1. 1
For a deterministic signal x(t), the spectrum is well defined: If
represents its Fourier transform, i.e., if
then represents its energy spectrum. This follows from
Parseval’s theorem since the signal energy is given by
Thus represents the signal energy in the band
(see Fig 18.1).
( )
X
( ) ( ) ,
j t
X x t e dt
2
| ( ) |
X
2 2
1
2
( ) | ( ) | .
x t dt X d E
(18-1)
(18-2)
2
| ( ) |
X
( , )
Fig 18.1
18. Power Spectrum
t
0
( )
X t
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0
2
| ( )|
X
Energy in ( , )
2. 2
However for stochastic processes, a direct application of (18-1)
generates a sequence of random variables for every Moreover,
for a stochastic process, E{| X(t) |2} represents the ensemble average
power (instantaneous energy) at the instant t.
To obtain the spectral distribution of power versus frequency for
stochastic processes, it is best to avoid infinite intervals to begin with,
and start with a finite interval (– T, T ) in (18-1). Formally, partial
Fourier transform of a process X(t) based on (– T, T ) is given by
so that
represents the power distribution associated with that realization based
on (– T, T ). Notice that (18-4) represents a random variable for every
and its ensemble average gives, the average power distribution
based on (– T, T ). Thus
.
( ) ( )
T j t
T T
X X t e dt
2 2
| ( ) | 1
( )
2 2
T j t
T
T
X
X t e dt
T T
(18-3)
(18-4)
,
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3. 3
represents the power distribution of X(t) based on (– T, T ). For wide
sense stationary (w.s.s) processes, it is possible to further simplify
(18-5). Thus if X(t) is assumed to be w.s.s, then
and (18-5) simplifies to
Let and proceeding as in (14-24), we get
to be the power distribution of the w.s.s. process X(t) based on
(– T, T ). Finally letting in (18-6), we obtain
T
1 2
( )
1 2 1 2
1
( ) ( ) .
2
T XX
T T j t t
T T
P R t t e dt dt
T
1 2
1 2
2
( )
*
1 2 1 2
( )
1 2 1 2
| ( ) | 1
( ) { ( ) ( )}
2 2
1
( , )
2
T
XX
T T j t t
T
T T
T T j t t
T T
X
P E E X t X t e dt dt
T T
R t t e dt dt
T
2
2
2 | |
2
2
1
( ) ( ) (2 | |)
2
( ) (1 ) 0
T XX
XX
T j
T
T j
T
T
P R e T d
T
R e d
1 2 1 2
( , ) ( )
XX XX
R t t R t t
1 2
t t
(18-5)
(18-6)
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4. 4
to be the power spectral density of the w.s.s process X(t). Notice that
i.e., the autocorrelation function and the power spectrum of a w.s.s
Process form a Fourier transform pair, a relation known as the
Wiener-Khinchin Theorem. From (18-8), the inverse formula gives
and in particular for we get
From (18-10), the area under represents the total power of the
process X(t), and hence truly represents the power
spectrum. (Fig 18.2).
( ) lim ( ) ( ) 0
XX T XX
j
T
S P R e d
F T
( ) ( ) 0.
XX XX
R S
1
2
( ) ( )
XX XX
j
R S e d
2
1
2 ( ) (0) {| ( ) | } ,
XX XX
S d R E X t P the total power.
0,
( )
XX
S
( )
XX
S
(18-9)
(18-8)
(18-7)
(18-10)
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5. 5
Fig 18.2
The nonnegative-definiteness property of the autocorrelation function
in (14-8) translates into the “nonnegative” property for its Fourier
transform (power spectrum), since from (14-8) and (18-9)
From (18-11), it follows that
( ) ( ) 0.
XX XX
R nonnegative- definite S
( )
* *
1 1 1 1
2
1
1
2
1
2
( ) ( )
( ) 0.
i j
XX XX
i
XX
n n n n
j t t
i j i j i j
i j i j
n j t
i
i
a a R t t a a S e d
S a e d
(18-11)
(18-12)
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0
represents the power
in the band ( , )
( )
XX
S
( )
XX
S
6. 6
If X(t) is a real w.s.s process, then so that
so that the power spectrum is an even function, (in addition to being
real and nonnegative).
( ) = ( )
XX XX
R R
0
( ) ( )
( )cos
2 ( )cos ( ) 0
XX XX
XX
XX XX
j
S R e d
R d
R d S
(18-13)
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7. 7
Power Spectra and Linear Systems
If a w.s.s process X(t) with autocorrelation
function is
applied to a linear system with impulse
response h(t), then the cross correlation
function and the output autocorrelation function are
given by (14-40)-(14-41). From there
But if
Then
since
( )
XY
R ( )
YY
R
( ) ( ) 0
XX XX
R S
h(t)
X(t) Y(t)
Fig 18.3
* *
( ) ( ) ( ), ( ) ( ) ( ) ( ).
XY XX YY XX
R R h R R h h
(18-14)
( ) ( ), ( ) ( )
f t F g t G
(18-16)
(18-15)
( ) ( ) ( ) ( )
f t g t F G
{ ( ) ( )} ( ) ( ) j t
f t g t f t g t e dt
F
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8. 8
(18-20)
(18-19)
(18-18)
Using (18-15)-(18-17) in (18-14) we get
since
where
represents the transfer function of the system, and
( )
{ ( ) ( )}= ( ) ( )
= ( ) ( ) ( )
= ( ) ( ).
j t
j j t
f t g t f g t d e dt
f e d g t e d t
F G
F
(18-17)
* *
( ) { ( ) ( )} ( ) ( )
XY XX XX
S R h S H
F
*
* *
( ) ( ) ( ),
j j t
h e d h t e dt H
( ) ( ) j t
H h t e dt
2
( ) { ( )} ( ) ( )
( ) | ( ) | .
YY YY XY
XX
S R S H
S H
F
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9. 9
From (18-18), the cross spectrum need not be real or nonnegative;
However the output power spectrum is real and nonnegative and is
related to the input spectrum and the system transfer function as in
(18-20). Eq. (18-20) can be used for system identification as well.
W.S.S White Noise Process: If W(t) is a w.s.s white noise process,
then from (14-43)
Thus the spectrum of a white noise process is flat, thus justifying its
name. Notice that a white noise process is unrealizable since its total
power is indeterminate.
From (18-20), if the input to an unknown system in Fig 18.3 is
a white noise process, then the output spectrum is given by
Notice that the output spectrum captures the system transfer function
characteristics entirely, and for rational systems Eq (18-22) may be
used to determine the pole/zero locations of the underlying system.
( ) ( ) ( ) .
WW WW
R q S q
(18-22)
(18-21)
2
( ) | ( ) |
YY
S q H
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10. 10
Example 18.1: A w.s.s white noise process W(t) is passed
through a low pass filter (LPF) with bandwidth B/2. Find the
autocorrelation function of the output process.
Solution: Let X(t) represent the output of the LPF. Then from (18-22)
Inverse transform of gives the output autocorrelation function
to be
2 , | | / 2
( ) | ( ) | .
0, | | / 2
XX
q B
S q H
B
( )
XX
S
/ 2 / 2
/ 2 / 2
( ) ( )
sin( /2)
sinc( /2)
( /2)
XX XX
B B
j j
B B
R S e d q e d
B
qB qB B
B
(18-23)
(18-24)
PILLAI
Fig. 18.4
(a) LPF
2
| ( )|
H
/2
B
/ 2
B
1
qB
( )
XX
R
(b)
11. 11
Eq (18-23) represents colored noise spectrum and (18-24) its
autocorrelation function (see Fig 18.4).
Example 18.2: Let
represent a “smoothing” operation using a moving window on the input
process X(t). Find the spectrum of the output Y(t) in term of that of X(t).
Solution: If we define an LTI system
with impulse response h(t) as in Fig 18.5,
then in term of h(t), Eq (18-25) reduces to
so that
Here
1
2
( ) ( )
t T
t T
T
Y t X d
(18-25)
( ) ( ) ( ) ( ) ( )
Y t h t X d h t X t
2
( ) ( ) | ( ) | .
YY XX
S S H
1
2
( ) sinc( )
T j t
T
T
H e dt T
(18-28)
(18-27)
(18-26)
PILLAI
Fig 18.5
T
T t
( )
h t
1/ 2T
12. 12
so that
2
( ) ( ) sinc ( ).
YY XX
S S T
(18-29)
Notice that the effect of the smoothing operation in (18-25) is to
suppress the high frequency components in the input
and the equivalent linear system acts as a low-pass filter (continuous-
time moving average) with bandwidth in this case.
PILLAI
(beyond / ),
T
Fig 18.6
( )
XX
S
2
sinc ( )
T
T
( )
YY
S
2 /T
13. 13
Discrete – Time Processes
For discrete-time w.s.s stochastic processes X(nT) with
autocorrelation sequence (proceeding as above) or formally
defining a continuous time process we get
the corresponding autocorrelation function to be
Its Fourier transform is given by
and it defines the power spectrum of the discrete-time process X(nT).
From (18-30),
so that is a periodic function with period
{ } ,
k
r
( ) ( ) ( ),
n
X t X nT t nT
( ) ( ).
XX k
k
R r kT
(18-30)
( ) 0,
XX
j T
k
k
S r e
( )
XX
S
( ) ( 2 / )
XX XX
S S T
2
2 .
B
T
(18-31)
(18-32)
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14. 14
This gives the inverse relation
and
represents the total power of the discrete-time process X(nT). The
input-output relations for discrete-time system h(nT) in (14-65)-(14-67)
translate into
and
where
represents the discrete-time system transfer function.
1
( )
2
XX
B jk T
k B
r S e d
B
(18-33)
2
0
1
{| ( ) | } ( )
2
XX
B
B
r E X nT S d
B
(18-34)
*
( ) ( ) ( )
XY XX
j
S S H e
2
( ) ( ) | ( ) |
YY XX
j
S S H e
( ) ( )
j j nT
n
H e h nT e
(18-35)
(18-37)
(18-36)
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15. 15
Matched Filter
Let r(t) represent a deterministic signal s(t) corrupted by noise. Thus
where r(t) represents the observed data,
and it is passed through a receiver
with impulse response h(t). The
output y(t) is given by
where
and it can be used to make a decision about the presence of absence
of s(t) in r(t). Towards this, one approach is to require that the
receiver output signal to noise ratio (SNR)0 at time instant t0 be
maximized. Notice that
h(t)
r(t)
y(t)
Fig 18.7 Matched Filter
0
t t
0
( ) ( ) ( ), 0
r t s t w t t t
(18-38)
(18-39)
( ) ( ) ( ), ( ) ( ) ( ),
s
y t s t h t n t w t h t
(18-40)
PILLAI
( ) ( ) ( )
s
y t y t n t
16. 16
represents the output SNR, where we have made use of (18-20) to
determine the average output noise power, and the problem is to
maximize (SNR)0 by optimally choosing the receiver filter
Optimum Receiver for White Noise Input: The simplest input
noise model assumes w(t) to be white noise in (18-38) with spectral
density N0, so that (18-41) simplifies to
and a direct application of Cauchy-Schwarz’ inequality
in (18-42) gives
(18-41)
( ).
H
0
2
0
2
0
( ) ( )
( )
2 | ( ) |
j t
S H e d
SNR
N H d
(18-42)
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0
2
0 0
0 2
2
2
0
2
1
2
1 1
2 2
Output signal power at | ( ) |
( )
Average output noise power {| ( ) | }
( ) ( )
| ( ) |
( ) ( ) | ( ) |
nn WW
s
j t
s
t t y t
SNR
E n t
S H e d
y t
S d S H d
17. 17
and equality in (18-43) is guaranteed if and only if
or
From (18-45), the optimum receiver that maximizes the output SNR
at t = t0 is given by (18-44)-(18-45). Notice that (18-45) need not be
causal, and the corresponding SNR is given by (18-43).
0
2
2 0
0
0 0
1
2
( )
( ) | ( ) | s
N
s t dt E
SNR S d
N N
(18-43)
0
*
( ) ( ) j t
H S e
0
( ) ( ).
h t s t t
(18-44)
(18-45)
PILLAI
Fig 18.8
(a) (b) t0=T/2 (c) t0=T
( )
h t
( )
s t ( )
h t
T
t
t
t
T
/ 2
T
t0
Fig 18-8 shows the optimum h(t) for two different values of t0. In Fig
18.8 (b), the receiver is noncausal, whereas in Fig 18-8 (c) the
receiver represents a causal waveform.
18. 18
If the receiver is not causal, the optimum causal receiver can be
shown to be
and the corresponding maximum (SNR)0 in that case is given by
Optimum Transmit Signal: In practice, the signal s(t) in (18-38) may
be the output of a target that has been illuminated by a transmit signal
f (t) of finite duration T. In that case
where q(t) represents the target impulse response. One interesting
question in this context is to determine the optimum transmit
(18-48)
0
( ) ( ) ( ) ( ) ( ) ,
T
s t f t q t f q t d
(18-47)
(18-46)
0
( ) ( ) ( )
opt
h t s t t u t
0
0
2
0 0
1
( ) ( )
t
N
SNR s t dt
( )
f t
T t
q(t)
( )
f t ( )
s t
Fig 18.9
PILLAI
19. 19
signal f (t) with normalized energy that maximizes the receiver output
SNR at t = t0 in Fig 18.7. Notice that for a given s(t), Eq (18-45)
represents the optimum receiver, and (18-43) gives the corresponding
maximum (SNR)0. To maximize (SNR)0 in (18-43), we may substitute
(18-48) into (18-43). This gives
where is given by
and is the largest eigenvalue of the integral equation
0
1 2
0
2
0 1 1 1
0 0
*
1 2 2 2 1 1
0 0 0
( , )
1 2 2 2 1 1 max 0
0 0
1
1
( ) |{ ( ) ( ) }|
( ) ( ) ( ) ( )
{ ( , ) ( ) } ( ) /
T
T T
T T
N
N
SNR q t f d dt
q t q t dt f d f d
f d f d N
max
1 2
( , )
*
1 2 1 2
0
( , ) ( ) ( )
q t q t dt
(18-49)
(18-50)
1 2 2 2 max 1 1
0
( , ) ( ) ( ), 0 .
T
f d f T
(18-51)
PILLAI
20. 20
PILLAI
If the causal solution in (18-46)-(18-47) is chosen, in that case the
kernel in (18-50) simplifies to
and the optimum transmit signal is given by (18-51). Notice
that in the causal case, information beyond t = t0 is not used.
and
Observe that the kernal in (18-50) captures the target
characteristics so as to maximize the output SNR at the observation
instant, and the optimum transmit signal is the solution of the integral
equation in (18-51) subject to the energy constraint in (18-52).
Fig 18.10 show the optimum transmit signal and the companion receiver
pair for a specific target with impulse response q(t) as shown there .
2
0
( ) 1.
T
f t dt
(18-52)
1 2
( , )
0 *
1 2 1 2
0
( , ) ( ) ( ) .
t
q t q t dt
(18-53)
t
(a)
( )
q t
(b)
t
T
( )
f t
0
t
( )
h t
t
(c)
Fig 18.10
21. 21
What if the additive noise in (18-38) is not white?
Let represent a (non-flat) power spectral density. In that case,
what is the optimum matched filter?
If the noise is not white, one approach is to whiten the input
noise first by passing it through a whitening filter, and then proceed
with the whitened output as before (Fig 18.7).
Notice that the signal part of the whitened output sg(t) equals
where g(t) represents the whitening filter, and the output noise n(t) is
white with unit spectral density. This interesting idea due to
( )
WW
S
( ) ( ) ( )
g
s t s t g t
(18-54)
PILLAI
Whitening Filter
g(t)
( ) ( ) ( )
r t s t w t
( ) ( )
g
s t n t
Fig 18.11
colored noise white noise
22. 22
Whitening Filter: What is a whitening filter? From the discussion
above, the output spectral density of the whitened noise process
equals unity, since it represents the normalized white noise by design.
But from (18-20)
which gives
i.e., the whitening filter transfer function satisfies the magnitude
relationship in (18-55). To be useful in practice, it is desirable to have
the whitening filter to be stable and causal as well. Moreover, at times
its inverse transfer function also needs to be implementable so that it
needs to be stable as well. How does one obtain such a filter (if any)?
[See section 11.1 page 499-502, (and also page 423-424), Text
for a discussion on obtaining the whitening filters.].
( )
nn
S
2
1 ( ) ( ) | ( ) | ,
WW
nn
S S G
2 1
| ( ) | .
( )
WW
G
S
(18-55)
PILLAI
( )
G
Wiener has been exploited in several other problems including
prediction, filtering etc.
23. 23
From there, any spectral density that satisfies the finite power constraint
and the Paley-Wiener constraint (see Eq. (11-4), Text)
can be factorized as
where H(s) together with its inverse function 1/H(s) represent two
filters that are both analytic in Re s > 0. Thus H(s) and its inverse 1/ H(s)
can be chosen to be stable and causal in (18-58). Such a filter is known
as the Wiener factor, and since it has all its poles and zeros in the left
half plane, it represents a minimum phase factor. In the rational case,
if X(t) represents a real process, then is even and hence (18-58)
reads
( )
XX
S d
(18-56)
(18-57)
2
| log ( ) |
1
XX
S
d
2
( ) | ( ) | ( ) ( ) |
XX s j
S H j H s H s
(18-58)
( )
XX
S
PILLAI
24. 24
Example 18.3: Consider the spectrum
which translates into
The poles ( ) and zeros ( ) of this
function are shown in Fig 18.12.
From there to maintain the symmetry
condition in (18-59), we may group
together the left half factors as
2 2
0 ( ) ( ) | ( ) ( ) | .
XX XX s j s j
S S s H s H s
(18-59)
2 2 2
4
( 1)( 2)
( )
( 1)
XX
S
2 2 2
2
4
(1 )(2 )
( ) .
1
XX
s s
S s
s
Fig 18.12
2
s j
2
s j
1
s
1
s
1
2
j
s
1
2
j
s
1
2
j
s
1
2
j
s
2
2
1 1
2 2
( 1)( 2 )( 2 ) ( 1)( 2)
( )
2 1
j j
s s
s s j s j s s
H s
s s
PILLAI
25. 25
and it represents the Wiener factor for the spectrum
Observe that the poles and zeros (if any) on the appear in
even multiples in and hence half of them may be paired with
H(s) (and the other half with H(– s)) to preserve the factorization
condition in (18-58). Notice that H(s) is stable, and so is its inverse.
More generally, if H(s) is minimum phase, then ln H(s) is analytic on
the right half plane so that
gives
Thus
and since are Hilbert transform pairs, it follows that
the phase function in (18-60) is given by the Hilbert
axis
j
( )
XX
S
PILLAI
( )
( ) ( ) j
H A e
(18-60)
0
ln ( ) ln ( ) ( ) ( ) .
j t
H A j b t e dt
0
0
ln ( ) ( )cos
( ) ( )sin
t
t
A b t t dt
b t t dt
cos and sin
t t
( )
( ) above.
XX
S
26. 26
transform of Thus
Eq. (18-60) may be used to generate the unknown phase function of
a minimum phase factor from its magnitude.
For discrete-time processes, the factorization conditions take the form
(see (9-203)-(9-205), Text)
and
In that case
where the discrete-time system
( ) <
XX
S d
(18-63)
(18-62)
ln ( ) > .
XX
S d
2
( ) | ( ) |
XX
j
S H e
0
( ) ( ) k
k
H z h k z
PILLAI
ln ( ).
A
( ) {ln ( )}.
A
H (18-61)
27. 27
is analytic together with its inverse in |z| >1. This unique minimum
phase function represents the Wiener factor in the discrete-case.
Matched Filter in Colored Noise:
Returning back to the matched filter problem in colored noise, the
design can be completed as shown in Fig 18.13.
(Here represents the whitening filter associated with the noise
spectral density as in (18-55)-(18-58). Notice that G(s) is the
inverse of the Wiener factor L(s) corresponding to the spectrum
i.e.,
The whitened output sg(t) + n(t) in Fig 18.13 is similar
h0(t)=sg(t0 – t)
1
( ) ( )
G j L j
0
t t
( ) ( )
g
s t n t
( ) ( ) ( )
r t s t w t
Whitening Filter Matched Filter
Fig 18.13
( )
G j
( )
WW
S
( ).
WW
S
2
( ) ( ) | | ( ) | ( ).
WW
s j
L s L s L j S
(18-64)
PILLAI
28. 28
to (18-38), and from (18-45) the optimum receiver is given by
where
If we insist on obtaining the receiver transfer function for the
original colored noise problem, we can deduce it easily from Fig 18.14
Notice that Fig 18.14 (a) and (b) are equivalent, and Fig 18.14 (b) is
equivalent to Fig 18.13. Hence (see Fig 18.14 (b))
or
0 0
( ) ( )
g
h t s t t
1
( ) ( ) ( ) ( ) ( ) ( ).
g g
s t S G j S L j S
( )
H
( )
H
0
t t
L-1(s) L(s) ( )
H
0
t t
( )
r t
( )
r t
0 ( )
H
(a) (b)
Fig 18.14
0 ( ) ( ) ( )
H L j H
PILLAI
29. 29
turns out to be the overall matched filter for the original problem.
Once again, transmit signal design can be carried out in this case also.
AM/FM Noise Analysis:
Consider the noisy AM signal
and the noisy FM signal
where
0
0
1 1 *
0
1 1 *
( ) ( ) ( ) ( ) ( )
( ){ ( ) ( )}
j t
g
j t
H L j H L S e
L L S e
(18-65)
0
( ) ( )cos( ) ( ),
X t m t t n t
(18-66)
(18-67)
0
( ) cos( ( ) ) ( ),
X t A t t n t
0
( ) FM
( )
( ) PM.
t
c m d
t
cm t
(18-68)
PILLAI
30. 30
Here m(t) represents the message signal and a random phase jitter
in the received signal. In the case of FM, so that
the instantaneous frequency is proportional to the message signal. We
will assume that both the message process m(t) and the noise process
n(t) are w.s.s with power spectra and respectively.
We wish to determine whether the AM and FM signals are w.s.s,
and if so their respective power spectral densities.
Solution: AM signal: In this case from (18-66), if we assume
then
so that (see Fig 18.15)
( ) ( ) ( )
t t c m t
( )
mm
S ( )
nn
S
~ (0,2 ),
U
0
1
( ) ( )cos ( )
2
XX mm nn
R R R
(18-69)
0 0
( ) ( )
( ) ( ).
2
XX XX
XX nn
S S
S S
(18-70)
( )
mm
S
0
0
0
( )
XX
S
0
( )
mm
S
(a) (b)
Fig 18.15 PILLAI
31. 31
Thus AM represents a stationary process under the above conditions.
What about FM?
FM signal: In this case (suppressing the additive noise component in
(18-67)) we obtain
since
2
0
0
2
0
0
2
0
( / 2, / 2) {cos( ( / 2) ( / 2) )
cos( ( / 2) ( / 2) )}
{cos[ ( / 2) ( / 2)]
2
cos[2 ( / 2) ( / 2) 2 ]}
[ {cos( ( / 2) ( / 2))}cos
2
{sin( ( / 2) ( / 2))
XX
R t t A E t t
t t
A
E t t
t t t
A
E t t
E t t
0
}sin ]
(18-71)
PILLAI
0
0
0
{cos(2 ( / 2) ( / 2) 2 )}
{cos(2 ( / 2) ( / 2))} {cos2 }
{sin(2 ( / 2) ( / 2))} {sin 2 } 0.
E t t t
E t t t E
E t t t E
0
0
32. 32
Eq (18-71) can be rewritten as
where
and
In general and depend on both t and so that noisy FM
is not w.s.s in general, even if the message process m(t) is w.s.s.
In the special case when m(t) is a stationary Gaussian process, from
(18-68), is also a stationary Gaussian process with autocorrelation
function
for the FM case. In that case the random variable
( , )
a t ( , )
b t
2
0 0
( / 2, / 2) [ ( , )cos ( , )sin ]
2
XX
A
R t t a t b t
(18-72)
(18-74)
(18-73)
( )
t
2
2
2
( )
( ) ( )
mm
d R
R c R
d
(18-75)
PILLAI
( , ) {cos( ( /2) ( /2))}
a t E t t
( , ) {sin( ( /2) ( /2))}
b t E t t
33. 33
where
Hence its characteristic function is given by
which for gives
where we have made use of (18-76) and (18-73)-(18-74). On comparing
(18-79) with (18-78) we get
and
so that the FM autocorrelation function in (18-72) simplifies into
(18-76)
2
2( (0) ( )).
Y
R R
(18-77)
2
2 2 ( (0) ( ))
/2
{ } Y
R R
j Y
E e e e
(18-78)
1
{ } {cos } {sin } ( , ) ( , ),
jY
E e E Y jE Y a t jb t
(18-79)
( (0) ( ))
( , )
R R
a t e
(18-80)
( , ) 0
b t (18-81)
PILLAI
2
( /2) ( /2) ~ (0, )
Y
Y t t N
34. 34
Notice that for stationary Gaussian message input m(t) (or ), the
nonlinear output X(t) is indeed strict sense stationary with
autocorrelation function as in (18-82).
Narrowband FM: If then (18-82) may be approximated
as
which is similar to the AM case in (18-69). Hence narrowband FM
and ordinary AM have equivalent performance in terms of noise
suppression.
Wideband FM: This case corresponds to In that case
a Taylor series expansion or gives
( )
t
2
( (0) ( ))
0
( ) cos .
2
XX
R R
A
R e
(18-82)
(0) 1,
R
( 1 , | | 1)
x
e x x
2
0
( ) {(1 (0)) ( )}cos
2
XX
A
R R R
(18-83)
(0) 1.
R
( )
R
PILLAI
35. 35
and substituting this into (18-82) we get
so that the power spectrum of FM in this case is given by
where
Notice that always occupies infinite bandwidth irrespective
of the actual message bandwidth (Fig 18.16)and this capacity to spread
the message signal across the entire spectral band helps to reduce the
noise effect in any band.
( )
XX
S
2 2
2
2
(0)
0
( ) cos
2
c
mm
XX
R
A
R e
(18-85)
(18-86)
PILLAI
0 0
1
2
( ) ( ) ( )
XX
S S S
2
2 2
1
( ) (0) (0) (0) (0)
2 2
mm
c
R R R R R
(18-84)
(18-87)
0
0
( )
XX
S
Fig 18.16
2 2
2
/ 2 (0)
( ) .
2
mm
c R
A
S e
~
36. 36
Spectrum Estimation / Extension Problem
Given a finite set of autocorrelations one interesting
problem is to extend the given sequence of autocorrelations such that
the spectrum corresponding to the overall sequence is nonnegative for
all frequencies. i.e., given we need to determine
such that
Notice that from (14-64), the given sequence satisfies Tn > 0, and at
every step of the extension, this nonnegativity condition must be
satisfied. Thus we must have
Let Then
0 1
, , , ,
n
r r r
0 1
, , , ,
n
r r r 1 2
, ,
n n
r r
( ) 0.
jk
k
k
S r e
(18-88)
0, 1, 2, .
n k
T k
1 .
n
r x
(18-89)
PILLAI
37. 37
so that after some algebra
or
where
2 2 2
1
1 1
1
| |
det 0
n n n
n n
n
x
T
(18-90)
2
2
1
1
| | ,
n
n n
n
r
(18-91)
PILLAI
1
1
* * *
1 0
=
, ,
n
n n
n
x
r
T T
r
x r r r
Fig 18.17
1
n
r
(18-92)
1
1 1 2 1 1
, [ , , , ] , [ , , , ] .
T T T
n n n n n
a T b a r r r b r r r
38. 38
PILLAI
Eq. (18-91) represents the interior of a circle with center and radius
as in Fig 18.17, and geometrically it represents the admissible
set of values for rn+1. Repeating this procedure for it
follows that the class of extensions that satisfy (18-85) are infinite.
It is possible to parameterically represent the class of all
admissible spectra. Known as the trigonometric moment problem,
extensive literature is available on this topic.
[See section 12.4 “Youla’s Parameterization”, pages 562-574, Text for
a reasonably complete description and further insight into this topic.].
2 3
, , ,
n n
r r
n
1
/
n n
