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

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.