Chapter_07 - CHAPTER 7 7.1 When M = 6 L = 4 we have for the...

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158 CHAPTER 7 7.1 When M = 6, L = 4, we have for the k th block For the ( k + j )th, with j = +1, +2,..., we have 7.2 (a) Using the 2 M -by-2 M constraint matrix G , we may rewrite the cross-correlation vector φ ( k ) as follows: Accordingly, the weight-update is rewritten in the compact form: where G = FG 1 F -1 u 4 k ( ) u 4 k -1 ( ) u 4 k -2 ( ) u 4 k -3 ( ) u 4 k -4 ( ) u 4 k -5 ( ) u 4 k +1 ( ) u 4 k ( ) u 4 k -1 ( ) u 4 k -2 ( ) u 4 k -3 ( ) u 4 k -4 ( ) u 4 k +2 ( ) u 4 k +1 ( ) u 4 k ( ) u 4 k -1 ( ) u 4 k -2 ( ) u 4 k -3 ( ) u 4 k +3 ( ) u 4 k +2 ( ) u 4 k +1 ( ) u 4 k ( ) u 4 k -1 ( ) u 4 k -2 ( ) w 0 k ( ) w 1 k ( ) w 2 k ( ) w 3 k ( ) w 4 k ( ) w 5 k ( ) y 4 k ( ) y 4 k +1 ( ) y 4 k +2 ( ) y 4 k +3 ( ) = u uk +4 j ( ) u uk +4 j -1 ( ) u uk +4 j -2 ( ) u uk +4 j -3 ( ) u uk +4 j -4 ( ) u uk +4 j -5 ( ) u uk +4 j +1 ( ) u uk +4 j ( ) u uk +4 j -1 ( ) u uk +4 j -2 ( ) u uk +4 j -3 ( ) u uk +4 j -4 ( ) u uk +4 j +2 ( ) u uk +4 j +1 ( ) u uk +4 j ( ) u uk +4 j -1 ( ) u uk +4 j -2 ( ) u uk +4 j -3 ( ) u uk +4 j +3 ( ) u uk +4 j +2 ( ) u uk +4 j +1 ( ) u uk +4 j ( ) u uk +4 j -1 ( ) u uk +4 j -2 ( ) w 0 k ( ) w 1 k ( ) w 2 k ( ) w 3 k ( ) w 4 k ( ) w 5 k ( ) y 4 k +4 j ( ) y 4 k +4 j +1 ( ) y 4 k +4 j +2 ( ) y 4 k +4 j +3 ( ) = φ T k ( ) 0 0 , , , [ ] T G 1 F 1 U H k ( ) E k ( ) = W ˆ k +1 ( ) W ˆ k ( ) µ FG 1 F 1 U H k ( ) E k ( ) + = W ˆ k ( ) µ GU H k ( ) E k ( ) + =
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159 The matrix operator F denotes discrete Fourier transformation, and F -1 denotes inverse discrete Fourier transformation. (b) With G 2 = [ 0 , I ] the error vector E ( k ) is readily rewritten as follows: (c) Using these matrix notations, the fast LMS algorithm may be reformulated as follows: Initialization: where M is the number of taps, and δ i is a small positive number. Computation: For each new block of M input samples, compute the following: E k ( ) F 0 0 e T k ( ) , , , [ ] T = FG 2 T e T k ( ) = W ˆ 0 ( ) 0 0 0 , , , [ ] T = P i 0 ( ) δ i , i 0 1 2 M -1 , , , = = U k ( ) diag F u kM - M ( ) u kM -1 ( ) u kM ( ) u kM + M -1 ( ) , , , , , [ ] T ( ) = Y k ( ) U k ( ) W k ( ) = y k ( ) G 2 F 1 Y k ( ) = e k ( ) d k ( ) y k ( ) = E k ( ) FG 2 T e k ( ) = P i k ( ) γ P i k -1 ( ) 1 γ ( ) U i k ( ) 2 , i + 0 1 2 M -1 , , , = = µ k ( ) diag P 0 1 k ( ) P 1 1 k ( ) P 2 M -1 1 k ( ) , , , ( ) =
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160 (d) When the constraint gradient G is equal to the identity matrix, the fast LMS algorithm reduces to its unconstrained frequency-domain form. 7.3 Removing the gradient constraint shown in Fig. 7.3 permits the adaptive filter to perform circular convolution instead of linear convolution. (Circular convolution is the form of convolution performed by the discrete Fourier transform.) This modification may be acceptable in those applications where there is no particular concern as to how the input signal is used to minimize the mean-square value of the error signal (estimation error). One such application is that of noise (interference) cancellation. 7.4 (a) From Fig. 7.1, we note that the z -transform of x k ( n ) is related to the z -transform of the input u ( n ) as follows: Cross-multiplying and rearranging terms: Taking inverse z -transforms: (b) We are given that (1) The error signal is (2) Hence, substituting Eq. (2) into (1), and rearranging:
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