Direct Batch Evaluation of Optimal Orthonormal Eigenvectors of the DFT Matrix

Abstract

The generation of Hermite-Gaussian-like orthonormal eigenvectors of the discrete Fourier transform (DFT) matrix F is an essential step in the development of the discrete fractional Fourier transform (DFRFT). Most existing techniques depend on the generation of orthonormal eigenvectors of a nearly tridiagonal matrix S which commutes with matrix F. More sophisticated methods view the eigenvectors of S as only initial ones and use them for generating final ones which better approximate the Hermite-Gaussian functions employing a technique like the orthogonal Procrustes algorithm (OPA). Here, a direct technique for the collective (batch) evaluation of optimal Hermite-Gaussian-like eigenvectors of matrix F is contributed. It is a direct technique in the sense that it does not require the generation of initial eigenvectors to be used for computing the final superior ones. It is a batch method in the sense that it solves for the entire target modal matrix of F instead of the sequential generation of the eigenvectors. The simulation results show that the proposed method is faster than the OPA.