time-frequency and waveletsJohn Cage, Concert for Piano and Orchestra (1957-58)
The analogy with musical notation is commonly used for introducing the idea of a time-frequency representation, each note of a piece of music being characterized by some localization in time (instant of occurrence, duration) as well as in frequency (pitch). While preserving this idea of associating a time-frequency plane with a "mathematical score", wavelet analysis adds the extra constraint that relies on the physical assumption according to which a note, to be perceived as such, should be played for a duration which is increased as much as its frequency is decreased (quoting N. Wiener in I am a Mathematician, "A fast jig on the lowest register of an organ is in fact not so much bad music but no music at all."). The time-frequency plane admits therefore a built-in tree structure rooting at lower frequencies.
The transform and the associated spectrum analysis, as it has been especially developed by , are ubiquitous tools in the processing of harmonic and/or stationary signals. However, they both rely on a notion of frequency that cannot accommodate for any time evolution, thus forbidding to give any sense to the intuitive notion of instantaneous frequency. This difficulty is fundamentally linked to the fact that time and frequency are "canonically conjugated" variables, with "uncertainty" relations as an unavoidable by-product for their associated representations. The first qualitative attention paid to this fact is attached to the name of (1925) and to a context of quantum mechanics (with position and momentum in place of time and frequency), whereas the precise mathematical form of classical inequalities is due to (1927) and to in signal theory (1946). Anticipating on a wish of ("Je cherche en même temps l'éternel et l'éphémère"), (1946) and (1948) first conciled time and frequency by proposing a definition of instantaneous frequency based on the notion of analytic signal, and hence on the use of the transform. Nevertheless, this approach is itself faced with a natural limitation as soon as the analyzed signals happen to be multi-component, since a single-valued function can by no way describe the simultaneous evolution of different time-frequency trajectories. In such cases, it becomes mandatory to call for a truely bidimensional approach, aimed at giving a representation (an "image") of a nonstationary signal in a time-frequency plane. In this case too, whereas intuitive and ad hoc procedures may have been used for ages on the basis of sliding transforms, it is within the framework of quantum mechanics that appeared the first truely joint distribution. It has been proposed by (1932) and rediscovered in the context of
signal theory by (1948). Even if it is not really unique, the distribution does possess a large number of remarkable theoretical properties and it has played a key role in most of the studies that followed. By construction, the distribution is a quadratic functional of the signal, making of it more an "evolutive spectrum density" than a time-dependent transform. Preserving the idea of linearity is another direction which led to variations upon harmonic analysis, with wavelet analysis in first place. In this approach, the concept of frequency is indeed replaced by that of scale, formalizing the idea of a local description of a signal at different resolution levels. As for the transform, the wavelet transform consists in projecting the analyzed signal upon a family of analyzing signals. This latter is in this case a two-parameter (time and scale) family, all of its elements being deduced from a unique waveform (the mother wavelet) by shifts and dilations. Even if the first construction of a wavelet basis goes back to (1911), the exponential growth of the modern theory (and of its applications) has started in the eighties, following the pioneering works of J. Morlet, A. Grossmann, Y. Meyer, S. Mallat, I. Daubechies, etc.
some general references
R. Carmona, H.L. Hwang, B. Torrésani, Practical Time-Frequency Analysis, Academic Press, 1998.
L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995.
I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.
P. Flandrin, Temps-Fréquence, Hermès, 1993-1998.
P. Flandrin, Time-Frequency/Time-Scale Analysis, Academic Press, 1999.
G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989.
C. Gasquet, P. Witomski, Analyse de Fourier et Applications : Filtrage, Calcul Numériques et Ondelettes, Masson, 1990.
F. Hlawatsch, Time-Frequency Analysis and Synthesis of Linear Signal Spaces, Kluwer, 1998.
T.W. Körner, Fourier Analysis, Cambridge Univ. Press, 1988.
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1997.
Y. Meyer, Ondelettes et Opérateurs I., Hermann, 1990.
Y. Meyer, Ondelettes, Algorithmes et Applications, Armand Colin, 1992.