Quadratic Time-Frequency Analysis III: The Affine Class and Other Covariant Classes

Paulo Gonçalvés, Jean Philippe Ovarlez, Richard Baraniuk

Research output: Chapter in Book/Report/Conference proceedingChapter

3 Scopus citations


Affine time-frequency distributions appeared towards the middle of the 1980s with the emergence of wavelet theory. The affine class is built upon the principle of covariance of the affine group, i.e., contractions-dilations and translations in time. This group provides an interesting alternative to the group of translations in time and in frequency, which forms the basis for the conventional time-frequency distributions of Cohen's class. More precisely, as the Doppler effect on "broadband" signals is expressed in terms of contractions-dilations, it is for the analysis of this category of signals that the affine class is particularly destined. The objective of this chapter is to present the various approaches for constructing the affine class and the associated tools devised over the past years. We will demonstrate how the latter supported the introduction of new mathematical concepts in signal processing - group theory, operator theory - as well as of new classes of covariant time-frequency distributions.

Original languageEnglish (US)
Title of host publicationTime-Frequency Analysis
Subtitle of host publicationConcepts and Methods
Number of pages34
ISBN (Print)9781848210332
StatePublished - Jan 26 2010


  • Affine group
  • Affine time-frequency analysis
  • Affine Wigner distributions
  • Bertrand distributions
  • Covariance principle
  • Hyperbolic class
  • Power classes
  • Unitary equivalence
  • Wavelets

ASJC Scopus subject areas

  • Engineering(all)


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