On the Construction and Frequency Localization of Finite Orthogonal Quadrature Filters

In this article we introduce a new method to construct finite orthogonal quadrature filters using convolution kernels. We show that every filter with value 1 at the origin can be obtained using 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 L p-norm between |m0| and the Shannon low-pass filter. For each N > 0 there is a filter m0 of length 2N minimizing the value of γp. We prove that for such a minimizing sequence we have γ p(m0 ) = 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.