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Handout3

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

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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 0 [n] h 1 [n] é 2 é 2 x[n] y 0 [n] y 1 [n] å 2 å 2 Analysis r 0 [n] r 1 [n] f 0 [n] f 1 [n] x[n] ^ Synthesis v 0 [n] v 1 [n] t 1 [n] t 0 [n] h 0 [n] = f 0 [n] = 0 1 1 µ 2 1 µ 2 1 µ 2 1 µ 2 -1 0 1

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3 h 1 [n] = f 1 [n] = -1 µ 2 1 µ 2 -1 µ 2 1 µ 2 0 1 -1 0 Analysis: r 0 [n] = (x[n] + x[n œ 1]) lowpass filter y 0 [n] = r 0 [2n] downsampler y 0 [n] = (x[2n] + x[2n œ 1]) ----------------- j Similarly y 1 [n] = (x[2n] œ x[2n œ 1]) ------------------ k 1 µ 2 1 µ 2 1 µ 2 4 Matrix form 4 = 4 y 0 [0] y 0 [1] : 4 : y 1 [0] y 1 [1] 1 µ 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 0 [n/2] n even t 0 [n] = upsampler 0 n odd v 0 [n] = ( t 0 [n + 1] + t 0 [n]) lowpass filter y 0 [n/2] n even y 0 [ ] n odd &)')( 1 µ 2 &)')( = 1 µ 2 1 µ 2 n + 1 2 6 Similarly v 1 [n] = & ) ' ) ( 1 µ 2 y 1 [n/2] n even 1 µ 2 - y 1 [ ] n odd n + 1 2 & ) ' ) ( ' 1 µ 2 (y 0 [n/2] + y 1 [n/2]) n even ^ So, the reconstructed signal is x[n] = v 0 [n] + v 1 [n] = 1 µ 2 (y 0 [ ] - y 1 [ ]) n odd n + 1 2 n + 1 2 3

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Ω Ω µ j k i.e. ^ x[2n-1] = 1 µ 2 (y 0 [n] œ y 1 [n]) = x[2n-1] from j and k ^ 1 x[2n] = 2 (y 0 [n] + y 1 [n]) = x[2n] ^ So x[n] = x[n] Ω Perfect reconstruction!
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Handout3 - Course 18.327 and 1.130 Wavelets and Filter...

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