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Handout10

# Handout10 - Course 18.327 and 1.130 Wavelets and Filter...

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φ φ φ φ φ Course 18.327 and 1.130 Wavelets and Filter Banks Refinement Equation: Iterative and Recursive Solution Techniques; Infinite Product Formula; Filter Bank Approach for Computing Scaling Functions and Wavelets Solution of the Refinement Equation N φ (t) = 2 h 0 [k] φ (2t-k) k = 0 First, note that the solution to this equation may not always exist! The existence of the solution will depend on the discrete-time filter h 0 [k]. If the solution does exist, it is unlikely that φ (t) will have a closed form solution. The solution is also unlikely to be smooth. We will see, however, that if h 0 [n] is FIR with h 0 [n] = 0 outside 0 n N then φ (t) has compact support: φ (t) = 0 outside 0 < t < N 2 1

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φ φ φ → ∞ φ φ (t) Approach 1 Iterate the box function φ (0) (t) = box function on [0 , 1] 0 1 t N φ (i + I) (t) = 2 h 0 [k] (i) (2t – k) k = 0 If the iteration converges, the solution will be given by lim (i) (t) i This is known as the cascade algorithm. 3 4 Example: 0 [k] {¼, ½, ¼} φ (i+1) (t) ½ φ (i) (2t) φ (i) (2t – 1) + ½ φ (i) (2t – 2) Then φ (0) (t) 1 0 1 2 t φ (0) (t) 1 0 1 2 t 2 3 2 1 2 1 φ (2) (t) 1 0 1 2 t 2 3 2 1 Converges to the hat function on [0, 2] φ (3) (t) 1 0 1 2 t 2 3 2 1 ½ suppose h = = + 2
φ φ φ φ Approach 2 Use recursion First solve for the values of φ (t) at integer values of t. Then solve for (t) at half integer values, then at quarter integer values and so on. This gives us a set of discrete values of the scaling function at all dyadic points t = n/2 i .

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Handout10 - Course 18.327 and 1.130 Wavelets and Filter...

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