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# sm8_30

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```Problem 8.30 Prove the following two properties of the autocorrelation function RX(\u3c4) of a random process
X(t):
(a) If X(t) contains a dc component equal to A, then RX(\u3c4) contains a constant component equal to A2.
(b) If X(t) contains a sinusoidal component, then RX(\u3c4) also contains a sinusoidal component of the same
frequency.

Solution
(a) Let Y and Y(t) is a random process with zero dc component. Then ( ) ( ) AtXt \u2212=
( )[ ] AtX =E

and

( ) ( ) ( )[ ]
( )( )( ) ( )( )( )[ ]
( )( ) ( )( )[ ] ( )( ) ( )( )
( ) 2
2
00 AR
AAtXAAtXAtXAtX
AAtXAAtX
tXtXR
Y
X
+++=
++\u2212++\u2212+\u2212=
+\u2212++\u2212=
+=
\u3c4
\u3c4\u3c4
\u3c4
\u3c4\u3c4
EEE
E
E

And thus RX(\u3c4) has a constant component A2.

(b) Let ( ) ( ) ( )tfAtYtX c\u3c02sin+= where Y(t) does not contain a sinusoidal component of
frequency fc.
( ) ( ) ( )[ ]
( ) ( )( ) ( ) ( )( ) ( ) ( )[ ][ ]
( )
( )
( )[ ]
( ) ( )\u3c4\u3c0\u3c4
\u3b8\u3c4\u3c0\u3c0\u3c4
\u3c4\u3c0\u3c0\u3c0\u3c4\u3c0
\u3c4\u3c4
cY
ccY
cccc
X
fAR
tftfAR
tftfAtfAtYtfAtY
tXtXR
2cos
2
22cos2cos
2
2sin2sin2sin2sin
2
2
2
+=
+++++=
+++++=
+=
K
EE
E

And thus RX(\u3c4) has a sinusoidal component at fc.

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page\u20268-36```