
5 Random Processes
Section 5.2: Ensemble Averaging
5.1. Let X (t) be a random process with autocorrelation function,
R
XX
(t
1
, t
2
) =
b
(t
1
+ c)
2
(t
2
+ c)
2
,
where b and c are positive constants, and let Y (t) be a random process defined by
Y (t) =
t
0
X (s) ds.
Derive the cross-correlation and autocorrelation functions, R
XY
(t
1
, t
2
) and R
Y Y
(t
1
, t
2
) .
Solution: The cross-correlation function is given by
R
XY
(t
1
, t
2
) = E {X (t
1
) Y (t
2
)}
= E
X (t
1
)
t
2
0
X (s) ds
!
=
t
2
0
E {X (t
1
) X (s)}ds
=
t
2
0
b (t
1
+ c)
−2
(s + c)
−2
ds
=
bt
2
c (t
1
+ c)
2
(t
2
+ c)
.
Similarly,
R
Y Y
(t
1
, t
2
) =
t
2
0
t
1
0
b (s
1
+ c)
−2
(s
2
+ c)
−2
ds
1
ds
2
=
bt
1
t
2
c
2
(t
1
+ c) (t
2
+ c)
.
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