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Problem 8.51 Consider a random process X(t) defined by ( )tftX c\u3c02sin)( = where the frequency fc is a random variable uniformly distributed over the interval [0,W]. Show that X(t) is nonstationary. Hint: Examine specific sample functions of the random process X(t) for, say, the frequencies W/4, W/2, and W. Solution To be stationary to first order implies that the mean value of the process X(t) must be constant and independent of t. In this case, [ ] ( )[ ] ( ) ( ) ( ) Wt Wt Wt wt dwwt W tftX W W c \u3c0 \u3c0 \u3c0 \u3c0 \u3c0 \u3c0 2 2cos1 2 2cos 2sin1 2sin)( 0 0 \u2212= \u2212= = = \u222b EE This mean value clearly depends on t, and thus the process X(t) is nonstationary. 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\u20268-63