Hermite–Gaussian-Like Eigenvectors of the Discrete Fourier Transform Matrix Based on the Singular-Value Decomposition of Its Orthogonal Projection Matrices

2004/11/01 by Magdy Tawfik Hanna, Nabila Philip Attalla Seif, Waleed Ahmed · 2 citations
Computer Science · Mathematics · #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Blind Source Separation Techniques

paper · doi:10.1109/tcsi.2004.836850

openalex publication_date 2004/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

A 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.

Citations

• The discrete fractional Fourier transform
• Eigenvalue and eigenvector decomposition of the discrete Fourier transform
• Eigenvectors and functions of the discrete Fourier transform

Cited by

• Analytic Expressions of Two Discrete Hermite–Gauss Signals
• Discrete Fourier transform [wikipedia]