sm8_29
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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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