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Problem 9.24. Equation (9.59) is the FM post-detection noise for an ideal low-pass filter. Find the post-detection noise for an FM signal when the post-detection filter is a second- order low-pass filter with magnitude response ( )( )1/ 24 1( ) 1 / H f f W = + Assume 2( )BP cH f f+ \u22481 for / 2Tf B< and 2TB W>> . Solution We modify Eq. (9.58) to include the effects of a non-ideal post-detection filter in order to estimate the average post-detection noise power: 2 2 20 0 2 2 20 2 40 1| ( ) | 1 ( / ) 2 1 1 ( / ) W W BPW W c c W c N N 4f H f df f dfA A N f W f df A f W \u2212 \u2212= \u22c5 + = \u22c5 + \u222b \u222b \u222b This can be evaluated by a partial fraction expansion of the integrand but for simplicity, we appeal to the formula: 2 2 2 1 4 4 2 2 2 2 1 1 2 2 2log tan , 0, 4 2 2 2 2 2 x dx x kx k kx aab k a bx bk x kx k k x b \u2212\u23a1 \u23a4\u2212 += + >\u23a2 \u23a5+ + + \u2212\u23a3 \u23a6\u222b = Using this result, we get the average post-detection noise power is Avg. post-detection noise power 33 0 0 2 2 2 2 2log 0.42 4 2 2 2c c N NW W A A \u3c0\u23a1 \u23a4\u2212= \u22c5 + =\u23a2 \u23a5+\u23a3 \u23a6 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.