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# Handout16 - Course 18.327 and 1.130 Wavelets and Filter...

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Course 18.327 and 1.130 Wavelets and Filter Banks Lifting: ladder structure for filter banks; factorization of polyphase matrix into lifting steps; lifting form of refinement equation 2 Basic idea: Lifting 2 H 1 (z) H 0 (z) 2 2 S(z) - S(z) + 2 F 1 (z) F 0 (z) + Filter bank is modified by a simple operation that preserves the perfect reconstruction property, regardless of the actual choice for S(z). 1

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3 Advantages: Leads to faster implementation of DWT Provides a framework for constructing wavelets on non-uniform grids. What are the effective filters in the modified filter bank? Synthesis bank F 1 (z) F 0 (z) 2 2 S(z 2 ) + + F 0 (z) 2 + F 1 (z)+S(z 2 )F 0 (z) 2 use 2 nd Noble identity F new (z) 1 ± ² ³ 4 So the effective highpass filter is F 1 (z) + S(z 2 )F 0 (z). The lowpass filter is unchanged. To modify the lowpass filter, add a second lifting step, e.g. Consider F new (z) = F (z) + S(z 2 )F 0 (z) i.e. f new [n] = f [n] + r[k] f 0 [n–k] where r[k] = F 1 (z) F 0 (z) 2 2 S(z) + T(z) + 2 + F 0 (z)+T(z 2 )F 1 (z) 2 F 1 (z) new new 1 k ± ² s[k/2] ; k even 0 ; k odd + 1 1 1 2
1 f r[2k] = s[k] r[2k + 1] = 0 So new [n ]=f[n ] + s[k] f 0 [n – 2k] 1 k w Then the corresponding wavelet is new (t) = f new [n] φ (2t – n) 1 n = f[n] φ (2t – n) + s[k] f 0 [n –2k] φ (2t – n) 1 n k n = w(t) + s[k] f 0 [ ± ] φ (2t - 2k - ± ) k ± = w(t) + s[k] φ (t – k) since φ (t) = f 0 [ ± ] φ (2t- ± ) k ± 5 Lifting for wavelet bases Lifting construction can be used to build a more complex set of scaling functions and wavelets from an initial biorthogonal set.

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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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Handout16 - Course 18.327 and 1.130 Wavelets and Filter...

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