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Handout7 - Course 18.327 and 1.130 Wavelets and Filter...

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Course 18.327 and 1.130 Wavelets and Filter Banks Orthogonal Filter Banks; Paraunitary Matrices; Orthogonality Condition (Condition O) in the Time Domain, Modulation Domain and Polyphase Domain 2 Unitary Matrices The constant complex matrix A is said to be unitary if A A = I example: A = 1 µ 2 1 -i i -1 A* = 1 µ 2 1 i -i -1 A -1 = -1 µ 2 -1 i -i 1 A † = A* T = 1 µ 2 1 -i i -1 Ω A = A -1 1
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ò w w w w w Paraunitary Matrices The matrix function H(z) is said to be paraunitary if it is unitary for all values of the parameter z H T (z -1 ) H(z) = I for all z 0 -----------------(1) Frequency Domain: H T (- w ) H( w ) = I for all or H* T ( w ) H( ) = I Note: we are assuming that h[n] are real. 3 4 Orthogonal Filter Banks Centered form (PR with no delay): y 0 [n] > > > > > > > > > > > > x[n] x[n] h 0 [n] h 1 [n] é 2 é 2 å 2 å 2 h 0 [-n] h 1 [-n] y 1 [n] Synthesis bank = transpose of analysis bank h 0 [n] causal Ω f 0 [n] ô h 0 [-n] anticausal + 2
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0 [n], h 1 [n], in the (i) time domain? (ii) polyphase domain? (iii)modulation domain? 5 6 Time Domain Analysis: N = 3 (filter length = 4) 4 y 0 [0] y 0 [1] y 0 [2] y 0 [3] 4 y 1 [0] y 1 [1] y 1 [2] y 1 [3] 4 = h 0 [3] h 0 [2] h 0 [1] h 0 [0] h 0 [3] h 0 [2] h 0 [1] h 0 [0] h 0 [3] h 0 [2] h 0 [1] h 0 [0] h 0 [3] h 0 [2] h 0 [1] h 0 [0] 3 h 1 [3] h 1 [2] h 1 [1] h 1 [0] h 1 [3] h 1 [2] h 1 [1] h 1 [0] h 1 [3] h 1 [2] h 1 [1] h 1 [0] h 1 [3] h 1 [2] h 1 [1] h 1 [0] 4 x [-3]
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Handout7 - Course 18.327 and 1.130 Wavelets and Filter...

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