sm8_30

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Problem 8.30 Prove the following two properties of the autocorrelation function RX(τ) of a random process 
X(t): 
(a) If X(t) contains a dc component equal to A, then RX(τ) contains a constant component equal to A2. 
(b) If X(t) contains a sinusoidal component, then RX(τ) 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 −=
 ( )[ ] AtX =E 
 
and 
 
 
( ) ( ) ( )[ ]
( )( )( ) ( )( )( )[ ]
( )( ) ( )( )[ ] ( )( ) ( )( )
( ) 2
2
00 AR
AAtXAAtXAtXAtX
AAtXAAtX
tXtXR
Y
X
+++=
++−++−+−=
+−++−=
+=
τ
ττ
τ
ττ
EEE
E
E
 
 
And thus RX(τ) has a constant component A2. 
 
(b) Let ( ) ( ) ( )tfAtYtX cπ2sin+= where Y(t) does not contain a sinusoidal component of 
frequency fc. 
 ( ) ( ) ( )[ ]
( ) ( )( ) ( ) ( )( ) ( ) ( )[ ][ ]
( )
( )
( )[ ]
( ) ( )τπτ
θτππτ
τπππτπ
ττ
cY
ccY
cccc
X
fAR
tftfAR
tftfAtfAtYtfAtY
tXtXR
2cos
2
22cos2cos
2
2sin2sin2sin2sin
2
2
2
+=
+++++=
+++++=
+=
K
EE
E
 
 
And thus RX(τ) has a sinusoidal component at fc. 
 
 
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