Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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The DTFT of the crosscorrelation function P
xy
! of two WSS signals x(n) and y(n) is
given by
P
xy
!
I
kÀI
r
xy
ke
Àj!k
,
8:1:22
or
P
xy
z
I
kÀI
r
xy
kz
Àk
:
8:1:23
This function is called the cross-power spectrum.
Example 8.3: The autocorrelation function of a WSS white random process can be
defined as
r
xx
k s
2
x
dk m
2
x
:
8:1:24
The corresponding PDS is given by
P
xx
! s
2
x
2pm
2
x
d!, j!j p:
8:1:25
An important white random signal is called white noise, which has zero mean.
Thus its autocorrelation function is expressed as
r
xx
k s
2
x
dk,
8:1:26
and the power spectrum is given by
P
xx
! s
2
x
, j!j < p,
8:1:27
which is of constant value for all frequencies !.
Consider a linear and time-invariant digital filter defined bythe impulse response
h(n), or the transfer function H(z). The input of the filter is a WSS random signal x(n)
with the PDS P
xx
!. As illustrated in Figure 8.1, the PDS of the filter output y(n) can
be expressed as
P
yy
!
H!
2
P
xx
!
8:1:28
or
P
yy
z
Hz
2
P
xx
z,
8:1:29
INTRODUCTION TO RANDOM PROCESSES
357
Summary :
Example 8.3: The autocorrelation function of a WSS white random process can be defined as r xx k s 2 x dk m 2 x : 8:1:24 The corresponding PDS is given by P xx !
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