Hermite--Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-value decomposition of its orthogonal projection matrices

A new technique is proposed for generating initial orthonormal eigenvectors of the discrete Fourier transform matrix F by the singular-value decomposition of its orthogonal projection matrices on its eigenspaces and efficiently computable expressions for those matrices are derived. In order to generate Hermite-Gaussian-like orthonormal eigenvectors of F given the initial ones, a new method called the sequential orthogonal procrustes algorithm (SOPA) is presented based on the sequential generation of the columns of a unitary matrix rather than the batch evaluation of that matrix as in the OPA. It is proved that for any of the SOPA, the OPA, or the Gram--Schmidt algorithm (GSA) the output Hermite-Gaussian-like orthonormal eigenvectors are invariant under the change of the input initial orthonormal eigenvectors. Index Terms--Discrete fractional Fourier transform (DFRFT), Gram-Schmidt algorithm (GSA), Hermite-Gaussian-like eigenvectors, orthogonal procrustes algorithm (OPA), sequential OPA (SOPA), singular-value decomposition.