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   -> Volume 7, Issue 1

Preprint: Systematized Collection of Wavelet Filters (Taswell)
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"Carl Taswell" (

PostPosted: Tue Dec 30, 1997 8:27 am    
Subject: Preprint: Systematized Collection of Wavelet Filters (Taswell)
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#4 Preprint: Systematized Collection of Wavelet Filters (Taswell)

A fully hyperlinked Adobe *.pdf document of 100 pages containing
extensive tables, multicolor figures, and text describing all
algorithms for the systematized collection of Daubechies wavelets is
now available, compatible with version 4.4b6 of the FirWav Filter

The Systematized Collection of Wavelet Filters Computable by Spectral
Factorization of the Daubechies Polynomial

Carl Taswell, 11 December 1997

Computational algorithms have been developed for generating minimum
length maximum flatness finite impulse response (FIR) filter
coefficients for a systematized collection of wavelet filters designed
by spectral factorization of a product filter. Both Lagrange and
Daubechies polynomials have been studied numerically as alternative
constructions for the required product filter which must be a halfband
autocorrelation filter.

The systematized collection obtained from the product filter comprises
real and complex orthogonal and biorthogonal wavelets in families
defined by optimization criteria for various filter parameters. The
main algorithm incorporates spectral factorization of the Daubechies
polynomial with a combinatorial search of spectral factor root sets
indexed by binary codes. The selected spectral factors are found by
optimizing the desired criterion characterizing either the filter
roots or coefficients.

Polynomial roots for the spectral factors are computed by a composite
conformal mapping with affine and inverse Joukowski
transformations. Filter coefficients are computed from the roots by
iterative convolution of linear root factors previously sorted in
increasing absolute value order. Experiments with higher order root
factors and other root sort orders, such as the Edrei- Leja order,
revealed no significant benefit to these alternatives.

Daubechies wavelet filter families have been systematized to include
those optimized for time-domain regularity, frequency-domain
selectivity, time- frequency uncertainty, and phase nonlinearity. The
latter criterion permits construction of the least and most asymmetric
and least and most symmetric real and complex orthogonal
filters. Biorthogonal symmetric spline and balanced length filters are
also computable by these methods.

All families have been indexed by the number K of roots at z = -1
corresponding to the number of vanishing moments on the wavelets. All
filters have been subjected to extensive numerical tests including all
of the filter parameters defining each of the different families as
well as the M-shift biorthogonality, M-shift orthogonality, and M-band
reconstruction errors. The numerically observed vanishing moments
number should equal K. However, it was observed to peak at
approximately 12 for each family tested when all computations were
done in double precision with tolerance for a vanishing moment set at

New filters with distinguishing features are demonstrated with
examples for each of the families. All of the filter families are
catalogued extensively for K = 1,2,...,24 with tables listing
numerical estimates of the filter parameters and figures displaying
plots of the filter zeros, impulse responses, and frequency responses.
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