630F09_hw4 - ENEE 630 Fall2009 Homework #4 Problem 1...

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ENEE 630 Fall’2009 Homework #4 Problem 1 Suppose the M-channel maximally decimated QMF bank is alias-free, and let T(z) be the distortion function. Suppose we define a new filter bank in which the analysis and synthesis filters are interchanged, that is F k ( z ) are the analysis filters and H k ( z ) are the synthesis filters. Show that the resulting system is free from aliasing and has the same distortion function T(z). So we can swap each F k ( z ) with the corresponding H k ( z ) without changing the input/output properties. (Hint: Use AC matrix formulation cleverly.) Problem 2 Consider Fig. P-2 with T = W * (a uniform DFT analysis bank). Suppose R k ( z ) are chosen as in Eq (1.1) below, so that the product R k ( z ) E k ( z ) is independent of k. This ensures that aliasing has been cancelled. R k ( z ) = Y l 6 = k E l ( z ) (1 . 1) Figure : P-2 a) First as a review, verify that the uniform-shift relations H k ( z ) = H 0 ( zW k ) and F k ( z ) = W - k F 0 ( zW k ) hold, where W = e - j 2 π M . b)
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630F09_hw4 - ENEE 630 Fall2009 Homework #4 Problem 1...

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