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Handout15 - Course 18.327 and 1.130 Wavelets and Filter...

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1 Course 18.327 and 1.130 Wavelets and Filter Banks Signal and Image Processing: finite length signals; boundary filters and boundary wavelets; wavelet compression algorithms. 2 Finite-Length Signals x[n] y[n] H( ω ) y -2 y -1 y 0 y 1 ± y N-1 ± = h 0 h 1 h 0 h -1 h 2 h 1 h 0 h -1 h 1 h 0 h -1 ²²² x -2 x -1 x 0 x 1 ± x N-1 ± ± ² ³ unknown length N (finite length) unknown
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2 3 ± ± -2 -1 0 1 2 3 artificial edge resulting from zero-padding 1) zero-padding n x[n] filtered by [1, 1] -2-1 0 1 2 3 ± n ± n y[n] y[n] filtered by [1, -1] 4 2) Periodic Extension -1 0 1 2 N-1N n x[n] x[N] = x[0] y 0 y 1 ² y N-1 h 1 h 0 h -1 h -2 ± h 2 h 1 h 1 h 0 h -1 ± h 3 h 2 h -1 h -2 ± h 2 h 1 h 0 ² x N-2 x N-1 x 0 x 1 x 2 ² x N-1 x 0 ² = wrap-around N-output circulant matrix = H
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3 5 What is the eigenvector for the circulant matrix? [1 e i ω e i2 ω ± e i(N-1) ω ] T We need e iN ω =1=e i0 ω N ω =2 π k, ω = discrete set of ω ’s For the 0 th row, H[k] = ¦ h[n] e -i n 2 π k N N-1 n=0 2 π k N 6 =[F ] w=e i 2 π N 11 1 ± 1 1ww 2 w N-1 1w 2 w 4 w 2(N-1) ² ²² ² N-1 [H] H[0]
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Handout15 - Course 18.327 and 1.130 Wavelets and Filter...

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