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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. 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\u20268-36