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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. Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. page\u20268-45