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# sm8_51

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Problem 8.51 Consider a random process X(t) defined by
( )tftX c\u3c02sin)( =

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
\u3c0
\u3c0
\u3c0
\u3c0
\u3c0
\u3c0
2
2cos1
2
2cos
2sin1
2sin)(
0
0
\u2212=
\u2212=
=
=
\u222b
EE

This mean value clearly depends on t, and thus the process X(t) is nonstationary.

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