6.1 Introduction

This chapter continues the time-frequency analysis of Chapter 5. We will introduce algorithms that allow us to decompose a given signal s into a linear combination of time-frequency atoms. The time-frequency atoms that we use are denoted by f _R and are "coded" by Heisenberg rectangles R with sides parallel to the axes and with area 1 or 2 ?, depending on the normalization. If R = [ a, b] [ ?, ?], we require that the function f _R be essentially supported on the interval [ a, b] and that its Fourier transform be essentially supported on [ ?, ?] and the opposite frequencies [ ? ?, ? ?]. We also want the algorithmic structure of f _R to be simple and explicit to facilitate numerical processing in real time. The decomposition

cannot be unique, and we take advantage of this flexibility by looking for optimal decompositions, which for our purposes means that they contain the fewest possible terms.

The point of view of Ville (and of numerous other signal-processing experts) is that it is first necessary to understand the physics of the process and that "the algorithms will follow." A careful reading of Ville's fundamental paper [254] suggests the following algorithm for finding the optimal decomposition (6.1): (1) Compute the Wigner Ville transform W ( t, ?) of f; (2) define the domains ? _j of...