sm8_13

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Problem 8.13 Let X(t) be a random process defined by 
 
)2cos()( ftAtX π= 
 
where A is uniformly distributed between 0 and 1, and f is constant. Determine the autocorrelation function 
of X. Is X wide-sense stationary? 
 
Solution 
( ) ( )[ ] [ ] ( ) ( )[ ] ( )( ) ( )[ ]21212 21
2
21
2cos2cos
2cos2cos
ttfttfA
ftftAtXtX
++−=
=
ππ
ππ
E
EE 
 
 
[ ] ∫ === 10
1
0
3
22
3
1
3
xdxxAE 
 
Since the autocorrelation function depends on 21 tt + as well as 21 tt − , the process is not 
wide-sense stationary. 
 
 
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