## Abstract

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 language | English (US) |
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Title of host publication | Time-Frequency Analysis |

Subtitle of host publication | Concepts and Methods |

Publisher | Wiley |

Pages | 193-226 |

Number of pages | 34 |

ISBN (Print) | 9781848210332 |

DOIs | |

State | Published - Jan 26 2010 |

## Keywords

- 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)