ps1 - (Ω = 2 α α 2 Ω 2 Ω ∈ R Determine the transfer...

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Problem Set #1 EECE-595, Section II Spring 2003, Adaptive Filtering Date Assigned: 02/10/03 Date Due: 02/17/03 Problem # 1.0 If A , B , C , D are square matrices show that the following relation is true: ( A + BCD ) - 1 = A - 1 - A - 1 B ( C - 1 + DA - 1 B ) - 1 DA - 1 provided that A , C , C - 1 + DA - 1 B are invertible. Use the result of this relation to further show that: ( A + vv H ) - 1 = A - 1 - A - 1 vv H A - 1 1 + v H A - 1 v . Problem # 2.0 If x [ n ] is a zero mean, WSS random signal with a power spectral density P xx ( e ) that satisfies the Payley-Wiener criteria, i.e., 1 2 π Z π - π log( P xx ( e )) dω < . show that the following inequality holds: 1 2 π Z π - π P xx ( e ) exp 1 2 π Z π - π log( P xx ( e )) . Remark: this inequality is sometimes referred to as the AM–GM inequality. This will come in handy when we study equalization. Problem # 3.0 Consider a zero mean, WSS random process x ( t ) that has been corrupted with white, zero-mean, unit-variance noise v ( t ) that is uncorrelated with the signal x ( t ). The power spectral density of the signal x ( t ) is of the form:

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Unformatted text preview: (Ω) = 2 α α 2 + Ω 2 , Ω ∈ R . Determine the transfer function of the: (a) optimal Wiener ﬁlter H opt ( s ), (b) optimal causal Wiener ﬁlter H cau ( s ). Determine the corresponding minimum mean-squared error associated with these ﬁlters. 1 Problem # 4.0 Let ² [ n ] and v [ n ] be two zero mean, independent white noise processes. In this problem we will look at a more general estimation problem where the observa-tions are given by: u [ n ] = ∞ X k ==-∞ h [ k ] ² [ n-k ] + v [ n ] , where h [ n ] represents a distortion operator. The desired signal d [ n ] is of the form: d [ n ] = ∞ X k =-∞ p [ n ] ² [ n-k ] . Determine the optimal Wiener smoother K ( z ) that estimates d [ n ] from the observations u [ n ] in terms of P ( z ) and H ( z ). Determine the corresponding MMSE. 2...
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