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Problem 8.29 A random process is defined by )2cos()( tfAtX c\u3c0= where A is a Gaussian random variable of zero mean and variance \u3c32. This random process is applied to an ideal integrator, producing an output Y(t) defined by \u222b= t dXtY 0 )()( \u3c4\u3c4 (a) Determine the probability density function of the output at a particular time. (b) Determine whether or not is stationary. Solution (a) The output process is given by ( ) ( ) ( )tf f A dfA dXtY c c t c t \u3c0\u3c0 \u3c4\u3c4\u3c0 \u3c4\u3c4 2sin 2 2cos )( 0 0 = = = \u222b \u222b At time t0, it follows that is Gaussian with zero mean, and variance ( )0tY ( ) ( )022 2 2sin 2 tf f cc \u3c0\u3c0 \u3c3 (b) No, the process Y(t) is not stationary as ( ) ( )10 tYtY FF \u2260 for all t1 and t0. 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\u20268-35