sm8_51
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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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