Time-frequency analysis and harmonic Gaussian functions

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 Τn, which associate to a function ψ, of the time variable t, a set of functions Ψn which depend on time, frequency and frequency (or time) standard deviation. Some properties of the transformations Τn and the functions Ψn are given. It is proved in particular that the square of the modulus of each function Ψn can be interpreted as a representation of the energy distribution of the signal, represented by the function ψ, in the time-frequency plane for a given value of the frequency (or time) standard deviation. It is also shown that the function ψ can be recovered from the functions Ψn.