Efficient computation of DRT commuting matrices by a closed-form infinite order approximation to the second differentiation matrix

Abstract

In order to define the discrete fractional Fourier transform, Hermite Gauss-like eigenvectors are needed and one way of extracting these eigenvectors is to employ DFT commuting matrices. Recently, Pei et al. exploited the idea of obtaining higher order DFT-commuting matrices, which was introduced by Candan previously. The upper bound of O(h(2k)) approximation to N x N commuting matrix is 2k+1 <= N in Candan's work and Pei et al. improved the proximity by removing this upper bound at the expense of higher computational cost. In this paper, we derive an exact closed form expression of infinite-order Taylor series approximation to discrete second derivative operator and employ it in the definition of excellent DFT commuting matrices. We show that in the limit this Taylor series expansion converges to a trigonometric function of second-order differentiating matrix. The commuting matrices possess eigenvectors that are closer to the samples of Hermite-Gaussian eigenfunctions of DFT better than any other methods in the literature with no additional computational cost. (C) 2010 Elsevier B.V. All rights reserved.