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Unformatted text preview: ENEE 630 Fall 2009 Homework #6 Problem 1 The tap-input vector of a transversal filter is defined by u ( n ) = α ( n ) s ( ω ) + v ( n ) where s ( ω ) = [1 ,e- jω ,...,e- jω ( M- 1) ] T v ( n ) = [ v ( n ) ,v ( n- 1) ,...,v ( n- M + 1)] T i.e. u ( n- k ) = α ( n ) · e- jkω + v ( n- k ) for k = 0 ,...,M- 1. For the tap-input vector at a given time n , α ( n ) is a complex random variable with zero mean and variance σ 2 α = E [ | α ( n ) | 2 ], and α ( n ) is uncorrelated with the w.s.s. process v ( n ). (a) Determine the correlation matrix of the tap-input vector u (n). (b) Suppose that the desired response d(n) is uncorrelated with u (n). What is the value of the tap-weight vector of the corresponding Wiener filter? (c) Suppose that the variance σ 2 α is zero, and the desired response is defined by d ( n ) = v ( n- k ) where 0 ≤ k ≤ M- 1. What is the new value of the tap-weight vector of the Wiener filter? (d) Determine the tap-weight vector of the Wiener filter for a desired response defined by d ( n ) = α ( n ) e- jwτ where τ is a prescribed delay....
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- Spring '10
- Signal Processing, Autocorrelation, UK, Wiener filter, tap-weight vector, tap-input vector