Synthesis of extremal wavelet-generating filters using Gaussian quadrature

Orthogonal wavelets can be generated from finite impulse response quadrature mirror filters; these filters are also used in perfect reconstruction filter banks. This paper addresses the problem of efficiently synthesizing such filters. A class of extremal filters is defined by the property that their magnitude spectrum maximizes an integral criterion. It is found that these filters are characterized by their zeros on the unit circle, which frequently can be obtained from a set of orthogonal polynomials. A family of filters is constructed that minimize the subband aliasing energy and can generate wavelets with an arbitrary number of vanishing moments. The algorithm for generating these filters makes use of the Levinson recursions, Gaussian quadrature and a fast version of Euclid´s algorithm. Similar to other methods for constructing quadrature mirror filters, the spectral factorization of a polynomial is the computationally expensive part of this algorithm