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Unformatted text preview: EE 378 Handout #11 Statistical Signal Processing Wednesday, May 9, 2007 Homework Set #5 Due: Wednesday, May 16, 2007. 1. Show that the frequency response of the optimal linear prediction filter for a regular WSS process { Y ( n ) } is given by A ( e jω ) = 1 S + Y Y ( e jω ) { e jω S + Y Y ( e jω ) } + . Let { Y ( n ) } be an ARMA(1,1) process described by Y ( n ) = αY ( n 1) + W ( n ) + βW ( n 1) , where { W ( n ) } is a zeromean, unitvariance white noise process, and  α  < 1. Express A ( e jω ) in terms of α and β for •  β  < 1, and •  β  > 1. 2. Let Y ( n ) = 1 2 Y ( n 1) + U ( n ) , where { U ( n ) } is a zeromean white noise process with variance σ 2 u = 1, and let Z ( n ) = Y ( n ) + W ( n ) , where { W ( n ) } is a zeromean WSS random process with the power spectral density S W ( e jω ) = 1 1 + α cos ω , π < ω ≤ π , and is independent of { U ( n ) } . (a) For α = . 8, find the optimal linear estimate of Y (0) given Z(k),∞ < k < ∞ . (b) For α = . 8, find the optimal linear estimate of Y (0) given Z(k),∞ < k ≤ 0. (c) How does the meansquare error of the estimates of parts (a) and (b) compare? (d) Would your answer to part (c) change if α 6 = . 8? Explain. 1 3. (a) Show that the frequency response of the optimal linear estimation filter of X ( n ) given Y ( k ),∞ < k ≤ n + d , is given by H d ( e jω ) = e jωd S + Y Y ( e jω ) S XY ( e jω ) e jωd S Y Y ( e jω ) + ....
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This note was uploaded on 03/03/2011 for the course EE 378 taught by Professor Weissman,i during the Spring '07 term at Stanford.
 Spring '07
 Weissman,I
 Frequency, Signal Processing

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