Time-frequency analysis and Harmonic Gaussian Functions

TL;DR

This paper introduces a novel time-frequency analysis method using harmonic Gaussian functions, Hermite polynomials, and Fourier analysis, enabling energy distribution representation and signal recovery in the time-frequency domain.

Contribution

It defines harmonic Gaussian functions and transformations T_n, providing a new framework for analyzing signals in the time-frequency plane with proven energy interpretation and reconstruction capability.

Findings

Functions { extPsi}_n represent signal energy distribution in time-frequency plane.

The method allows signal reconstruction from the transformed functions.

Properties of the transformations T_n are established.

Abstract

A method for time-frequency analysis is given. The approach utilizes properties of Gaussian distribution, properties of Hermite polynomials and Fourier analysis. We begin by the definitions of a set of functions called harmonic Gaussian functions. Then these functions are used to define a set of transformations,noted T_n, which associate to a function {\psi},of the time variable t, a set of functions {\Psi}_n which depend on time, frequency and frequency (or time) standard deviation. Some properties of the transformations T_n and the functions {\Psi}_n are given. It is proved in particular that the square of the modulus of each function {\Psi}_n can be interpreted as a representation of the energy distribution of the signal, represented by the function {\psi}, in the time-frequency plane for a given value of the frequency (or time) standard deviation. It is also shown that the function {\psi}, can be recovered from the functions{\Psi}_n.