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Handout9 - Course 18.327 and 1.130 Wavelets and Filter...

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Course 18.327 and 1.130 Wavelets and Filter Banks Multiresolution Analysis (MRA): Requirements for MRA; Nested Spaces and Complementary Spaces; Scaling Functions and Wavelets 2 Scaling Functions and Wavelets Continuous time: φ (t) Box function 1 0 1 t 1 0 1 1 0 t t 1/2 1/2 φ (2t) Scaling φ (2t - 1) Scaling + Shifting 1
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φ φ φ φ φ φ φ φ < φ φ φ φ φ τ τ φ For this example: φ (t) = (2t) + (2t – 1) More generally: N Refinement equation (t) = 2 h 0 [k] φ (2t – k) or k=0 Two-scale difference equation φ (t) is called a scaling function The refinement equation couples the representations of a continuous-time function at two time scales. The continuous-time function is determined by a discrete- time filter, h 0 [n]! For the above (Haar) example: h 0 [0] = h 0 [1] = ½ (a lowpass filter) 3 Note: (i) Solution to refinement equation may not always exist. If it does… (ii) (t) has compact support i.e. (t) = 0 outside 0 t N (comes from the FIR filter, h 0 [n]) (iii) (t) often has no closed form solution. (iv) (t) is unlikely to be smooth. Constraint on h 0 [n]: N (t)dt = 2 h 0 [k] (2t – k)dt k=0 N = 2 h 0 [k] • ½ φ ( τ )d k=0 So N h 0 [k] = 1 Assumes (t)dt 0 k=0 4 2
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φ 5 Now consider: 1 0 1/2 1 t 1 0 1/2 t w(t) Square wave of finite length - Haar wavelet 1 0 1/2 t φ (2t) Scaled - φ (2t – 1) Scaled + shifted + sign flipped w(t) φ (2t) φ (2t – 1) = - More generally: N w(t) = 2 h 1 [k] (2t – k) Wavelet equation k=0 For the Haar wavelet example: h 1 [0] = ½ h 1 [1] = -½ (a highpass filter) 6 3
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7 Some observations for Haar scaling function and wavelet 1. Orthogonality of integer shifts (translates): 1 0 2 1 0 t t 1 1 φ (t) φ
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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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Handout9 - Course 18.327 and 1.130 Wavelets and Filter...

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