Time-Frequency Analysis and Harmonic Gaussian Functions

Tokiniaina Ranaivoson; Raoelina Andriambololona; Rakotoson Hanitriarivo

doi:doi:10.11648/j.pamj.20130202.14

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Time-Frequency Analysis and Harmonic Gaussian Functions

Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 2)

Published: 2 April 2013

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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.

Published in	Pure and Applied Mathematics Journal (Volume 2, Issue 2)
DOI	10.11648/j.pamj.20130202.14
Page(s)	71-78
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2013. Published by Science Publishing Group

Keywords

Time-Frequency Analysis, Signal, Energy Distribution, Gaussian Distribution, Hermite Polynomials

References

[1]	F.Hlawatsch, G. F.Boudreaux-Bartels, « Linear and quadratic time frequency signal representations », IEEE Signal Process. Mag., vol. 9, n_2, p. 21.67, avril 1992.
[2]	L. Cohen, "Time-Frequency Analysis", Prentice-Hall, New York, 1995.
[3]	L. Cohen, "Time-Frequency Distributions—A Review," Proceedings of the IEEE, vol. 77, no. 7, pp. 941–981, 1989
[4]	A. J. E. M. Janssen, "Bilinear time-frequency distributions", Wavelets and Their Applications, NATO ASI Series Volume 442, pp 297-311, 1994.
[5]	F. Hlawatsch, R.L. Urbanke, "Bilinear Time-Frequency Representations of Signals: The Shift-Scale Invariant Class", doi:10.1109/78.275608, IEEE transactions on signal processing, vol. 42, no. 2, february 1994
[6]	S. Qian and D. Chen, "Joint Time-Frequency Analysis: Methods and Applications", Prentice-Hall, 1996
[7]	L. Debnath, "Wavelet Transforms and Time-Frequency Signal Analysis", Birkhäuser, Boston, 2001.
[8]	E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev. 40,749-759. 1932
[9]	J. Ville, "Théorie et Applications de la Notion de Signal Analytique", Câbles et Transmission, Vol.2,pp 61–74, 1948
[10]	K. Grochenig, "Foundations of Time-Frequency Analysis", Birkhauser, Boston, 2001
[11]	W.Mecklenbräuker, F.Hlawatsch, Eds., The Wigner Distribution.Theory and Applications in Signal Processing, Elsevier, Amsterdam, The Netherlands, 1997.
[12]	Raoelina Andriambololona, "Mécanique quantique", Collection LIRA, INSTN Madagascar.pp 25.387-394, 1990

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APA Style

Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo. (2013). Time-Frequency Analysis and Harmonic Gaussian Functions. Pure and Applied Mathematics Journal, 2(2), 71-78. https://doi.org/10.11648/j.pamj.20130202.14

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ACS Style

Tokiniaina Ranaivoson; Raoelina Andriambololona; Rakotoson Hanitriarivo. Time-Frequency Analysis and Harmonic Gaussian Functions. Pure Appl. Math. J. 2013, 2(2), 71-78. doi: 10.11648/j.pamj.20130202.14

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AMA Style

Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo. Time-Frequency Analysis and Harmonic Gaussian Functions. Pure Appl Math J. 2013;2(2):71-78. doi: 10.11648/j.pamj.20130202.14

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@article{10.11648/j.pamj.20130202.14,
  author = {Tokiniaina Ranaivoson and Raoelina Andriambololona and Rakotoson Hanitriarivo},
  title = {Time-Frequency Analysis and Harmonic Gaussian Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {2},
  pages = {71-78},
  doi = {10.11648/j.pamj.20130202.14},
  url = {https://doi.org/10.11648/j.pamj.20130202.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130202.14},
  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.},
 year = {2013}
}

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TY  - JOUR
T1  - Time-Frequency Analysis and Harmonic Gaussian Functions
AU  - Tokiniaina Ranaivoson
AU  - Raoelina Andriambololona
AU  - Rakotoson Hanitriarivo
Y1  - 2013/04/02
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130202.14
DO  - 10.11648/j.pamj.20130202.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 71
EP  - 78
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130202.14
AB  - 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.
VL  - 2
IS  - 2
ER  -

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Author Information

Tokiniaina Ranaivoson

Theoretical Physics Dept., Antananarivo, Madagascar
Raoelina Andriambololona

Theoretical Physics Dept., Antananarivo, Madagascar
Rakotoson Hanitriarivo

Theoretical Physics Dept., Antananarivo, Madagascar

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APA Style

Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo. (2013). Time-Frequency Analysis and Harmonic Gaussian Functions. Pure and Applied Mathematics Journal, 2(2), 71-78. https://doi.org/10.11648/j.pamj.20130202.14

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ACS Style

Tokiniaina Ranaivoson; Raoelina Andriambololona; Rakotoson Hanitriarivo. Time-Frequency Analysis and Harmonic Gaussian Functions. Pure Appl. Math. J. 2013, 2(2), 71-78. doi: 10.11648/j.pamj.20130202.14

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AMA Style

Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo. Time-Frequency Analysis and Harmonic Gaussian Functions. Pure Appl Math J. 2013;2(2):71-78. doi: 10.11648/j.pamj.20130202.14

Copy | Download

@article{10.11648/j.pamj.20130202.14,
  author = {Tokiniaina Ranaivoson and Raoelina Andriambololona and Rakotoson Hanitriarivo},
  title = {Time-Frequency Analysis and Harmonic Gaussian Functions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {2},
  pages = {71-78},
  doi = {10.11648/j.pamj.20130202.14},
  url = {https://doi.org/10.11648/j.pamj.20130202.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130202.14},
  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.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - Time-Frequency Analysis and Harmonic Gaussian Functions
AU  - Tokiniaina Ranaivoson
AU  - Raoelina Andriambololona
AU  - Rakotoson Hanitriarivo
Y1  - 2013/04/02
PY  - 2013
N1  - https://doi.org/10.11648/j.pamj.20130202.14
DO  - 10.11648/j.pamj.20130202.14
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 71
EP  - 78
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130202.14
AB  - 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.
VL  - 2
IS  - 2
ER  -

Copy | Download