Daubechies wavelet filter computation
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)
% P the "Lagrange à trous" filter for N=2 is explicit % and given by: P = [ -1/16 0 9/16 1 9/16 0 -1/16] P = -0.0625 0 0.5625 1.0000 0.5625 0 -0.0625 % The db2 Daubechies scaling filter w, is a % solution of the equation: P = conv(wrev(w),w) * 2. % % This filter P is symmetric, easy to generate, and w is % a minimum phase solution of the previous equation, % based on the roots of P. rP = roots(P); % Retaining only the root inside the unit circle (here it % is the sixth value of rP), and two roots located at -1, % we obtain the Daubechies wavelet of order 2: ww = poly([rP(6) -1 -1]); % filter construction ww = ww / sum(ww) % normalize sum ww = 0.3415 0.5915 0.1585 -0.0915 % Check that ww is correct and equal to % the db2 Daubechies scaling filter w. w = dbaux(2) w = 0.3415 0.5915 0.1585 -0.0915
The algorithm used is based on a result obtained by Shensa (see "References"), showing a correspondence between the "Lagrange à trous" filters and the convolutional squares of the Daubechies wavelet filters.
The computation of the order N Daubechies scaling filter w proceeds in two steps: compute a "Lagrange à trous" filter P, and extract a square root. More precisely:
wrev(w),w) where w is a filter of length 2N.
Note that other methods can be used; see various solutions of the spectral factorization problem in Strang-Nguyen (p. 157).
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,
Strang, G.; T. Nguyen (1996), Wavelets and Filter Banks, Wellesley-Cambridge Press.