TABLE OF CONTENTS Time-frequency analysis for signal processing is an active field of research. Here, as in many domains, heuristic concepts structure and guide the work. The heuristic notions that will serve us in this and the following three chapters are (1) time-frequency atoms, (2) the optimal decomposition of a signal into time-frequency atoms, (3) instantaneous frequency, (4) the time-frequency plane, (5) the optimal representation of a signal in the time-frequency plane, and (6) optimal partitioning of the time-frequency plane. In this and the following chapters, we will try to give precise scientific meaning to these heuristic ideas. We add, however, that this is a large field of research and that our exposition is by no means exhaustive.
Dennis Gabor [124] and Jean Ville [254] both addressed the problem of developing a mixed representation of a signal in terms of a double sequence of elementary signals, each of which occupies a certain domain in the time-frequency plane. In the following sections we will define what is meant by time-frequency plane and mixed representation, and we will suggest several choices for the elementary signals, or atoms.
Roger Balian tackled the same problem and expressed the motivation for his work in these terms [17, p. 1357]:
One is interested, in communication theory, in representing an oscillating signal as a superposition of elementary wavelets, each of which has a rather well defined frequency and position in time. Indeed, useful information is often conveyed by both the emitted frequencies and the signal's...
