630F09_hw3 - ENEE 630 Fall’2009 Homework#3 Problem 1 In a...

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Unformatted text preview: ENEE 630 Fall’2009 Homework #3 Problem 1 In a two-channel QMF bank, if the filters are related as H 1 ( z ) = H (- z ) ,F ( z ) = H ( z ) ,F 1 ( z ) =- H 1 ( z ) and if H ( z ) is chosen to have real coefficients and linear phase, then the distortion function is given by T ( e jw ) = 1 2 e- jwN ( | H ( e jw ) | 2- (- 1) N | H ( e j ( π- w )) | 2 ) (1) If filter order N is even, this implies T ( e j π 2 ) = 0, so in order to avoid strong attenuation on ω = π 2 , the filter order N for this filter bank structure has to be odd for many applications. Now consider a modified QMF bank shown in Figure P-1 where the filters could be FIR or IIR. Express ˆ X ( z ) in terms of X(z). With H 1 ( z ) = H (- z ), show that the choice F ( z ) = H ( z ) and F 1 ( z ) = H 1 ( z ) cancels aliasing. With this choice write down the distortion T(z) in terms of H ( z ).-1 z-1 z x[n] H(z) H(z) 1 F(z) F(z) 1 x[n] ^ 2 2 2 2 Figure : P-1 a) Now let H ( z ) be a real coefficient linear phase FIR lowpass filter of order N. Simplify) be a real coefficient linear phase FIR lowpass filter of order N....
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630F09_hw3 - ENEE 630 Fall’2009 Homework#3 Problem 1 In a...

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