sm8_51

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Problem 8.51 Consider a random process X(t) defined by 
 ( )tftX cπ2sin)( = 
 
where the frequency fc is a random variable uniformly distributed over the interval [0,W]. Show that X(t) is 
nonstationary. Hint: Examine specific sample functions of the random process X(t) for, say, the frequencies 
W/4, W/2, and W. 
 
Solution 
To be stationary to first order implies that the mean value of the process X(t) must be 
constant and independent of t. In this case, 
 [ ] ( )[ ]
( )
( )
( )
Wt
Wt
Wt
wt
dwwt
W
tftX
W
W
c
π
π
π
π
π
π
2
2cos1
2
2cos
2sin1
2sin)(
0
0
−=
−=
=
=
∫
EE
 
 
This mean value clearly depends on t, and thus the process X(t) is nonstationary. 
 
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