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Handout3 - Course 18.327 and 1.130 Wavelets and Filter...

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Unformatted text preview: Course 18.327 and 1.130 Wavelets and Filter Banks Filter Banks: time domain (Haar example) and frequency domain; conditions for alias cancellation and no distortion 2 Simplest (non-trivial) example of a two channel FIR perfect reconstruction filter bank. Haar Filter Bank h [n] h 1 [n] 2 2 x[n] y [n] y 1 [n] 2 2 Analysis r [n] r 1 [n] f [n] f 1 [n] x[n] ^ Synthesis v [n] v 1 [n] t 1 [n] t [n] h [n] = f [n] = 0 1 1 2 1 2 1 2 1 2 -1 0 1 3 h 1 [n] = f 1 [n] = -1 2 1 2 -1 2 1 2 0 1 -1 0 Analysis: r [n] = (x[n] + x[n 1]) lowpass filter y [n] = r [2n] downsampler y [n] = (x[2n] + x[2n 1]) ----------------- j Similarly y 1 [n] = (x[2n] x[2n 1]) ------------------ k 1 2 1 2 1 2 4 Matrix form 4 = 4 y [0] y [1] : 4 : y 1 [0] y 1 [1] 1 2 4 3 1 1 0 0 3 3 0 0 1 1 3 3-1 1 0 0 3 3 0 0 -1 1 3 4 : x[-1] x[0] x[1] x[2] : :------------------- l y o y 1 = L B x 2 5 Synthesis y [n/2] n even t [n] = upsampler 0 n odd v [n] = ( t [n + 1] + t [n]) lowpass filter y [n/2] n even y [ ] n odd &)')( &)')( &)')( &)')( 1 2 &)')( &)')( &)')( &)')( = 1 2 1 2 n + 1 2 6 Similarly v 1 [n] = & ) ' ) ( & ) ' ) ( & ) ' ) ( & ) ' ) ( 1 2 y 1 [n/2] n even 1 2 - y 1 [ ] n odd n + 1 2 & ) ' ) ( & ) ' ) ( & ) ' ) ( & ) ' ) ( 1 2 (y [n/2] + y 1 [n/2]) n even ^ So, the reconstructed signal is x[n] = v [n] + v 1 [n] = 1 2 (y [ ] - y 1 [ ]) n odd n + 1 2 n + 1 2 3 j k i.e....
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This note was uploaded on 12/04/2011 for the course ESD 18.327 taught by Professor Gilbertstrang during the Spring '03 term at MIT.

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Handout3 - Course 18.327 and 1.130 Wavelets and Filter...

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