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Daubechies wavelet filter computation
W = dbaux(N,SUMW)
W = dbaux(N)
W = dbaux(N,0)
W = dbaux(N,SUMW) is the order N Daubechies scaling filter such that sum(W) = SUMW. Possible values for N are 1, 2, 3, ...
W = dbaux(N) is equivalent to W = dbaux(N,1)
W = dbaux(N,0) is equivalent to W = dbaux(N,1)
This example shows to determine the Daubechies' extremal phase scaling filter with a specified sum. The two most common values for the sum are and 1.
Construct two versions of the 'db4' scaling filter. One scaling filter sums to and the other version sums to 1.
NumVanishingMoments = 4; h = dbaux(NumVanishingMoments,sqrt(2)); m0 = dbaux(NumVanishingMoments,1);
The filter with sum equal to is the synthesis (reconstruction) filter returned by wfilters and used in the discrete wavelet transform.
[LoD,HiD,LoR,HiR] = wfilters('db4');
max(abs(LoR-h))
ans = 4.2613e-13
For orthogonal wavelets, the analysis (decomposition) filter is the time-reverse of the synthesis filter.
max(abs(LoD-fliplr(h)))
ans = 4.2613e-13
The computation of the dbN Daubechies scaling filter requires the extraction of the roots of a polynomial of order 4N. Instability may occur when N is too large.
Daubechies, I. (1992), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics, SIAM Ed.
Shensa, M.J. (1992), "The discrete wavelet transform: wedding the a trous and Mallat Algorithms," IEEE Trans. on Signal Processing, vol. 40, 10, pp. 2464-2482.
Strang, G.; T. Nguyen (1996), Wavelets and Filter Banks, Wellesley-Cambridge Press.