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Unformatted text preview: 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 (t) = 2 h [k] (2tk) k = 0 First, note that the solution to this equation may not always exist! The existence of the solution will depend on the discretetime filter h [k]. If the solution does exist, it is unlikely that (t) will have a closed form solution. The solution is also unlikely to be smooth. We will see, however, that if h [n] is FIR with h [n] = 0 outside 0 n N then (t) has compact support: (t) = 0 outside 0 < t < N 2 1 (t) Approach 1 Iterate the box function (0) (t) = box function on [0 , 1] 0 1 t N (i + I) (t) = 2 h [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: [k] {, , } (i+1) (t) (i) (2t) (i) (2t 1) + (i) (2t 2) Then (0) (t) 1 0 1 2 t (0) (t) 1 0 1 2 t 2 3 2 1 2 1 (2) (t) 1 0 1 2 t 2 3 2 1 Converges to the hat function on [0, 2] (3) (t) 1 0 1 2 t 2 3 2 1 suppose h = = + 2 Approach 2 Use recursion First solve for the values of (t) at integer values of t. Then solve for (t) at half integer values, then at quarter integer values and so on....
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 Spring '03
 GilbertStrang

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