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On Time.Frequency and \Mavelet Methods 1 .9.1 .03 ON TIME.FREQUENCY AND \MAVELET METHODS Patrick Flandrin Ecole Normale Supörieure de Lyon Laboratoire de Physique (URA 1325 CNRS) 46 all6e d'Italie, 69364 Lyon Cedex 07, France B-mail: f landrin0physique. ens-Iyon. f r ABSTRACT a time-dependent power,. ), thus making intuitively hope- less the existence of a unique tool which would work univer- Time-frequency and wavelet methods have been introduced sally well. This claim admits of course a precise mathemat- and extensively studied in a recent past for dealing - - ical formulation and the result is that trying to incorporate with "nonstationary" signals. They offer both a natural some time dependence in a Fourier-like description may ap- language for many situations encountered in practice, and pear either as a problem without a solution in its greater a large number of new tools whose relative merits are to generality (pessimistic viewpoint) or as a problem with as be compared before any use in a specific situation. Basic many solutions as specific types of (op- questions about the construction of such tools are addressed "nonstationarity" timistic viewpoint). The central question .is therefore to and some keys are proposed for tailoring adapted solutions. construct classes of solutions, given specific constraints. 1. INTRODUCTION 3. *ATOMIC' DECOMPOSITIONS VS. In most cases of practical interest, data exhibit some form ENERGY DISTRIBUTIONS of "nonstationarity" and it has now become common prac- Starting from the interpretation of the Fourier transform tice to make use of "time-frequency" methods for analyzing according to which any waveform can be represented as- a them. The problem of how to do it in an efficient way is superposition of sine waves (which have a perfect localiza- however far from being resolved in an unambiguous way and tion in frequency and no localization at all in frequency) -, the field is still a very active area of research. The main rea- a first possibility is to retain the idea of expressing a signal son for this situation is that, unlike the Fourier transform as a linear superposition of elementary "building blocks", which is the unique tool for describing a waveform in terms while replacing the delocalized sine waves by more local- of frequencies, no unique approach can be advocated when ized objects in both time and frequency, which would play time and frequency are simultaneously considered. The con- the role of time-frequency "atoms". The most prominent sequence is that, given a specific problem, potential users examples of such an approach are lhe short-time Fourier are faced with a number of different possible methods be- transform (where the atoms are deduced from an elemen- tween which they have to choose, thus making mandatory tary low-pass waveform by shifts in time and shifts in fre- to have a clear picture of the potentialities of the different quency) and the wauelet transform (where the atoms are approaches and of their relative merits for the considered deduced from an elementary band-pass waveform by shifts problem. It is the purpose of this contribution to briefly in time and dilations), but many more possibilities are of- address some basic questions aimed at better positioning fered, which can be interpreted in terms of tilings of the problem" and propose some keys the "time-frequency to time-frequency plane [0, Z, to]. for tailoring adapted solutions. As far as spectrum analysis is considered, the Fourier transform is a linear operation which is most useful (e.g., for 2. BEYOND FOURIER reconstruction purposes) but, in many cases, informations The starting point for any "time-frequency" approach is to about energetic (i.e., second-order) properties of a signal recognize that, in many cases, the Fourier transform - al- can be preferred. This is especially true with the concept of though mathematically correct cannot be considered as spectrum density for which the question of an extension to providing a satisfactory description- of the physical reality, time-dependent situations naturally arises. In this respect because it only amounts to trading time versus frequency, too, general classes of solutions can be constructed, based seeing the world as a pure superposition of monochromatic on the requirement of a quadratic dependence upon the sig- waves without any localization properties in time. Loosely nal and on the fulfillment of specific constraints imposed by speaking, the signals for which such a difficulty arises can the user. In parallel with the linear case, the most promi- be referred to as "nonstationary" in the sense that some of nent classes of solutions are the so-called Cohen's closs and their essential features are varying with respect to time. It affine class - which extend, respectively, the spectrogram has however to be noticed that this terminology can encom- (squared short-time Fourier transform) and the scalogram 'pass very different situations (e.g., quasi-monochromatic (squared wavelet transform) -, but many more classes have gliding tones, impulsive transients, broadband noise with recently appeared [5, 7, 9]. Medical & Biological Engineering & Computing Vol. 34, Supplement 1, Part 1, 1996 The 1Oth Nordic-Baltic Conferenc€ on Biomedical Engineering, June 9-13, 1996, Tampere, Finland 421 A basic element of quadratic energy distributions is the and efficient algorithms, but at the expense of a relatively Wigner-Ville distribution which outperforms - in terms restricted flexibility in the choice of the analysis wavelet. of localization on individual components distributions Conversely, no such limitation exists for continuous (highly based - on linear transforms, such as specrograms or scalo- redundant) transforms, but at the expense of an intense grams. This has however to be paid at the price of a reduced computational burden. However, a way out to this alter- readability when multiple components are simultaneously native has been recently proposed, which combines the ad- present, a situation for which a large number of modifica- vantages of both approaches (to any prescribed degree of tions have been proposed (adaptive or not), each with its accuracy) by fully exploiting the framework of multiresolu- own advantages and design rules [b, T, 9] (among the most tion analysis [f]. recent developments, one can cite, e.g., the reassignment Multi-analysis. Since no "uniformly best,, tool exists for method [a]). time-frequency in general, and rather than using some ,.less worst" solution for a given problem, it can be more appro- 4. TrME-FREQUENCY VS. TIME-SCALE priate to combine partial informations from different meth- Whatever the nature of the transform (linear or quadratic), ods, each with its own advantages (to be retained) and ,,multi-analysis" it is in fact not that clear why the description of a ,,nonsta- drawbacks (to be ignored). Such a ap- tionary" signal should be based on time and frequency: this proach offers many interesting perspectives of a data fusion again depends on what type of is con- type. "nonstationarity', ,,nonstationarity,', sidered. Clearly, time-frequency representations are well- Paradigm. Whatever the type of one adapted to all those situations for which a time-dependent of the key features of time-frequency or wavelet methods spectral analysis is needed (typically, signals which consist is that they offer a natural language for describing signals in a superposition of components modulated in amplitude and formulating their processing. In this respect, one of and frequency) and, in such cases, short-time Fourier trans- the deep reasons for their increasing popularity is maybe forms or members of Cohen's class are good candidates. to be found in the fact that, with them, a new and conve- However, there are many other instances in which a ,,non- nient paradigm has emerged, with all of the various methods stationary" character is much more associated with tran- appearing mostly as technical details. sients of short duration, abrupt changes,. making , more REFERENCES efficient an approach able to look at fine details in a signal. This is especially the case of the wavelet transform (and of [1] P. Asny and A. Ar,onousy, ,,Designing multiresolu- its quadratic extensions within the affine class), for which tion analysis-type wavelets and their fast algorithms,,' "nonstationary" signals are not considered from the point J. Fourier Anal. AppI., Yol. 2, No. 2, pp. 135-15g, of view of a time-dependent spectral analysis, but in rnul- 1995. tiresolution terms according to which, at any level of resolu- [2] P. AnRy, P. GoNqar,vös and p. FlaNouN, ,(Wavelets, tion, a signal can be decomposed into an approximation at spectrum analysis and ll f processes,', in Wauelets and a coarser resolution and a detail which exactly corresponds Statistics (A. Antoniadis, ed.), Lecture Notes in Statis- to the diference in information between the two resolu- tics 103, pp. 15-29, Springer-Verlag, 1995. tion levels [6]. Strictly speaking, a wavelet transform is a P. Asny and P. Fr,aNoRnr, ,,Point time-scale transform, although it may have a possible time- [3] processes, long- range dependence and wavelets," Wauelets frequency interpretation as a by-product in restricted cases in in BioI- ogy and Medicine (A. of well-behaved wavelets. Whereas time-frequency methods Ar,onousr and M. Urqspn, eds.), CRC Press. 1996. are primarily concerned with, let us say, the time evolu- ,,Improving tion of spectral lines, time-scale methods have the ability of [4] F. Aucpn and P. FLaNoRn{, the readabil- "zooming in" on singularities by looking at the way a signal ity of time-frequency and time-scale representations is structured across scales at a given point. This property, by the reassignment method," IEEE Trans. on Signal which is a key feature of time-scale methods, becomes of an Proc., Vol. SP-43, No. b, pp. l0O8-1089, 1995. extreme importance in the case of self-similar (or fractal) [5] L.
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