Formulas for orthogonal IIR wavelet filters

Explicit solutions are given for the rational function P(z) for two classes of IIR orthogonal two-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 (1992) 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, a closed-form solution is provided by the solution to an older problem in group delay approximation by digital allpole filters