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 frequency localization of finite filters. The frequency localization gamma (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(0)(N) of length 2N minimizing the value gamma (p). We prove that for such a minimizing sequence we have gamma (p)(p)(m(0)(N)) = O(1 N), 1 less than or equal to p less than or equal to 2, and this estimate is optimal. We construct several new families of both MRA and non-MRA filters with optimal asymptotic frequency localization. (C) 2000 Academic Press.