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
Document source : notes.ump.edu.my
h(n)
H(w)
x(n)
P
xx
(w)
P
yy
(w)
y(n)
Figure 8.1 Linear filtering of random processes
where H! is the frequencyresponse of the filter. Therefore the value of the output
PDS at frequency ! depends on the squared magnitude response of the filter and the
input PDS at the same frequency.
Another important relationships between x(n) and y(n) are
m
y
E
I
lÀI
hlxn À l
4
5
I
lÀI
hlExn À l m
x
I
lÀI
hl,
8:1:30
and
r
yx
k Eyn kxn E
I
lÀI
hlxn k À lxn
4
5
I
lÀI
hlr
xx
k À l hk à r
xx
k:
8:1:31
Taking the z-transform of both sides, we obtain
P
yx
z HzP
xx
z:
8:1:32
Similarly, the relationships between the input and the output signals are
r
xy
k
I
lÀI
hlr
xx
k l hk à r
xx
Àk
8:1:33
and
P
yx
z H
Ã
zP
xx
z:
8:1:34
If the input signal x(n) is a zero-mean white noise with the autocorrelation function
defined in (8.1.26), Equation (8.1.31) becomes
r
yx
k
I
lÀI
hls
2
x
dk À l s
2
x
hk:
8:1:35
This equation shows that bycomputing the crosscorrelation function r
yx
k, the impulse
response h(n) of a filter (or system) can be obtained. This fact can be used to estimate an
unknown system such as the room impulse response used in Chapter 4.
358
ADAPTIVE FILTERING
Summary :
Another important relationships between x(n) and y(n) are m y E I lÀI hlxn À l 4 5 I lÀI hlExn À l m x I lÀI hl, 8:1:30 and r yx k Eyn kxn E I lÀI hlxn k À lxn 4 5 I lÀI hlr xx k À l hk à r xx k: 8:1:31 Taking the z-transform of both sides, we obtain P yx z HzP xx z: 8:1:32 Similarly, the relationships between the input and the output signals are r xy k I lÀI hlr xx k l hk à r xx Àk 8:1:33 and P yx z H à zP xx z: 8:1:34 If the input signal x(n) is a zero-mean white noise with the autocorrelation function defined in (8.1.26), Equation (8.1.31) becomes r yx k I lÀI hls 2 x dk À l s 2 x hk: 8:1:35 This equation shows that bycomputing the crosscorrelation function r yx k, the impulse response h(n) of a filter (or system) can be obtained.
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lài,filter,input,response,function,8131,impulse,output,between,hlr,frequency,equation,used