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   -> Volume 7, Issue 5

Answer: Parametrization of wavelets (WD 7.3 #19)
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Algirdas Bastys (

PostPosted: Mon Apr 20, 1998 8:36 pm    
Subject: Answer: Parametrization of wavelets (WD 7.3 #19)
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#24 Answer: Parametrization of wavelets (WD 7.3 #19)

Answer to Question 19 "Parametrization of wavelets", WD, Vol.7, Nr 3

Dear Senthil,

To parametrize 2N-length filter h_n, n=0,...,2N-1, that correspond to
a tight (as a rule to orthogonal) scaling function $phi$ you can
choose N-1 parameters 0lealpha_1,alpha_2,ldots,alpha_{N-1}<1, and
use the following recursion:

1. Initialization

h_0^0=1/sqrt2; h_1^0=1/sqrt2;
g_0^0=-h_1^0; g_1^0=h_0^0;

2. Recursion

h_i^k = cosalpha_k (cosalpha_k h_i^{k-1} + sinalpha_k g_i^{k-1}) +
sinalpha_k (sinalpha_k h_{i-2}^{k-1} - cosalpha_k g_{i-2}^{k-1});
g_i^k = (-1)^{1-i} h_{2k+1-i}^{k-1};
k=1,...,N-1; i=0,...,2k+1

At any k-th step of the recursion you will have an admissible filter
h_n^k_{n=0}^{2k+1}. For example, the initial step (k=0) gives the Haar
filter h_0,h_1; choosing alpha_1=1/12 after the first step (k=1) you
will have the Daubechies filter h_0,h_1,h_2,h_3.

Inversely, for any admissible filter h_n, n=0,...,2N-1, you can find
N-1 parameter alpha that recursions above give h_n^{N-1}=h_n.

Similar parametrization can be written for biorthogonal wavelets.

Sincerely yours,

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