The Wigner Distribution

1st Edition

Theory and Applications in Signal Processing


  • W. Mecklenbräuker
  • Franz Hlawatsch
    • Print ISBN 9780444888563
    The Wigner Distribution

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    In the last two decades, the field of time-frequency analysis has evolved into a widely recognized and applied discipline of signal processing. Besides linear time-frequency representations such as the short-time Fourier transform, the Gabor transform, and the wavelet transform, an important contribution to this development has undoubtedly been the Wigner distribution (WD) which holds an exceptional position within the field of bilinear/quadratic time-frequency representations.

    The WD was first defined in quantum mechanics as early as 1932 by the later Nobel laureate E. Wigner. In 1948, J. Ville introduced this concept in signal analysis. Based on investigations of its mathematical structure and properties by N.G. de Bruijn in 1967, the WD was brought to the attention of a larger signal processing community in 1980. The WD was soon recognized to be important for two reasons: firstly, it provides a powerful theoretical basis for quadratic time-frequency analysis; secondly, its discrete-time form (supplemented by suitable windowing and smoothing) is an eminently practical signal analysis tool.

    The seven chapters of this book cover a wide range of different aspects of the WD and other linear time-frequency distributions: properties such as positivity, spread, and interference term geometry; signal synthesis methods and their application to signal design, time-frequency filtering, and signal separation; WD based analysis of nonstationary random processes; singular value decompositions and their application to WD based detection and classification; and optical applications of the WD. The size of the chapters has been chosen such that an in-depth treatment of the various topics is achieved.

    Table of Contents

    Positivity and Spread of Bilinear Time-Frequency Distributions
    (A.J.E.M. Janssen).
    Introduction. Cohen's class of bilinear time-frequency distributions. Time-frequency distributions and spread. Positivity properties of the Wigner distribution. Theorems of the Hudson type for Wigner-Rihaczek distributions. Wigner type theorems for bilinear time-frequency distributions. Smoothing bilinear time-frequency distributions. Some additional results and further developments. References.
    The Interference Structure of the Wigner Distribution and Related Time-Frequency Signal Representations
    (F. Hlawatsch, P. Flandrin).
    Part I - Wigner Distribution
    Outer interference. Interference geometry. Energetic interpretation of interference terms. Inner interference . The stationary-phase approximation. Singularities of interference terms. Sampling and aliasing.
    Part II - Related Time-Frequency Representations. Generalized Wigner distribution. Ambiguity function. The shift-invariant class. The shift-scale-invariant class. Smoothed versions of Wigner distribution. Conclusion. References.
    Signal Synthesis Algorithms for Bilinear Time-Frequency Signal Representations
    (F. Hlawatsch, W. Krattenthaler)
    Introduction. Subspace-unitary bilinear time-frequency representations. Wigner distribution. Smoothed versions of Wigner distribution. Spectrogram. On-line synthesis algorithms for pseudo Wigner distribution. Phase matching algorithms. Applications to time-frequency filtering. Conclusion. References.
    The Wigner-Ville Spectrum of Nonstationary Random Signals
    (P. Flandrin, W. Martin)
    Introduction. Stationarity and nonstationarity. Definitions of time-dependent spectra. The Wigner-Ville spectrum. Estimation of the Wigner-Ville spectrum. Wigner-Ville based optimum detection. Conclusion. References.
    Time-Varying Signal Proce


    © 1997
    Elsevier Science
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    About the authors

    Franz Hlawatsch

    Affiliations and Expertise

    Vienna University of Technology, Vienna, Austria


    "An invaluable source if you are involved with functions of both time and frequency, or of course of both space and spatial frequency" --Ultramicroscopy, Vol 75

    " excellent overview of the theory and applications of the Wigner distribution in signal analysis...written by some of the leading researchers in this area..." --International Journal of Electronics & Communications

    "...Thanks to its content, which has now become more or less classical material, this book should be a fine addition to the library of anyone interested in time-frequency signal analysis." --Mathematical Reviews