Time-Frequency Analysis and Harmonic Gaussian Functions
Pure and Applied Mathematics Journal
Volume 2, Issue 2, April 2013, Pages: 71-78
Received: Mar. 7, 2013; Published: Apr. 2, 2013
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Authors
Tokiniaina Ranaivoson, Theoretical Physics Dept., Antananarivo, Madagascar; Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar), Antananarivo, Madagascar
Raoelina Andriambololona, Theoretical Physics Dept., Antananarivo, Madagascar; Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar), Antananarivo, Madagascar
Rakotoson Hanitriarivo, Theoretical Physics Dept., Antananarivo, Madagascar; Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar), Antananarivo, Madagascar
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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 ψ, of the time variable t, a set of functions Ψ_n which depend on time, frequency and frequency (or time) standard deviation. Some properties of the transformations T_n and the functions Ψ_n are given. It is proved in particular that the square of the modulus of each function Ψ_n 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 Ψ_n.
Keywords
Time-Frequency Analysis, Signal, Energy Distribution, Gaussian Distribution, Hermite Polynomials
To cite this article
Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo, Time-Frequency Analysis and Harmonic Gaussian Functions, Pure and Applied Mathematics Journal. Vol. 2, No. 2, 2013, pp. 71-78. doi: 10.11648/j.pamj.20130202.14
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