ReviewProblems1_sol - ENEE630 ADSP REVIEW PROBLEMS I w/...

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Unformatted text preview: ENEE630 ADSP REVIEW PROBLEMS I w/ solution December 7, 2009 1. Consider the backward prediction error sequence b ( n ) ,b 1 ( n ) ,...,b M ( n ) for the observed sequence { u ( n ) } . (a) Define b ( n ) = [ b ( n ) ,b 1 ( n ) ,...,b M ( n )] T , and u ( n ) = [ u ( n ) ,u ( n- 1) ,...,u ( n- M )] T , find L in terms of the coefficients of the backward prediction-error filter where b ( n ) = Lu ( n ). (b) Let the correlation matrix for b ( n ) be D , and that for u ( n ) be R . Is D diagonal? What is relation between R and D ? Show that a lower triangular matrix A exists such that R- 1 = A H A . (c) Now we are to perform joint estimation of a desired sequence { d ( n ) } by using either { b k ( n ) } or { u ( n ) } , and their corresponding optimal weight vectors are k and w , respectively. What is relation between k and w ? Solution: (a) Since b m ( n ) = ∑ m k =0 a m,m- k u ( n- k ), the matrix is L = 1 a 1 , 1 1 a 2 , 2 a 2 , 1 1 . . . . . . . . . a M,M a M,M- 1 ... ... 1 (b) Due to the orthogonality of b m ( n ), i.e., E ( b m ( n ) b * k ( n )) = P m δ km . Therefore, D is a diagonal matrix with diagonal entries P ,P 1 ,...,P M . D = E ( b ( n ) b H ( n )) = E ( Lu ( n ) u H ( n ) L H ) = LRL H . Since det( L ) = 1, L is invertible. R = L- 1 DL- H . R- 1 = ( L- 1 DL- H )- 1 = L H D- 1 L = ( D- 1 / 2 L ) H ( D- 1 / 2 L ) where D- 1 / 2 L is a lower-triangle matrix. (c) w = R- 1 E ( u ( n ) d * ( n )) = L H D- 1 L E ( u ( n ) d * ( n )). On the other hand, k = D- 1 E ( b ( n ) d * ( n )) = D- 1 L E ( u ( n ) d * ( n )). We can conclude that w = L H k ....
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ReviewProblems1_sol - ENEE630 ADSP REVIEW PROBLEMS I w/...

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