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Problem 8.17 A discrete-time white noise process {Wn} has an autocorrelation function given by RW(n) = N0δ(n). (a) Using the discrete Fourier transform, determine the power spectral density of {Wn}. (b) The white noise process is passed through a discrete-time filter having a discrete- frequency response k Nk W WkH α α − −= 1 )(1)( where, for a N-point discrete Fourier transform, W = exp{j2π/N}. What is the spectrum of the filter output? Solution The spectrum of the discrete white noise process is ( ) ( ) ( ) 0 1 0 0 1 0 N WnN WnRkS N n nk N n nk = = = ∑ ∑ − = − = δ The spectrum of the process after filtering is ( ) ( ) ( ) 2 0 2 1 )(1 k Nk Y W WN kSkHkS α α − −= = 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…8-19
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