Analytic Expressions of Two Discrete Hermite–Gauss Signals

Abstract

This brief presents the analytical expressions for the discrete zeroth-and first-order Hermite-Gauss functions, which are normally obtained by numerical methods. These two ldquoGaussian-typerdquo functions have the following interesting properties. (a) They have simple analytic forms (form of a product) when the lengths of the functions satisfy certain conditions. (b) They are the eigenvectors of discrete Fourier transforms (DFTs). The zero points of the functions and their respective DFTs are all located on the real axis. These discrete functions are compared with the continuous zeroth and first Hermite Gaussians. They resemble very well to the continuous functions, and the coincidence of the shapes with the continuous cases is remarkable.