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 1e-4.
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.