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Time-Frequency Analysis and Harmonic Gaussian Functions

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
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    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
  • Theoretical Physics Dept., Antananarivo, Madagascar

  • Theoretical Physics Dept., Antananarivo, Madagascar

  • Theoretical Physics Dept., Antananarivo, Madagascar

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