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Wavelets and Signal Processing pp 177–231Cite as

Birkhäuser

Covariant Time-Frequency Analysis

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Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

We present a theory of linear and bilinear/quadratic time-frequency (TF) representations that satisfy a covariance property with respect to “TF displacement operators” These operators cause TF displacements such as (possibly dispersive) TF shifts and dilations/compressions. Our covariance theory establishes a unified framework for important classes of linear TF representations (e.g., the short-time Fourier transform and continuous wavelet transform) as well as bilinear TF representations (e.g., Cohen’s class and the affine class). It yields a theoretical basis for TF analysis and allows the systematic construction of covariant TF representations.

The covariance principle is developed both in the group domain and in the TF domain Fundamental properties of the displacement function connecting these two domains and their far-reaching consequences are studied, and a method for constructing the displacement function is presented.

We also introduce important classes of operator families (modulation and warping operators, dual and affine operators), and we apply the results of the covariance theory to these operator classes. It is shown that for dual operator pairs the characteristic function method for constructing bilinear TF representations is equivalent to the covariance method.

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Authors
  1. Franz Hlawatsch
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  2. Georg Tauböck
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  3. Teresa Twaroch
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Editors and Affiliations

  1. Department of Mathematics, The University of Texas-Pan American, Edinburg, TX, 78539-2999, USA

    Lokenath Debnath

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Hlawatsch, F., Tauböck, G., Twaroch, T. (2003). Covariant Time-Frequency Analysis. In: Debnath, L. (eds) Wavelets and Signal Processing. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0025-3_7

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  • DOI: https://doi.org/10.1007/978-1-4612-0025-3_7

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6578-8

  • Online ISBN: 978-1-4612-0025-3

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