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Problem 8.39 Assume the narrow-band process X(t) described in Problem 8.38 is Gaussian with zero mean and variance . 2 X\u3c3 (a) Calculate . 2 X\u3c3 (b) Determine the joint probability density function of the random variables Y and Z obtained by observing the in-phase and quadrature components of X(t) at some fixed time. Solution (a) The variance is given by ( ) ( ) watts hbhb dffSRX 3 11 2 112 2 12 2 1 2 12 0 2211 2 = \u239f\u23a0 \u239e\u239c\u239d \u239b \u22c5\u22c5+\u22c5\u22c5= \u239f\u23a0 \u239e\u239c\u239d \u239b += == \u222b\u221e\u221e\u2212\u3c3 (b) The random variables Y and Z have zero mean, are Gaussian and have variance . If Y and Z are independent, the joint density is given by 2 X\u3c3 ( ) \u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b +\u2212= \u239f\u23a0 \u239e\u239c\u239d \u239b\u2212\u22c5\u239f\u23a0 \u239e\u239c\u239d \u239b\u2212= 2 22 2 2 2 2 2 , 2 exp 2 1 2exp2 1 2exp2 1, XX XXXX ZY zy zyZYf \u3c3\u3c0\u3c3 \u3c3\u3c3\u3c0\u3c3\u3c3\u3c0 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\u20268-49