### Orthogonal IIR Wavelets

#### Abstract

Explicit solutions are given for the rational function P(z) for two classes of IIR orthogonal 2-band wavelet bases, for which the scaling filter is maximally flat. P(z) denotes the rational transfer function H(z) H(1/z) where H(z) is the (lowpass) scaling filter. The first is the class of solutions that are intermediate, between the Daubechies and the Butterworth wavelets. It is found that the Daubechies, the Butterworth, and the intermediate solutions are unified by a single formula. The second is the class of scaling filters realizable as a parallel sum of two allpass filters (a particular case of which yields the class of symmetric IIR orthogonal wavelet bases). For this class, an explicit solution is provided by the solution to an older problem in group delay approximation by digital allpole filters.

I. W. Selesnick, Fomulas for IIR Wavelet Filters,
*IEEE Trans. on Signal Processing*,
46(4):1138-1141, April 1998.

#### Matlab code

- idabusfi.m - Intermediate Daubechies-Butterworth scaling filter
- apassfi.m - Allpass sum scaling filter
- maxflat2.m - Generalized digital Butterworth filter
- flatdlay.m - Maximally flat delay digital allpole filter
- maxflaps.m - Maximally flat allpass sum
- choose.m - Binomial coefficients

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