On the Construction and Frequency Localization of Finite Orthogonal Quadrature Filters

Abstract

We introduce a new method to construct finite orthogonal quadrature filters using convolution kernels and show that every filter with value 1 at the origin can be obtained from an even nonnegative kernel. We apply the method to estimate the optimal frequency localization of finite filters. The frequency localization γp of a finite filter m0 is given by the distance in Lp-norm between |m0|2 and the Shannon low-pass filter. For each N>0 there is a filter mN0 of length 2N minimizing the value of γp. We prove that for such a minimizing sequence we have γpp(mN0)=O(1/N), 1⩽p⩽2, and this estimate is optimal. We construct several new families of both MRA and non-MRA filters with optimal asymptotic frequency localization.