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.
Thetransform 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"),
(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
(1932) and rediscovered in the context of
signal theory by(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.