input of a lowpass filter with magnitude response
Then the resulting (output) signal would have the spectrum
27
The filter can be represented by the algebraic sum
Also,
)( fH L
ccL fffffH sgnsgn2
1)(
ccccc
ccc
ccccc
ccccc
ccc
L
DSBLLSSB
ffffMffffMA
ffMffMA
ffffMffffMA
ffffMffffMA
ffMffMA
fH
fSfHtS
sgnsgn4
4
sgnsgn4
sgnsgn4
2
)(
28
To shed more light on the last equation, let us introduce the concept of the
Hilbert transform.
Definition
A Hilbert transform filter is a filter that simply phase shifts all frequency
components by π/2 radians, i.e., its transfer function is described by
where the signum function is defined by
Observations:
1.
2.
fjfH sgn
0,1
0,0
0,1
sgn
f
f
f
f
1fH
0,2/
0,2/arg
f
ffH
29
In the time domain, the impulse response of the filter is given by
Therefore, the Hilbert transform of a function g(t) can be interpreted as the
convolution of the impulse response of the filter h(t) and g(t), i.e.,
In the frequency domain,
Therefore,
Using the exponential modulation theorem,
tth
1
)(
d
t
g
tgthtg
)(1
)()()(~
fGfjfG sgn~
fGfjtgF
sgn)(~
cc
Ftfj ffffjGetg c sgn)(~ 2
30
In the time domain, we can synthesize lower SSB mathematically as follows:
where is the Hilbert transform of m(t)
Likewise, the upper SSB signal is described by
tftmAtftmA
etmAj
etmAj
tftmA
ffffMffffMAF
ffMffMAF
fSFts
cccc
tfjc
tfjccc
ccccc
ccc
LSSB
LSSB
cc
2sin)(~2
12cos)(
2
1
)(~4
1)(~
4
12cos)(
2
1
sgnsgn4
1
4
1
)(
22
1
1
1
)(~ tm
tftmAtftmAts ccccUSSB 2sin)(~
2
12cos)(
2
1)(
31
A widely used form of SSB is vestigial sideband, which is used in TV broadcasts.
SSB is a special case of VSB, since the filter that is used to obtain SSB is a
limiting case of the filter used to generate VSB.
Generation of VSB
Instead of the ideal filter used in SSB, let us use the following filter:
can be analytically described by fH LVSB
cQcQL
VSB ffHffHjfH
32
The transfer function has the following characteristics:
Observations about :
• Odd symmetry about the origin
• Linear phase
Mathematically,
and
The ‘+’ sign means that a vestige of the upper sideband is transmitted and a ‘-’ sign means that a vestige of the lower sideband is transmitted.
fH Q
fH Q
tftmAtftmAts ccccLVSB 2sin)(
2
12cos)(
2
1)(
tftmAtftmAts ccccUVSB 2sin)(
2
12cos)(
2
1)(
33
Example: Now, if the magnitude spectrum of the DSB signal is as described
before, then the magnitude spectrum of is
Vestigial sideband magnitude spectrum (upper sideband vestige).
)(ts LVSB
34
To help in the demodulation process, let us insert a pilot signal to the VSB signal
or
where
Therefore, the complex envelope of this pilot aided VSB signal is
tfAtftmAtftmAkts ccccccaPVSB 2cos2sin)(
2
12cos)(
2
1)(
,2sin)(2
12cos)(
2
11)( tftmAktftmkAts ccacac
PVSB
tftmtftmts cQcIPVSB 2sin)(2cos)()(
)(2
1)(
)(2
11)(
tmAktm
tmkAtm
caQ
acI
,)()()()(~ )(tjQI
PVSB etatjmtmts
35
where the real envelope a(t) is described by
and its phase is
If we use an envelope detector to demodulate the pilot-aided VSB, its output would be
2/122
22
22
)(2
1)(
2
11
)()()(
tmkAtmkA
tmtmta
acac
QI
)(
)(tan)( 1
tm
tmt
I
Q
1/ 22 2
1/ 22
1 1( ) 1 ( ) ( )
2 2
1( )1 21 ( ) 1
12 1 ( )2
c a a
a
c a
a
a t A k m t k m t
k m tA k m t
k m t
36
Clearly, if m´(t) is small, then we can extract the message signal m(t) by using
a DC blocking capacitor.
In practice, is minimized by increasing the width of the vestigial
sideband.
Angle Modulation
Let i(t) be the instantaneous angle of a modulated sinusoidal carrier, i.e.,
where Ac is the constant amplitude.
The instantaneous frequency is
Observation: The signal s(t) can be thought of as a rotating phasor of length Ac
and angle i(t).
)(tm
),(cos)( tAts ic
.)(
)(dt
tdt i
i
37
If s(t) were an unmodulated carrier signal, then the instantaneous angle would be
where c Constant angular velocity in rad/s
c Constant but arbitrary phase angle in radians
Let i(t) be varied linearly with the message signal m(t) and = 0, then
where kp Phase sensitivity of the modulator in rad/volt
In this case we say that the carrier has been phase modulated.
The phase modulated waveform is given by
Let the instantaneous frequency i(t) be varied linearly with the message signal
m(t), i.e.,
cci tt )(
),()( tmktt pci
))(cos()( tmktAts pccPM
)()( tmkt ci
38
where k Frequency sensitivity of the modulator in rad/s/volt
In this case we say that the carrier has been frequency modulated and the
instantaneous angle is obtained by integrating the instantaneous frequency, i.e.,
The modulated waveform is therefore described by
Observation: Both phase and frequency modulation are related to each other and
one can be obtained from the other. Hence, we could deduce the properties of one
of the two modulation schemes once we know the properties of the other.
t
c
t
c
t
ii dmktdmkdt000
)()()()(
t
ccFM dmktAts0
)(cos)(
39
FM modulation
40
PM modulation
41
The figure below shows a comparison between AM, FM and PM modulation of the same message waveform:
42
Frequency Modulation
Consider the frequency modulation of a message signal (frequency tone)
The instantaneous frequency (in Hz) of the FM signal is
Define the maximum frequency deviation as
The instantaneous phase angle of the FM signal is
),2cos()( tfAtm mm
).2cos()( tfAkftf mmfci
.mf Akf
)2sin(2
)2sin(2
)(2)(0
tftf
tff
Aktf
dft
mc
mm
mfc
t
ii
43
where is known as the FM modulation index (for a tone) or the
maximum phase deviation (in rad) produced by the tone in question
The FM modulated tone is therefore given by:
The signal sFM(t) is nonperiodic unless fc = nfm, where n is a positive integer.
For the general case,
where the complex envelope of the FM signal is described by
Observation: Unlike sFM(t), se(t) is periodic with period 1/fm.
,/ mmf fAk
( ) cos 2 sin(2 )
cos(2 )cos( sin(2 )) sin(2 )sin( sin(2 )) .
FM c c m
c c m c m
s t A f t f t
A f t f t f t f t
,)(Re
Re)(2
)2sin(2
tfje
tftfjcFM
c
mc
ets
eAts
)2sin()( tfjce
meAts
44
Since se(t) meets the Dirichlet conditions, we can compute its Fourier series, i.e.,
where the Fourier series coefficients are given by
,)( 2 tfnj
nne
mects
.dtefA
dteeAf
dte)t(sf
f/T,dte)t(sT
c
m
m
mm
m
m
mm
m
m
m
m
f/
f/
tfn)tfsin(jmc
f/
f/
tfnj)tfsin(jcm
f/
f/
tfnjem
m
/T
/T
tfnjen
21
21
22
21
21
22
21
21
2
2
2
2 11
45
Let
Then,
and
where is the Bessel function of the first kind of order n.
Therefore,
and the FM tone waveform is described by
.tfx m2
dxf
dtm2
1
),(JA
dxeA
c
nc
nxxsinjcn
2
dxe)(J nxxsinj
n 2
1
n
tnfjnce
meJAts 2)()(
.)(2cos)()(
n
mcncFM tnffJAts
46
In the frequency domain,
Average power of the FM waveform:
The power delivered to a 1 ohm resistor load by the FM waveform is
But,
is also the power of the FM waveform.
( ) ( )
( ) cos 2 ( )
( ) cos 2 ( )
( )( ) ( )
2
FM FM
c n c mn
c n c mn
nc c m c m
n
S f s t
A J f nf t
A J F f nf t
JA f f nf f f nf
2/2cAP
n
nc J
AP )(
22
2
47
48
Let us now take a look at the properties of the Bessel function.
1.
2.
Hence, the average power of an FM tone is
Suppose is small, i.e., 0 < ≤ 0.3, then
•
•
•
Under the assumption that is small, the Fourier series representation of the FM
waveform can be simplified to three terms.
)()1()( nn
n JJ
1)(2
nnJ
.2/2cA
1)(0 J
2/)(1 J
2,0)( nJ n
49
Thus, for small, the FM tone may be described by
In the frequency domain,
)t)ff(cos(A)t)ff(cos(A)tfcos(A
)t)ff(cos()t)ff(cos()tfcos(A)t(s
mccmcccc
mcmcccFM
22
22
2
22
22
2
mcmcc
mcmcc
ccc
FM
ffffffA
ffffffA
ffffA
fS
4
42)(
50
A plot of the magnitude spectrum of the FM tone with small is shown below
The time domain FM waveform can be represented in phasor form as follows:
For arbitrary t = t0, and small , we can illustrate graphically the phasor
representation and arrive at some conclusion.
tfAtfAAS mcmccFM 22
12
2
10
51
The following figure shows an example of the phasor representation
Observation:
The resultant phasor , has magnitude and is out of phase with
respect to the carrier phasor
Analytically,
FMS
,cFM AS
.0cA
)2sin()2cos()2sin()2cos(2
1 tfjtftfjtfAAS mmmmccFM
52
But,
and
Consequently, the resultant phasor is given by
The magnitude of the resultant may be approximated by
since
)2cos(
)2sin(sincos)2cos()2cos(0
tf
tftftf
m
mmm
)2sin(
)2cos(sincos)2sin()2sin(0
tf
tftftf
m
mmm
)2sin( tfjAAS mccFM
,)2(sin2
11
)2(sin
22
2222
tfA
tfAAS
mc
mccFM
1,1)1( nxnxx n
53
Finally, the magnitude and phase of the resultant are found to be
Observation:
• For an FM tone, the spectral lines sufficiently away from the carrier may be
ignored because their contribution (amplitude) is very small.
FM Transmission Bandwidth:
For an FM tone, as becomes large Jn() has significant lines only for
All significant lines are contained in the frequency range
where f is the peak frequency deviation.
)4cos(
441)(
22
tfAtS mcFM
)2sin(tan)( 1 tftS mSFM
.// mmmf fffAkn
,ffff cmc
54
Let be small, i.e., 0 < ≤ 0.3, then
which means that only the first pair of spectral lines is significant, i.e., the significant
lines are contained in the range fc ± fm
Observation: The previous analysis of an FM tone suggests that
1. For large the FM bandwidth is
2. For small the FM bandwidth is
In general, the FM transmission bandwidth may be approximated by
This is known as the Carson’s rule.
Observation: Carson’s rule underestimates the transmission bandwidth by about
10%.
0),()(0 nJJ n
fBFM 2
.2 mFM fB
)/11(2
)/1(2
22
f
fff
ffB
m
mT
55
Alternative definition of FM tone transmission bandwidth:
A band of frequencies that keeps all spectral lines whose magnitudes are greater
than 1% of the unmodulated carrier amplitude Ac, i.e.,
where
,2 max mT fnB
.01.0)(:maxmax nJnn
56
General Case:
Let an arbitrary message signal m(t) have bandwidth Wm.
Define the peak frequency deviation and the deviation ratio by
and
Then Carson’s rule can be used to define the transmission bandwidth of an
arbitrary FM signal, i.e., when m(t) is arbitrary.
Specifically, the FM transmission bandwidth can be defined by
)/11(2
)/1(2
22
Df
fWf
WfB
m
mT
)(maxˆ tmkft
f
./ˆ mWfD
57
Example: In commercial FM in the US, f = 75 kHz, Wm = 15 kHz.
Therefore, the deviation ratio is D = 75 kHz/15 kHz = 5.
Using Carson’s rule, the transmission bandwidth is
Using the Universal curve, the transmission bandwidth is
In practice, FM radio in the US uses a transmission bandwidth of BT = 200 kHz.
Generation of FM
The frequency of the carrier can be varied by the modulating signal m(t) directly or
indirectly.
,180)/11(2 kHzDfBT
.kHzf.BT 24023
58
Direct generation of FM
If a very high degree of stability of the carrier frequency is not a concern, then we
can generate FM directly using circuits without external crystal oscillators. Examples
of this method are VCO’s, varactor diode modulators, reactance modulators, Crosby
modulators (modulators that use automatic frequency control), etc..
Reactance FM modulator
m(t)
+
-
R2
R1
R3
C1
RFC1
C2
R2 C3
R6
R5RFC2
R7 C4
C7
C6
C5
L1 L2
+
-
+VCC
)(tsFM
59
Indirect generation of FM
Commercial applications of FM (as established by the FCC and other spectrum
governing bodies) require a high degree of stability of the carrier frequency. Such
restrictions can be satisfied by using external crystal oscillators, a narrowband
phase modulator, several stages of frequency multiplication and mixers.
Let us begin with the synthesis of narrow-band FM.
Narrowband frequency modulator
The narrow band FM signal is given by
t
fccNB dmktfAts0
)(22cos)(
60
with kf (and thus fNB) small
Let us now consider a technique to increase the FM signal bandwidth.
Let sNB(t) be input to a nonlinear device with transfer characteristic y(t) = axn(t),
where x(t) is its input, namely.
Nonlinear device.
Let , then at the output of the nonlinear device, we
observe
Let us expand this last equation to infer the effect of this nonlinear device.
t
fci dmktft0
)(22)(
)(cos)( taAty inn
c
61
cosni(t) can be expanded as follows:
Likewise,
Thus,
Expanding the last term of the previous equality, we get
)(cos)(2cos2
1)(cos
2
1
)(cos)(2cos12
1
)(cos)(cos)(cos
22
2
22
ttt
tt
ttt
in
iin
in
i
in
iin
)(cos)(2cos2
1)(cos
2
1)(cos 442 tttt i
nii
ni
n
)(cos)(2cos4
1)(cos)(2cos
4
1)(cos
2
1)(cos 4242 tttttt i
nii
nii
ni
n
).(cos)(4cos14
1)(cos)(2cos 442 tttt i
nii
ni
62
Rewriting the equation before the last one, we get
The last term in the expansion of cosni(t) is given by
)(cos)(6cos32
1
)(cos)(2cos32
1)(cos)(4cos
16
1
)(cos)(2cos4
1)(cos
8
1)(cos
2
1
)(cos)(4cos8
1)(cos
8
1
)(cos)(2cos4
1)(cos
2
1)(cos
6
66
442
44
42
tt
tttt
tttt
ttt
tttt
in
i
in
iin
i
in
iin
in
in
iin
in
iin
in
).(cos)(cos2
11
ttk ikn
ik
63
Let n be an even number, then, when k = n, the last term is
If, on the other hand, n is an odd number, then when k = n-1, the last term is
Therefore, the last term in the expansion of cosni(t) is
So, can be expanded as
)(cos2
11
tn in
)(cos2
1)()2cos(
2
1)(cos)()1cos(
2
1112
tntnttn ininiin
)(cos2
11
tn in
)(cos2
)(2cos)(cos)(1210 tn
Aatctccty in
nc
ii
64
Example: Consider the cases when n = 2 and n = 3.
Let n = 2, then
or
Let n = 3, then
)(cos)( 22 taAty ic
)(2cos222
)(2cos1)(
222 t
aAaAtaAty i
ccic
)(3cos4
)(cos4
3
)(cos2
1)(3cos
2
1
2
1)(cos
2
1
)(cos)(2cos2
1)(2cos
2
1)(
33
3
3
taA
taA
tttaA
tttaAty
ic
ic
iiic
iiic
65
Finally,
Let y(t) be input to an ideal bandpass filter with unity gain, bandwidth wide enough to accommodate spectrum of wide band signal and center frequency nfc, i.e.,
Ideal bandpass filter
Then,
t
fcn
nc
t
fc
t
fc
dmkntfnA
a
dmktfcdmktfccty
01
0
2
0
10
)(22cos2
)(44cos)(22cos)(
t
fcn
nc
WB dmkntfnaA
ts0
1)(22cos
2)(
66
The instantaneous frequency of sWB(t) is
Observations about sWB(t) :
1. The carrier frequency is nfc
2. The peak frequency deviation is nfNB
These are the desired properties of the WB FM signal.
The overall frequency multiplier device is shown below:
Complete frequency multiplier
)()( tmnknftf fci
67
Example: Noncommercial FM broadcast in the US uses the 88-90 MHz band and
commercial FM broadcast uses the 90-108 MHz band (divided into 200 kHz
channels). In either case f = 75 kHz. Suppose we target a station with fc = 90.1
MHz. Then the indirect FM generation method suggested by Armstrong enables
us to achieve our goals.
68
Armstrong indirect method of FM generation
69
Demodulation of FM signals
Consider the following receiver architecture
Frequency discriminator implementation of an FM demodulator
The slope circuit is characterized by a purely imaginary transfer function
Let H1(f ) be described by
.2,1),( isH i
elsewhere,0
2/2/),2/(2
2/2/),2/(2
)(1 TcTcTc
TcTcTc
BffBfBffaj
BffBfBffaj
fH
70
where a > 0 is a constant that determines the slope of H1(s ).
Define G1(f ) H1(f )/j, then g1(t ) is the impulse response of a real bandpass
system described by G1(f ).
In the time domain,
where g1,I(t) and g1,Q(t) are the in-phase and quadrature components of g1(t).
Therefore, the complex envelope of g1(t) is described by
which implies that
Using this information, we get
)2cos()()2cos()()( ,1,11 tftgtftgtg cQcI
)()()(~,1,11 tjgtgtg QI
tfj cetgtg 211 )(~Re)(
)(2
)2sin()(2)2cos()(2
)()()()()(~)(~
1
,1,1
2,1,1
2,1,1
21
21
tg
tftgtftg
etjgtgetjgtgetgetg
cQcI
tfjQI
tfjQI
tfjtfj cccc
71
But,
or
which implies that has a lowpass frequency response limited to
and that
Observations: can be obtained by taking the part of G1(f) that corresponds
to positive frequencies, shifting it to the origin and then scaling it by a factor of 2.
In the next figure is replaced by and replaced by
tfjtfj cc etgetgFtgF 21
211 )(~)(~)(2
))((~
)(~
)(2 111 cc ffGffGfG
)(~
1 fG 2/TBf
.0),()(~
11 ffGffG c
)(~
1 fG
)(1 fG jfH /)(1 )(~
1 fG
./)(~
1 jfH
72
Frequency responses of H1(f ), and H2(f )),(
~1 fH
73
From the previous derivation,
But,
If s1(t) is the output of the slope filter H1(f ) when the input is sFM(t) then the complex
envelope of the output is
elsewhere,0
2/),2/(4)(
~1
TT BfBfajfH
.)(~Re
Re
)(cos)(
)(
tfjFM
tfjdmkjc
t
fccFM
c
ct
f
ets
eeA
dmktfAts
2
22
0
0
22
)(~)(~
2
1)(~
11 tsthts FM
74
In the frequency domain,
But,
which implies that
elsewhere
BffSBfaj
fSfHfS
TFMT
FM
,0
2/),(~
)2/(2
)(~
)(~
2
1)(
~11
)(2)(
ffXjdt
tdxF
tf
tf
tf
dmkj
T
fTc
dmkjTc
dmkjfc
FMTFM
etmB
kBaAj
eBaAetmkaAj
tsBjadt
tsdats
0
00
)(2
)(2)(2
1
)(2
1
2)(2
)(~)(~)(~
75
Therefore,
Observation: s1(t ) contains both AM and FM.
However, if
2)(22cos)(
21
)(22sin)(2
1
)(2
1Re
)(~Re)(
0
0
)(22
211
0
t
fcT
fTc
t
fcT
fTc
dmkjtfj
T
fTc
tfj
dmktftmB
kBaA
dmktftmB
kBaA
etmB
kBaAj
etsts
tfc
c
ttmB
k
T
f ,1)(2
76
Then a distortionless envelope detector can extract m(t) plus a bias, i.e.,
Finally, if
then
Moreover,
The cascade of a slope circuit and an envelope detector is known as a frequency
discriminator.
.)(2
1)(1
tm
B
kBaAts
T
fTce
)(~
)(~
12 fHfH
.)(2
1)(2
tm
B
kBaAts
T
fTce
)(4
)()()(ˆ 21
tmaA
tststm
c
ee
77
Frequency discriminator
FM demodulation via phase-locked loops
We consider the Phase-locked loop (PLL) FM detector shown below
Phase-locked loop FM detector
78
From the previous block diagram,
Let the phase detector be described by
then,
))(sin()(
))(cos()(
0 ttAte
ttAtx
cv
ccr
)()(sin)()(2sin2
))(sin())(cos()(1
tttttAAk
kttAttAte
cvcd
dcvcc
79
with the proper choice of lowpass filter, the output of the phase detector is
A VCO is an FM modulator with peak frequency deviation
where implies that
An equivalent nonlinear model is now shown
Nonlinear model of PLL FM demodulator
)()(sin2
)( ttAAk
te vcdd
dt
tdf
t
)(max
)(ˆ)(
tmkdt
tdvco
.)(ˆ)(
0t
vco dmkt
80
Let the PLL operate in the lock condition, i.e., , or that
is small. Then,
and the linear approximation of the PLL is given by
Linear model of the phase-locked loop
Let the loop filter have the transfer function HLF(s) = 1, then
)()( tt )()( tt
)()()()(sin tttt
)()(
)()(2
)(ˆ
ttk
ttAAk
tm
t
vcd
81
Thus, the output of the vco is given by
Let k0 = ktkvco, then
or
In the s-domain, assuming zero initial conditions,
The closed-loop transfer function is therefore given by
t
vcot
t
vco dkkdmkt00
)()()(ˆ)(
)()()(
0 ttkdt
td
),()()(
00 tktkdt
td
),()()( 00 sksks
0
0
)(
)()(
ks
k
s
ssH cl
82
The corresponding impulse response is
Let us now find out what happens when the loop gain k0 is increased, i.e.,
Clearly,
or , for large k0.
Example: Let the message be a step function, i.e., m(t) = Au(t), then
In this case,
In the s-domain, and
)()( 00 tuekth tk
cl
1lim)(lim0
0
00
ks
ksH
kcl
k
)()( ss
)()( tt
t
ccFM dAuktAtx0
)(cos)(
tAkduAktt
0
)()(
2)(
s
Aks )(
)(0
20
kss
kAks
83
The Laplace transform of is then given by
Let k1 = kω/kvco, then in the time domain,
Clearly, as t , the estimate
Observation: This result is valid when the initial phase error is small.
Remark: A large loop gain k0 results in practical difficulties, hence, a different loop
filter has to be used.
Consider the loop filter described by
Then the output of the vco is given by
)(ˆ tm
.)(
)()()(ˆ
0000
0
11111
kssk
Ak
ksskkAk
kss
kAkssksM
vco
tt
)()()()()(ˆ 001111 tueAktmktueAktuAktm tktk
).()(ˆ 1 tmktm
0,/)( asassH LF
)()()(
)( 0 sss
sHks LF
84
Let be small, then the closed-loop transfer function is
Define then
where
Consider again the step function message m(t) = Au(t). Then
)()( tt
aksks
ask
sHks
sHk
s
ssH
LF
LFcl
002
0
0
0
)(
)(
)(
)()(
),()(ˆ)( sss
)(2
)()()(
)()()(
22
2
002
2
0
0 sss
ss
aksks
ss
sHks
sHkss
nnLF
LF
akn 0
02 kn
a
k
ak
kk
n
0
0
00
2
1
22
tAkdAkdAukttt
00
)()(
85
In the complex frequency domain,
Let kA, then (s) = /s2
and
If 0 < < 1, then
and
Hence the steady-state phase error is zero.
A typical value of is 0.707.
2)(
s
Aks
22 2)(
nn sss
tet nt
n
n 2
21sin
1)(
.0)(lim
tt
86
Superheterodyne Receiver
Definition: To heterodyne means to combine a radio frequency wave with a locally
generated wave of different frequency in order to produce a new frequency equal to
the sum or difference of the two.
Specifically, a superheterodyne receiver is one that performs the operations of
carrier frequency tuning of the desired signal, filtering it to separate it from
unwanted signals, in most instances, amplifying it to compensate for signal power
loss due to propagation medium.
Generic superheterodyne receiver
87
Example: Let m(t) modulate a sinusoidal carrier with frequency fc = 10 MHz. Let the
bandwidth of the modulated carrier be BT = 200 kHz and let fIF = 1 MHz. Let the
local oscillator local frequency be fLO = 11 MHz and let an interferer have its
spectrum located at 11.95 ≤ f ≤ 12.05 MHz. If no filter is used in the RF section,
then, at the output of the RF filter and of the mixer we have (for f 0 ).
Spectra when no RF filter is used