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# sm8_29

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```Problem 8.29 A random process is defined by

)2cos()( tfAtX c\u3c0=

where A is a Gaussian random variable of zero mean and variance \u3c32. This random process is applied to an ideal
integrator, producing an output Y(t) defined by

\u222b=
t
dXtY
0
)()( \u3c4\u3c4

(a) Determine the probability density function of the output at a particular time.
(b) Determine whether or not is stationary.

Solution
(a) The output process is given by
( )
( )
( )tf
f
A
dfA
dXtY
c
c
t
c
t
\u3c0\u3c0
\u3c4\u3c4\u3c0
\u3c4\u3c4
2sin
2
2cos
)(
0
0
=
=
=
\u222b
\u222b

At time t0, it follows that is Gaussian with zero mean, and variance ( )0tY
( ) ( )022
2
2sin
2
tf
f cc
\u3c0\u3c0
\u3c3

(b) No, the process Y(t) is not stationary as ( ) ( )10 tYtY FF \u2260 for all t1 and t0.

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page\u20268-35```