sm8_35
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sm8_35


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Problem 8.35 Consider a wide-sense stationary process X(t) having the power spectral density SX(f) 
shown in Fig. 8.26. Find the autocorrelation function RX(\u3c4) of the process X(t). 
 
Solution 
The Wiener-Khintchine relations imply the autocorrelation is given by the inverse 
Fourier transform of the power spectral density, thus 
 
( ) ( )
( ) ( )\u222b
\u222b
\u2212=
= \u221e\u221e\u2212
1
0
2cos1
)2exp(
dfftf
dfftjfSR X
\u3c0
\u3c0\u3c4
 
 
where we have used the symmetry properties of the spectrum to obtain the second line. 
Integrating by parts, we obtain 
 
( ) ( )
( )
( )
( )2
1
0
2
1
0
1
0
2
2cos1
2
)2cos(0
2
2sin
2
2sin)1()(
\u3c0\u3c4
\u3c0\u3c4
\u3c0\u3c4
\u3c4\u3c0
\u3c0\u3c4
\u3c4\u3c0
\u3c0\u3c4
\u3c4\u3c0\u3c4
\u2212=
\u2212+=
+\u2212= \u222b
f
dffffRX
 
 
Using the half-angle formula sin2(\u3b8) = ½(1-cos(2\u3b8), this result simplifies to 
 
( )
( )
)(sinc
2
sin2)(
2
2
1
2
2
\u3c4
\u3c0\u3c4
\u3c0\u3c4\u3c4
=
=XR 
 
where sinc(x) = sin(\u3c0x)/\u3c0x. 
 
 
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