On the Construction and Frequency Localization of Finite Orthogonal Quadrature Filters

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 @c"p of a finite filter m"0 is given by the distance in L^p-norm between |m"0|^2 and the Shannon low-pass filter. For each N>0 there is a filter m^N"0 of length 2N minimizing the value of @c"p. We prove that for such a minimizing sequence we have @c^p"p(m^N"0)=O(1/N), 1=