inverse problems and theoretical imaging I.M. Combes A. Grossmann Ph. Tchamitchian (Eds.) Wavelets Time-Frequency Methods and Phase Space Proceedings of the International Conference, Marseille, France, December 14-18, 1987 With 88 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Professor Jean-Michel Combes Professor Alexander Grossmann Professor Philippe Tchamitchian Centre National de la Recherche Scientifique Luminy - Case 907, F-13288 Marseille Cedex 9, France ISBN-13: 978-3-642-97179-2 e-ISBN-13: 978-3-642-97177-8 DOl: 10,1007/978-3-642-97177-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks, Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Softcover reprint of the hardcover 1st edition 1989 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2157/3150-543210 - Printed on acid-free paper Preface The last two subjects mentioned in the title "Wavelets" are so well established that they do not need any explanations. The first is related to them, but a short introduction is appropriate since the concept of wavelets emerged fairly recently. Roughly speaking, a wavelet decomposition is an expansion of an arbitrary function into smooth localized contributions labeled by a scale and a position parameter. Many of the ideas and techniques related to such expansions have existed for a long time and are widely used in mathematical analysis, theoretical physics and engineering. However, the rate of progress increased significantly when it was realized that these ideas could give rise to straightforward calculational methods applicable to different fields. The interdisciplinary structure (R.c.P. "Ondelettes") of the C.N .R.S. and help from the Societe Nationale Elf-Aquitaine greatly fostered these developments. This conference was held at the Centre National de Rencontres Mathematiques (C.I.R.M) in Marseille from December 14 to 18, 1987 and brought together an interdisciplinary mix of participants. We hope that these proceedings will convey to the reader some of the excitement and flavor of the meeting. In the preparation of the conference we have benefited from the help and support of the following organisations: the Societe Mathematique de France and the C.I.R.M.; the Universite Aix-Marseille IT, Faculte de Luminy; the Universite de Toulon et du Var; the Conseil Regional Provence-Alpes-Cote d' Azur; the Laboratoire de Mecanique et Acoustique and Centre de Physique Theorique, both at the C.N.R.S., Marseille. The company DIGILOG kindly provided the signal processor SYTER for demonstration purposes. The editors are extremely grateful to all of them, to the participants and to all other people who helped in various ways to make this meeting a real success. Marseille, December 1988 l.-M. Combes A. Grossmann Ph. Tchamitchian (received: March 16, 1989) v In Memoriam We have learned with shock the news of the sudden death of Professor Franz B. Tuteur His absence is keenly felt by those of us who had the privilege of knowing him and working with him. VI Contents Part I Introduction to Wavelet Transforms Reading and Understanding Continuous Wavelet Transforms By A. Grossmann, R. Kronland-Martinet, and I. Morlet (With 23 Figures) 2 Orthonormal Wavelets By Y. Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Orthonormal Bases of Wavelets with Finite Support - Connection with Discrete Filters By I. Daubechies (With 9 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Part II Some Topics in Signal Analysis Some Aspects of Non-Stationary Signal Processing with Emphasis on Time-Frequency and Time-Scale Methods By P. Flandrin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Detection of Abrupt Changes in Signal Processing By M. Basseville (With 1 Figure) . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 The Computer, Music, and Sound Models By I.-C. Risset (With 2 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Part III Wavelets and Signal Processing Wavelets and Seismic Interpretation By I.L. Larsonneur and I. Morlet (With 3 Figures) 126 Wavelet Transformations in Signal Detection By F.B. Tuteur (With 4 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Use of Wavelet Transforms in the Study of Propagation of Transient Acoustic Signals Across a Plane Interface Between Two Homogeneous Media By S. Ginette, A. Grossmann, and Ph. Tchamitchian (With 7 Figures) .. 139 VII Time-Frequency Analysis of Signals Related to Scattering Problems in Acoustics Part I: Wigner-Ville Analysis of Echoes Scattered by a Spherical Shell By J.P. Sessarego, J. Sageloli, P. Flandrin, and M. Zakharia (With 4 Figures) ........... . . . . . . . . . . . . . . . . . . . . . . . . . . .. 147 Coherence and Projectors in Acoustics By J.G. Slama ........................................ 154 Wavelets and Granular Analysis of Speech By J.S. Lienard and C. d' Alessandro (With 4 Figures) ............. 158 Time-Frequency Representations of Broad-Band Signals By J. Bertrand and P. Bertrand (With 2 Figures) ................. 164 Operator Groups and Ambiguity Functions in Signal Processing By A. Berthon ........................................ 172 Part N Mathematics and Mathematical Physics Wavelet Transform Analysis of Invariant Measures of Some Dynamical Systems By A. Arneodo, G. Grasseau, and M. Holschneider (With 15 Figures) .. 182 Holomorphic Integral Representations for the Solutions of the Helmholtz Equation By J. Bros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 197 Wavelets and Path Integral By T. Paul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Mean Value Theorems and Concentration Operators in Bargmann and Bergman Space By K. Seip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Besov Sobolev Algebras of Symbols By G. Bohnke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Poincare Coherent States and Relativistic Phase Space Analysis By J.-P. Antoine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 221 A Relativistic Wigner Function Affiliated with the Weyl-Poincare Group By J. Bertrand and P. Bertrand ............................. 232 Wavelet Transforms Associated to the n-Dimensional Euclidean Group with Dilations: Signal in More Than One Dimension By R. Murenzi ........................................ 239 Construction of Wavelets on Open Sets By S. Jaffard (With 8 Figures) ............................. 247 Wavelets on Chord-Arc Curves By P. Auscher ........................................ 253 VIII Multiresolution Analysis in Non-Homogeneous Media By R.R. Coifrnan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 About Wavelets and Elliptic Operators By Ph. Tchamitchian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 263 Towards a Method for Solving Partial Differential Equations Using Wavelet Bases By V. Perrier (With 7 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Part V Implementations A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform By M. Holschneider, R. Kronland-Martinet, J. Morlet, and Ph. Tchamitchian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 An Implementation of the "algorithme a trous" to Compute the Wavelet Transform By P. Dutilleux (With 7 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 An Algorithm for Fast Imaging of Wavelet Transforms By P. Hanusse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 SUbject Index 313 Index of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 315 IX Introduction to Wavelet Transforms Reading and Understanding Continuous Wavelet Transforms A. Grossmann 1, R. Kronland-Martinet 2 , andJ. Morlet 3 1Centre de Physique Theorique, Section II, C.N.R.S., Luminy Case 907, F-13288 Marseille Cedex 09, France 2Faculre des Sciences de Luminy and Laboratoire de Mecanique et d'Acoustique, C.N.R.S., 31, Chemin J. Aiguier, F-13402 Marseille Cedex 09, France 3TRAVIS, c/o O.R.I.C. 371 bis, Rue Napoleon Bonaparte, F-92500 Rueil-Malmaison, France 1. Introduction One of the aims of wavelet transforms is to provide an easily interpretable visual representation of signals. This is a prerequisite for applications such as selective modifications of signals or pattern recognition. This paper contains some background material on continuous wavelet transforms and a description of the representation methods that have gradually evolved in our work. A related topic, also discussed here, is the influence of the choice of the wavelet in the interpretation of wavelets transforms. Roughly speaking, there are many qualitative features (in particularly concerning the phase) which are independent of the choice of analyzing wavelet; however, in some situations (such as detection of "musical chords") an appropriate choice of wavelet is essential. We also briefly discuss the finite interpolation problem for wavelet transforms with respect to a given analyzing wavelet, and give some details about analyzing wavelets of gaussian type. 2. Definitions The continuous wavelet transform of a real signal s(t) with respect to the analyzing wavelet g(t) (in general, g(t) is complex) may be defined as a function: (2.1) S(b,a)= fa-fg ((t~b))S(t) dt (gdenotes the complex conjugate of g) defined on the open "time and scale" half-plane H (b E R, a>O). We shall find it convenient to use a somewhat unusual coordinate system on H, with the b-axis ("dimensionless time") facing to the right and the a-axis ("scale") facing downward (Fig 2.1). The a-axis faces downward since small scales correspond, roughly speaking, to high frequencies, and we are used to seeing high frequencies above low frequencies. 2 The function (2.1) can also be written in terms of the Fourier transforms g(w), ~(w) of sIt) and g(t). The expression is: (2.1 ') S(b,a)= raf~ (aw) eibro g(w) dw We impose on g the "admissibility condition" cg=27tfl~(W)1 ~; < 00. If ~(w) is differentiable (which we assume here), this implies: ~(O) = 0 i.e Jg(t)dt = 0 a- 1/2 gC~b) then (2.1) can be written as a scalar product: S(b,a) = <g(b,a)1 s> The main motivation for the admissibility condition convergence of: is that it implies the (weak) If we define g(b,a)(t) as g(b,a)(t) = (2.2) If Ig(b.ab<g(b,a)1 d~~b This operator (in the space L2(R ,dt) of signals of finite energy) is then easily shown to be Cg 1, where 1 is the identity. 3. Graphical conventions We want to display complex-valued functions such as (2.1) in a way which will allow us to gather -visually- a certain amount of useful information about the signal sIt). Two preliminary comments are in order here: The qualitative (and visual) information gathered from our pictures is certainly not the end of all desire of signal analysis. We believe however that it supplements in a non-trivial way the information obtained by inspection of the signal itself, of its Fourier transform or of one of its time-frequency representations such as Wigner-Ville. We shall not attempt here a comparison of various methods, and refer e.g to [3]. The expression (2.1) depends manifestly on the choice of the analyzing wavelet g; as a matter of fact, it is essentially symmetric in s and in g. In order to obtain full quantitative information about s from its tranform S, we need to know the analyzing wavelet g. There are however many features of the signal which can be seen on (2.1) and which are independent of the choice of g. It will turn out that such features often involve the phase of the complexvalued function (2.1). After these remarks, we get down to business: The {b,a}-half-plane can be either displayed as in Fig. 2.1, or it can be mapped on the full plane (b,-Iog(a)} (Fig. 3.1). 3.1 3 This second representation is indispensable if we want to display on a single picture information in a wide range of scale parameters. Such is the case when one is concerned with sound signals in the audible range, where a spread of 10 octaves is not excessive. A disadvantage of these representation is that straight lines of the open {b,a} half-plane, if they are not parallel or perpendicular to one of the axes, become exponential curves in the logarithmic representation. Voices: We shall often consider restrictions of S(b,a) to fixed discrete values of the scale parameter. Such a restriction S(b,aj) (aj fixed) is called a voice. In agreement with our preceding discussion, two consecutive voices correspond to a fixed ratio ...!L. The most common aj+1 situation is aj = aD 2jiv (j integer). where the integer v, the number of voices per octave, defines a well-tempered scale in the sense of music. The value v=12 (well-tempered scale of Western music) gives, in practice, a continuous picture. How should the values of S(b,a) be represented: Here we use two alternative representations. The first one, simpler to implement, consists in plotting, say, the real part (or sometimes the modulus or the phase) of each voice, and place such plots one above the other (see e.g Fig. 0). Such plots can carry quite detailed information, but they do not give a truly two-dimensional picture of S(b,a). A two-dimensional picture is provided by Figs. 1 to 16 (in color), and we shall now describe the conventions used in these representations. On each one of the pictures 1 to 16, the {b,a}half-plane is represented in the logarithmic coordinates of Fig. 3.1. The quantities displayed are the modulus and the phase of S(b,a); they are both shown on one and the same picture: S(b,a) = I S(b,a) I ej<p(b,a) (0 ~ <p(b,a) < 2rr) The modulus, I S(b,a) I , is color-coded in accordance with the palette visible on the pictures. The actual coding is done as follows: Let x = Isl!~xl ~ 1 be the value of lSI, normalized to its maximum within the picture. The "true colors" (as distinguished from black or white) are used in an interval sat.min ~ x ~sat.max; the values of sat.min and sat.max are chosen for each picture so as to emphasize the features of interest. They are given, together with other relevant information, in Table 1. If x<sat.min, then the color is white (small modulus). If x>sat.max, then the color is black (large modulus saturation). Notice that sat.max can be greater than 1; this only means that black will never appear on the pictures. The progression of "true colors" can be seen in the horizontal stripes of Fig. 1. In that picture, for any fixed b, the function a -> IS(b,a)1 is a gaussian. The asymetry of the stripes is due to the use of log(a) as a coordinate. The local value of the phase <p(b,a) is given by the density of black dots on the picture. As the simplest example, consider Fig.1. From left to right, in a period, one can follows an increase in density, corresponding to a regular increase of the phase of S(b,a) from 0 to 2rr. When the phases reaches 2rr, it is wrapped around to the value 0; these lines where the density of dots drops abruptly to zero are clearly visible on the pictures and will play an important role in the interpretation, as highly visible lines of constant phase. We have adopted one further convention in order to increase the legibility of the pictures. If at a point {b,a} the modulus I S(b,a) I is smaller than a cutoff cutoff.ph , we decree that the phase shall not be represented (Le., equivalently from the point of view of the graphical representation, that it shall be set equal to zero). The value of cutoff.ph may be equal to sat.min as in Fig. 1, but this is not necessary. For instance, in Fig. 2, one has cutoff.ph < sat.min, so that the lines of constant phase can be followed also for very small values of the modulus. The first five columns of Table 1 give, from left to right, the number of the picture, the quantities displayed (e.g. in fig. 13 and 14 only the modulus is shown), and the values of the parameters just discussed. The sixth column (duration) gives the time interval that would correspond to the picture if the continuous signal were sampled 32000 times per second. 4 It is sometimes convenient to display a slightly modified form of (2.1) namely the function: T(b,a) = Ja S(b,a) = a- 1 fg ((t~b)) s(t) dt . This is the function shown if the column "normal." contains 1/a. Finally, the last column gives the number of voices per octave and the total number of voices. 4. Localization a) Locality in time The correspondance (2.1) is, in general, non-local. The value of S(b,a) at a point H depends on s(t) for all t. Assume however that g(t) vanishes outside some interval [tmin ,tmaxl . We may ask two questions: 1) Which domain of the {b,a} plane can be influenced by the value of s(t) at to (i.e in an arbitrary small neighbourhood of a point to)? The answer is obvious from (2.1). The "domain of influence" of the point to is the cone to - b E aA with vertex at the point b=to on the edge of the {b,a} half-plane (Fig. 4.1). {b,a} E In logaritmic representation, the b-axis is sent to infinity, and the cone of Fig (4.1) becomes the domain shown in Fig. 4.2. b a Figure 4.1 Log a Log a = 0 --I-----+-~-------1~ Figure 4.2 The second question is: which values s(t) can influence the transform S(bo,ao) at a given point of the {b,a}-plane? The same equation as above, namely to E aoA + bo , now gives an interval determined by a cone facing upward from {bo,ao} (Fig.4.3). 5 b Figure 4.3 a b) Locality in frequency We now change the assumptions about the wavelet g, and assume that its Fourier transform ~(Ol) vanishes outside an interval r=(Olmin(g),Olmax(g)). We ask now the same questions as above: 1) Which domain of the {b,a} plane can be influenced by the value of a Fourier component g(OlO) ? There is no loss in generality in supposing 000>0. The answer comes now from (2.1'); if we restrict g(Ol) to a small neighbourhood of 000, then (2.1') vanishes if Oloa is not in a small neighbourhood of (Olmin(g),Olmax(g)). So the domain of influence of a Fourier component ~(0l0) of the signal is the horizontal strip: Olmin(g) Olmax(g) - - - < a < - - - of the (b,a}-half-plane (Fig.4.4). 000 000 • b w Figure 4.4 2) Which Fourier components of the signal are felt at the point {bo,ao} of the (b,a}-halfplane? The answer is : The components ~(Ol) such that: Olmin(g) Olmax(g) ---<00< - - - a a 5. Covariance, progressivity A very simple but fundamental property of the continuous wavelet transform is its covariance with respect to shifts and dilations of the signal. Fix the analyzing wavelet g. If S(b,a) is the transform of s(t), then S(b-to ,a) is the 1 t b a . transform of s(t-to), and S( ~;):" ) IS the transform of s(~) (1..>0). ."fi: 6 - Complex monochromatic signals: The covariance under time shifts has an immediate consequence. Assume for a moment that s(t) is an eigenvector of the shift operator : s(t-to) = A.(to) s(t), which can be satisfied only by s(t) = exp(iwot). Then, S(b,a) is satisfies S(b-to , a) = A.(to) S(b,a), which means that S(b,a) is of the form: (5.1 ) S(b,a) = exp(iwob) f(a) (f(a) may be complex-valued). The frequency of a complex monochromatic signal can be read off from the phase of any restriction of its wavelet transform to a horizontal line a=constanl. This fact is independent of the choice of the wavelet and a consequence of nothing else than shift covariance. The function f(a) in (5.1) can be calculated. If we put &(00) = 6(00 - roo) in (5.1), we obtain from (2.1'): (5.2) S(b,a) = -Va exp(iwob) 9' (awo) The modulus of (5.2) is constant along lines of constant scale, and varies as -Va ~ (awo) along line of constant time. A "spectral line" in the signal gets translated in a horizontal pattern in the transform, with constant modulus and linear phase. - Real monochromatic signal : Progressive wavelets The above discussion does not apply if the complex monochromatic signal is replaced by a real monochromatic signal such as cos(wot). The transform (2.1 ') of this function is: (5.3) 1 '2 exp(iwob) - 9' (awo) + '21 exp(-iwob) -9' (-awo) The modulus of this function is not constant on lines of constant scale because of interference between the two summands. This is a serious disadvantage for graphical interpretation. It is, however, clear that this problem does not arise if the one the terms of (5.3) vanishes, i.e if §(w) vanishes on a half-axis. If §(w) = 0 for 00 < 0, we shall say that g(t) is a progressive wavelet. That is, a progressive wavelet is defined as a complex-valued function that satisfies the admissibility condition of Sec.2 and does not have Fourier components on the negative frequency axis. All the examples shown in this paper were calculated with progressive wavelets. Figs 1 and 2 show the transform of a real monochromatic signal with respect to two different progressive wavelets. The relevant feature here is that the colored domains, which represent the modulus, are horizontal strips of constant width. On any horizontal line, the phase varies linearly. Its rate of variation is the frequency of the signal, which can thus be accurately measured from the phase picture. This fact is independent of the choice of analyzing wavelet. The frequency of a monochromatic signal can also be read off from the modulus of its wavelet transform. If g is progressive, this modulus is 1 g(awo) I. We see that the relationship here is less intuitive and that it depends on the choice of g. - Homogeneous signals A function f(t) is said to be homogeneous of order ex at the point t=o (ex arbitrary, it may even be complex) if: f(A.t)= A.(1f(t) 7 In other words. f is an eigenfunction of the dilation operator. Since dilations do not interchange the positive and negative axis. the natural example (analogous to complex exponential for shift operators) lives on one side of 0: fit) = { ~a (t>O) (t$O) It is convenient to introduce a normalization factor. and define: __ 1_ta (bO) 1) u(i(t) = { q:+ (t$O) Considered as a distribution. ut(t) is entire analytic in its dependence on ex (see e.g [4]). If ex is a negative integer. then ut(t) is a derivative of the S-function: (n=1.2 •..... ). The wavelet transform of an homogeneous signal is fully determined by its restriction to any line a=const. 6. Reproduci ng kernel The transform (2.1) is a correspondance between the function sit) of one variable and the function S = Lgs of two variables. It is reasonable to expect that S is not arbitrary. One set of equations satisfied by S is deduced easily from the expression (2.2) for the identity operator in L2. Since S(b.a) = < g(b,a) Is>. we have: 1 S(b.a) = Cg i.e S(b.a) = f If < g(b,a) I g(b',a') > da'db' ~ < g(b'.a') Is> pg(b.a;b·.a') S(b'.a·) pg(b.a;b'.a·) = (6.1 ) = ~ 9 d~~2b' where < g(b,a) I g(b',a') > ~g ~fg (a't-~+b') rK"J e '(bl - b')1 a' "\Ja 1 _ = Cg g(t) dt A am .Co g (~\:j(m)dm Equation (6.1) says that pg is the reproducing kernel for the space of functions S(b,a) that are wavelet transforms. with respect to g. of signals sit) of finite energy. We shall also say that Pg is the reproducing kernel associated to g. From expression (6.1) one sees that Ipg (b.a;b· .a·)1 attains its maximal value when {b.a}={b·.a·}. With the wavelets that we use. Ipg(b.a;b·.a')1 decays very fast when. say. {b'.a'} moves away from {b.a}. In other words. for fixed {bo.ao}, pg(bo.ao; .•. ) is a function on H that is localized around {bo.ao}. In the following section. we shall use finite families of such functions to obtain local approximation to S(b.a). As an illustration. we give here the scalar product < g(b,a) I g(O. 1) > where g(t)=eictexp(- ~ t cp 2} The phase of this scalar product is: bc(1+a) = (1 +a2) while its modulus is: 8 m= _ {2rta ( !.. b 2+c( 1-a)2) -\J ~a2 exp - 2 (1+a2) This function and a function of the type < g(b,a) I g(b O,1/2) > are displayed on Fig.9. Another example is shown on Fig.10. This example will be discussed later. If F(b,a) is an arbitrary function on the {b,a}-half-plane, such that: If ff IF(b,a)i2 d~~b < (finite energy). then the function 00 pg(b,a;b',a') F(b',a') da'db' ~ is the transform of some signal s(t) of finite energy. 7. Local approximations to S(b,a) It is known that a wavelet transform S(b,a) is fully determined by its values on a suitable grid of the {b,a}-half-plane; this grid depends on the choice of the analyzing wavelet (see the article of I. Oaubechies in these proceedings). We are now caught in a dilemma: On the one hand, the continuous function S(b,a) has many desirable properties (full covariance with respect to shifts and dilation, simple interpretation, etc.), on the other hand, computing and storing this function on very fine grids is clearly wasteful of computer time and memory. We shall now derive a very simple "local interpolation" formula which does the following: We start with n arbitrary points Pl={bl,alL ..... ,Pn={bn,an} of the {b,a}-half-plane. We assume that the points are distinct; Pj 7' Pj if i7'j. We assume that an analyzing wavelet g is given, and that the wavelet transform S=Lgs of a signal s is known at the points Pj; the value of S at Pj is the complex number Sj. S(bj,aj) = Sj (i=1 ... n) We shall approximate S(b,a) (on an appropriate compact subset of arguments b,a) by a linear combination of the functions ej(b,a) = pg(bj,aj;b,a) introduced in the preceding section: n (7.1) Sappr(b,a) = LYj ej(b,a) j=1 We shall determine the coefficients Yj by the requirement that Sappr should take the "correct" values Sj = S(bj,aj) at the points Pj (i=1 .. n). It should be stressed that the basic "Ansatz" (7.1) can be wildly wrong as an approximation of S, e.g if {b,a} is taken to be "far away" from all the points Pj. Notice however that at such a point all the functions ej(b,a) are very small, by the basic concentration properties of our wavelets (and consequently of reproducing kernels). If the points Pj are not spaced too far from each other (e.g if they are adjacent elements of a grid giving rise to a good frame) and if P={a,b} is chosen inside the convex hull of these points, the approximation (7.1) can be excellent. The determination of the coefficients Yj is easy. The interpolation conditions are: n Sj = Sappr(bj,aj) = LYj ej(bj,aj) j=1 n = L, Yj j=l pg(bj,aj;bj,aj) n =L,AjjYj j=1 Where A = (Ajj) is the n by n Gram matrix: 1 (bi,ai) (b',a') Ajj = pg(bj,aj;bj,aj) = - < g i g J J > Cg (Notice the order of i and j) which is known to be hermitean and positive definite. 9 Introducing the inverse B A-1, we find: n 'Yj = L B jj S j j=1 and the final local approximation formula: (7.2) Sappr(b,a) = n n ;=1 j=1 L L with ej(b,a) = pg(bj,aj;b,a) Sj = S(bj,aj) ej(b,a) Bjj S j (i=1 .. n) 0=1 .. n) Covariance of the interpolation-approximation formu la. The result (7.2) would be of little use if the matrices A and B had to be re-calculated whenever the interpolation nodes P1 ... P n are changed. This is in fact not necessary. The formula (7.2) is invariant with respect to the two basic families of transformations which define the natural geometry of the {b,a}-half-plane H : the time shifts and the rescalings. In order to visualize the content of these statements, it is useful to think of H in the linear (rather than logarithmic) representation of sec. 3. A time shift (by to E R) of the points {P1· ... Pn}={{b1,a1} .... {bn,an}} brings them into the "congruent" family of points {{b1+tO,a1} ... {bn+to,an}}. Similarly, a re-scaling (by bO, and at the point b=O on the boundary a=O of the half-plane) brings them into the "congruent" family of points {{Ab1,Aaj) .... {Abn,Aan}}. The re-scaling at a different point b=bo of the boundary can be written in terms of the time shifts and of the re-scalings at b=O; such general re-scalings together with time-shifts are the most general transformations in the natural geometry of H. The covariance statement is then: If one transforms simultaneously (i) the interpolation nodes P1 ... Pn (ii) the points P={b,a} by one of the geometrical transformations of H, then the only item to be changed in (7.2) are the numbers Sj (which will of course correspond to different values of S(b,a) ). This remark is useful in the practical implementation of the "fleshing out" of the transform starting from its skeleton on a grid. 8. Admissible and almost progressive gaussians Gaussians (shifted in time, in frequency and re-scaled) have many properties which recommend them as analyzing wavelets. They have the best p<>ssible simultaneous concentration in time and in frequency. The set of their finite linear combinations is closed under Fourier transform, pointwise multiplication and convolution. The scalar product of any two members of this set is given by an explicit formula. They are among the very few classes of functions where the transition from one to more dimensions is immediate. We have, however, to reconcile this praise of gaussians with our requirements that an analyzing wavelet be admissible and progressive. While a finite linear combination of gaussians may be admissible, no such combination can be progressive, because the tail of any gaussian extends to infinity. In the words of W.C. Fields, the time has come to take the gaussian bull by the tail and face the situation. Progressivity and admissibility may of course be enforced by the simple expedient of "cutting the tail" of a gaussian in the frequency space. This is however best done on a linear combination of gaussians, at a point where this linear combination has a zero of sufficently high order. We now describe the construction of such linear combinations, which also keep some of the good properties described above. 1) We shall start by introducing a linear combination of gaussians of different widths, all centered at x=O, that vanishes at ~, where c is a preassigned positive number. It is useful to require that our linear combination be invariant under Fourier transform (like the basic gaussian). Define: 10 ho(x) = exp( -t x2 ) Choose a number A>O, and consider the dilated gaussian with the same L2-norm: (i") (DAho)(x) = A- 1/2 h o Then the Fourier transform of DAh o is D11Ah o. Consequently, for any real y, the function ho(x)- y[(DAho)(x) + (D 11Aho)(x)] is invariant under Fourier transform, real and symmetric under x -> -x. We can make it vanish at X=!C by choosing: ho(c) (8.1 ) (i} y=-------~~------- A- 1/2 h o A1/2ho(AC) We define consequently h1(c;A;x) = ho(x) _y(A- 1I2 hO(i") + (8.2) A1/2 hO(AX)) where y is given by (8.1). Since h 1(c;A;x) = hdc;A· 1;x), there is no loss of generality in assuming that A~1. We have h 1(tc;1 ;x) = 0. The n-th derivative of h1 (C;A;X) with respect to x is: h(~)(c;A;x) = (-1)n[He n(X)h o(X) - y (A· n·tHe n(A· 1X)h o(i") + An+~Hen(AX)ho(AX))] Here the Hen(x) are modified Hermite polynomials: Hen(x) = 2· n/2 H{~) , and the Hn the usual Hermite polynomials. It is now easy to construct functions hn = hn(C;A1 ,.... An ;x) that: (i) are finite linear combinations of gaussians (ii) are invariant under Fourier transform : An = hn (iii) have a zero of order n at te. Take n distinct numbers A1, ... ,An > 1, and define hn as the determinant h1(C;A1 ;x) ........ h1(C;An;x) h; (C;Ap) ........ h; (C;An;X) 2) Consider now the function gn(C;A1 .... An;x) = eicx hn(C;A1···. An;x) by the above, the Fourier transform of gn has a zero of n-th order at ~=O. With reasonable values of C,A1 , .... ,An the function gn will be practically progressive, and suitable for numerical work. - Gaussian chirp The wavelets that we just considered are (cosmetic) improvements on the basic wavelet eiCX.exp(-t x2 ) introduced by one of us a while ago. If c~5 this wavelet is practically admissible 11 and progressive. Its "instantaneous frequency· (derivative of the phase with respect to time) is independent on the time x. We shall now contruct a related wavelet with instantaneous frequency that increase linearly in time. Such "gaussian chirps" are known in various fields. In order to save time, we shall not repeat here the discussion of the preceding section concerning the enforcement of strict admissibility and progressivity. We consider now the wavelet: (8.3) 21 1 . kx 2 elCXexp(i exp (-2" x2) where c is as above, and bO. This k is the rate of increase of instantaneous frequency c+kx. Some of the examples described in the next section have been computed with the wavelet (8.3). - The "two-humped" wavelet The humps here are in frequency space, and the wavelet is of the form: (8.4) gh(X) = (exp(ic 1x) + exp(ic2X)) exp(-} x2) with Fourier transform: (8.4') ~h(~) = exp(-t (~-C1)2) + exp(-t (~-C2)2) where both C1 and C2 are sufficiently large so that (8.4) is practically progressive. The motivation for introducing this and similar wavelets is the detection, in the signal, of contributions that correspond to a given "chord". This is a variation on the "matched filter" theme. The transform of a monochromatic signal of frequency C3 with respect to the wavelet (8.4) is: (8.5) S(b,a) = ~ exp(ibc3) (§h(a(c3-C1)) + ~h(a(C3-c2))) The modulus of S is the same as the modulus of the transform associated to the sum of two monochromatic components, taken with respect to the one-humped wavelet (9.1). However, in contrast to that case, the rate of change of the phase is independent of the scale parameter a. Examples of "octave detection" will be seen on Figs. 13 and 14. 9. Exampl es Fig. 0 : The signal to be analyzed is shown at the bottom of the picture. It corresponds to the sound "e" of the word "person". The total duration is about 23 ms. Just above the signal, one can see its reconstruction from seven voices, with the help of the one-dimensional reconstruction formula; (see e.g [9]). The real part of the voices are shown in the upper part of the picture. There is one voice per octave. The highest voice is centered at 4000Hz, and the lowest one at 62.5Hz. Fig. 1 : This figure is a representation of the wavelet transform of the real monochromatic signal discussed in Sec. 5. One can see on it the features described here: horizontal strips of constant modulus, and phase in step with the phase of the signal. The analyzing wavelet is the modulated gaussian : e icx exp(-t x2) with c=5.0. Fig. 2 : This figure shows the transform of a monochromatic wave with respect to the wavelet (8.3). It should be compared with Fig. 1 where a monochromatic wave is analyzed with the help of the wavelet (9.1). The important difference between the two pictures lies in the behaviour of lines of constant phase. These lines are straight and vertical in Fig. 1 and parabolas with horizontal axis in Fig. 2. A simple calculation shows that the maximum modulus of the transform is obtained at points where these parabolas have vertical tangent. This can also be seen clearly on the figure. The phase pictures made with the help of wavelets (8.3) can thus be used for the detection of spectral lines in a signal. (9.1) 12 Fig. 3 : The signal is now the superposition of two monochromatic waves with frequencies f and 2f. The ratio of the two frequencies is clearly visible on the phases; notice also the points of zero modulus which correspond to the appearance of new phase lines. The wavelet chosen here is (9.1). Fig. 4 : In this figure the signal is a localized pulse that approximates a delta function. One can see the lines of constant phases pointing toward its location. The modulus increases toward the top of the picture (small scale-parameters) in accordance with the discussion at the end of Sec. 5, with U= -1. The wavelet is (9.1). Fig. 5 : The signal here is the same as in Fig. 4, but the wavelet transform is computed with (8.3). The general appearance of the picture is not very different from Fig. 4. Nevertheless, a closer look along a horizontal line (constant scale) shows the increase of the rate of change of the phase. Fig. 6 : The signal is 0 for kO and eot for 1>0. This is, locally, situation discussed in Sec. 5, with u = O. Notice that, in contrast to Fig. 4, the modulus on a line of constant phase does not increase indefinitely as the scale parameter goes to zero. The wavelet is (9.1). Fig. 7 : The signal is the same as in Fig. 3 except for its values at two points. The modification on one of the points is visible on graph of the signal; the other peak is much smaller and does not appear on the graph. It is however very clearly visible on the wavelet transform as the second peak of phase lines. The behaviour of the moduli of the transform makes it clear that the discontinuities are of "delta-function type". The stronger of the two peaks manages to get through the domain of the two monochromatic signals, and is visible at the bottom of the picture. The wavelet is (9.1). Fig. 8 : Here the signal itself is not discontinuous but its first derivative is. The signal vanishes for t<to and consists of three sinusoids of frequencies f, 2f and 4f entering respectively at times to, to+T and to+t- . Both the frequencies and the beginings of the components are very clearly visible on the picture. The lines of constant phases are not vertical in the the part of the picture corresponding to the periodic signal. This is due to our choice of analyzing wavelet, which is (8.3). Fig. 9 : This picture represents the reproducing kernel associated to the wavelet (9.1) for two values of {bo,ao}. It is discussed in Sec. 6. Fig.10 : Reproducing kernel associated to the wavelet (8.3). Notice again the acceleration of the rate of change of phases. Fig. 11 : The signal here is computer-generated noise and nothing else. Even though consecutive values of the signal are not correlated, the transform shows local order (i.e correlations). The size of the ordered regions is given but the reproducing kernel and depends on a (see e.g [6]). The wavelet is (9.1). Fig. 12 : The signal here is the sum of the one shown on Fig. 6 and the one of Fig. 11. The signal-to-noise ratio is 1 (0 db). The features of Fig. 6 are readily recognizable in the noisy background. The wavelet is (9.1). Fig. 13 : On this figure the phases are not shown. The signal is the sum of a monochromatic component and a contribution with time-increasing frequency. The analyzing wavelet is the two-humped gaussian (8.4) with C2/C1 =2. Each of the two components of the signal gives rise to two lines of maximum modulus (Sec. 8). Thus the generic number of maxima on any vertical line of the picture is four. However, at the time where the frequency sweep differs by an octave from the monochromatic component, this number of maxima is three. Such an intersection can be seen twice on the figure, corresponding, first, to the sweep one octave below the fixed frequency component, and then one octave above. Between these two times, one can notice the more obvious but less interesting point where the signals frequencies are equal and where consequently the transform shows only two maxima. We wish to stess that the detection of a chord in a signal with the help of time-and-scale methods can be reduced to the search for a qualitative pattern rather than the search for a maximum. Fig. 14 : The signal here is described in musical notation shown in the picture. The wavelet is given by (8.4) with C2/C1 =2. The four time intervals where the signal is an octave are caracterized by a recognizable pattern. Fig. 15 : The signal is the begining of the sound "tik" analyzed with the wavelet (9.1). We notice in the lower left corner converging phase lines that correspond to the begining of the sound; in the upper left region, a noise like pattern generated by the explosion of the "t". On the 13 right-hand side of the picture, the formants due to the resonances of the vocal tract are clearly visible. Fig. 16 : Wavelet transform of the begining of a clarinet sound analyzed with the help of the wavelet (9.1) . Again, the harmonics are clearly visibles and can be mesured with the help of phase information. Table 1 .£jgua W!IllL H1Jllill UlJD.in ~ 14 ~ ~ ~ ~ 1 mod.phs 0.99 0 .047 0 .0047 16ms l/a 8 58 2 mod.phs 1.36 0 .025 0.0018 16ms l/a 10 60 3 mod.phs 1.01 0 .22 0 .005 l/a 8 58 4 mod.phs 0.75 0 .019 0 .001 16ms 1/a 8 58 5 mod.pha 1.00 0.055 0.002 16ms l /a 10 60 6 mod.pha 0.99 0.04 0 .04 16ms l / -1a 8 59 7 mod.pha 0.90 0.018 0 .001 16ms l /a 8 58 8 mod.phs 1.00 0 .077 0 .004 32ms l /a 10 60 16ms 9 mod.pha 0.87 0.05 0 .0025 16ms l /a 12 58 10 mod.pha 1.00 0 .066 0 .008 16ms l /a 10 60 11 mod.pha 1.00 0 .25 0 .0017 16ms l/-1a 8 58 12 mod.pha 1.00 0 . 138 0 .002 16ms l / -1a 8 59 13 mod. 0.88 0 .033 1 0 .44s l/a 10 60 14 mod. 0.57 0 .23 1 3 .36s l/a 10 60 15 mod.pha 0.61 0.03 0.006 48ms l /a 8 60 16 mod.pha 1.00 0 .076 0 .076 32ms l/a 8 60 N 15 16 m u: '"01 u: 18 References [1] I. DAUBECHIES: Time-frequency localization operators: A Geometric approach. Preprint, Courant Institute, New York University. phase space [2] I. DAUBECHIES: The wavelet transform, time-frequency localization and signal analysis. Preprint, AT&T Bell Laboratories, Murray Hill, N.J. Submitted to IEEE Information Theory. [3] DUTILLEUX P., GROSSMANN A., KRONLAND-MARTINET R. : Application of the wavelet transform to the analysis, transformation and synthesis of musical sounds. Preprint of the 85TH A.E.S Convention, Los Angeles, 1988. [4] GUELFAND 10M , CHILOV G.E, generalized functions vol1, Dunod paris ed. 1965. [5] A. GROSSMANN: Wavelet transforms and edge detection. In: Stochastic Processes in Physics and Engineering, Ph. Blanchard, L.Streit and M. Hazewinkel, Editors. Reidel Publishing Co. [6] A.GROSSMANN, M.HOLSCHNEIDER, R.KRONLAND-MARTINET and J.MORLET : Detection of abrupt changes in sound signals with the help of wavelet transforms. In: Inverse Problems: An Interdisciplinary Study. Advances in Electronics and Electron Physics, Supplement 19, Academic Press,1987. [7] A. GROSSMANN and R. KRONLAND-MARTINET: Time-and-scale representation obtained through continuous wavelet transform. In Signal Processing IV, Theories and Applications, vol.2 Elsevier Science Pub. B.V, 1988, pp.475-482. [8] A. GROSSMANN and J. MORLET: SIAM J. Math. Analysis 15, 1984, pp.723-736. [9] A. GROSSMANN and J. MORLET: Decomposition of functions into wavelets of constant shape, and related transforms. In: "Mathematics and Physics, Lecture on Recent Results· L. Streit, Editor. World Scientific Publishing (Singapore) 1985. [10] A. GROSSMANN, J.MORLET and T. PAUL: Transforms associated to square integrable group representations I. General results. J. Math. Phys. 27, 1985, pp.2473-79. [11] A. GROSSMANN, J.MORLET and T. PAUL: group representations II : Examples. Ann. Inst Henri Poincare 45, 1986, pp. 293-309. Transforms associated to square integrable [12] S.JAFFARD, P.G. LEMARIE, S. MALLAT and Y.MEYER: Multiscale analysis. Preprint, CEREMADE, Universite Paris-Dauphine, Paris, France. [13] S. JAFFARD and Y.MEYER: Bases d'ondelettes dans les ouverts de Rn. To appear in Journal de MatMmatiques Pures et Appliquees. [14] R. KRONLAND-MARTINET: The use of the wavelet transform for the analysis, synthesis and processing of speech and music sounds. Preprint, LMA CNRS, 31 Chemin Joseph Aiguier, 13402 Marseille CEDEX 9 France. [15] KRONLAND-MARTINET, R. ,MORLET, J. and GROSSMANN, A. (1987): Analysis of sound patterns through Wavelet Transforms. International Journal of Pattern Recognition and Artificial Intelligence, Special issue on expert systems and pattern analysis. Vol 1 n02, World Scientific Publishing Company, 97-126. 19 (16) S. G. MALLAT: A compact multiresolution representation: the wavelet model. Preprint. GRASP lab. Dept of Computer and Information Science. U. of Pennsylvania. Philadelphia. PA. (17) Y. MEYER: Ondelettes. fonctions splines et analyses graduees. Preprint. CEREMADE. Universite Paris-Dauphine. Paris. France. (18) TH. PAUL: Functions analytic on the half-plane as quantum-mechanical states. Math. Phys. 25 • 1984. pp. 3252-3263. 20 J. Orthonormal Wavelets Y.Meyer Ceremade, Universite Paris Dauphine, F-75775 Paris Cedex 16, France 1. Introduction. The aim of this survey on the theory of wavelets is to help the scientific community to use wavelets as an alternative to the standard Fourier analysis. This survey is a written and extended version of a lecture I have been asked to give at the international conference held at Marseille on ondelettes, methodes temps-frequences et espaces des phases (December 14-18, 1987). This conference was a remarkable success. People with distinct scientific educations could communicate and interact, using the new born concept of wavelets. As often the flexibility of the concept helped a lot and during the lunches and dinners, physicians, physicists and even mathematicians were surprised and delighted to understand each other. Now my task is less pleasant and I have to do my job, giving precise definitions and describing specific algorithms. The advantage will be to prepare the ground for programming these algorithms. We all agreed that wavelets should be , at least in the one dimensional case, generated by one function named the analyzing wavelet. This function of the real variable x is denoted w(x) and should have a finite energy (energy= L2 norm). Moreover we also agreed on the so called compatibility condition 00 f I ~ (x) 12 dx/lxl o < 00. Here and in what follows 1\ will denote the Fourier transformation. This condition expresses in a very loose sense that the integral of a wavelet should vanish (it will be the case if the wavelet happens to be integrable). These two conditions mean that the wavelet should oscillate like a short wave. The functions sin(x) or cos(x) cannot be called wavelets since they do not have a finite L2 norm. As it became clear during the conference, distinct people are using distinct analyzing wavelets but they all agreed on the above mention ned properties. The analyzing wavelet w(x) is «the mother» of the wavelet-family. The other members of the family are generated by the following algorithm. We translate and dilate our analyzing wavelet w(x) by b ERa n d a E (0,00) and we obtain the collection Wa,b(X) = a- 1/2 w(a- 1 (x-b)) of the other wavelets. The game consists in writing any function as a linear combination of these wa,b. It is obvious that finite sums do not suffice and infinite sums or integrals should be used. We will face the fundamental problem of deciding what 21 a general function (any function) is allowed to be and what type of convergence can be expected. These problems are not of academic nature since the quality of convergence will immediately affect the speed of the numerical computations. My interest in wavelets arose when J.Lascoux convinced me that the sostudied by Coifman and Weiss could be called «atomic decompositions» connected to some recent work by A.Grossmann and J.Morlet on wavelets. I jumped to Marseille and had the opportunity of meeting I.Daubechies, A.Grossmann, J.Morlet and T.Paul. Through informal and friendly discussions, I became fascinated by the scope of this new theme and was ready to invest much time and efforts in wavelets. My mathematical education originated from Antoni Zygmund's book, «Trigonometric Series» and from several papers by A.Calderon. With this background I could easily recognize that Morlet's formula was similar t~ < < Calderon's reproducing identity», with a subtle linguistic change. To be more precise, in Calderon's formula one begins with two functions g(x) and h(x) defined on R n and satisfying the following identity 00 f ~(tu)A(tu)dt/t (1 .1 ) o =1 for all u in R n distinct from zero. One denotes by Gt and Ht the convolution operators with gt and ht where gt(x)= t- n g(t- 1 x) and ht is defined similarly. 00 Finally one obtains the identity f o GtHtdtlt = I which is Calderon's reproducing formula (if A and B are two operators AB means that A is acting on the result of the action of B). In Morlet's approach, h(x)=g(-x) and (1.1) is precisely the compatibility condition he imposed upon a wavelet. The emphasis is given to the Hilbert space structure of L2(Rn). As in the one dimensional case, the functions t- n / 2 g(t- 1 (x-xo» are called wavelets, the funtion g being the analyzing wavelet. Given any function f, one first calculates its wavelet transform W(a,b)= < f(x)19a,b(X) > where < I > denotes the inner product in our Hilbert space L2(Rn). The parameter a is strictly positive and plays the role of t, b is a vector in Rn and ga,b(x)=a- n / 2 g(a- 1 (x-b». One recovers our original function f(x) through J.Morlet's inversion (1.2) formula f(x)=Hp ga,b(x)W(a,b)a- n - 1dadb where P= (0, 00) x Rn. Roughly speaking the wavelets mimic an orthonormal basis for L2(Rn), the wavelet transform playing the role of the coefficients in this basis and the reproducing identity being like the expansion of an arbitrary vector of the Hilbert 22 space in this specific basis. These simply minded observations were quite challenging and I wanted to know whether this approach could lead to a true orthonormal basis. Was it possible to select some specific set S of pairs (a,b) such that the corresponding subcollection ga,b(X), (a,b) S, would be an orthonormal basis for L 2 (R n) ? An obvious solution to that problem has been known for a long time, since the so called Haar system 2i/2h (2ix-k), i and k being integers, is an orthonormal basis when h(x) =1 on (0,1/2) , -1 on (1/2,1) and 0 elsewhere. This function h(x) fulfils the requirements for being a wavelet but its lack of smoothness and cancellation has some bad effects we would like to explain in detail. If instead of a general function in L2(R n), one wants to analyze a function with much less or much more regularity, the solution given by the Haar system is inappropriate, the reason being that either the coefficients do not make any sense or their decay is awfully bad. Let us explain why. When f(x) is a generic smooth function and when g(x) is supported by [ -1,1], then as n tends to infinity the integral JR f(x)g(nx)dx is O(n-k-1) if and only if the first k moments of g(x) vanish. In the case of the Haar system the integral of h(x) vanishes but the first moment Jxh(x)dx does not and for that reason the Haar coefficients of a smooth function have a poor decay at infinity. Our goal will be to replace the function h(x) of the Haar system by a substitute w(x) which would be much better behaved in terms of smoothness and cancellation (the cancellation being expressed by the number of vanishing moments ), this goal being achieved while keeping the good localization of the Haar system. A function w(x) of the real variable x is called an analyzing wavelet of order r if we can impose upon w(x) the competing (but fortunately compatible !) properties (1.3)-(1.4) (1.3) l(d/dx)qw(x)1 (1.4)the collection ~ em (1+ Ixlj-m for all x, all m and O~ q ~r 2i/ 2 w(2ix-k), je Z, ke Z is an orthonormal basis of L2(R ). These two properties imply that all moments of w(x) of order k~r should vanish. If r=O, the Haar system is a solution but no-one knew if it was possible to find a solution when r=1, this case being as difficult as the general case (r large). A first solution to this problem was found by J.O.Stromberg , using spline functions ([13]). A more systematic approach was later discovered by S.Maliat ([9]). Mallat's algorithm is being described in the next section and his construction of an orthonormal basis of wavelets is reasonably simple in the one dimensional case. Finally 1.0aubechies ([4]) discovered that for any r there exists a compactly supported w(x)=wr(x) such that (1.3) and (1.4) hold. 23 Before ending these preliminaries, let us write down explicitely the analysis and synthesis one can achieve with wavelets forming an orthonormal basis. Instead of 2j/2 w (2ix-k), let us write Wj,k(X). Then the wavelet coefficients of f(x) are aj,k=fRf(x)Wj,k(X)dx and f(x) is recovered through f(x)= LLaj,kwj,k(X) in full similarity with Fourier series. It will be shown that wavelet expansions are better behaved than Fourier expansions in the sense that in the standard Fourier analysis there is no relationship between the local behavior of a function f(x) and the size of its Fourier coefficients while the wavelet coefficients of f will provide a rich and deep information on this local behavior. On the other hand Fourier series are limited to periodic functions. Finally the success of wavelets comes from the fact that they are both local with respect to the «space» variable and to the <<frequency» variable, the only limitation coming from Heisenberg uncertainty principle. If the analyzing wavelet w(x) happens to be compactly supported, the wavelets expansions will provide a local Fourier analysis with a remarkable flexibility in the scaling. The construction of a local Fourier analysis with dyadic scalings has been a challenging problem for a long time. If the wavelet analysis is aimed to be truly local, the analyzing wavelet w(x) needs to be compactly supported and if this condition is satisfied the behavior at infinity of the signal should not play any role. It is unrealistic to have to wait for centuries in order to check that the signal is square integrable before beginning to write its wavelet expansion. If w(x) is compactly supported we are led to ignore the behavior at infinity of the signal f(x). Then new problems have to be faced. If for example f(x)=1 identically, all wavelet coefficients vanish and 1 is not the sum of the wavelet expansion since 1;to. One might laugh and just observe that 1 is not square integrable but a more interesting viewpoint would be to use a better orthonormal basis permitting to expand elementary functions like 1 or polynomials. This new basis will be a slight modification on the orthonormal wavelets basis and consists in the collection Wj,k(X), je N, ke Z, together with the collection q(x-k), ke Z, where q(x) is a new smooth function with a rapid decay at infinity. If both q(x) and w(x) are compactly supported with r continuous derivatives, then any distribution S(x) of order not exceeding r, whatever be its growth at infinity, can uniquely be written as (1.5) where Ck=<S,qk> ,qk(X)=q(x-k), je Nand aj,k=<S,Wj,k> which makes sense since the smooth and compactly supported wavelets Wj,k can play the role of testing functions. Such an expansion means that S, whatever be its growth at infinity, is the sum of a smooth part Eo(S) (giving a sketchy approximation to S) which needs 24 to be corrected by a sequence of finer and finer fluctuations given by the double sum. When S is a polynomial of degree not exceeding r, (1.5) is valid since Eo(S)=S and Dj(S)=O. Let Vj be the closed linear span of q(2jx-k), ke Z, in L2 (R). Then Vj is contained in Vj+1 and these spaces satisfy a few more properties which will be listed in the next section. The game will then consist in extracting q(x) and next w(x) from a given sequence Vj , je Z. 2.Multiresolution analysis. We would like to explain what a multiresolution analysis of L2(Rn) could be , using a metaphor. From a subtle and complicated image, one may extract a blurred version or a schematic version, the former being a smoothened approximation in which the edges or contrasts have been weakened, the latter being a sketchy approximation resembling the pictures one can find in cartoons. These two kinds of simplifications have their analogues in mathematics or physics. Smoothing has been used for a long time and is usually achieved by convolution. From the numerical analyst viewpoint a smoothed appoximation may have a complicated numerical description, while sketchy approximations can be written exactly using specific building blocks which are generally spline functions. We now consider sequences fj, je N of better and better sketchy approximations to a given function f(x). Each fj will belong to a vector space Vj which is as simple as possible and generally is a space of splines of order rand fj will then be the best L2-approximation to f(x). We are now so close to the precise definition of a multiresolution analysis that we leave heuristics for mathematics. Definition 1. A multiresolution analYsis of L2(Rn) is an increasing sequence of closed linear subs paces Vj ,je Z, Qi L2(Rn) with the following properties (2.1 ) (2.2) n je Z Vj - 0 , U Vj is dense in L2(Rn) je Z whenever f(x) belongs to Vj , f(2x) belongs to Vj+1 (2.3) whenever f(x) belongs to Vo arui k belongs to Zn , f(x-k) belongs to Vo (2.4) there exists a function g(x) in Vo such that the collection g(x-k), ke zn, is a Riesz basis for VO. 25 A collection ej, je J, of vectors in a Hilbert space H is a Riesz basis if x = I,aje j where any vector x in H can be written in a unique way as a sum I,lajl2 is finite, the norm of x in Hand (I,lajI2)1/2 being equivalent. A multiresolution analysis is called r-regular if the function g(x) defined by (2.4) has the following property (2.5) I aa g(x)1 ~ Cm (1 +Ixl}-m for lal ~ r, all x in Rn and all integers m. Let us list some examples in which the precise definition of the space Vo will be specified, the spaces Vj will be constructed using (2.2) while the verification of (2.1) -- (2.4) is left to the reader. Example1.The space Vo consists of all step functions f(x) which are in L2(R), with jump discontinuities at ke Z. In that case g(x)=90(x) can be chosen as being 1 on [0,1], 0 elsewhere. Example2. This time, Vo is composed with continuous functions f(x) , belonging to L2(R), whose restriction to any interval [k,k+1] is affine. A natural choice for g(x) is the standard triangle function T(x) in Vo which is supported by [-1,1] and is normalized by T(0)=1. Example3. We begin with an integer m. This time, Vo consists of all functions with m-1 continuous derivatives whose restrictions to any interval [k,k+1] coincide with a polynomial of degree less than or equal to m. Then a natural choice for g(x) is given by the so called basic spline gm (x) which is the m-fold convolution of the function 90(x) of the first example. One easily cheks that this basic spline gm(x) is a non-negative function supported by [O,m+1] and verifies I, gm(x-k)-1. Example4. We denote by Vo the subspace of L2(R) defined by the condition that the Fourier transform of f(x) be supported by [-x,x]. A natural candidate for g(x) is the cardinal sine defined by g(x)= sinxxixx. Example5. This example is an improvement of the preceding one, in order to obtain a function g(x) belonging to the Schwartz class ( i.e. with .a rapid decay at infinity). We start with a function 9(x) which will be smooth on the real line , with compact support, even, 9 (x)-1 on [O,2x/3], 9 (x).O if 4x/3 ~ x, with o ~ 9(x) S:1 everywhere and finally 92 (x) + 92 (2x-x)-1 on [2x/3,4x/3]. Let q(x) be the inverse Fourier transform of 9(x). The functions q(x-k),ke Z, form, by construction, an orthonormal sequence and Vo is defined as the closed linear span of those q(x-k). Then Vo and (2.2) yield a multiresolution analysis. 26 Example 6. This example concerns L2(R2). It is a special case of a general procedure for building a multiresolution analysis in the product setting, which is given by a tensor product between two multi resolution analysis on each factor. To be more specific, let Vj be a multiresolution analysis of L2 (R). Then a multi resolution analysis of L2 (R 2) is given by V j = Vj ~ V j. If q(x-k) is an orthonormal basis of Va , then q(x-k)q(y-I) ,kE Z, IE Z, will be an orthonormal basis of Va. We stop the list of examples. We do not know if there is any way of describing all multiresolution analysis of L2(R). S.Maliat attacked this problem and found a partial answer which is flexible enough to cover all the cases which are needed in the applications. In particular Mallat's procedure which is explained in the next section is used in the construction by I.Daubechies of the compactly supported wavelets with a preassigned regularity. 3. How to construct wavelets. We begin with the one-dimensional case where the construction is much simpler. If g(x-k), kE Z, is a Riesz basis of Va, then an orthonormal basis q(x-k) can be explicitely computed with a well known general algorithm due to H.C.Schweinler and E.P .Wigner ([12]) and attributed to Poincare by mathematicians. Let us describe this orthogonalization method. If H is a Hilbert space and ej , jE J, is a Riesz basis of H, we then first calculate the Gram matrix G whose entries are g(j,j')=<ej ,ef>. This matrix is obviously positive-definite and its square root G 112 makes sense. Then the required orthonormal basis is given by fj= G-1/2(ej). The advantage of this method when compared to the more traditional Gram-Schmidt construction is to preserve all the additional stucture of the given sequence ej , jE J. In our case the original sequence is invariant under a group action and the Gram algorithm yields a sequence of the form q(x-k), kE Z. We have (3.1 ) In our examples 2 or 3, the corresponding function q(x) will have an exponential decay at infinity. In order to construct the wavelets, we denote by Wj complement of Vj in Vj+ 1. Then we obviously have (3.2) f(x) E Wj <===> f(2x) E W j+ 1 the orthogonal and 27 f(x) e Wo (3.3) <===> f(x-k) e Wo , k e Zn . In order to give simple formulas for the wavelets generated by our multiresolution analysis, we have to stick to the one dimensional case. The general case has recently been treated by K.Grochenig ([6]). It is not needed in our applications since we can construct the multidimensional wavelets by the method of the tensor product which will be described in this section. In the one dimensional case we proceed as follows. The «symbol» of the multiresolution analysis is the function mO (u) which is Coo, 21t-periodic and satisfies a(2u) = mO(u)~(u). Then m1 (u) = e iu mo(u+1t) is also a COO function which is 21t-periodic and finally the Fourier transform of the wavelet w(x) generated by our multiresolution analysis is given by (3.4) " " w(2u) = m1 (u)q(u) . It is interesting to observe that the symbol mO (u) information needed to calculate the function q(x). We have contains all the (3.5) This basic observation is due to S.Maliat ([9]) and is the keypoint in I.Daubechies' theorem ([4]) She carefully selects a trigonometric polynomial mO(u) such that ImO(u)12 + ImO(u+1t)12 -1, mO(O) -1 and with a few more properties of technical nature. Then (3.5) yields a multi resolution analysis and the main difficulty in I.Daubechies theorem is to fix mO(u) in such a way that q(x) be smooth with a rapid decay at infinity. But this can be achieved. It is not from q(x). We necessary to use the Fourier transform in order to compute w(x) define the coefficients ak by 1/2 q(x/2) - 2. akq(x-k) or ak .. 1/2 JR q(x/2) q(x-k)dx and we obtain 1/2 w(x/2) .. 2. (-1)k akq (x-k). Nevertheless it is still necessary to use the Fourier transform in order to compute q(x) from g(x). The wavelet w(x) has the property that w(x-k), ke Z, is an orthonormal basis of W00 By a simple rescaling we obtain that, for any fixed j and k running along Z, 2i/2 w(2ix-k) is an orthonormal basis of Wj. Since L2(R) is the direct orthonormal sum of these Wj, then the full collection 2i/2 w(2ix-k) is an orthonormal basis of L2(R) These Wj can be labelled «channels» and the distinct channels are orthogonal. The first decomposition of an arbitrary function consists in writing f(x)- 2. fj(x) where each fj belongs to the corresponding channel Wj. Furthermore, inside each channel, fj(x) is decomposed into an orthonormal sum 2. aj,k2j/2 w(2ix-k). Finally we have 28 f(x)- (3.6) l":l": aj,k2j/2 w(2jx-k). We are in a good position for building the twodimensional wavelets. We define Vo = Vo ~ Vo and this implies Wo .. (VO ~ WO) + (WO ~ VO) + (WO ~ WO)· Since q(x-k) is an orthonormal basis of Vo and w(x-I) does the same for WO, then q(x-k) w(y-I), (k,l) E Z2, is an orthonormal basis for Vo ~ Wo ; similarly w(x-k) q(y-I) is an orthonormal basis for Wo ~ Vo and w(x-k) w(y-I) is an orthonormal basis for Wo ~ WO. Putting all together it implies that the full collection q(x-k)w(y-I), w(x-k)q(y-I). w(x-k)w(y-I). (k,I)E Z 2, is an QIthonormal By a simple rescaling we deduce that 2jq(2jx-k)w(2jy-I), basis for W00 2iw(2ix-k)q(2iy-I), 2jw(2jx-k)w(2iy-l) is an orthonormal basis for Wj and finally the union of these three twodimensional wavelet basis. 4. Littlewood-Paley theory, collections filtering and when j runs over Z is the sampling. In the thirties, Littlewood and Paley among others tried to develop some algorithms for computing or at least estimating the LP norm of a Fourier series f(x)= l":akeikx when 1<p< 00. The case p=2 is trivial and given by the Plancherel theorem. When p ~ 2, they observed that the information given by the full collection of Fourier coefficients is inadequate for estimating the LP norm, the phases of these coefficients playing a role which is too subtle for being tractable. The main discovery has been that it is better to forget this full information given by all the terms of the Fourier series and to proceed to some regroupings. These regroupings, named dyadic blocks are defined by di(x)=l":iakeikx , where l":j means that we sum over the k's such that 2j~ Ikl ~2i+1. Then the fundamental discovery has been that, for 1<p< 00, the LP norm of f(x) is equivalent to the LP norm of (l":1 dj(x)12)1/2+laOI. This theorem was immediately used by Marcinkievicz to prove his celebrated multiplier theorem which was published in one of the last issues of Studia Mathematica which appeared before 2 n d world war. For studying other spaces than LP we need to change the definition of the dyadic blocks. We use a function t(x) which is smooth, real valued, even, supported by 1~lxl~3 and satisfies Lt(2-jx)=1 for x ~ O. With this function, we define the dyadic block of order i of the Fourier expansion of f by Dj(f)(x)=l":akt(2-ik)eikx. Then f(x) belongs to CS if and only if lDi(f)(x)I~C2-is , uniformly in x and j. 29 This story seems unrelated to wavelets or multiresolution analysis. Indeed there is a straightforward connection and some people even say that the two theories are the sames. Let us begin with recalling a few facts about filtering and sampling. Let f(x) be a continuous function with a polynomial growth at infinity whose Fourier transform is supported by [-T,T] (we then write f(x)E ET) . Then the sampling f(kB),kE Z, B> 0, will completely define f(x) if and only if B < nIT. When we a priory know that f(x) belongs to L2, then it suffices to assume B ~ nIT. This theorem, known as Shannon's rule, has the following weak point. When B < nIT, it is always true that the sampling is redundant and the sequence f(kB) satisfies infinitely many linear relations of the form L Ckf(kB) =0. On a practical level, this means that it is difficult to guess if a given sequence ak is of the form f(kB) for some fE ET. There are other approaches to filtering or smoothing and a very convenient one is to start with a specific linear space Vo of nicely behaved spline functions and to decide that the filtering of a given function f(x) will be defined as the best L2-approximation of f(x) by a function fo in Vo This viewpoint gives an elegant solution to the problem of defining a new notion of filtering and obtaining a sharper relationship between filtering and sampling.The space ET is replaced by ET which is defined by the following construction. We denote by Il(x) a smooth and compactly supported function of the real variable x such that Lkll(X+2kT) never vanishes. Then the substitute ET for ET is defined as the collection of all tempered distributions f(x) whose Fourier transforms are products of Il(x) by arbitrary 2T-periodic distributions S(x). The intersection ETn L2 = Vo is the space defined in our example 5, if T = n and if Il(x) satisfies the additional requirements imposed to 8(x). Finally the sampling of f(x) E ET on 1tT-1 Z yields an isomorphism between ET and the space of all numerical sequences with a polynomial growth at infinity. Let us return to the special case of our example 5 and denote by Si the corresponding filtering operator defined by convolution with the function 2iq(2ix) where the Fourier transform of q(x) is Il(x). We know that the functions q(x-k) , kE Z , form an orthonormal basis of VO. Let us consider the best L2-approximation foE V 0 to a function f(x) in L2(R) and relate the coefficients ak of fo in this basis to the filtering and sampling of this function f(x). (4.1) We have fo(X)=Lakq(x-k) if and only if ak=SO(f)(k). This observation comes from ak =ff(x)q(x-k)dx = (2n)-1 f So (f)(k) by Fourier's inversion formula. 30 t (u) ~(u)eikudu = Next we consider the best L2-approximation fj(x)E Vj to f(x) and expand it in the basis of 2i /2 q (2ix-k). The coefficients of the expansion are given by ak=2-j/2 Sj(f)(2-jk). The operators Sj play the role of low pass filters tuned to frequencies not exceeding C2j and the sampling of Sj(f)(x) at k2-j is compatible with Shannon's rule. Similar considerations apply to the spaces Wj, the function ~(x) being replaced by dh(X) which will be defined now. We assume 0<h:;;21t/3 and dh(X) will be a real valued even function such that dh(X) =0 outside [1t-h,21t+2h], dh(x)=1 on [1t+h,21t-2h]. Furthermore dh(X) will be smooth, dh(X)=dh(21t-x) and dh 2 (X)+dh 2 (2x)=1 on [1t-h,1t+h]. We forget the dependance in h and define the operator OJ by the condition that the Fourier transform of OJ (f) be the product f(u)d(2-ju). Then we have :EDj2=1 and the Dj 2(f) are, in the periodic case, a smoothed version of the dyadic blocks :Ej of the Fourier series of f. The wavelet coefficients of f are given by aj,k=2-j/2D i(f)(2-jk+2-j-1) and this remarkable identity explains the deep relation between the wavelet-analysis and the classical Littlewood-Paley theory. The wavelet coefficients are given by a sampling of blocks similar to the ones used in the Littlewood-Paley theory. It also explains the characterizations of the classical Banach spaces by the order of magnitude of the wavelet coefficients. These characterizations can be deduced from previous characterizations of these spaces by the LittlewoodPaley-Stein theory. However there are two main differences between the two theories. The Littlewood-Paley theory provides only a localization on the Fourier transform side while the wavelets are localized on both sides. Therefore the wavelets give a more accurate local analysis. The second observation is that the identity :EDj2=1 of the Littlewood-Paley theory does not suffice to build an orthonormal basis of wavelets and the Littlewood-Paley analysis might be more flexible than orthonormal wavelets. 5.How to use wavelets. For easing our notations, the one dimensional wavelets have been written Wj,k instead of 2j/2 w(2jx-k). Then by construction, any function f(x) in L2 (R) can be uniquely written f(x)=:E:Eaj,k Wj,k(X) where aj,k= fR f(x)Wj,k(X)dx. The fundamental problem is to know whether this algorithm can be applied to functions which are not in L2(R). This problem is not of academic nature and the following counter example shows that serious troubles can arise from the behavior of f(x) at infinity. If f(x)=1 identically, then all the wavelet coefficients of f(x) vanish since the integral of a wavelet is zero. The above identity would give 1=0 , which is absurd. The reason of this paradox is the simple observation that 1 is not square-integrable. In order to give a deeper explanation, one denotes by E(x,Y) the kernel 31 L. q(x k)q(y-k) and observe that Ej(x,y)= 2j E (2jx, 2jy) is the kernel of the orthogonal projection on Vj. Finally Dj(x,y)=Ej+1 (x,y) - Ej(x,y) is the kernel of the orthogonal projector on Wj. Then the series L. Dj(f) is similar to a Littlewood-Paley decomposition of f(x) and converges to the given function f if and only if Ej(f) tends to 0 when j tends to - 00 and to f(x) when j tends to + 00. This simple criterion is tautological and explains what is happening if f(x)=1 since we have Ej(f)=1 for all j's. The kernel E(x,Y) has the following property (5.1) l(a/ax)p(a/ay)qE(x,y)1 ~ C m (1 +Ix-yltm for O~p~r , O~q~r , m~1 where r denotes the regularity of our wavelets and (5.2) From these estimates it is trivial to deduce that Ej(f) tends to f as j tends to infinity when f belongs to the classical Banach spaces: for example the space of functions of class CS where O<s<r or the Lebesgue spaces LP for 1~p~ 00, the usual norm topology being replaced by the weak-star topology in the latter case. When j tends to - 00, the situation is distinct and Ej(f) gives a kind of mean-value of f(x) on the interval [-2-j,2-j]. Therefore Ej(f) tends to zero if, in some weak sense, f(x) tends to zero at infinity. Assuming this property on f(x), we can conclude that f=L. Dj(f). Then we decompose each Dj(f) into a sum L.aj,kwj,k and this decomposition resembles a localization. For that reason there is no further problem of convergence. One can observe that we imposed an ordering on the double sum L.L.aj,kwj,k since we have to sum first in k and then in j for rebuilding f(x). But for most classical spaces, the ordering is irrelevant, the exceptions being L1 or L00 • If we want a wavelet series L.L.aj,kw j,k to be unconditionally convergent in L 1 (unconditionally meaning that permutations do not affect the convergence) then the sum of the series should belong to H1 which is the real version by Stein and Weiss of the classical Hardy-space. The classical functional spaces for which the wavelets are unconditional are characterized by size conditions on the wavelet coefficients. Let us give a few examples. A function f(x) belongs to LP(R) ,1 <p< 00, if and only if its wavelet coefficients aj,k satisfy laj,kl ~ Cj,k where the growth of measured by the condition that the auxiliary function c j,k is [L.L.(Cj,k)22jE j,k(X)]1/2 should belong to LP ,Ej,k(X) denoting the indicator funtion of [k2-j,(k+1 )2-j]. The Holder spaces CS are characterized by even simpler conditions when s is not an integer ( in the latter case one has to replace 32 C 1 by the Zygmund class, C2 by the primitives of the functions in the Zygmund class ... ), the characterization being laj,kl~C2-j/22-js for some constant C. These characterizations play a fundamental role. Together with the good localization of the wavelets they imply that in any region where the function to be analyzed is smooth, the wavelet coefficients are extremely small and can be neglected. On the opposite, any abrupt change in the behavior of a function strongly affects the coefficients of the wavelet expansion around the singularity. This local analysis is impossible to achieve with a traditional Fourier type analysis since in that case the localization of a singularity is related to the phases of all the coefficients. As it was mention ned above there is a second way of using wavelets expansions which is much more convenient when there is no a priori knowledge about the behavior of f(x) at infinity. We consider the collection q(x-k), k e Z, together with all the wavelets Wj,k(X) where je Nand ke Z. This collection is also an orthonormal basis of L2(R) but is much more flexible when applied to other functional settings. The convergence depends only on the behavior of Ej(f) when j tends to + 00. This means that the wavelet expansion of a function with a polynomial growth at infinity converges to that function. Morever the wavelet series of f(x) converges to f(x) even if f(x) is a distribution , the only limitation being that the order of the distribution should not exceed the regularity of the wavelets. We all know that distributions can only be created with oscillating functions since a positive distribution would be a measure. Wavelets provide a precise version of this statement and any distribution S is the sum of a smooth part which describes a «sketchy» image of S and of finer and finer corrections, all being built through the same algorithm. These corrections are selfsimilar since they are given by aj,kwj,kWhich are deduced from the analyzing wavelet by similarity. This kind of fractal description of distributions had been anticipated by B.Mandelbrot «<generic irregular functions are fractals»). A convenient labelling for the n-dimensional wavelets will be given now. Define an odd vector u in R n as being k2-j-1 where j belongs to Z and k to Z n \ 2 Z n. The collection of all odd vectors will be denoted U. Then the ndimensional wavelets are indexed by these ue U. If u""k2-j-1, then k=21H where r=(r1 ,... ,rn) and rj=O or 1, the sequence (0, ... ,0) being excluded and I belonging to zn. We denote by R the finite set of these r and we have wu(x) =2 n j/2 wr (2jx-I). The full collection of all n-dimensional wavelets is then characterized by the finite set of the 2n-1 wavelets wr , re R. When the n-dimensional wavelets are given by a tensor type construction, the meaning of r is clearer. We decide that w 0 (x) = q(x) and that w1 (x)=w(x). When r=(r(1 ),r(2), ,r(n)), then wr(x) = Wr(1 )(X1 ) ... wr(n)(xn) and since r is never (0, ... ,0), we recover the n-dimensional wavelets of the tensor product construction. 33 The wavelet coefficients are c(u)=if(x)wu (x)dx and we are expecting to rebuild f(x) through f(x) = IIc(u)wu(x). In full similarity with the one dimensional case, this is true whenever f(x) belongs to L2(Rn). It is also true when f(x) belongs to LP and 1<P<oo. The same paradox we described in the one dimensional case would appear here if we would try to apply our identity to the function 1. This paradox disappears if we consider the orthonormal basis consisting of q(x-k), ke zn , and wu(x) where u=k2-j-1 and j~O. When the n-dimensional wavelets are built through the compactly supported one dimensional wavelets of I.Daubechies, this latter orthonormal basis permits to analyze any function or distribution , whatever be its behavior at infinity, the only limitation being the obvious condition that the order of the distribution should not exceed r. Everything we said about the characterization of classical Banach spaces in the one dimensional case remains true here. 6. Is orthogonality needed? As we said in the introduction, few people are using orthogonal wavelets. Let us examine the nearest case where a wavelet g(x) has the property that the family 9j,k(X)=2j/2g (2jx-k) is a Riesz basis for L2(R). By abstract non-sense, there exists a second family hj,k(X) such that the coefficients of the expansion in the first basis are given by the scalar products with the functions of the second one. Unfortunately two bad things may happen, even in the case when g(x) belongs to the class of Schwartz. The functions hj,k may not be given by hj,k(X)=2j/2h(2jx-k) and they may be bad in terms of size and regularity. This means that the validity of the wavelet expansion might be confined a ro u n d L 2 (R). Some counter-examples have been constructed by Ph.Tchamitchian where for any p > 2 , there is a choice of g , in the class of Schwartz, such that the collection 9j,k is not total in LP while being a Riesz basis for L2. The same pathology appears with the space CS for any s>O. That is the worst possible situation since any hope for a better approximation than L2 has disappeared. If that is the case, wavelets are uninteresting since the Haar system would do it as well. Wavelets should have the property that the wavelet have a rapid decay insuring the rapid coefficients of a smooth function convergence of the corresponding wavelet series. Now let us examine what is happening with the wavelets which are commonly used like the second derivative of the Gauss function exp(-x 2 /2). We do not know if it is possible to form a Riesz basis of L2 with translates and dilates of this analyzing wavelet g(x). The theory of frames remedies this problem and allows to expand any function in L2 with an over complete collection ga,b (x) where a=2- d j, d>O, and b=Bk2- d j, B>O, d and B being small enough. But we do not know if the corresponding expansions would converge in other functional settings like CS or LP. No needs to say that all these 34 problems disappear entirely with the orthonormal basis of r regular wavelets we constructed since the convergence is given for free as long as 1<p< in the LP case or O<Sd in the CS case. 7.Prospects for the future. All the preceding constructions were somehow based on the fundamental group action of Zn on R n and this observation leads to the important question of knowing what can be done in contexts where this group action does not exist. Such contexts are very natural. One can try for example to construct wavelets on a sphere. We can also raise a similar problem on some geometrical domain with a boundary. These wavelets should first be correctly defined so that the problem would make some good sense. Let us stick to the case of the sphere for the sake of simplicity. On a sphere one could start with a sequence of finer and finer meshes providing an increasing sequence of spaces of splines resembling our old friends Vj of a multiresolution analysis. But what is obviously missing are the dyadic dilations . The other missing action is the rotation invariance. Nevertheless S.Jaffard ([7]) proved a theorem showing that in some sense the construction which has been so successful in the flat case can be carried over the sphere. The geometry is given by an increasing sequence Fj. jE N .of finite subsets of the sphere M with the following properties. If dj= inf Ix-yl. XE Fj. yE Fj. x "# Y . then any ball of radius Cdj. centered on the sphere. should intersect Fj. Moreover we assume that dj~Cdj+1. Then there are many ways of defining increasing sequences Vj. j~O. of spaces of splines of order r with nodes belonging to Fj- P.G.Lemarie and S.Jaffard have proposed two distinct constructions. They both end with wavelets Wu • UE Fj+1\Fj • jE N. and wu. u E F O. enjoying the following properties: each Wu is centered around u. with width dj. height ~ Cdrn/2 where n is the dimension of the sphere and enjoys the best possible regularity compatible with the localization. It means that. in local l(d/dx)CX Wu I ~Cm drn/2-lcxl (1 +Ix-ylldjt m for In I~r • coordinates. we have XE M. yE M. These wavelets are adapted to the analysis or synthesis of functions on the sphere. An other fascinating problem is the connection between wavelets and P.D.E.'s. The difficulty seems to connect a specific orthonormal basis of wavelets with a specific differential operator. The reason could be that the ordinary sine and cosine functions are so well adapted to differential operators that there is no room left for the wavelets. On the other hand wavelets seem to be useful in scientific computing. They can be fixed so that they give a good description of far more complicated operators than the ordinary pseudo-differential operators. For example some of the involved operators of Calderon's program are nearly 35 diagonal in suitable wavelet basis but we are unable to construct such a basis in which the Cauchy operator be diagonal. 8.References. [1] R.Balian. Un principe d'incertitude fort en theorie du signal ou en mecanique quantique. C.R.Acad.Sc.Paris, t.292, Serie 11,1357-1361,1981. [2] G.Battie. A block spin construction of ondelettes. Part I: Lemarie functions. Commun. Math. Phys. 110, (1987), 601-615. [3] G.Battle. A block spin construction of ondelettes. connection. Commun. Math. Phys. 114, (1988), 93-102. [4] I.Daubechies. Orthonormal bases of compactly supported wavelets. (Preprint from AT&T Bell Laboratories, 600 Montain Avenue, Murray Hill, NJ 07974. [5] I.Daubechies. The wavelet transform, time-frequency localization and signal analysis (AT&T Bell). [6] K.Grochenig. Analyse multiechelles et C.R.Acad.Sci.Paris, t.305, Serie 1,(1987),13-17. [7] S.Jaffard et Y.Meyer. Bases d'ondelettes dans des ouverts de Rn. A paraltre au Journal de Math. Pures et Appliquees. [8] P.G.Lemarie et Y. Meyer. Ondelettes et bases hilbertiennes. Revista Matematica Iberoamericana 2, (1986),1-18. [9] S.Mallat. Multiresolution approximation and wavelets.( Preprint from G RASP lab, Dept. of Computer Science, University of Pennsylvania Philadelphia, PA 19104-6389. [10] Y.Meyer. Wavelets and operators. Cahiers mathematiques de la decision nO 8704. Part II: the OFT bases d'ondelettes. [11]------.Ondelettes, fonctions splines et analyses graduees. Cahiers n0 8703. 36 [12] H.C. Schweinler and E.P. Wigner. Orthogonalization methods. Journal of Mathematical Physics. VoLlI, (1970), 1693-1694. [13] J.O.Stromberg, A modified Franklin system and higher order spline systems on Rn as unconditional bases for Hardy spaces, Conference in Harmonic Analysis in honor of Antoni Zygmund, Vol.lI, 475-493, edited by W.Beckner and aI., Wadworth math. series. 37 Orthonormal Bases of Wavelets with Finite Support Connection with Discrete Filters I. Daubechies* AT & T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ07974, U.S.A. Abstract: We define wavelets and the wavelet transform. After discussing their basic properties, we focus on orthonormal bases of wavelets, in particular bases of wavelets with finite support. Contents: 1. The wavelet transform: continuous and discrete versions - frames. 2. Orthonormal bases of wavelets and multiresolution analysis. 3. S. Mallat's algorithm: the connection with discrete filters. 4. Orthonormal bases of wavelets with finite support. 5. Fractal properties. 1. The wavelet transform: Continuous and discrete versions. As proposed by J. Morlet [1], the wavelet transform of a function Jis given by c!>Wav,f(a, b) = 1 ~ f dx h (X-b) -a- J(x) , (1.1) where h is the basic "wavelet". The parameters a, b can be chosen to vary either continuously (a, b E IR, with a "* 0), or in a discrete way (a = a3', b = nb oa3' , with m, nElL., and ao > I, bo > 0 fixed). J. Morlet proposed the discrete version of (1.1) as an alternative to the windowed Fourier transform, which computes, for a given function c!>Wind.F.T.,t(P, q) = f dx e- ipx g(x J, (1.2) - q)J(x) , where g is a fixed window function. In Gabor's approach, the function g is chosen to be Gaussian, g(x) The (P * 38 = 1T- 1I4 exp(-x 2 /2), parameters p, q can = mpo, q = nqo, again but many other window functions can be (and are) used. vary either continuously (p, q E IR), or discretely with m, nElL., and Po, qo > 0 fixed). The wavelet transform (1.1) and "Bevoegdverklaard Navorser" at the Belgium National Foundation for Scientific Research (on leave); on leave also from Department of Theoretical Physics, Vrije Universiteit Brussel, (Belgium). the windowed Fourier transform (1.2) have many features in common. Provided the basic functions h, g and their Fourier transforms are reasonably well concentrated, the two transforms analyze the frequency content of the signal f, locally in time (if the variable x is to be understood as "time"). This is clear from the fact that (1.1), (1.2) are scalar products of f lal-1h (X:b), with ha.b(x) = transform g are gP.q(x) = eipxg(x - q) respectively. If g and its Fourier both concentrated around 0 (as for g Gaussian), then gp,q is concentrated around q, while gp~q is concentrated around p. The scalar product (gp,q' j) therefore analyzes f in a neighborhood of the time-frequency point (q, p). A similar argument holds for the scalar products (ha,b, j); note however that the frequency analysis performed by the wavelet transform is different from the windowed Fourier transform (see also below). Another feature that (1.1) and (1.2) have in common is the reconstruction formula for from <PI f (continuous version). We have f (x) = II 2~ dp dq <PWind.F.T.,f(P, q) e ipx g (x - q) (1.3) and f( x ) = _1_ 2 C n h II dadb '" ( b ) Ia 1-1/2h(X-b) 2 '+'Wav.T,f a, a (1.4) a In (1.4) the constant Chis defined by (1.5) where ir(~) lirw I = = (2n)-1/2 lir(-~) I I dxeix'~h(x) is the Fourier transform of h. We assume that for all ~ (otherwise (1.4) has to be replaced by a more complicated formula), and that Ch < amounts to requiring 00. For reasonably nice h (e.g. Ih(x) I dx h (x) I :S C(1 + Ix I)-a, ex > 1), this = o. The similarity between (1.3) and (1.4) is due to the fact that (1.1)-(1.4) and (1.2)-(1.3) are both special cases of square integrable representations, as pointed out by A. Grossmann. The reconstruction formula (1.4) and the associated reproducing kernel Hilbert space enabled A. Grossmann and J. Morlet to analyze in detail the continuous wavelet transform [2]. The paragraphs above list a few analogies between wavelet transform and windowed Fourier transform. illustrated by Even more interesting are their crucial differences. They can easily be looking at the graphs for gmn(x) = e impoX g(x - nqo) and 39 ,I ~ h- 2 •30 R. Q.. 3 \ \ ., II / h II II "" """ II " It I: u:~ ,, IIII 1111 It tI Figure 1. Examples of functions gmn' h mn corresponding to resp. the windowed Fourier transform and the wavelet transform. hmn(x) = ao m/2 h(aomx - nb o ), corresponding to the discrete versions of both transforms. Figure 1 shows gmn' h mn for a few values of m, n, for the choices g(x) = 'I1"-1/4 e -x 212, h(x) = 2/Y3 '11"-1/4(1 - x2)e-X2/2. In both cases different values of m correspond to different frequency ranges. The high frequency gmn are high frequency oscillations with an amplitude modulation given by g (x - nq 0). The envelope function of all the gmn is therefore always the same function g, translated to the relevant time interval (indexed by n). The high frequency h mn look very different, however. As contracted (and translated) versions of the basic function h they have variable width, adapted to their frequency range: the higher that range, the more narrow they are. This difference in time-resolution for high versus low frequency wavelets, in contrast to the fixed time resolution for all frequency components of the sliding windowed Fourier transform, is illustrated very clearly by Fig. 2. For both transforms, the centers of localization in the time-frequency plane are plotted, corresponding to the gmn or hmn' respectively. For the wavelet transform, this discrete lattice shows the differences in time-resolution as the frequency bounds change. Note that while for higher frequencies the time resolution becomes better, the frequency resolution becomes worse, as was to be expected from the Heisenberg uncertainty principle. The better time resolution for high frequency components enables the wavelet transform to perform better than the sliding window Fourier transform for signals which typically have short-lived high frequency components superposed on longer-lived lower frequency parts, as in e.g. music, or speech. The exponential rather than linear treatment of frequency is also more closely related to our auditory perception. It should be noted that techniques related to the wavelet transform, based on the use of dilations and translations, have been used in many different fields. Written in a different way, the reconstruction formula (1.4) appears in the pioneering work of A. Calderon in harmonic 40 ........ ........... . __ ~ ______ Po __ ~ ~ __ ~ ________ __________~------------~x ~x 90 .... . . .. . .......... . Figure 2. The centers of localization in the time-frequency plane (x = time, k = frequency) corresponding to the gmn (windowed Fourier transform), resp. hmn (wavelet transform) . analysis [3]. In this field also, the possibility to zoom in on short-lived high frequency phenomena was important, for applications to the study of singular integral operators (see, e.g. [4]). Both the wavelet transform and the windowed Fourier transform, with their respective reconstruction formulas (1.3), (1.4), are examples of coherent state decompositions used in quantum physics (for a review, see [5]). The affine coherent states, as the wavelets are called in this framework, were first introduced in [6]. They are shown to playa particular role for the hydrogen atom in [7]. The idea of decomposing into building blocks of constant "shape" but different size (and this is essentially what the wavelet transform does) is also central to the renormalization group theory, important in statistical mechanics and quantum field theory [8]. It is therefore not altogether surprising that new developments in wavelets have led to an elegant application in quantum field theory [9]. Finally, the same ideas are also related to certain filter banks used in acoustic signal analysis. We shall come back to this later. The reconstruction formulas (1.3), (1.4) use the continuously labelled windowed Fourier transform or wavelet transform of f in order to reconstruct f. When discretely labelled gmn or hmn are used, different reconstruction algorithms apply. In both cases, we define the map by (Cj)m,n = (4)mn,j) , where 4>mn is either gmn (windowed Fourier transform) or hmn (wavelet transform), and where ( , ) denotes the standard L 2 -inner product 41 (f, g) = f dx f(x)g(x) . The map C depends of course on the chosen function h or g, and on the parameters ao, b o or Po, qo which determine the density of the lattices in Figure 2. If h or g and its Fouier transform are reasonably well concentrated (Le. in all the cases of practical interest), then the operator C is bounded. In order to have a "good" characterization of signals f by their coefficients (Cf)mn, we require that *" 12, then Cf1 *" 1. C is one-to-one: if f1 2. C has a bounded inverse on its range: if Cf1 and Ch are "close", then so are f1 and fz. This means that there exist A > 0, B < A co Ch such that, for all f E L2(1R), If(x) 12 :S}: 1(<I>mn.J)1 2 :s B fdx m,n fdx If(x) 12 . (1.6) The set of vectors {<I>mn;m,n E I} is then called a "frame" for L2(1R). Note that a frame is not necessarily a basis; in many cases it is "overcomplete". A simple example in 1R2 (not related to wavelets or windowed Fourier transform) illustrates this. e 2 = (- ± t, V;), I(ej' v) 12 j=l = t e 3 = (- t, - V;). //v//2. The {ej; j Define, in 1R2, e 1 = (1, 0), It is easy to check that for = 1,2, 3} constitute all v E 1R2, therefore a frame for 1R2, while they are clearly not a basis, since they are not linearly independent. If the two frame bounds A, B in (1.5) are equal, then we call the frame "tight". For a tight frame we have hence m,n For tight frames, this is the desired inversion formula, allowing to reconstruct f from the (Cf)mn' For general, non-tight frames, one can still write f 42 = 2(A + B)-I}: m,n <l>mn(Cj)mn + Rf, (1.7) where the remainder term Rf is bounded by I dx Since (BA -1 - 1)/(BA -1 1(Rf)(x) 12 :s BA -: - 1 +1 BA - + 1) < 1, I dx If(x) 12 it is clear that (1.6) can be iterated to obtain a reconstruction formula for ffrom the (Cf)mn with any desired accuracy. For a given precision, the number of terms required depends on BA -1 - 1. The closer the two frame bounds are to each other, the more "snugly" the frame fits, and the fewer iterations are required. It is therefore important to have good estimates for A, B. Such estimates can be computed by means of the Poisson sum formula. The following table lists values of the frame bounds for the wavelet transform, for various choices of ao and boo Frame bounds for the wavelet transform, with h(x) = 2tV37T- 1/4 (l - x 2 )e- x212 • ao bo 2. .5 1. V2 .5 1. A 6.546 B 7.092 BA -1 - 1 0.083 3.223 3.596 0.116 13.637 13.639 0.0002 6.768 6.870 0.015 It is clear that even for relatively large values of ao, b o the frame constants can be so close that one can drop the remainder term Rf in (1.6) to obtain an approximate reconstruction formula that is extremely accurate. For other examples, variations on the same theme, and more details, we refer the reader to [10]. Note that other inversion formulas than (1.6) may be used. One easily checks that For values of ao, b o close to 1,0 respectively, one can then use a discrete approximation of this formula as a reconstruction algorithm for f. Regardless of which algorithm is used, the condition (1.5) is necessary to ensure that the inversion is numerically stable. The concepts of frame and frame bounds, and formula (1.6) can also be applied to the windowed Fourier transform. In fact, frames are essential in windowed Fourier transform analysis if one is also interested in good time-frequency localization. This is due to the following theorem, first stated, in a more restricted version, by Balian [11] and Low [12], and 43 extended and rigorously proved by Coifman and Semmes [10]. A simpler and much more elegant proof for bases was subsequently found by Battle [13]. Theorem: Po . qo If g E L 2 (1R), Po, qo > 0 are such that the gmn constitute a frame, and = 2'11", then either xg or g' i L 2 (1R). In practical applications, good time and frequency localization of g is required (in fact, much stronger conditions than xg, g' E L 2 are usually satisfied). In order to have numerically stable inversion formulas recovering f from the (gmn, fl, one is therefore forced to consider time-frequency lattices with mesh size Po . qo < 2'11". Lattices with Po . qo = 2'11" correspond to "sampling" with the Nyquist density, which is the reason why such lattices have been proposed for windowed Fourier transform analysis (starting with Gabor [14], who proposed g(x) = Po . qo 'I1"-1/4 e -x 2 /2, = Po = 2'11", qo = 1). The above theorem tells us that for reasonable g, 2'11" always leads to numerically unstable inversion algorithms. This confirms the fact that the Gabor transform has no bounded inverse [15] [16]. Figure 3 illustrates this fact in a dramatic way. For the windowed Fourier transform (1.6) can be written as (1.8) m,n where gmn(x) = e impoX g(x - nqo), and g is a "dual" function, completely determined by g, Po and qo, g(x) = g= 'I1"-1/4 e -x 2 /2 1. For A = + B)-lg + O(BA- 1 - 1). Figure 3 shows this dual function g, for and Po = qo = V 2'11" A , for the different values A = .25, .5, .75, .95 and 2(A 1, the gmn do not constitute a frame, which is expressed by the singularities of g; .J. .'i A:.2.5 ., I I\~. S- .5 . .2. o. -10. -5. •3 .~ O. O. A,=.315 O. 5. 10. -10. .6 -5. O. 5. 10. -10. 5. .4 .J .2 O. O. -5. O. 5. 10. >-'=1. O. -5. Figure 3. The dual function g corresponding to the lattice gmn' with g(x) = 'I1"-1/4 exp (-x 2 /2), and Po = qo = (2'11"A)1I2, for A = .25, .375, .5, .75, .95 and the singular case A = 1. 44 (1.7) has to be understood in a distributional sense in this case. In fact for A = 1 [16]. For the four smaller values of A, how g evolves, for increasing A, g is still g is not even in L2 eX, with gaussian decay; it is clear from a gaussian profile to the singular function for A = 1. On the other hand, one can easily show that a family of gmn can only constitute an orthonormal basis if Po . qo = 2'TT. The Balian-Low theorem excludes therefore the existence of any orthonormal basis for the windowed Fourier transform with reasonable localization properties. This leads us to another fundamental difference between the windowed Fourier transform and the wavelet transform: there do exist suitably chosen functions h and constants ao, b o such that both h and its Fourier transform have fast decay (e.g. faster than any inverse polynomial power) and such that the corresponding hmn constitute an orthonormal basis of L 2(1R). The first example of such an orthonormal basis was constructed by Y. Meyer [17], and extended to higher dimensions by P. G. Lemarie and Y. Meyer [18]. Other examples were constructed shortly after by G. Battle [19] and P. G. Lemarie [20]. These first constructions were rather mysterious, and relied on "miraculous" cancellations. The introduction of multiresolution analysis by S. Mallat and Y. Meyer led to a deeper understanding of these bases. 2. Orthonormal bases of wavelets and muItiresolution analysis. The papers by S. Mallat and Y. Meyer in this same volume no doubt discuss multiresolution analysis in greater detail than possible here. This paragraph is therefore restricted to a short review. A "multiresolution analysis" of a function j consists in a hierarchy of approximations of j, defined as averages on different scales. The finer the scale, the better the approximation. More precisely, one has a hierarchy of subspaces of L 2(1R) ... C V -2 C V -I C Vo C VI C V 2 C (2.1) such that n Vj = {O}, U Vj = L2(1R) . jEl jEl The "scaling" aspect is translated by the condition g E Vj - g(2.) E Vj + 1 (2.2) The space V 0 thus determines the whole ladder of spaces. A typical but not very sophisticated example is the case where V 0 consists of piecewise constant functions, 45 Vo = {g E L 2(~); g is constant on every half open interval [k, k + 1[, for all (2.3) k E I} . The spaces Vj then contain functions which are piecewise constant on the intervals [k2 j , (k + 1)2j [; they clearly satisfy (2.1). We also impose some translation invariance: g (. -k) E Vo for all k E I . g E Vo - (2.4) The final requirement is that there exists <l> E V 0 such that its integer translates are an orthonormal basis for V 0, i. e. for all g E v o , I dx Ig(x) 12 = ~ k II dx <l>(x - k)g(x) 12 . (2.5) (In fact, it is sufficient to require that the <l>(. -k) constitute a Riesz basis for Vo; one can prove that this entails the existence of <l> E Vo satisfying (2.5) [21]). In the example (2.3) above, one can choose <l>(x) = 1 if 0 For a given function ! :S x < 1,0 otherwise; this clearly satisfies (2.5). the successive multiresolution approximations are defined as the orthogonal projections onto the Vj Pj! = ~ <l>jk (<l>jbf> , kEl where <l>jk(X) = 2jI2 <l>(2j x - k); the <l>jk constitute an orthonormal basis for Vj by (2.5) and (2.2). The "difference in information" between two successive approximations P j ! and P j + 11 is given by the orthogonal projection Qj! of ! onto the orthogonal complement Wj of Vj in Vj + 1, The four requirements (2.1), (2.2), (2.4), (2.5) imply that the spaces Wj are also scaled versions of one space W 0, (2.6) that they are translation invariant for the discrete translations 2- jl, 46 g E Wo ~ g(' -k) E Wo and that they are mutually orthogonal spaces generating all of L 2(1R), (2.7) Moreover [21] there exists a function $ E Wo such that the $(' -k) constitute an orthonormal basis for W 0, for all g E W o , fdx Ig(x)1 2 = ~ Ifdx $(x-k)g(x) HZ 12 . (2.8) By (2.6) it follows that the $jk(X) = 2j!2 $(2 j x - k), for fixed j, constitute an orthonormal basis for W j . Hence by (2.7), the {$jko j, k E Z} are an orthonormal basis of wavelets for L 2(1R). In the example (2.3) it is easy to guess $. The space Wo is constituted by those functions that are piecewise constant on the intervals [kI2, (k + 1)/2[, and are orthogonal to the functions constant on [e, e + 1[. It is easy to convince oneself that Wo is therefore spanned by the function $(x) = 1 for 0 :5 X < 112, -1 for 1/2:5 x < 1, 0 otherwise, and its integer translates. Since the integer translates of $ are mutually orthogonal, $ satisfies (2.8), and the $jk are the orthonormal wavelet basis associated with the multiresolution analysis defined by (2.3). This basis was in fact well-known long before wavelets existed: it is called the Haar basis, and is known to be an unconditional basis for all LP -spaces, 1<P < 00. Multiresolution analysis allows one to construct orthonormal bases, such as the Battle-Lemarie bases, which generalize the Haar basis, but are smoother (the Battle-Lemarie bases are C k and have exponential decay). They are therefore suitable for other spaces (Sobolev, Besov, .:.) than only the LP -spaces. There exists an explicit algorithm for the construction of $, once <\l is known. <\l E V 0 C V I, there exist Cn Since such that <\l(x) = ~ Cn (2.9) <\l(2x - n) . Cn = v'2 (<\lIn, <\l»). $(x) = ~(-W Cn+1 <\l(2x + n) . (Since the <\lIn are an orthonormal basis in V I, one has Then [21] (2.10) 47 3. Stephane Mallat's algorithm: the connection with discrete filters. s. Mallat uses multiresolution analysis and orthonormal bases of wavelets in a discrete algorithm applied to vision analysis [22]. In fact the concept of multiresolution analysis owes it existence to inspiration drawn from concrete schemes for analysis and reconstruction of vision data. To illustrate how related ideas were being used in vision analysis, we sketch one such scheme, the "Laplacian pyramid" of Burt and Adelson [23]. While visual data are of course 2-dimensional, we shall restrict our discussion to 1 dimension; the generalization to higher dimensions is easy. The initial data are a sequence of numbers (Cn)nEZ, indicating the grey levels at different, equally spaced grid points. We shall identify the original data with the subscript 0, (C~)nEZ. The idea behind the Laplacian pyramid scheme is to define successively more and more blurred versions c t of the data cO (corresponding to low-pass filters applied to co), and to encode the "difference in information" between successive ct. The scheme proposed by Burt and Adelson [23] achieves this goal in the following elegant way. Define d = ~ w(n - 2k)c~, (3.1) n where the w (j) are weighting factors satisfying = 1. Typically only a small number of ~ w (j) j w (j) are different from zero; an example used in [23] is w(O) = a, w(1) = w(-I) = .25, w(2) = w(-2) = .25 - a12, w(j) = 0 otherwise ,(3.2) with a E ].125, .625[ (see [24]). The operation (3.1) consists in a convolution and a decimation; the resulting c 1 "lives" on a larger scale than cO, containing less information. To encode the "difference in information" between cO and c 1 , we first compute, from c 1 , a sequence cO that lives on the same scale as cO, using the same weight coefficients, c~ = ~ W(lI - 2k) d, (see Figure 4b) (3.3) k and we subtract, ~ = c~ - c~ . (3.4) The information in cO is thus split up into dO and c 1, and is fully recoverable from these two new sequences. The same process, using the sequence c 1 as starting point rather than cO, can be repeated, leading to sequences c 2 and d 1 • After L iterations we have decomposed cO into dO, d 1 , 48 •.• , d L and c L + 1 • b. ... 1 I c;:.-2 Figure 4. Schematic representation of the operations (3.1) (a) and (3.3) (b). Let us introduce the filter operator F: e2 cl.) - e2 (i1.) (Fah = ~w(n n defined by (3.5) 2k)an . F consists of a convolution, followed by a decimation (only one out of every two samples is retained after the convolution). We also define the dual operation F* by n where a, b are arbitrary sequences in n e2 cl.). Since the w(n) are real, this leads to (F*b)n = ~w(n - 2k)bk • (3.6) k The equations (3.1), (3.3) and (3.4) and their iterates can then be rewritten as The filters F, F* are extremely easy to implement numerically, and the decomposition of CO into the "difference" sequences dO, ... , d L and a much smoothed out version c L + 1 can be computed as fast as an FFT (fast Fourier transform). Reconstruction of cO from the dO, ... , d L and c L + 1 is just as easy, since one only needs to iterate (3.7) starting from e = L. 49 One last remark concerning the Laplacian pyramid scheme. The reconstruction formula (3.7) effectively writes cO as Since the filter operator F· is iterated. it is worthwhile to ensure that it does not look "messy" when applied to. e.g .• a sequence with only one non-zero entry. The filters (3.2) satisfy this requirement. as shown in Figure 5. In this figure we visualize the successive sequences e (with eo = 1. en = 0 otherwise). F· e. (F·)2 e .... by piecewise constant functions. with levels given by the entries of the sequence. The stepwidth of the intervals is adapted to the iteration level e in (F·)ee. since band F·b "live" on different scales (see Fig. 4b). It is clear that the (F·)te converge to a "nice" function in Fig. 5; for a mathematical proof of this convergence we refer to [24]. S. Mallat's algorithm defines "averages" c t and "differences" d t • from an initial sequence co. via multiresolution analysis in the following way. From co. he defines 10 E Vo by 10 As an element of V 0 = = ~ c~ (3.8) <\>Ok • k V -I EEl W -I , 1 can be decomposed into I - I E V -I and g -I E W -I • 10 =1-1 + where we have used that the <\> jb g-I = ~d<\>-Ik + ~dll\l-Ik. k (3.9) k 1\1jk are orthonormal bases for V j • Wj respectively (see §2). Here 1 -I corresponds to an "averaged" version of 10. and g -I to the difference in information between 10 and this average. Since (3.8). (3.9) are expansions with respect to orthonormal basis. the coefficients o d. dl are easy to compute. -1 0 -1 -1 0 -1 0 o 1 Figure 5. Representation of the sequences e. F· e. (F·)3 e by piecewise constant functions, and the limit function. The stepwidth for each (F·)t e is 2- t . The filter coefficients for this figure are given by (3.2). with a = .375. 50 d = (<P-lbj) = ~ e~ (<P-lb <POn) = dl = (l\!-lk,j) = ~ e~ (l\!-lb <POn) n where hen) = (<P-IO, Consequently both e1 <Pan) and = 2- 112 f d1 ~h(n - 2k)e~ n n = ~ g(n - 2k)e~, n dx <p(x/2) <p(x - n), g(n) = 2- 112 f dx l\!(x/2) <p(x - n). are derived from cO by the application of filters H, G of type (3.5), (3.10) On the other hand = ~d (<POn' <P-l,k) + ~dk (<POn,l\!-l.k) k = k ~ [hen - 2k)d + g(n - 2k)dkJ. k or (3.11) with H*, G* defined analogously to (3.6). The whole decomposition process can of course be iterated: f -1 decomposes into f -2 + g -2, corresponding to sequences e 2 = He 1 and d 2 = Gel, etc. Schematically, decomposition and reconstruction can be represented as in Figure 6. Figure 6. Schematic representation of the tree-algorithms for the decomposition and reconstruction in S. Mallat's scheme. The tree-structure, together with the easy convolution and decimation structure of H, G, makes that this algorithm works very fast; the whole decomposition can be done as fast as an FFT. Note that at every level e t is replaced by a roughly equivalent number of entries: if the e~ are zero except for N consecutive entries, then, apart from edge effects, only NI2 entries of e t+ 1, d t + 1 will be non-vanishing. The total number of relevant entries in d 1, d 2 , ..• , d L , eL is therefore essentially the same as in the original sequence eO. This is in opposition to the 51 Laplacian pyramid scheme, where the total number of entries after L iterations is essentially (2 - 2- L ) times the original number. The multiresolution-based algorithm of S. Mallat is thus more efficient than the Laplacian pyramid scheme. Moreover, when generalized to 2 dimensions, it also turns out to be orientation selective, at no extra cost, which is another advantage over the Laplacian pyramid scheme [22]. In fact, for the implementation of S. Mallat's algorithm, one only needs the two filters G, H; their multiresolution analysis origins are not used explicitly. One may therefore try to isolate the relevant properties of the filters, and design filters satisfying all these properties directly, without multiresolution analysis. From (3.10) and (3.11) a first condition can be derived. H*H (C1) + G*G = Id . The orthogonality of V -1 and W -1 results in (C2) HG* = 0; this expresses the fact that the two terms in the decomposition (3.11) of cO are always orthogonal sequences. A third condition expresses the fact that H is an "averaging operator", i.e. a low pass filter, while G measures the difference between a sequence and its average, and is therefore a band pass filter. This results in (C3) ~g(n) = 0 n ~h(n) = V2 n V2 -normalization where the is due to the decimation 2: 1 in the definition of the filter H (see [24]). Finally, we also impose a regularity condition, similar to the condition on the Laplacian pyramid d 1, d2, d L , c L is ... , scheme. The complete reconstruction formula for We shall therefore require that the operator H* which is iterated satisfies (C4) the piecewise constant functions representing (H*)e e (where en = 0 for n '" O,eo 52 1) converge to a "nice" function as e_ 00 • cO from For a more precise formulation of this condition, see [24]. Filters H, G which are derived from a multiresolution analysis automatically satisfy conditions C1-C3. Moreover, one can show that in this case the piecewise constant functions representing (H"/ e converge to the averaging function <I> itself [24], so that C4 is also satisfied. It is possible to construct filters H, G of type (3.5) which satisfy C1-C3, but not C4. An example is given in Figure 7. In this case the (H"/e converge, for e- 00, to a distribution which is singular at every dyadic rational between 0 and 3, i.e. every point of the form k2- m , with 0 :5 k < 3 2m . This example shows that condition C4 is necessary to avoid "messy" iterations. It turns out [24] that conditions C1-C4 ensure that the filters H, G are associated to a multiresolution analysis. The "averaging function" <I> of that multiresolution analysis is exactly the "nice" function to which the (H")f e-piecewise constant functions converge. The proof in [24] of this equivalence between filters and orthonormal bases of wavelets essentially uses this "graphical" construction of a limit of piecewise constant functions representing sequences. <I> as The orthogonality of the <1>(. -k) e.g. can easily be proved in the following way. Let T be the translation by one unit step for sequences, (Ta)n = an _). Then e and Tf e are obviously orthogonal if e "* 0, On the other hand, the special structure (3.5) of the filters H, G together with the conditions C1-C2 imply HH" = Id = GG". It follows that (H")me and (H")mTfe are orthogonal for all L ___ _ .1 _ __ _ o o (H ')~ e 3 Figure 7. 3 An example of a pair of filters H, G which do not satis~ the regularity condition C4. In this case h(O) = 15/(13Y2), h(1) = 10/(13V2), h(2) = -2/(13Y2), h(3) = 3/(13Y2), with all the other h(n) = O. The g(n) are defined by (3.12), g(n) = (-l)nh(-n + 1). One readily checks that C1-C3 are satisfied. 53 m ~ O. The piecewise constant functions representing them are therefore also orthogonal for all m, and so are their limits <I>(x) and <I>(x - e). Defining ljI(x) = V2 ~ g(n)<I>(2x - n), n one proves similarly that the 1jI(' -k) are orthogonal, and that the <1>(' -k) are orthogonal to the 1jI(' - e). The property <I>(x) = V2 ~ hen) <I>(2x - n) n follows immediately from the construction, while Cl implies that for all J>O ~ n l(f, <l>J+l,m) 12 = ~ k l(f, <I>-J,k) 12 + ±~ j=-J k I E L2(~), and all l(f, IjIjk) 12 It is not hard to prove that ~ l(f, <I>-J,k) 12 - 0, while ~ l(f, <l>J+l,mW n k J - 00. f dx II (x) 12 for It then follows that the IjIjk are an orthonormal basis of wavelets [24]. Remarks. 1. The conditions CI-C2 are, in a different form, the "unitarity conditions" imposed by Y. Meyer (see e.g. [25], or his paper in this volume). 2. A different proof of the equivalence filters - wavelet bases can be found in [22], where the regularity condition C4 is replaced by the condition inf ~EIR, 1~ls1T/2 I~ hen) ein~ n 1> O. This positivity condition is sufficient to ensure that the IjImn are an orthonormal basis; it does not guarantee any regularity for <I> or 1jI, however. The "messy" example in Fig. 7, e.g., satisfies this positivity condition. 3. Using the special form (3.5) of Hand G one can show [24] that C2 is already implied by C1. 4. The condition C 1 can easily be rewritten in z-transform-language. Let us associate, to any sequence C = (Cn)nEZ, the function c(z) = ~ cnz n . n Then the definition of the operator H as a convolution followed by a decimation implies (Hc)(z2) = where ~(z) = ~ h(-n)zn. n 54 '21 [~(z)c(z) + ~(-z)c(-z)], Similarly the z-transform of B" e is = We also define '8(z) e(z) = (B"Be ~ g(-n)zn. n Condition C1 gives then + G·Ge)(z) 1 2[1~(z)12 + 1'8(z)1 2]e(z) - + [~(z) ~(-z) + '8(z)'8(-z)]e(-z) , or 1~(z)12 + 1'8(z)12 = 2, ~(z)~(-z) + '8(z)'8(-z) = O. This amounts to requiring that the 2 x 2 matrix 1 is unitary. Note that this implies v'2 ~(-z) 1 v'2 '8(-z) ~(z)'8(z) + ~(-z)'8(-z) = 0, which can easily be shown to be equivalent with C2. This is another way of proving that C1 implies C2. 5. Filters satisfying conditions C1-C3 had been constructed before by Smith and Barnwell [26]. They call these filters "conjugate quadrature filters" (CQF) as a special case of the "quadrature mirror filters" (QMF) of Esteban and Galand [27]. CQF give exact reconstruction, without any aliasing, as all QMF do, but also without any amplitude or phase distortion. For their purposes, they do not impose the regularity condition C4, and their filters are therefore not equivalent to an orthonormal wavelet basis in general. 6. While other solutions to C1 exist, it is convenient to choose the g(n) such that g(n) = (-1)nh(-n + 1). (3.12) This choice reduces condition C1 to an equation for Lh(n)ein~ [24]. It is the analog of n the correspondence (2.9)-(2.10) between <!> and \fl. 55 7. One way to ensure that the regularity condition C4 is satisfied is to impose that, for some N ~ 2, (3.13) where sup IQ(E) 1 < 2N ~ER 1/2 • In order for (3.12) to satisfy CI-C3 it is necessary and sufficient that (3.14) where R is a real, odd function (see [24]). In order to derive (3.14) we assume that the g (n) are defined by (3.12). 4. Orthonormal bases of wavelets with finite support. In the preceding paragraph we saw that every CQF, i.e. every pair of filters satisfying C1C3, which also satisfies the regularity condition C4, automatically defines an associated orthonormal wavelet basis for L2(1R); where the function <I> is the limit of the piecewise constant functions representing the (H*)te, for f _ 00. On the other hand, it is clear from Figure 5 that <I> will have a compact support, i.e. will vanish outside a finite interval, if only finitely many h (n) are different from zero. More concretely, if the h (n) are *0 only for O:s n :s N, then the piecewise constant function + 114], the piecewise constant function + NI4 + 118], .... In general (H*)te corresponds to a piecewise constant function on [_2- t - 1 , N(1 - 2- t ) + 2- e- 1 ]; the limit function <I> is therefore representing H* e will be concentrated on [-114, NI2 for (H*)2e on [-1I8,NI2 concentrated on [0, N] (see Fig. 5). If we define the g (n) by (3.12), then only finitely many g(n) will be nonvanishing as well, implying that 1\1, as a finite linear combination of the compactly supported <I>(2x - n), has compact support too. One checks that if hen) for 0 :s n :s N, then 1\1 is concentrated on [(1 - N) 12, (1 * 0 only + N) 12] (see [24]). In order to construct orthonormal bases of wavelets with finite support, it is therefore sufficient to construct filters with only finitely many coefficients or "taps" which satisfy the conditions C1-C4. A family of examples is given by the hen) defined by (3.13), with R ... 0 in (3.14). One also needs a procedure to determine the polynomial Q(E) from its squared modulus IQm 12 (3.14); this procedure is given by a lemma of Riesz' [28][24]. Figure 8 shows a few examples of compactly supported wavelet bases obtained in this way. In each case both <I> and 1\1 are plotted. The figure shows clearly that <1>,1\1 become more 56 .t<P I. .?Y' I- O. O. -I. .t. I. 3. O. -I. \. 6'+ 61> I. .2. I. o. O. -I. 5. O. O. -5. 10. 5. I. 10 <P 10 '+ O. 5. Figure 8. 10. 10. increases. 10. The case N = 1 (not plotted) corresponds to the discontinuous Haar basis, where ljJ(x) = 1 for 0 next 5. A few examples of functions q" IjJ giving rise to orthonormal bases of wavelets, corresponding to (3.13), (3.14), with R == O. It is clear that Nq"NIjJ become more regular for larger values of N. regular as N (see (3.13» The o. -5. case, N = 2, h(2) = (3 - Y3)/4V2, leads to :s; x h(O) = (1 < + Y3)/4Y2, h(3) = (1 - Y3)/4V2, h(1) = < 1, 0 otherwise. (3 + Y3) 14Y2 , other hen) = O. 112, -1 for 112 :s; x and all The corresponding q, and IjJ are continuous but not C 1; they turn out to be Holder continuous with exponent 'Y = In(1 + Y3)/ln2 = .5500... (see § below). One can prove [24] that the regularity of q" IjJ in this family of examples increases linearly with N, i.e. there exists Il. > 0 such that Nq" NIjJ E CtJN for all N ~ 2. In this family of examples the size of the support of q" IjJ is thus determined by the desired regularity. It turns out that this is a general feature, and that a linear relationship between 57 these two quantities (regularity and support width) is the best one can hope for. More precisely, one can prove [29a] Theorem: If <I> E C k , support <I> C [0, N] and <I>(x) = N ~ an <l>(2x - n), then N 2: k + 2. n=O The proof is so simple that we include it here. Proof. 1. Let Vo E IIlN - 1 be the vector (vo)j = <I>(j). The equation for <I> implies the existence of a matrix A, completely determined by the an, such that Vo 2. Since <I>'(x) VI = 2Av 1. none of the = 2 n ~ an n=O <I>'(2x - n), the vector Analogously one defines Vj can be zero, since Vj = V2, ... , Vb VI = Avo. defined by (Vl)j each satisfying Vj = <I>'(j) = 2j AVj. satisfies Moreover 0 would imply <I>(j)(x) = 0 for all x of the type k2- t (by iteration of the equation for <1», which leads to <I>(j) == O. This is, however, incompatible with the finite support condition on <1>. 3. It follows that the (N - 1) x (N - 1) matrix A has at least the (k 1/2, , ... ,2- k • Hence N 2: k + 1) eigenvalues 1, + 2. • 5. Fractal properties. The graph of 2<1> (see Figure 8) exhibits a certain "jaggedness" that seems to repeat itself in a self-similar way at smaller scales. This is made even clearer by the blowups in Figure 9. t. .5 .3. Figure 9. 58 The function 2<1>(X) (see Fig. 8) and two successive blow-ups of its behavior around x = 1. Analogous self-similar patterns repeat itself, on smaller and smaller scales, near every dyadic rational point, i.e. near every x of the form k2- t , 0:$ x < 3. A closer study of 2<1> reveals a very rich structure, although the function is not C 1, it is differentiable almost everywhere. In fact, if the binary expansion of x E [0, 3] contains more than (roughly) 25% of digits 1, then <I> is differentiable in x. (We shall make this statement more precise below). Since almost all numbers have 50% of the digits in their binary expansion equal to 1, this implies that <I> is almost everywhere differentiable. Let us see how such properties can be derived. We know that support <I> = [0,3] and that + <I>(x) = ao<l>(2x) where aj = Y2 hU), or ao al<l>(2x - 1) = (1 + + a2<1>(2x - 2) V3)/4, al = + a3<1>(2x - 4), (5.1) = (3 - V3)/4, (3 + V3)/4, a2 a3 = (1 - V3)/4 (see §4). It follows that { + a2 = al + a3 a 1 + 3a 3 = 2a 2 . ao (5.2) For x E [0, 1], we define v (x) E [R3 by vex) = 1 <I> (x ) [ <I>(x + 1) <I>(x + 2) From (5.1) one easily checks that for x :S 112: vex) = Tov(2x) for x ~ 112: vex) = T 1v(2x - 1), (5.3) where To = [:: ° ~l ~ol and Tl a3 a2 The matrices To, Tl have very special properties. In particular, they both have eigenvalue 1, with a common left eigenvector, (1,1,1) To = (1,1,1) = (1,1,1) Tl (5.4) Similarly one computes (use (5.2» 59 1 1 + ("2 + (1,2,3)T o = "2(1,2,3) a2)(1, 1, 1) (5.5) 1 + (1,2,3) Tl = "2 (1,2,3) a2(1, 1, 1) . This implies that To, T 1 both have eigenvalue 112, and that they have a common left invariant subspace, associated to the eigenvalues 1 and 112. Note. The functions N<I> obtained from (3.13) with R == ° in (3.14), are associated to (2N - 1) x (2N - 1) matrices To, T 1 in exactly the same way as in the case N = 2. For general N, the martices To, Tl have N common eigenvalues 1,1/2, ... , 2- N + 1 . If we define the row vectors eigenvectors for Uj = span {uo, U1, Uj' j both = 0, ... , N To, T 1 ... , Uj}. - 1 for by the Uj = (1j, 2j , ... , eigenvalue 2- j (2N - l)j), always lie in then the the left subspace (Full details are given in [29b]). These spectral properties of To, Tl have several consequences. It follows, e.g., that, for all x E [0, 1], <I>(x) + <I>(x + + 1) <I>(x + (5.6) 2) = 1 Proof· 1. We prove this only for x of the type k2- t , with k, e E N. By continuity the result then follows for all x. 2. For e = 0, we have <1>(0) = <1>(3) = 0. <I>(x) for all x 3. = Take any + k2- t E [0,1], with e> + 1) + <I>(x e= ° Hence <I>(x (k = + 2) = <1>(1) + <1>(2) 0,1 are the only possibilities). 0. Then <I>(x) + = <I>(x { + 1) + <I>(x + 2) = Uo . v (x) uo . Tov(2x) = Uo . v(2x) Uo' T 1 v (2x - 1) = Uo if x . v (2x - 1) if x :S 112 ~ 112 . Since 2x or 2x - 1 are of the type k 2 - e+ 1, we conclude by induction that Uo . v (x) = <1>(1) for all x E [0, 1]. 60 + <1>(2) for all x of type k2- t . By continuity Uo . v (x) = <1>(1) + <1>(2) 3 4. 1 It then follows that f dx<l>(x) o this implies <1>(1) = fdx[uo· vex)] = <I>(x) + <1>(2). Since fdx <I>(x) = 1, 0 + <1>(2) = 1, hence (5.6). • Similarly, using (5.5) one proves, for all x E [0, 1] <I>(x) + 2<1>(x + 1) + 3<1>(x + 2) = -x + (1 + (5.7) a2) . Note. In the general case (N v(x),j Uj· = ~ 2), we find that N<I> satisfies N such sum rules, one for each 0, ... , N - 1. One can also use (5.3) to study the local behavior of <I> in the neighborhood of a point x. For any x E [0, 1], we write the binary expansion of x, e.g. x = .1011001011100 ... Define then TX to be given by the same binary expansion, except for the first digit, which is dropped, TX It follows that TX = 2x if x = .011001011100 ... < 112 and TX = 2x - 1 if x > 1/2. Consequently (5.3) can be rewritten as (5.8) where d/x) denotes the j-th digit in the binary expansion of x. Note that the binary expansion is not ambiguously defined for dyadic rationals x, i.e. for x of the type k2- t . For x = 112, e.g., both the expansions .0111111... and .1000000 ... are admissible. Consequently T 112 is not well-defined, giving the answer 1 or 0 according to the chosen binary expansion. One easily checks, however, that TIV(O) = Tov(1) (use (5.6) and (5.7», so that (5.8) holds, even for x = 112, regardless of the choice of binary expansion. It is easy to convince oneself that (5.8) and its iterates never lead to contradictions at dyadic rationals x. Iterating (5.8) leads to (5.9) where 61 Similarly, for t small enough so that the binary expansion of x +t has the same m first digits as the expansion for x, (5.10) In order to estimate the difference vex + t) - vex), we use the spectral decomposition of Tm(x). From (5.4) it follows that Uo = (1,1,1) is a left eigenvector for Tm(x), el(m, x) = Uo with eigenvalue 1, Because of (5.5) one finds that Tm(x) also has eigenvalue 2- m . The corresponding left eigenvector e2(m, x) is a linear combination of Uo and Ul = (1,2,3), with The rm(x) third = eigenvalue of Tm(x) can be computed from its determinant. Defining m m- 1 ~ dj(x) to be the average number of digits 1 in the first m digits of the j=l expansion for x, we find It follows that the third eigenvalue of Tm(x) is Am(X) = a~rm(X) a~(l-rm(X)). One can find explicit expressions for the corresponding left eigenvector e3(m, x), as well for the three right (column) eigenvectors ej(m, x) of Tm(x), but these are not really necessary. It is sufficient to know (this is proved in [29b]) that they are all uniformly bounded in m and x. For any v E ~3 we have Applying this to (5.8), (5.9), and using (5.6), (5.7) we find thus, for sufficiently small t, 62 Hence For all x E [0, 1] such that there exists a limit for the average incidence of digits 1 in the binary expansion, r(x) = lim rm(x) m-'" and such that 0 < r (x) < 1, one can easily show that "sufficiently small t" means t :$ 2- m(1+E), where E > 0 can be chosen arbitrarily small, for large enough m. Choosing t such that 2-(m+1)(1+E) :$ t :$ 2- m(1+E), we find then t- 1 [v(x + t) - v(x)] = -e2(m, x) + (5.11) R(m, x), where the remainder term R(m, x) is bounded by IIR(m,x)II:$ C2m(1+E) /Am(X)/. f(x) > (log 2ao)/(logao - log /a3/) == .2368 If m 2 (1+E) /Am(X) / - [2 1+. ab-r(x) /a3/ r (x)]m tends to zero for m - co, if m-'" then E is small enough. The second term in (5.11) can therefore be neglected for large enough m. On the other hand e2(m, x) tends towards a limit as m - co (see [29b]). It follows that v(x) is differentiable for < all x such that r(x) is well-defined and .2368 <I>(x + 1), <I>(x r(x) < 1, which implies that <I> (x) , + 2) are differentiable as well. The same technique, i.e. the spectral analysis of Tm(x), can be used to prove that <I> is Holder continuous, with exponent 2 - In (V3 + 1) lIn 2 == .5500.... This exponent is the best possible one. One can also analyze the behavior of <I> near dyadic rational points. As approached from below, the binary expansion near a dyadic point has a tail of only 1 - s; as approached from above, the expansion has only 0 - s (see above). The result is that <I> is leftdifferentiable but not right-differentiable at every dyadic rational point. This can clearly be seen of Figure 9, at x = 1. A similar analysis can be carried out for the N<I> corresponding to higher values of N; see [29b]. 63 References. 1. See e.g. J. Morlet, G. Arens, 1. Fourgeau and D. Giard, "Wave propagation and sampling theory," Geophysics 47 (1982) 203-236. 2. A. Grossman and J. Morlet, "Decomposition of Hardy functions into square integrable wavelets of constant shape", SIAM J. Math. Anal. 15 (1984) 723-736. P. Goupillaud, A. Grossmann and J. Morlet, "Cycle-octave and related transforms in seismic signal analysis", Geoexploration 23 (1984) 85. 3. The wavelet transform is implicitly used in A. Calderon, "Intermediate spaces and interpolation, the complex method", Studia Math. 24 (1964) 113-190. It appears more explicitly in e.g. A. Calderon and A. Torchinsky, "Parabolic maximal functions associated to a distribution, I", Adv. Math. 16 (1975) 1-64. 4. An application to a singular integral operator relevant for quantum mechanics can be found in C. Fefferman and R. de la Llave, "Relativistic stability of matter," Rev. Mat. Iberoamericana 2 (1986). 5. J. R. Klauder and B.-S. Skagerstam, "Coherent States", World Scientific (Singapore) 1985. 6. E. W. Aslaksen and J. R. Klauder, "Unitary representations of the affine group," J. Math. Phys. 9 (1968) 206-211; "Continuous representation theory using the affine group", J. Math. Phys. 10 (1969) 2267-2275. 7. T. Paul, "Affine coherent states and the radial Schrodinger equation 1. Radial harmonic oscillator and the hydrogen atom", to be published. 8. K. G. Wilson and J. B. Kogut, Physics Reports 12C (1974) 77. J. Glimm and A. Jaffe, "Quantum physics: a functional integral point of view", Springer (New York) 1981. 9. G. Battle and P. Federbush, "Ondelettes and phase cell cluster expansions: a vindication", Comm. Math. Phys. 109 (1987) 417-419. 10. 1. Daubechies, "The wavelet transform, time-frequency localisation and signal analysis", to be published in IEEE Trans. Inf. Theory. 11. R. Balian, "Un principe d'incertitude fort en the'orie du signal ou en mecanique quantique", C. R. Acad. Sc. Paris 292, serie 2 (1981) 1357-1362. 64 12. F. Low, "Complex sets of wave-packets" in "A passion for physics - Essays in honor of G. Chew", World Scientific (Singapore) 1985, pp. 17-22. 13. G. Battle, "Heisenberg proof of the Balian-Low theorem", to be published in Lett. Math.Phys. 14. D. Gabor, "Theory of communication", J. Inst. Elec. Eng. (London) 93 III (1946) 429457. 15. M. J. Bastiaans, "A sampling theorem for the complex spectrogram and Gabor's expansion of a signal in Gaussian elementary signals", Optical Eng. 20 (1981) 594-598. 16. A.J.E. M. Janssen, "Gabor representation of generalized functions," J. Math. Appl. 80 (1981) 377-394. 17. Y. Meyer, "Principe d'incertitde, bases hilbertiennes et algebres d'operateurs", Seminaire Bourbaki, 1985-1986, nr.662. 18. P. G. Lemarie and Y. Meyer, "Ondelettes et bases hilbertiennes", Rev. Mat. Iberoamericana 2 (1986) 1-18. 19. G. Battle, "A block spin construction of ondelettes. I: Lemarie functions," Comm. Math. Phys. 110 (1987) 601-615. 20. P. Lemarie, "Ondelettes a localisation exponentielle", to be published in J. de Math. Pures et Appl. 21. 22. S. Mallat, "Multiresolution approximation and wavelets", to be published. S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation", to be published in IEEE Trans. on Pattern Analysis and Machine Intelligence. 23. P. Burt and E. Adelson, "The Laplacian pyramid as a compact image code", IEEE Trans. Comm. 31 (1983) 582-540, and "A multiresolution spline with application to image mosaics", ACM Trans. on Graphics, 2 (1983) 217-236. 24. I. Daubechies, "Orthonormal bases of compactly supported wavelets", Comm. Pure & Appl. Math. 49 (1988) 909-996. 25. Y. Meyer, "Ondelettes et fonctions splines", Seminaire E.D.P., Ecole Poly technique, Paris, France, December 86. 65 26. M. J. Smith and D. P. Barnwell, "Exact reconstruction techniques for tree-structured subband coders", IEEE Trans. on ASSP 34 (1986) 434-441. 27. D. Esteban and C. Galand, "Application of quadrature minor filters to split band voice coding schemes", Proc. Int. Conf. ASSP (1977) 191-195. 28. G. Polya and G. Szego, "Aufgaben und Lehrsiitse aus der Analysis" Vol. II, Springer (Berlin) 1971. 29a. I. Daubechies and J. Lagarias, "Two-scale difference equations: I. Global regularity of solutions", preprint AT&T Bell Laboratories. 29b. __ , "Two-scale difference equations: II Infinite products of matrices, local regularity and fractals.", preprint AT&T Bell Laboratories. 66 Part " Some Topics in Signal Analysis Some Aspects of Non-Stationary Signal Processing with Emphasis on Time-Frequency and Time-Scale Methods P. Flandrin Laboratoire de Traitement du Signal, UA 346 CNRS, ICPI, 25, Rue du Plat, F-69288 Lyon Cedex 02, France The analysis and the processing of non stationary signals call for specific tools which go beyond Fourier analysis. This paper is intended to review most of the Signal Processing methods which have been proposed in this direction. Emphasis is put on time-frequency representations and on their time-scale versions which implicitly make use of "wavelet" concepts. Relationships between Gabor expansion, wavelet transform and ambiguity functions are detailed by considering signal decomposition as a detection-estimation problem. This permits one to make more precise some of the links which exist between time-frequency and time-scale. 1. Introduction The analysis and the processing of nonstationary signals call for specific tools which go beyond Fourier analysis. This is clear from the definition itself of the Fourier transform, which does not preserve any time dependence and, hence, which cannot provide any information regarding either a time evolution of spectral characteristics or a possible occurrence of time localized events. Numerous approaches have been proposed in the Signal Processing literature to overcome these limitations. Most of them are concerned with time-frequency representations and with the problems related to their definition, estimation and interpretation. A number of modifications have also been proposed, which transform them into time-scale methods and Which, in fact, make implicitly use of "wavelet" concepts. Therefore, this paper is intended to review most of these methods and to make more precise the links which exist between time-frequency and time-scale. The paper is organized as follows. Section 2 is devoted to recall basic concepts related to stationarity and nonstationarity. Section 3 is concerned with bilinear time-frequency distributions. It addresses the time-frequency problem and it summarizes the different possible approaches : adaptive, evolutive, nonparametric, parametric. It also presents time-scale modifications related mostly to constant-Q analyses. Section 4 reviews some linear (time-frequency and time-scale) signal decompositions in both the discrete and the continuous case. At last, Section 5 discusses some relationships between time-frequency and time-scale by considering signal decomposition as a detection-estimation problem, which permits one to inter prete formal equivalences between short-time Fourier or wavelet transforms and ambiguity functions. It should be mentioned that no presentation of the wavelet transform itself is provided in this paper, the reader being referred e.g. to [32, 33, 54, 69) for thorough discussions. 68 2. Stationarity and non-stationarity Since non-stationarity is a negative property, its simplest definition refers to the corresponding positive property: that of stationarity Although this concept is theoretically well-defined only in a stochastic framework, intuition attaches to it some sense for both deterministic and stochastic signals : loosely speaking, a signal is considered to be stationary as soon as its relevant properties remain the same throughout all time. In a deterministic context, these relevant properties are mostly related to the instantaneous behavior of the signal regarding amplitude and frequency. Given a deterministic real-valued signal x(t) with Fourier transform (1) Xlv) = I- x(t) e -i2Kvt dt, the study of such properties needs the introduction of the analytic signal zx(t) associated to x( t), and defined as [47, I 0 11 I-X(V) ,""'" dv. (2) ',(l). 2 o The instantaneol/s amplitl/de ax(t) and freql/ency vx(t) follow then from eq.(2) by the definitions [10 1J : (3b) v (t) x = - 1 d - 2n dt Arg[z (t)J . x In accordance with what intuition suggests, a deterministic signal can then be considered as stationary if it consists in a superposition of components such that their instantaneous behavior, as described by eqs.(3), does not depend on time. In such a case, Fourier analysis is sufficient for a satisfactory and meaningful description. In a stochastic context, stationarity is concerned with the probabilistic behavior and corresponds to the fact that all the finite-dimensional probability distributions are timeinvariant and, thus, only depend on time differences [18, 68, 89, 931. From a practical point of view, stationarity refers usually to weak (or wlde-sense) stationarity; understood as time unchangingness only up to the second order. Given a stochastic signal x(t), this means that the mean value mx(t) and the covariance function rx(t, s) are of the form: (4a) m (t) x = (4b) r (t, s) x E[x(t)J = = constant, E[(x(t) - m (t))(x(s) - m (s))*J x x = y (t - s) , x where E stands for the expectation operator, the star for the complex conjugation and Yx is a non-negative definite function. Stationary stochastic signals are therefore fully 69 characterized by a unique time-independent spectral description. their power spectral density: (5) r (v) = x I- -i2AY't V (,t) e x dt . Departing now from stationarity. non-stationary signals are those for which the aforementioned requirements are no more met. As a consequence. Fourier-based spectral descriptions (such as the Fourier transform or the power spectral density which are. by definition. non-localized in time). appear now as not sufficient for a physically meaningful description. This inability of conventional spectral analysis to deal with non-stationary signals calls naturally for extensions which would be able to encompass time variations of spectral properties: this is precisely what time-frequency representations are aimed at. Beside the description of non-stationary signals by using suitable extensions of spectral concepts. other characterizations are possible. which are more aimed at detecting non-stationary features by evidencing the occurrence of more or less localized reference signals and. hence. at synthesizing waveforms by means of elementary building blocks. Such a decomposition approach leads generally to drop the concept of strict frequency (which is in contradiction with that of time localization) but it allows to examine the signal behavior relatively to different parameters such as frequency band or scale : this is now what signal decompositions and time-scale representations are aimed at. Keeping in mind that both aspects of description and detection are of interest in the analysis of non-stationary signals. we will now discuss in more detail the available tools which all share a common feature: that of being time-dependent. 3. Time-frequency 3.1. The time-frequency problem In the stationary case. spectral descriptions are uniquely defined from the Fourier transform. On the contrary. in the non-stationary case. there is a priori no unique way to associate a time-dependent spectral description to a given signal. Intuitively. a natural extension of classical spectral analysis should be one that preserves all the physical properties of a spectral density function. while incorporating an explicit time-dependence. Unfortunately. there is no solution to this general requirement. imposing as a consequence the existence of a multiplicity of possible candidates. each with its own advantages and drawbacks 117. 19.35.42.52.53.78.84.93. 104]. Before discussing in more detail the possible definitions. we can provide brief justifications for their non-uniqueness. If we consider first deterministic signals. an energy description which is sufficient for a stationary signal x(t) of finite energy Ex is the energy spectral density, defined as the squared modulus of the Fourier transform (1) and such that: (6) I-IXlV)I' dv - Ex' Its natural extension to non-stationary signals should thus be a non-negative quantity Cx(t. v). function of both time and frequency. such that: 70 It appears then that such a requirement (with non-negativity imposed for an energy density interpretation) is incompatible with numerous other requirements 161, 104). For instance, given two deterministic signals x(t) and y(t), a proper transformation x(t) ~ Cx(t, v) should preserve inner products when passing from the time-domain to the timefrequency plane. This means that a relation of the type (8) fI'- C,I t, v) C;lt, v) dt dv - f~lit) y'lt) dt 2 should hold for any two signals. Considering then x(t) and y(t) as being two orthogonal signals on the real line, the left-hand side of eq.(8) must vanish, which is clearly incompatible with the non-negativity of the associated time-frequency distributions II 06J. If x(t) is now supposed to be stochastic, the characterization of its stationary character, as depicted by its power spectral density, stemms from the existence of a doubly orthogonal decomposition 118, 68J: (9) x(t) - f- e i211vt dX(v) . Double orthogonality refers to the fact that the basis functions of the decomposition are orthogonal with respect to the usual inner product of functions on the real line : whereas the decomposition increments are uncorrelated : This characterization of stationary processes is important from a physical point of view since the basis functions are directly associated, through the complex exponentials, with the usual concept of frequency. In the case of non-stationary signals, eqs.( 10) do not hold any more, giving rise to a multiplicity of possible extensions: in a very general manner, decompositions of non-stationary stochastic signals lead necessarily either to relax the choice of complex exponentials as basis functions, and, hence, to loose the concept of frequency, or to accept some correlation between spectral increments. 3.2. Main approaches Since stationary signals can be viewed as a special case of the more general class of non-stationary signals, a first possible approach to the analysis of non-stationary signals is to preserve the classical tools aimed at stationary signals while trying to adapt them to non-stationary situations. This corresponds to adaptive metnods 152J relying mostly on 71 deviations from stationarity. It is clear that such methods can provide informations related to both description and detection of non-stationarities. Description can be achieved, for instance, by reducing observation durations in order to satisfy quasi-stationarity assumptions. On another hand, detection can result from the observation of changes when updating the descriptors. Nevertheless, this type of approach has also clear shortcomings due merely to the necessity of a prioriinformations concerning the signal evolution, and to unachievable accuracy trade-offs between time and frequency. Beside these adaptive methods, derived from the stationary case, there is then a need for specific methods considering non-stationary signals as such, without any special reference to stationary tools. This corresponds to evolutive methods [52), which are derived in a more satisfactory way, since their definition results from systematic attempts to fulfill necessary requirements aimed explicitly at non-stationary concepts. In both adaptive and evolutive approaches, numerous methods are available, which can be schematically classified in two large families, referred to as parametric and nonparametric Non-parametric methods are those for which no assumption on a possible structure of the analyzed signal is made: they can be viewed, in some sense, as the non-stationary counterparts of Fourier-based methods and they lead directly to time-frequency distributions(of energy or power). On the other hand, parametric methods make usually assumptions on the analyzed signal and they consider it as the output of a system which can be efficiently modeled by means of a few parameters which are to be identified. This results in time-dependent models from which, however, time-frequency distributions can be obtained as a byproduct. Clearly, the applicability of parametric methods is less general than that of nonparametric ones and their relevance to the analyzed signals must be asserted before their use. Nevertheless, when well-suited models are chosen, the corresponding introduction of a priori information allows generally to obtain informations sharper than those resulting from "blind" non-parametric methods. 3.3. Time-frequency distributions As mentioned previously, time-frequency distributions are supposed to provide a natural time-dependent extension to the stationary concepts of energy and power spectral density. Although other choices are possible [3D), it seems thus reasonable to impose to the wanted distributions a bilinear dependence with respect to the signal. This is in fact a very common assumption for all the definitions that we will now review. 3.3.1. Adaptive methods The most intuitive (and widely spread) way to perform a time-dependent spectral analysis is certainly to consider a non-stationary signal as the concatenation of quasistationary segments for which stationary methods are relevant. The prototype of this approach reads f - x(u) h(t - u) e-i2xvu du 72 2 (where h(t) is some short-time window), and is referred to as a spectrogram [2, 6, 36, 91, 951. Given an unknown non-stationary signal, the limitations of this method come from the a priori choice of the arbitrary window h(t) : a good frequency resolution can only be achieved by means of a large window, which results in a poor time resolution and, hence, smoothes out brief non-stationarities; conversely, a good time resolution implies a short window, which results in a poor frequency resolution. A dual distribution can be obtained from the frequency representation of the signal by passing it through a filter bank and by observing the output power of each of the bandpass filters. The result, which is referred to as a sonagram [67, 91, 921 takes then on the form: (12) \(t,v)- f - X(n)H*(n-v)e i2llnl dn 2 It should be remarked that eqs.(l1) and (12) define exactly the same quantity if the transfer function H is chosen as the Fourier transform of the window h. Consequently, the sonagram admits shortcomings similar to those of the spectrogram. Apart from the intrinsic trade-off between time and frequency behaviors, the spectrogram and the sonagram do not permit one to get access to relevant non-stationary features such as instantaneous amplitude and frequency, and their structure render them incompatible with a number of usual transformations in signal processing such as linear filtering, modulation or scale changing operations [421. Considering this latter point. several modifications have been proposed for improving the definitions (11) and (12). In fact, it is known from the Fourier theory that a scale change upon a time signal affects its frequency representation by an inverse scale change: therefore, it would be natural to expect a corresponding behavior from a short-time Fourier analysis. More precisely, if we introduce the scaled signal (I 3) x (t) : x(at) a ; a >0 , the question can be formulated: is there a window h(t) such that v a (14) S (t, v): Ha). S (at, -)? xa x (where f is some real-valued function). The answer is no, expect if we accept the window to be frequency dependent [23, 48, 491. In this case, all the windows of the form (IS) h(t, v): g(tv). t b (where g is some real-valued function and b some real exponent) are convenient. This permits one to deal with stable filters such as the one of (frequency-dependent) impulse response 2 (16) h(t, v) : (2ntv) . e -lIlv . u(t) , 73 (where u(t) is the unit step function) and which is known to be one of the simplest possible models for the time-frequency analysis performed by the peripheral auditory system [37, 911. More generally, if we choose for the window: (17) h( t, v) = w( ~) , (where w is some real-valued function and q a positive parameter), we end up with the formulation (18) S'x(t, v) = f - x(u) _f wL (t - u) qv] e-i2KYU du 2 operation which realizes exactly a constant-(J short-time spectral analysis [1071. As compared to eq.( 11), we see that, in such an analysis, the time resolution (which is governed by the equivalent width of the window w) is increased at high frequencies, whereas, at the same time, the absolute frequency resolution is decreased, the relative bandwidth being kept constant : this defines exactly a constant -Q analysis [911. The parameter Q refers classically to the inverse of the relative bandwidth of a filter, taking then on high values for sharply tuned filters. It can be shown that eq.(18) admits the equivalent frequency formulation: 2 ( 19) S' (t v) x ' = -q- 2 2 f - X(n) W'[(n - v) ~q] ei2Knt dn 2 4n v which is, in the case q ~ 1 and up to the factor v- 2, the expression of the analysis referred to as FTANfFrequency Til11eANalyzer) (74). Although all the aforementioned methods do present great improvements with respect to a crude spectral analysis, and have been widely used in many applications where "soft" non-stationarities are involved (e.g. vowels or slowly-evolving formants in speech processing [37]), their essential weakness comes from their definition which relies too much on stationary concepts and, hence, which render them unable to deal with "hard" non-stationarities (e.g. stop consonants or plosives in speech processing). We will see now how this can be overcome by introducing, in a comprehensive way, new definitions. 3.3.2. Evolutive methods Instead of starting from stationarity and trying to adapt classical tools for taking into account time evolutions, it is possible to choose as starting point a number of requirements that should be fulfilled by a time-frequency distribution, and then to build, in a constructive and comprehensive way, the corresponding definition. Extensive lists of such requirements are available in the literature [17, 28, 42, 52, 53, 58, 78. 84) and. here. we will just mention the basic ideas upon which they rely : in a general manner, most of the required properties are derived from those of spectral density functions by adding an explicit time-dependence. More specifically, they can be classified as follows: 74 1. Nature of the distribution: the wanted time-frequency distribution should be a real-valued, non-negative function of time and (physical) frequency; 2. Theoretical properties: the distribution should satisfy a number of invariances : shift invariance, scale invariance, invertibility, support preservation in time and frequency, inner product conservation, compatibility with linear filtering and modulations, ... It should also permit one to get access to typical non-stationary features such as instantaneous amplitude and frequency. 3. Practical properties: the distribution should clearly evidence the non-stationary structure of a signal by providing a readable picture in the time-frequency plane. It should also be possible to efficiently estimate it from a single observation. As it has been previously mentioned in Section 2.1, all such requirements cannot be simultaneously fulfilled and, therefore, there is no chance to obtain one unique and welldefined distribution. It follows from this negative statement that the choice of a definition is matter of trade-offs, and that it can be achieved only relatively to a class of constraints. In this respect, if we consider bilinear distributions of deterministic signals, it is remarkable to point out [42, 70] that the only shift-invariance is sufficient for restricting admissible definitions to those which belong to the so-called Cohen:<; class defined as [28, 29] (201 C,(t, v, nJ - JJ- nIt - u, v - nl W,(u, nl du dn, where (21) Wx(t, v) = f - I (t + '2t) 1* (t - '2t) e -i21tY't dt is the Wigner-Ville distribution [2 1, 26, 27, 28,41, 101, 103], and where 11 is an arbitrary time-frequency function normalized such that (221 JJ- nIt, vi dt dv - I , in order to satisfy the energy distribution relation (7). This general formulation admits as special cases numerous definitions which have been proposed independently in the past. Among these, we can mention the complex energy density or Rihaczek dJ:<;tribution defined as [94J (23) R (t, v) x = I(t) X*(v) e -i2ltvt and corresponding to eq.(20) with (24) 11R(t, v) - 2 e -i4ltvt 75 Another example is provided by the (causal) Page d!strJoution(88) : (25) P (t, v) = -a x at 2 t J x(u) e-i2KYU du associated to a function II whose two-dimensional Fourier transform f is of the form (26) f(n, t) = e i2Kn I~ I . At last, it is worthwhile to point out that the spectrogram (11) is itself a member of Cohen's class, the corresponding arbitrary function reducing then to the Wigner- Ville distribution of the analyzing window [28, 35) : (27) 5,(1, v) • U- Wb(1 - U, v - n) W,( u, n) du dn Cohen's class is thus parameterized by means of an arbitrary function, which permits one to translate requirements concerning the wanted distribution in terms of properties of the associated function. Systematic investigations have been carried out in this direction [19, 28, 35, 39, 42, 58, 84) : their conclusion is that, regarding theoretical properties and accepting the posssible occurrence of negative values, the definition which fulfills the greatest number of requirements is the Wigner- Ville distribution. However, from a point of view of practical interpretation and estimation, the choice of this definition presents some shortcomings which render necessary slight modifications. In terms of interpretation first, the Wigner- Ville distribution relies on a bilinear transformation which tends to obscure the time-frequency picture by interference phenomena (and which is also responsible for the existence of negative values) [19, 40, 42, 571. This results from the bilinear superposition principle according to Which, given two signals x(t) and y(t), (28) W ~y (t, v) - W (t, v) + W (t, v) + 2 Re (W (t, v) ) , where (29) WQ(t,v)= x Y Q J- x(t+'2t) y* (t-'2t) e -i211~ dt is the tTos9-Wigner-Ville distribution between x and y. By construction, such interference terms appear midway between signal components and, contrarily to them, they present an oscillatory structure depending on the time-frequency distance between the interfering components. This suggests a natural improvement by means of a suitable time-frequency smoothing aimed at reducing the influence of oscillatory cross-terms without affecting significantly the relevant signal terms [401. Such a smoothing operation is a special case of eq.(20) if the arbitary function is chosen as being sufficiently regular and localized in the time-frequency plane. 76 In fact, an implicit application of this idea is present when using a spectrogram, by virtue of eq.(27). Nevertheless, this choice is too restrictive since the double timefrequency smoothing operation is governed, in this case, by only one degree of freedom: the short-time window h. As it is known from practical experience, time and frequency behaviors are then conflicting ones, leading to unachievable trade-offs in really nonstationary situations. A natural way out is therefore to come back to eq.(20), but to force the independence between the time and frequency behaviors by imposing to the arbitrary function to be separable in time and frequency, i.e. to be of the type (30) IJ(t, v) = g(t) . H(v) , +-iHe-i2·~ leading to the so-called smootbed pseudo-Wigner-Ville distribution [40-42, 831: <311 SPW,lt, vl- (hltl [('It - ul I( u • i) dt This versatile and easily computable distribution allows also to overcome some of the shortcomings of the Wigner- Ville distribution concerning estimation problems. In fact, in the case of stochastic non-stationary signals, it turns out that a satisfactory definition of a time-dependent spectrum is provided by the generalization of eq.(20) : t t) e f ....r (t + - , t (32) W (t, v) x 2 x - - -i211Y'C 2 dt, referred to as the Wigner- Ville spectrum [81-83). Given now one observed realization of a stochastic signal. a general class of non-parametric estimators of (32) is again of the form (20), where the analyzed (deterministic) signal is the observed realization. For reasons similar to those mentioned above, the most versatile estimator is that defined as in eq.(31 ), and which allows to control in an independent way the statistical properties in the time and frequency directions [831. Smoothed pseudo-Wigner-Ville distributions provide then a unique and coherent framework for the (shift-invariant) time-frequency analysis of both deterministic and stochastic non-stationary signals. If we remember that Cohen's class stemms out from requmng invariances with respect to shifts in the time-frequency plane, it is clear that other classes of admissible distributions can be obtained by considering other transformations. An example of such an approach, parallel to that leading to Cohen's class (and, hence, to the Wigner- Ville distribution) via shift invariances, is the one which is associated to invariances with respect to the affine group, i.e. shifts and dilations in the time direction. This approach is especially relevant for physical situations for which dilation effects (due e.g. to Doppler effect) cannot be reduced to frequency shifts. Moreover, its consideration is necessary for operating on the Hardy space of analytic signals. The central distribution resulting from the above requirement is the "affine" distribution f. . (33) B (t,v)=v x e [2 Sh(e12J 2(n>1) z [ eve -9/2] x 2 sh(el2) eve +9/2] -i2119vt e de, x 2 sh(e/2) z· [ in which the dimensionless parameter n depends on the nature of the signal x( t) [14-161. 77 It can be justified that. in the case of narrowband signals. the integration of eq.(33) reduces to the neighborhood of a - O. which leads to an approximation of the Wigner-Ville distribution. The affine distribution can be then viewed as a generalization of the WignerVille distribution and it shares with it most of its properties. when suitably translated in terms of the underlying affine group. For instance. in a way similar to that by which the Wigner- Ville distribution can be forced to be everywhere non-negative by application of a suitable smoothing operation (c[. eq.(27)). the affine distribution can also be "regularized" by a different. but similar. operation which will be discussed lateron. At last. it is worthwhile to point out that other possible generalizations of the WignerVille distribution are offered when considering it as the expectation value of a parity operator (90). 3.3.3. Usefulness of time-frequency distributions All the aforementioned methods present some specific interest for both the description of non-stationary signals and the detection of possible non-stationary features. First. time-frequency distributions achieve a time-dependent spectral analysis from which non-stationary characteristics (such as evolution of amplitude or frequency. occurrence of localized components .. J can be efficiently evidenced and estimated : this permits one to characterize non-stationary signals by means of parameters extracted from such quantities. Second. time-frequency distributions provide us with time-frequency patterns or signatures associated to non-stationary signals. Decisions can therefore be inferred from their observation by making use of either pattern recognition (and/or image processing) techniques or time-frequency optimum processing methods (1. 6. 20. 24. 42. 43. 65. 92. 106). In this latter case. a key role is played by all the distributions (and. among them. by the Wigner-Ville distribution) which verify eq.(8) (referred to as Moyllf's formu/a(28. 87)) : this allows to preserve a classical correlation-based approach when passing to the timefrequency plane. which matches optimality with intuition (43J. Despite of all these advantages. it should be mentioned that the bilinear structure of the considered distributions render intricate the synthesis problem which consists in associating a signal to a given time-frequency energy distribution (22). 3.4. Time-dependent models Given a signal. a possible way of analysis is to model it as the output of a system characterized by a few parameters which are then to be identified. In order to have most of the signal information conveyed by the system itself. the input must be as neutral as possible: in the case of stochastic signals. this corresponds to white noise-driven systems. Considering then discrete-time stationary stochastic signals x(n). a powerful and versatile model is the so-called Auto-Hegressive Moving Average (ARMA) model defined as (18. 55. 64. 93) p (34) q ~atx(n-k)-~b.e(n-jl ; n- .... -l.O.l .... toO j-O J where e(n) is discrete-time white noise and the ak (resp. bj) are the AR (resp. MA) coefficients. 78 Such a model presents multiple interesting properties. For instance. it is naturally associated to systems whose transfer function is rational. the poles (resonances) being associated to the AR part and the zeroes (nulls) to the MA part: this allows an efficient modeling of power spectral densities in terms of peaks and valleys. In another respect. fitting an AR model to an observed signal is equivalent to extrapolate its autocorrelation function: this is of special interest for achieving better spectral resolution than Fourierbased methods when only short data are available. Moreover. fast and efficient algorithms exist for estimating the model parameters. especially in the AR case for which equations are linear [641. Of course. the improved performance comes out from a priori assumptions (concerning the signal) which have been incorporated in the analysis. and which must be checked before drawing conclusions. Among these assumptions is that of stationarity which requires. when it is not fulfilled. some modifications of the considered modeling approach. Again. these modified approaches are of an either adaptive or evolutive type. 3.4. L Adaptive models Given a non-stationary signal. a first possibility is to analyze it through short-time windows over which a quasi-stationarity assumption is reasonable. The situation is therefore similar to that observed for adaptive time-frequency distributions except that the trade-off between quasi-stationarity and frequency resolution may be less drastic : this follows from the fact that. if e.g. a short-time AR model is well-suited. the window is allowed to be shortened (as compared to a spectrogram) for a given frequency resolution. reducing then deviations from stationarity within the window. Moreover. this kind of modeling can also be modified for providing a frequency-dependent resolution [251. such as the one required e.g. for a constant-Q analysis. In such approaches. the characterization of a non-stationary process is achieved by sets of parameters for each position of the window. Furthermore. the form of the linear equations from which the parameters are estimated allows to update the solution at the arrival of each new data point. which gave rise to a number of powerful recursive algorithms. which are especially well-suited for the tracking of slowly time-varying systems [12. 52. 77. 80]. Nevertheless. since all the adaptive methods need a running observation window (explicit or implicit by the use of a forgetting factor) upon which they perform local second order estimations. they are faced with a classical trade-off between the jointly wanted accuracy and tracking capability. This calls naturally for models which could encompass. in their definition. non-stationary behaviors: evo/utive models offer such a possibility. 3 ..f.2. Evolutive models Instead of using local approximations of non-stationary signals by means of stationary ARMA models. it is possible to adapt directly the definition (34) by making the parameters time-dependent. This can take on the form [51-531 p (35) I toO q atr:(n - k) x{n - k) - I b.{n - j) e{n - j) ; n - .... -1. 0.1 .... j-O J where the time-dependence of the parameters is chosen such that the considered signal admits the following "synchronous" state-space formulation: 79 -a (n-I) 1 1. \ 1 yen - I) ... (36a) y(n)- e(n) ; -a (n-I) n (36b) x(n) = 11 0 ------ 0) yen) . Within this formulation, the time-dependent model admits at each time instant n - t a tangential stationary model, allowing hence the definition of a natural time-dependent spectrum called rational relief (5 1-53) : (37) Gx(t, v) B(t, z) B(t, Z-I) _I = A(t, z) A(t, z ) zoe i2KY with: P (38a) A(t, z) - I ak(t - 1) £ k-O k , and: q . (38b) B(t, Z) - I b.W z-J . j _0 J This quantity possesses most of the wanted spectrum. By means of this time-dependent ARM A characterized by the trajectories of its parameters. trajectories can furthermore be achieved if they can be of the type: M M (39) a..(n) - I a.. f (n) • m- 0 .m m properties for a time-dependent model, a non-stationary signal is An efficient estimation of these themselves parameterized by laws bien) = Iob im fm(n) , m- where the f m are some (a priorichosen) basis functions over the observation interval (5153!. This results from the fact that the identification of the scalar non-stationary model (35) becomes then equivalent to that of a vector stationary model for which classical algorithms are available. It should be noticed that more elaborate modeling is possible for non-stationary signals, by using for instance generaliZations of (39) which consider the time-dependent parameters as stochastic and modeled themselves by ARM A models (SO, 76). This covers an eItremely wide class of situations but at the eIpense of requiring an intricate identification. 80 3.4.2. Usefulness of time-dependent models The main interest of time-dependent models for both the description and the detection of non-stationarities is related to their ability of summarizing in a parsimonious representation (the set of a few parameters) the relevant features of a signal. From a point of view of description, slowly time-varying signals are mainly characterized by the trajectories (time evolution) of their fitted parameters, providing hence a time-dependent spectral analysis as a by- product of the model identification. In any case, concentrating the relevant features of a signal in a few parameters is of course of great interest for any post-processing such as recognition or classification [55). Moreover, the description of a signal as the output of a generating system permits one to synthesize waveforms in a natural way. From a point of view of detection, it is clear that time-dependent models offer also key tools [9, 341. Apart from inspecting directly changes in the signal parameters, other possibilities of detection exist, e.g. in terms of prediction. In fact, it is known that fitting an AR model to a given signal is equivalent to perform linear prediction (79), i.e. to find the best (in a least-squares sense) set of coefficients such that the signal can be approximated at each data point by a linear combination of a number of its past values. Therefore, if a stationary model is identified over a given window, this identification allows to make a prediction about what the signal should be just after. As long as the identified stationary model remains the same, the prediction error (difference between the predicted value and the actual one) follows fluctuations within an acceptable range. On the contrary, as soon as the signal deviates from stationarity (or switches to another stationary model), the predicted values have no relation with the actual ones, which results in large prediction errors. 4. Signal decompositions By choosing a time-frequency (or time-dependent modeling) approach for the study of non-stationary signals, emphasis is mainly made on time variations concerning spectral characteristics. Other points of view are however possible, which rather consider nonstationary signals as a superposition of a number of elementary components which are more or less localized. In this class of approaches, the choice of frequency as auxiliary variable is not mandatory and can be replaced e.g. by that of scale. 4.1. Discrete decompositions Instead of looking for a distribution of signal energy in the time-frequency plane, one can imagine to build decompositions of the signal itself over a discrete set of basis functions which would be as localized as possible: +- (40) x(t) - +- L L cnt xnt(t) . n--II:-- The functions Xnk(t) being a priori arbitrary (assuming that they form a basis), a reasonable choice is to deduce all of them from one unique waveform. The first proposal in this sense was that of Gabor [471. who dealt with shifted versions in time and frequency of a Gaussian function, which is known to be most concentrated in both time and frequency : 81 () 41 Xnt () t - e -(,tl2)l(tIAt) - nl2 .e -i~t(tI At) This corresponds to decompose any signal via Gaussians of equivalent duration .M and hence of equivalent bandwidth lIflt, shifted in both time and frequency. Unfortunately, this choice does not ensure an orthogonal expansion, which render the computation of the decomposition coefficients intricate, although feasible [10, 11. 32, 59, 601. Considering a sharp localization of the basis functions in time and frequency as a necessary requirement, an alternative to Gaussian waveforms is given by prolate spheroidal wave functions, which are known to be eigenfunctions of localization operators in time and frequency [72, 92). Given a time frequency domain flt.flv, these are defined as the solutions of the eigenequation : f (n.1/2)At (42) A 'I' nt (t) m m (n-1/2)At f (t.II2)Av 'I' nt (u) m e -i2Itv(u - t) du dv, (k-1/2)Av which leads to an expansion of the type (71) : (43) x(t) M-I [N/2 = 2 2 K/2 2 cnt(m) m - 0 n - -N/2 t - -KI2 '1':t (t) ] . Provided that the analyzed signal is of approximate dimension BT, only NK = BT / .M.flv coefficients are necessary for each m, the error associated to the truncation at the order M being measured by the neglected eigenvalues in eq.(42). The two aforementioned expansions are in fact special cases of general decompositions involving shifted versions, in both time and frequency, of a unique basic waveform. This corresponds to expressions of the type (40) with and where the major constraint is that of the completeness of the corresponding family [731. However, although not strictly necessary [311. another constraint is generally to be added: that of orthogonality. A number of different solutions have been proposed in this respect, and their inspection reveals that imposing orthogonality to (44) is generally achieved at the expense of loosing good localization properties for the generating function. A first example of this is provided by the choice (which stemms from a low timefrequency dispersion constraint) (8) : J 8V (45) 4I~(t) ~ ~ .j with 2n &v £(t, n) dn , o Z(P &v t + i &t n, P) (46 ) f (t,n ) = --:-----~ I Z(P &v t + i &t n, P) I 82 where 83 is a Jacobi theta function and p a real-valued, non-negative, dimensionless parameter. The basis functions (which vary in between a (sin x)/x function for p = 0 and an indicator function for p = 00) present a mainlobe around t = 0, but also sidepeaks at all the points t = (m + 112) ~t (with m integer), and these peaks can be viewed as responsible for the orthogonality of the expansion. A second approach [62] to the same problem justifies that, if ~v = 1I~t, the orthogonality requirement leads to the condition: (47) +00 1n~_~$(nl)t+t)e i2xn 6t v 12 -1; Od,;l)t. Imposing then to the basic waveform to be absolutely summable and of finite energy, a convenient solution is provided by the following piecewise continuous function: (48) ~t + n I)t) = l~(t) ; n < -1 ; n - -1 2 [1 - a (t))[-a(tl] n ; n > -1 with (49) a(t) = 1 - (I - t/l)t)6 / 5 ; 0,; t,; I)t. Again, orthogonality is achieved by means of a non-localized generating function. Given (48) and (49), the expansion coefficients correspond to +00 (SO) cnk = f x(t) ~t - n I)t) e -i211(k/6t)t dt, which is of the form of a local Fourier transform. All the previously considered decompositions make use of shifts in both time and frequency. Nevertheless, other methods have been developed, which retain the idea of dealing with only one generating function, but which consider e.g. form invariance with respect to scale changes. This is for instance the starting point of an approach (suggested by a physical modeling of echo formation mechanism [4S]) developed by Altes [4], and according to which a signal is supposed to be expanded under the form where y(k) stands for the derivative (resp. primitive) of a basic waveform y when k is positive (resp. negative). The waveform y is chosen in order to ensure a form invariance with respect to scale changes in a way such that derivation (resp. integration) of order k is equivalent to a compression (resp. dilation) with factor gk , k > 1. It can then be shown that such a constraint leads to admissible waveforms defined in their frequency representation by 83 (52) Y(v)=e - LolUV/VO)/2 Log gi .e -i2l1b Log(v/v O) Log g . U(v) , where U is the unit step function, vo' g and b being arbitrary constants. Such signals do not constitute an orthogonal basis but, nevertheless, a satisfactory approximation of the associated coefficients can be found by computing the wideband cross-ambiguity functions [3, 66, 98) between the analyzed signal x(t) and the dilated (or compressed) versions of y(t) [4, 108): (53) cnk :: gkl2 J- ( x(t) y gk (t - t n ) ) dt. Signals defined by eq.(52) correspond to linear period modulations and they provide a very efficient way of modeling natural signals such as those emitted by animal sonar systems (bats echolocation calls or dolphin clicks) [4, 5). Moreover, it is worthwhile to point out that, for a proper choice of the arbitrary parameters, the corresponding waveform presents a striking resemblance with compactly supported wavelets leading to orthonormal bases [33l. At the end of this section devoted to discrete decompositions, it is worthwhile to point out their relationships with other classical methods such as Prony's, in which signals are modeled as a superposition of exponentially damped sinusoids which are to be identified [64). From a related, but somewhat different, point of view, "classification-substraction" methods have also been proposed, which consider a sequential identification of longer and longer wave trains [71. 4.2. Continuous decompositions If we start back from the expansion (40), a natural extension is possible when replacing the discrete summation by an integral of the type (54) I(t) - JJ~g(t, v) 'I'(t; t, v) dt dv . The representation of x(t) is then provided by the function g(l, v) where, again, the two decomposition variables have II priori no reason to be interpreted as time and frequency. The first application of this idea goes back to Helstrom [56) who preserved Gabor's idea of using Gaussian elementary waveforms: . ) _ ( ( (55) 'I't,l,V - 2na 2)-1/4 e -It - ~)2 1402 i211vt .e.e -ill~ The major interest of this approach is to give access to an exact expression for g according to: (56) g(t, v) - 84 J~X(t) 'I"(t; t. v) dt. In a way similar to that which held in the discrete case, a generalization is then possible when replacing the Gaussian kernel by another (regular enough) function [86]. In fact, considering any square integrable function h(t), admissible decomposition functions are given by (57) \fI(t; t, v) = h(t - t) e i2ltvl .e -iltV't which leads to (- (58) g(t, v) = e-iltV't J x(t) h*(t - t) e -i2ltvl dt . Up to a pure phase term, this is exactly the short-time Fourier transform (with window h) of the signal x. It appears then that a spectrogram can be thought of as the squared modulus of the generalized Helstrom's decomposition elements: 2 (59) S(t,v)=lg(t,v)l. x In this interpretation, the squared coefficients of the original Gabor's expansion can be viewed as an approximation of a spectrogram with a window whose time-frequency spreading is minimum. In a second interpretation, eq.(58) is the narrowband cross-ambiguity function (in the sense of Woodward [105]) between the analyzed signal x( t) and the analyzing waveform h(t), the basic underlying operation being that of a correlation with a time delayed and frequency shifted elementary signal. A natural modification is therefore to replace frequency shifts by dilations or compressions. This corresponds to make use of the wlO'ebandcross-ambiguity function which reads [3, 13, 63, 66, 98] : (60) A/t. q) • j;j J- I(t) h'[q(t - til dt , where the variable 11 plays generally the physical role of a Doppler factor. It follows clearly from eq.( 60) that relabelling this variable by 1la, where a stands now for a scale factor, leads directly to the general wavelet transform [32, 54, 69] (611 T/t. a)· A,(t. ~). ; . f,(,) h'('; d, t) provided that the usual admissibility conditions H(O) = 0 hold. 85 An interesting possibility is to choose [54, 102) 2 -a Log v (63) H(v) = e . U(v) , which is of course to be compared with eq.(52). Another fundamental connection is to be made between such an analysis and what was said about constant-Q spectral analysis (eL Section 2.3.1.). In fact, given an analyzing wavelet h(t), which is supposed to be localized in some sense along the time axis, the choice of a scale parameter a determines implicitly an equivalent bandpass filter whose dominant frequency is roughly related to l/a. Changing now the scale parameter affects simultaneously, and in a similar way, the bandwidth and the dominant frequency. This means that the ratio bandwidth/dominant frequency is kept constant, which characterizes a constant-Q analysis. 4.3. Usefulness of signal decompositions The main advantage of the aforementioned signal decompositions (and of related ones, see e.g. [75]) is their linear character. This advantage is in fact twofold: on one hand, linearity of the involved transformations does not create spurious interaction contributions like those which appeared in (Wigner-like) energetic distributions; on the other hand, it provides a trivial way of performing signal synthesis by means of suitable summations of the form (40) or (54). Apart from these analysis and synthesis aspects, signal decompositions are also of importance for handling decision problems since the expansion coefficients can be used as a signature for detection, estimation, recognition or classification. Among different successful applications in such directions, one can mention the detection (in noise) of transient signals of imperfectly known shape: in this case, detection is achieved by thresholding a statistics which is built on a very few Gabor coefficients (46). Depending on the class of signals to detect, it is clear that companion approaches could be provided, with similar performance, by using other decompositions such as, e.g., wavelet coefficients. 5. Some relationships between time-frequency and time-scale Although emphasis has been put on spectral representations (based on a frequency variable), it has been already mentioned that signal decompositions can equally involve other variables such as scale. Time-frequency and time-scale representations are not to be directly compared but, nevertheless, relationships exist between them, which have already been suggested and which will be investigated further in this Section. 5.1. Signal decomposition as a detection-estimation problem Given an expected signal y(t) and an observation dt) known on a time interval (T), a classical detection problem, which stemms from radar/sonar theory, is to choose at best between the two following hypotheses: (64) 86 dt) = n(t) dt) = n(t) + L(8 o)y(t) 1t E (Tl , where n(t) is additive noise and L an operator (depending on unknown parameters described by the vector 90 ) which modifies the expected signal [100 J. The problem is then to decide whether the expected signal is present or not, and by which parameters it is affected. In the radar/sonar case, y(t) is the emitted signal and L characterizes the modifications resulting in the (noiseless) returning echo. In a first approximation, L depends upon two physical parameters: delay (related to the range of the target) and lJoppler effect (related to its velocity). In the radar situation, for which the emitted signal is generally narrowband, Doppler effect can be approximated by a frequency shift of the signal spectrum and, hence, the associated operator LR corresponds to transformations of the Weyl-Heisenberg group: where t is the delay and $ the Doppler shift. In the sonar situation, the emitted signal is generally sufficiently wldeband for forbidding the approximation of Doppler effect by a simple frequency shift. The returning echo must be modelled as a dilated (stretched or compressed) version of the emitted signal and, hence, the associated operator LS corresponds to transformations of the affine group (or 'ax+b' group) : where 11 is now the (scaling) Doppler factor. In both cases, if the additive noise is supposed to be white, the key quantity upon which the decision relies expresses as (67) Ala)· f rlt) ILia )ylt))' dt , (T) where 9 is an hypothesized vector parameter [1001. Detection is achieved by comparing eq.( 67) (or its squared modulus if L takes also in account a random phase factor) with a threshold. The value of 9 for which eq.(67) (or its squared modulus) is maximum, when it exceeds the threshold, is then used as an estimate of the true parameter. The decision procedure is therefore of the form of an inner product between the observation and hypothesized modified versions of the emitted signal used as a reference. This corresponds to a cross-correlation operation which can also be understood as a matched filter in the sense that the response is supposed to be maximum when the observation is best fit to the (suitably modified) shape of the expected signal. It follows from eq.(67) that these responses in the radar and sonar situations are respectively: 168.) 1,1",) - f rlt) Y'lt - ,) .-;>w, dt (T) and 87 (68b) ,,(q) - f j;j r(t) y'(~(t - ,)) dt. (T) The first expression is referred to as a narrowband(cross-) ambiguity function [lOS) and the second one as a wideband(cross-)ambiguityfunction (66). This terminology comes from the fact that. given an emitted signal y(t) and a returning echo r(t). there exists an infinity of pairs ('t.~) or ('t. '1) which yield an identical result for respectively AR or AS : this results in an ambiguity concerning range and velocity. The key point is that eqs.(68) can be also interpreted in terms of signal decompositions. In fact. if the "observation" r(t) is understood as a signal x(t) to analyze and the "emitted" signal y(t) as a window (or analyzing signal) h(t). it is clear that the narrowband ambiguity function (68a) is identical (up to a pure phase term) to the shorttime Fourier transform (58). whereas the wideband ambiguity function (68b) reduces to the wavelet transform (61), provided that the Doppler factor is relabelled as IIa. In both cases. a strict equivalence requires to consider the integration interval as being infinite: in practice. this holds as long as all the shifted and/or dilated versions of the (localized) analyzing signal remain strictly included within (Tl. The meaning of the above interpretation is that. when a signal is supposed to result from the superposition of elementary waveforms. its decomposition can be viewed as a detection-estimation problem relatively to each of its hypothesized components. According to the discussion concerning the two typical cases of time-frequency and time-scale. the properties of the corresponding decompositions can then be deduced from those of ambiguity functions. Nevertheless. if wideband cross-ambiguity functions and wavelet transforms are exactly of the same mathematical form. it must be emphasized that they drastically differ regarding the orders of magnitude of the scale parameters: in the case of ambiguity functions. Doppler factors always remain near from unity. whereas wavelet transforms are generally intended to analyze signals over a large number of octaves. 5.2. ApproIimations In order to precise the links which exist between time-frequency and time-scale analyses. a possible approach is to make use of the known connections between narrowband and wideband ambiguity functions. In fact. although some general (and formal) exact relationships are known to hold (38). physically meaningful features express rather in terms of approximations. First. it has been mentioned that a frequency shift can be viewed as an approximation of the true Doppler effect for narrowband signals. In accordance with that physical significance. it follows that the narrowband ambiguity function appears as a natural approximation of the wideband ambiguity function. provided of course that the range of variation of the scale parameter remains compatible with a Doppler interpretation. This type of behavior is to be compared with that which occurred between the "affine" distribution (33) and the Wigner-Ville distribution (21) (cr. the end of Section 3.3.2.). In some sense. this is no surprise since it is known that the (suitably symmetrized) narrowband ambiguity function and the Wigner- Ville distribution form a two-dimensional Fourier transform pair. namely [28. 35) : (69) 88 ff W,(t. v) .''''(''' wl dt dv - fx( u , ;-)x'( u - ~) .'''''' du An open question is therefore to know if there would exist a similar relation between the "affine" distribution and the wideband ambiguity function, understood as generalizations of respectively the Wigner-Ville distribution and the narrowband ambiguity function. 5.3. Regularizations On one hand, we know from eq.(59) that a spectrogram can be viewed as the squared modulus of Gabor-Helstrom's coefficients. On the other hand, the same quantity is known to result from a suitable smoothing of a Wigner- Ville distribution (cf. eq.(27)), which results in a regularization of this latter in the sense that the obtained quantity is everywhere non-negative. The minimum amount of smoothing which is required for such a regularization corresponds to the choice of Gaussian elementary waveforms, or coherent states. In a similar way, affine distributions can be regularized by making use of affine coherent states. More precisely, we can define the quantity [14-16): (70) B '(t,V)=f-f-B (u,n)B x t (u,n)dudn, x . ; .v - 0 where the second term under the integral refers to the affine distribution associated to the elementary signal: (71) • t, v (n) - Fn $(~) e-i2Knt , v with $ a function whose closed form expression can be found in [151. In such a case, calculation of eq.(70) yields (72) B '(t, v) x f-z 2 x (n), t, v *(n) dn o or, in an equivalent way (73) B '(t, v) = x ./v f-z x(u) q,*[v(u - t)) du 2 where q, stands for the Fourier transform of $. Inspection of eq.(73) reveals that such a regularized affine distribution is nothing else than the squared modulus of wavelet coefficients as defined in eq.(61), when the scale parameter is formally identified to the inverse of frequency. 5A. Matching signals and analyses It has been said in Section 5.1. that optimum detection-estimation can be achieved by means of a squared inner product between the observation and modified versions of the signal to detect. However, when frequency modulated signals are used, intuition suggests that decision should also be inferred from the comparison of the associated time-frequency structures, e.g. by means of a correlation-type operation in the time-frequency plane. This 89 can be shown to hold (43) as soon as the considered time-frequency distributions satisfy an equality which is referred to as MoyaJ's formula (87). This is the case for the WignerVille distribution for which Moyal's formula reads (74) f-I1t) y'!t) dt 2 - If """ Wxlt, v) W/t, v) dt dv . A companion formula is provided for the affine distribution, which takes on the form (14) 2 (75) = f"""f"""Bx(t, m) B/t, m) m 2(2n+l) dt ... dm . 0 In both cases, it is clear that simplifications would occur if the considered signals were such that their associated distributions would be perfectly concentrated on some time-frequency curve. In the Wigner-Ville case, this happens with linear chirps (linearly frequency modulated signals) such that and for which eq.(74) reduces to (77) f- Wxlt, v, •mtl dt , i.e. to a simple path integration along straight lines in the plane (65). In the case of affine distributions, a similar simplification occurs when replacing linear frequency modulations by linear period modulations, i.e. when dealing with signals whose instantaneous frequency law is hyperbolic. Optimum detection-estimation results therefore from integration along hyperbola in the plane. This time-frequency formulation of optimum decision shows that specific signals can be naturally matched with analysis systems. A remarkable feature is that, in the affine case, i.e. in the case of constant-Q analysis, the "natural" signal involves a linear period modulation: this is in close accordance with the stucture of echolocation calls emitted by a number of bats whose auditory system is known to be of a constant-Q type over a wide range of frequencies [4, 85, 96). 6. Conclusion Although the concept of wavelet is extremely unifying in different areas and has led to original and substantial developments (see e.g. [32, 33)), it has been shown that related methods have already been proposed in the Signal ProceSSing literature for improving Fourier analysis. This is no surprise since, from a physical point of view, the two main instances where implicit wavelet ideas were present are those for which the introduction of the affine group is natural either for the signals to process (wideband Doppler effect) or for the processing system (constant-Q analysis). Moreover, the Signal Processing approach 90 (and especially the detection-estimation point of view) has provided natural connections between time-frequency and time-scale. 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Basseville IRISNCNRS, Campus de Beaulieu, F-35042 Rennes Cedex, France The problem of detecting abrupt changes in the dynamical properties of a signal appears to be a natural complement of most of the adaptive techniques for nonstationary signal processing which track only slow variations of model parameters. The approach then consists in using statistical parametric models where abrupt changes -or discontinuities- in (some of) the parameters may occur and between which the signal characteristics are to be considered as constant or slowly varying. The main problem is to detect the changes and to estimate their location (time or space) and magnitude. This type of framework and tools aer useful for various purposes, among which we distinguish: • gains updating in adaptive algorithms for tracking quick variations of the parameters [10] [14] ; • segmentation of signals (and images) for recognition or monitoring [1] [2] [12] [6]. It turns out that a significant amount of methodological tools and experimental results is available now, as can be seen from the sophistication of the algorithms which are used and the numerous fields of application which are concerned. Hereafter we give a list of relevant references, which may be completed with the aid of the survey papers [7] and [15] and the book [5]. The work presented at the Conference is entirely described in [3], [4], [1] and in several chapters of [5]. An example of segmentation of continuous speech is shown below. Among the open problems which remain in this framework of change detection are the following: • on-line segmentation of signals when the segments to be found are short and quite nonstationary (e.g. transients in speech) ; • automatic hierarchical segmentation in scale (e.g. of geophysical signals). Techniques based upon classical parametric (ARMA) models are probably not convenient for these tasks. An attempt to derive another type of detection tool may be found in [11] and starts from Gabor transform. Similar efforts to add a stochastic framework to the wavelet transform theory should give rise to new useful tools for these open problems. 99 ~ o o The vertical lines denote the detected boundaries. Figure no 1 : Segmentation of a continuous speech signal REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] Andre-Obrecht, R. (1988). A new statistical approach for the automatic segmentation of continuous speech signals. IEEE Trans. on A.S.S.P., 36,29-40. Appel, D., AV. Brandt (1983). Adaptive sequential segmentation of piecewise stationary time series. Information sciences, 29. Basseville, M., A Benveniste (1983a). Design and comparative study of some sequential jump detection algorithms for digital signals. IEEE Trans. on A.S.S.P., 31, 521-534. Basseville, M., A. Benveniste (1983b). Sequential detection of abrupt changes in spectral characteristics of digital signals. IEEE Trans. on Inf. Th., 29, 709-724. Basseville, M., A. Benveniste (Ed. (1986). Detection of abrupt changes in signals and dynamical systems, LNCIS n077, Springer-Verlag, Berlin. Basseville, M., A Benveniste, G. Moustakides, A Rougee (1987). Detection and diagnosis of changes in the eigenstructure of nonstationary multivariables systems. Automatica, 23, 479489. Basseville, M. (1988). Detecting changes in signals and systems. A survey. Automatica, 24. Benveniste, A, M. Basseville, G. Moustakides (1987). The asymptotic local approach to change detection and model validation. IEEE Trans. on Aut. Cont., 32, 583-592. Deshayes, 1, D. Picard (1986). Off-line statistical analysis of change-point models using non parametric and likelihood methods, in [5], ch.5, 103-168. [10] Favier, G., A. Smolders (1984). Adaptive smoother-predictors for tracking maneuvering targets. Proc. 23rd Conf. on Dec. and Contr." Las Vegas, NV, 831-836. [11] Friedlander, B., B. Porat (1987). Detection of transient signals by the Gabor representation. Proc. ICASSP, 24, 380-391. [12a] Gustavson, D.E., AS. Willsky, J.Y. Wang, M.C. Lancaster, 1H. Triebwasser (1978a). ECGNCG rhythm diagnosis using statistical signal analysis. Part I : Identification of persistent rhythms. IEEE Trans. on Biomed. Eng., 25, 344-353. [12b] Gustavson, D.E., A.S. Will sky, LY. Wang, M.C. Lancaster, J.H. Triebwasser (1978b). ECGNCG rhythm diagnosis using statistical signal analysis. Part II : Identification of transient rhythms. IEEE Trans. on Biomcd. Eng., 25, 353-361. [13] Nikiforov, LV. (1986). Sequential detection of changes in stochastic systems. In [5], ch.7, 216258. [14] Perriot-Mathonna, D. (1984). Recursive stochastic estimation of parameters subject to random jumps. IEEE Trans. on Aut. Cont., 29, 962-969. [15] Willsky, AS. (1976). A survey of design methods for failure detection in dynamic systems. Automatica, 12, 601-611. 101 The Computer, Music, and Sound Models J.-C. Risset Laboratoire de Mecanique et d'Acoustique, C.N.R.S., et Faculte des Sciences de Marseille-Luminy, France It may seem surprising to find a presentation on the computer and music in a scientific meeting dedicated to new developments in the field of time-frequency methods. But music is both a demanding and a rewarding field: it has benefited from science and technology, but it has also stimulated several scientific and technical developments. In 1957 , Max Mathews pioneered at Bell Laboratories digital recording and synthesis of sound: his primary interest was the development of novel musical instruments. The exploration of the virtually unlimited resources of synthesis and processing has involved research that has completely transformed our understanding of musical sound and how it is perceived. It is not surprising that the wavelet transform was first applied to sound signals in a computer music team, namely our "Equipe d'informatique musicale"; to implement this application efficiently, Richard Kronland-Martinet took advantage of the SYTER audioprocessor, developed specially for music. In this presentation, I shall only give brief indications about the uses of the computer in music, specially computer assisted composition, which do not involve sound. I shall mostly deal with the problem of musical sound synthesis and processing, which relates directly with the theme of the meeting. The written presentation will lack the sound examples, an essential part of the talk. Speaking about novel sound effects can be as irrelevant as discussing the taste of a fruit the listener has never tried: "the proof of the cake is in the eating, not in the cooking". Hence the references of some recordings are provided. The reader looking for further information is referred to a bibliography. Computer assisted composition "The Engine's operating mechanism might act upon other things besides number, were objects found whose mutual relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine. Supposing, for instance, that the fundamental relations of pitched sound in the signs of harmony and of musical 102 composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent." This was written around 1840 by Lady Ada Lovelace, Lord Byron's daughter (the programming language ADA has been named in her honor), who helped Charles Babbage with his "Analytical Engine", a large mechanical machine design to compute mathematical tables, and in several respects a forerunner of the computer. One can find some early examples of automated music composing, from the "Dice Game composition" attributed to Mozart to the "stochastic music" experiments of John Pierce and Elizabeth Shannon, reported in 1950. Lady Lovelace had a clear vision of the logical possibilities of machines: her fantasy has come of age with the digital computer. Starting around 1956, several people worked on computer-assisted composition, specially Lejaren Hiller and his collaborators, who described them in a book and also published the "ILLIAC Suite", a string quartet "composed" by the ILLIAC computer of the University of Illinois. Basically, the computer was called upon to perform a random choice of musical symbols and then to screen them so that the sequence selected would be "grammatical" according to a prescribed set of rules. Statistical rules were easy to implement, but not well adapted to imitate classical music. Hiller programmed the rules of contrepoint, according to a reputed XVIII century treatise. The results were informative but musically disappointing: they seem to respect the letter but not the spirit of the rules. As the composer Milton Babbitt put it, "the rules of counterpoint tell what not to do, they do not tell what to do." Computer composition has been used effectively by lannis Xenakis, James Tenney, and Denis Lorrain in certain of their works. Some composers such as Pierre Barbaud and Gottfried-Michael Koenig put so much weight on rigor in their compositional activity that they make explicit their compositional choices in a set of rules which a computer can then follow automatically. However, these are exceptional cases: generally the composer cannot completely specify in advance the criteria of his compositional decisions. As Debussy stated, "works of art make rules, rules do not make works of art". Computer composition seems primarily interesting as an experiment in artificial creativity. Music is a highly complex and connotated field: artificial intelligence is far from being advanced enough to provide substitutes for composers. However, the computer can aid the composer in a more ancillary role. Computer-aided composition is becoming more wide-spread now that one can find computers that are inexpensive and user-friendly, with their sonic and graphic outputs. Moreover the development in computer languages is 103 changing the picture. Programming languages like ALGOL or FORTRAN were able to implement procedures by specifying them completely, step by step. Now object-oriented languages facilitate the manipulation of "objects" that are characterized by specific data as well as by specific procedures for transforming them. An example is the musical language "FORMES", developed at IRCAM, Paris. Logical programming, introduced by Alain Colmerauer, who developed PROLOG at Marseille-Luminy, makes it convenient to experiment upon a set of rules; it also lends itself to programming expert systems that can react to orders expressed in a "declarative" form - much easier to formulate than orders that must prescribe every step the computer has to take. Although these developments are still in their infancy, they are very promising for music - and music composition is a good field to test them in .. Other non-sonic applications of the computer in music The computer has been used in musicology: it can help in the analysis of musical scores. This implies the preliminary coding of music in a computer-compatible form. One can then use the computer to help transcribe scores from one notation to another one, as Helene Charnasse has shown for XVIIth century music originally written in a format only lute players could read. A lot of work on computer music analysis has been done in the United States and in Sweden. Here again, it is easy to perform statistical analyses, but harder to have the computer reveal musically significant features. This field relates to computer composition, since it is useful to check the relevance of an analysis by using its conclusions as premises for an imitative composition. It also relates to artificial intelligence and pattern recognition. The computer is often used for programmed instruction of music, specially in conjunction with computer synthesis of sound - which is also useful to check that music to be analyzed has been properly coded. The computer has already begun to change the field of music printing. Simple musical scores can now be printed by inexpensive computer systems. Superb programs like SCORE developed by Leland Smith are now available on personal computers, although good printing equipment for large size sheets is still expensive. The person coding the music into the computer must be somewhat expert - but so is the present copyist. With such programs, errors are likely to become fewer, specially in the parts for transposing instruments. computer synthesis of sound The use of the computer to generate musical sound has developed considerably since its introduction thirty years ago. Computer music is the 104 most lively branch of electronic music; it is also merging with instrumental music. It was long used primarily by avant-garde musicians, looking for novel sound materials lending themselves to novel sonic architectures. It has recently invaded popular music. As we shall see, developing the potential of sound synthesis is a scientific and technical challenge as well as a musical one. Starting in 1957, Max Mathews implemented direct digital synthesis, also called software synthesis, whereby the computer directly computes samples of the waveform. The numbers are turned into a variable voltage by a digital to analog converter. This voltage can be recorded in "analog" form on a tape recorder; it yields sound when it is sent to the input of a loudspeaker. The production of sound from the samples is much like in a compact disc: however the samples here come from computations, rather than from the recording of sounds. The computer, so to say, controls directly the loudspeaker: its possibilities are in principle unrestricted- as though it directly engraved a record groove. It should be clear that direct digital synthesis can produce a rich polyphony, just as an orchestra can be recorded on a single groove. Unlike using a real-time synthesizer, there is in principle no limit to the complexity of the sounds that can be produced this way: more complex sounds will just take more computer time. To use this process efficiently, the user must provide a program to instruct the computer how to manufacture the sounds he desires. To avoid rewriting a new program for each new sound, Mathews has designed convenient modular compilers, the MUSIC programs, (MUSIC III (1959), MUSIC IV, MUSIC V (1967)), which can produce a wide variety of sounds, simple or complex (Cf. Mathews, 1969). The user of such programs must basically specify the physical structure of the desired sounds. To do so, he must first specify "instruments", that is, combination of basic modules adders, multipliers, random number generators, oscillators, filters... (these are in fact segments of programs). Now an ordinary instrument can play different notes: similarly, the MUSIC V "instruments" are triggered by "note" instructions that start and stop them at prescribed times. The 'note" instructions also prescribe the undetermined parameters of the instrument, such as loudness and frequency for an instrument yielding pitched sounds. The program also allows for the specification of functions - through mathematical formulas (for instance as the sum of sine waves), or as piecewise functions for which the end points are selected, or by drawing on a computer tablet, or through sampling an analog function ... Depending on the instrument design, such fonctions can specify a waveshape, or a temporal envelope, or something else, for instance the way one parameter depends upon another one. 105 All these specifications - forming the computer "score" - must be made according to specific conventions, which in effect define an operational language for sound description and synthesis. A given MUSIC V score is both a recipe requested by the computer to "cook up" the sounds, and a thorough description of these sounds, which may be usefully communicated to other users. This modular approach, as well as other principles of these programs, have been replicated in the design of analog synthesizers (Moog, 1964) and of recent digital processors (Di Giugno & Kott, 1981). Programs such as MUSIC V have proven useful and powerful; they can be made portable, and they have been adapted to many kinds of computers. In our Marseille team, Daniel Arfib is completing a version for the popular PC type, with a view to provide the individual user with the flexibility of software synthesis. With programs such as MUSIC V, the programming problem may be considered as basically solved: extremely varied sounds can be manufactured with unprecedented reproductibility and precision, granted a proper description of their physical structure. To manufacture a sound, the user must specify the MUSIC V "instrument" by describing a "patch" of basic modules, and then activate this patch at the proper time with the desired parameters. Now a given MUSIC V instrument specifies a combination of processes which in effect embodies a specific model of sound synthesis: to generate sounds, the user has to Select a generative model of sound and evaluate the parameters for this model. Additive, substractive and non-linear methods for synthesis Although many processes can be combined for implementing sound synthesis models, it may be enlightening to mention three basic categories of methods. The idea of additive synthesis seems to be associated with a Fourier theorem: a periodic sound can be made up as the sum of sine wave components. Actually, the XVllth organ builders already practiced empirical additive synthesis to alter the timbre of pipes in the so-called mutation stops. Daniel Arfib likens additive synthesis to the process of building a solid object from elementary bricks: sine waves are the most popular sound bricks, although the Morlet wavelet will probably also become very popular, as suggested by the synthesis experiments presented in this meeting by Richard Kronland. Now musical sounds are quasi-periodic rather than strictly periodic: it has been demonstrated that one could obtain a huge variety of musical sounds by adding sine wave components controlled in amplitude and frequency by appropriate time-varying functions. This process is powerful and general, but computationally expensive; one has to specify a lot of information to control evolving sounds, although the description can often be simplified. The time envelope can be schematized 106 (Risset & Mathews, 1969; Grey & Moorer, 1977; Charbonneau, 1981). Fourier additive synthesis can also be simplified by grouping several harmonics together: one does not actually require very detailed information on the high-order harmonics. Additive synthesis performed from the analysis data of a wavelet transform (Kronland-Martinet & ai., 1987) is more economical than that using phase vocoder data (Dolson, 1986), since the analysis operates at a constant 6. flf resolution. Additive synthesis permits to control quasi-periodic components in a very detailed way, and to generate unusual sounds. For instance, it can produce faithful imitations of bells. It permits to generate paradoxical sounds going up for ever, going up and down at the same time, or speeding up and slowing down at the same time (Risset, 1978) . Subtractive synthesis consists of eliminating undesired components, like the sculptor who takes away the stone surrounding his statue. The process starts with a rich signal and rejects the unwanted components through filtering. The process is specially adequate for certain kinds of sounds, for instance bowed string sounds (Cf. below) or speech sounds. In speech, articulation shapes the vocal tract so that it filters in a specific way the spectrally rich source produced by the vocal chords. One must mention that filters include all-pass filters (which change only the phase spectrum), useful to achieve colorless reverberation. Non-linear methods, also called global methods, proceed by modification of a simple wave, for instance a sine wave: this distorsion generates a complex spectrum. Here the analogy is that of a chunk of clay distorted into the proper shape by the sculptor. Indeed, the technique of non-linear distorsion, using a non-linear function f to generate a complex wave f(A sin2nft), is also called waveshaping (Arfib, 1979; Le Brun, 1979). This technique can yield a prescribed spectrum at a given intensity, but it makes it hard to predict the spectral variations, which are very important to the ear, as will be indicated below. The outstanding example of a non-linear method is John Chowning's frequency modulation (FM) technique (1973), using FM at audio rates. This technique has proven extremely powerful and economical: it concentrates in a few parameters the control of salient aspects of the sound. Chowning's FM is used in the Yamaha DX7 and the Synclavier digital synthesizers. Perceptual, physical and signal processing models The above methods can be further elaborated and combined. For instance additive synthesis can use other elements than slowly modulated sine waves, as in the FOF method (Rodet, 1984) or in multiple carrier FM. As was mentioned above, programs such as MUSIC V require a model of the sound to be synthesized. The desired sound may well be novel and unlike existing 107 sounds. However the discussion may be clarified by considering the case where one attempts to simulate an existing sound by synthesis. When the task at hand is to replicate sounds, the appropriate goodnes-of-fit evaluation technique is to make auditory comparisons between the original sound and its synthetic replica: information reduction must be performed in ways which the ear will ignore or tolerate. This criterion of validity defines a "perceptual model". In contradistinction, a "physical model" would attempt to simulate the physical mechanisms that give rise to the sound (Cf. Hiller & Ruiz, 1971, Cadoz & al.,1984). Physical models can be very appealing: however they often require crude simplifications and perilous numerical approximations; they may not be the way to getting as close as possible to the original signal. Whether one resorts to a perceptual or a physical model, practical synthesis implementation requires that such models be translated into adequate "signal processing models". Such models in fact describe the digital implementation, specifying what signal processing operations (addition, multiplication, tablelook-up, or more elaborate operations such as oscilating or filtering) must be performed to synthesize the sound. The models can then be implemented on any computer equipped with a program such as MUSIC V (possibly incremented with additional modules corresponding to operations not available in the original program). They may also be put at work on special digital processors, provided these processors support the required signal processing operations. Specialized digital audio processors tend to be faster than general purpose computers, often making possible to operate in real time. They have some drawbacks, however: they are less general and more difficult to program or to reconfigure; real-time operation imposes a limit on the maximal processing complexity; and the processors tend to become obsolete and be replaced by more powerful ones with different characteristics, which makes portability a major problem. Software synthesis is slower, but the recipes found for synthesis on a program like MUSIC V twenty years ago are still valid and operational today. It is an important challenge for computer music to develop sound models that can fulfill musical demands, capable of yielding sounds with enough life, interest and identity, and that can at the same time be implemented in an efficient and economical fashion with the available computers and/or special purpose digital circuits. The economical aspect here is contingent upon the evolution of technology. For instance certain combinations of elementary operations can become useful primitives for Signal processing models if they are integrated onto popular and inexpensive chips. 108 The failure of traditional sound models Digital synthesis of sound thus requires that one select a generative model of sound. When computer sound synthesis was implemented some thirty years ago, it was first tried to resort to a traditional model for musical sound - a model still described as "the" model in reputed treatises such as the Feynman Lectures on Physics. According to this model, musical sounds are periodic - or rather quasi-periodic. The amplitude of the wave varies throughout the tone; the curve describing this variation is called the envelope: it comprises an attack portion, a steady state and a decay. The musical pitch is determined by the frequency of the wave; the loudness depends upon the amplitude, or rather the power of the wave, which can be calculated from the Fourier spectrum, according to the Parseval formula. Different periodic waveshapes correspond to different timbres. Ohm's acoustical law states that the ear is insensitive to the phase relations between the harmonics of a periodic sound: thus the timbre should be determined by the frequency spectrum alone. Indeed, Helmholtz stated that the spectrum was the essential feature characterizing the timbre, enabling us to identify the musical instrument playing a given pitch. Herman argued with Helmholtz that timbral constancy throughout the frequency range of an instrument was obtained not with a constant spectrum moving up with the pitch, but rather with a fixed spectral envelope - a so-called "formant" structure (for the speech signal, the formants are the spectral peaks corresponding to the fixed vocal tract resonances associated to a given vowel). Certainly sounds with very different spectra have very different timbres. However the spectrum is not the only aspect which determines timbre. Temporal factors are important. One does not recognize the timbre of a piano when it is played backwards on a tape recorder. Synthesis can easily demonstrate that a sharp attack followed by a regular, exponential-like decay, gives a plucked quality to any waveform. Such temporal aspects are grossly taken care of through the specification of the envelope in the traditional model mentioned above. Now the first computer synthesis experiments explored this model: it quickly turned out that it did not yield sounds as varied and as interesting as one could have hoped. The sounds were reminding of an electronic organ; they were often dull and they did not have a strong identity. It was tried to imitate the sounds of specific musical instruments by using their description in terms of spectra and envelopes attack and decay times as could be found in Acoustics treatises. The results were very disappointing. Some timbres could be crudely evoked, although not in a very convincing way, but others - specially trumpet and violin sounds - resisted the efforts of imitation. One must conclude that the traditional model of musical sound 109 is oversimplified, insufficient, characterize certain timbres. and even completely inadequate to Analysis by synthesis : trumpet and violin studies Specific studies were undertaken to try to understand why one could not imitate the trumpet and the violin (Risset & Mathews, 1969). Instrumental sounds were analyzed, and the relevance of these analyses was checked by synthesis : if they retain the aurally significant parameters, they must allow a reasonably good imitation of the analyzed sound. I shall go into some detail about these instrumental studies because they reveal important aspects of timbre perception and they indicate some requisites for sound models. I began the study of the trumpet by performing pitch-synchronous analyses of trumpet, in order to follow the evolution of the parameters. The sound was first separated into pitch quasi-periods, and the line Fourier spectrum was calculated for each pitch period. The plots of the successive spectra indicated the variation of amplitude for each harmonic. The trumpet tones have a complicated and varying physical structure. The attack lasts about 30 milliseconds, but it is complex. The frequency components are not synchronous: the high order harmonics appear later. Synthesis demonstrates that this non-synchrony can be heard: it has to be taken in account in a proper model. The spectrum varies a lot from one sound to another and also within one sound, even during the "steady state". This case illustrates the fact that most natural sounds are extremely complex: it is unthinkable to specify all the details for a full physical replication. Fortunately, one cannot hear all the details. Hence one must try to simplify the description in a way that does not create a big loss for the listener. From the analysis plots, curves were extracted approximating the evolution of the amplitudes for harmonic 1,2, etc (depending on the pitch of the tone and its intensity, 5 to 15 harmonics had significant amplitude). If the spectrum had been constant, these curves would be parallel in log amplitude scale, or affine in linear scale.: this is not so (Cf. fig. 1). Such curves were taken as envelopes for synthetic tones obtained by additive synthesis: adding up the harmonics controlled by such envelopes did give a faithful resynthesis of the initial trumpet tone. As was mentioned above, the envelopes could often be simplified substantially, and some components could be grouped without much damage to the sound: this resulted in a substantial reduction of the amount of information necessary to resynthesize the tone. while 110 an Thus a constant spectrum model is not valid for brass sounds, additive synthesis model - where separate harmonics are Fig. 1. Functions approximating t he evolution in t m i e of 13 harmonics lor additive synthesis 01 a D4 trumpet tQnes lasting 0.2 s (Irom Risset & Mathews,1969). This shows the wealth of details but also the complex ity which additive synthesis can entail. Such functions can often be simplified: in this case, substractive synthesis - with a variable band-pass filter -and especially frequency modulation permit more economical syntheses. 0.04 0 .03 0.16 controlled by separate envelopes -requires a lot of information (Cf. fig. 1): the set of envelopes needed to control the harmonics' amplitudes is different for different tones, depending upon their pitch and their amplitude. Imitating a gamut of trumpet tones thus seems to demand a prohibitive amount of specifications. I looked for characteristic properties that would permit to synthesize trumpet tones automatically, "by rule": instead of repeating the analysis "ad hoc" for each tone, one should try to find rules which describe how to change the parameters of the synthesis tone according to its desired pitch and intensity. For tones of the same loudness, I found that the "formant structure" hypothesis advocated by Herman one century was essentially valid: going up or down the pitch scale, the spectrum changes so as to preserve a fixed spectral envelope. However the most salient feature was the following: the louder the tone, the richer the spectrum in high harmonics energy. This was true within a single tone: during a loudness increase ("crescendo"), the high order harmonics increased more in amplitude than the low order ones. This also accounted for the non-synchrony of the components during the attack. Now synthesis demonstrated that this feature - the fact that the spectrum was correlated with the intensity so as to get richer in high frequencies 111 when the intensity increased - was indeed the essential cue to the brassy quality: by stipulating the rule that the spectrum varies with the intensity in this fashion, one did get a brassy sound. This characteristic property - an increase of the spectrum bandwidth when the loudness inCreases- can be implemented by additive synthesis. The envelopes of the component harmonics can be determined automatically, "by rule", making the amplitude of each harmonic a prescribed increasing function of the amplitude of the first one - the higher the order of the harmonic, the bigger the increase. Now it turns out that non-linear methods permit to implement this type of spectral variation in a most elegant and economical fashion, by coupling the modulation index (for FM) or the distorsion index (for waveshaping) to the instantaneous amplitude. The synthesis of brassy sounds was one of the first demonstrations of the power of the method of frequency modulation (Chowning, 1973). To get a proper imitation of, say, a trumpet, one had to be careful about specific details: the proper placement of the spectrum at a given intensity, with a broad formant around 1000 - 1500 Hz; some inharmonicity giving more " bite" to the attack of the tone; and specific frequency contours, with occasional "vibrato" (a slow frequency modulation of a rate around 6 Hz and a range inferior to one per cent). Beyond such details -important for the listener, to be sure- this study emphasized the need for a model with a variable spectrum, and it indicated that the typical signature of a given timbre can be ascribed to a specific relation between different parameters - here, spectrum and intensity - rather than to a given spectrum, as in the traditional conception. Max Mathews studied the case of the violin. He found that the spectrum was quite rich, and that it changed from pitch to pitch, and also during the vibrato - an important ingredient of classical violin playing. Now the vibration of the bowed string has a triangular waveshape: hence the spectral richness and variability must come from the frequency response of the violin box. Mathews simulated this response using a battery of resonant filters (Mathews & Kohut, 1973) : he showed that although the details of the jagged frequency response curve are not aurally critical, its peak-to-valley ratio has to be proper: if the curve is too flat, the sound is dull and reminds of a triangular wave; if it is too irregular, with very deep valleys, the sound is hollow and uneven. Now such a frequency response ensures a rich spectrum, critically depending on the pitch: the harmonic components can fall near a peak or a valley, on an ascending or a descending slope. In particular, the vibrato is accompanied by a complex spectral modulation. Indeed, it is not possible to imitate a good violin sound with vibrato by imposing a slow frequency modulation on a wave with a fixed spectrum. On 112 the other hand, even a crude simulation of a vibrato with a spectral variation correlated with the frequency excursion gives a strong hint of a bowed string. Thus subtractive synthesis has proved very useful in the imitation of the violin. Mathews developed an electronic violin: four ordinary strings are played with a bow, but an electronic circuit replaces the violin box. If the violin response is simulated by a battery of appropriate resonant filters, it can sound much like an ordinary violin. On the other hand, the electronic circuit can implement other sonic behaviors - for instance the feature found typical of brassy sounds. In order to implement in the analog world the spectral variation which I found to be characteristic of brass sounds, Moog developed a voltage-controlled filter which admits more high frequencies when the input voltage increases: it suffices to derive this input voltage from an envelope detection to link the spectral widening to the amplitude. Using such a filter, one can obtain brassy sounds by bowing Mathews' This demonstrates that the features mentioned above electronic violin! indeed characterize timbres and instrumental identities. Some factors of timbre I have dwelt in some detail on these instrumental studies because they reveal important aspects of the correlates of timbre: variability of spectrum, importance of the evolution of parameters in time, correlation between different parameters. One should add typical idiosyncrasies, which are often important to identify the origin of the sound: for instance the frequency glides at the onset of the tone for the slide trombone; or the erratic frequency variations at the onset of a violin tone, which suggests bow scratches. Such details are very revealing to the ear: they are often very specific to a given sound source. This makes it difficult to sketch a simple universal model for musical sounds. It also suggests to attempt modeling the physical behavior of the instruments, as was exemplified by Hiller and Ruiz, Cadoz and Weinreich, despite the difficulties encountered. Temporal aspects can be essential for timbre, specially during the attack and early decay. Transient details of the attack can be quite audible and even characteristic: rise time, rise asynchrony for different partials, burst of noise or inharmonicity during the attack. Rise times can vary with pitch. The spectrum is essential to the tone quality: but it can be either approximately constant or largely variable during a tone. In the latter case that of the brass, of the violin, or of percussion instruments - the spectral variation and the way it depends upon other parameters can be a strong cue to timbral identity. Laws of variation of the spectrum throughout the pitch 113 register are also of significance. Timbre subjective invariance is associated with a formant structure, which may be explained by the fact that hearing attempts to analyse the sound in terms of excitation and resonance. The spectrum can be harmonic or nearly so, inharmonic, or noisy. Hearing seems to be less discriminating for "saturated" spectra, when the amplitude of the frequency components decays by 6 dB/octave or less (12 or 18 dB/octave are more typical for instruments). The precise spectral composition can be important for inharmonic sounds. When the first strong components are spaced wider than a critical bandwidth of hearing - approximately 1/3 octave- the sound can be close to a bell rather than to a gong, if the envelope is appropriate. The ear is not equally sensitive to all spectral parameters. For instance, for speech-like sounds, the listener is very sensitive to the frequencies of the spectral peaks or formants - a given vowel can be characterized by the frequencies of the three first formants but it is much less sensitive to the amplitude of those peaks. Various types of "micromodulation" are often significant: a similar modulation applied to a number of components help these components fuse into one single auditory entity. The "chorus" effect produces a complex amplitude modulation, from which the ear infers a multiplicity of sound sources.The vibrato, a frequency modulation around 6 Hz, can be accompanied by a spectral variation. Pitch irregularities can be very significant, and also amplitude modulation, irregular or regular (the latter case can often be ascribed to beats, for instance in piano or bell tones). Fine adjustments of the temporal behavior of certain parameters can be essential: for instance, to simulate a singing voice, one must delay the vibrato with respect to the onset of the tone and make it irregular. Listening in a reverberant environment enriches the sound while generally preserving its identity, despite gross changes in the spectrum. The idiosyncrasies of perception Little can be said here about aural perception, but the above review indicates that it is much more complex and specific in its modalities than one generally believes. Although many auditory attributes such as pitch - are correlated to physical parameters - like frequency - the psychoacoustic relation between them can be quite involved in certain cases. I have demonstrated this in a provocative manner by generating a sound that will seem to go down in pitch when its frequencies are doubled; similar oddities can be demonstrated with rhythm (Risset, 1986). Such paradoxes, or auditory illusions, evidence some specific mechanisms of hearing; as Purkinje stated long ago, the illusions are deceptions of the senses but truths of perception. (Wessel & Risset, 1979). Is there an Ariane clue that helps us understand the raison-d'iHre of the idiosyncrasies of perception ? It seems that they make sense only if one adopts an 114 "ecological" viewpoint, following and extending the research of Gibson (1966): the senses are not physical detectors, but perceptual systems optimized to extract, from the physical signals they capture, information about the environment that can be of survival value. Many aspects of perceptual organization, in particular in the auditory domain, seem to have developed to fullfill that goal (cf. Shepard, and Bregman, in Kubovy & Pomerantz, 1981; Risset, 1973, 1988). Digital sound processing So far we have dealt only with sound synthesis. Digitized sounds can also be processed digitally in a very precise way, using the computer or special purpose processors. This is frequently used to add reverberation to recordings that sound too dry (Cf. Moorer, 1979). Such cosmetic fixes can be very important to make the sound musically interesting. The very precision and reproductibility of the computer permits to perform interesting transformations. It is easy to impose envelopes on the digitized sounds: digital mixing can of course be very precise, but it can also lead to unexpected -and possibly useful- effects. For instance, adding a sound to itself with a delay around 1 ms gives a spectral coloration - this is a comb-filter effect. By adding stable quasi-periodic tones with a slight frequency difference - Mlf possibly smaller than 1/10000 - one obtains intriguing varying textures : the different harmonics beat at different rates (Hartmann, 1985). The mastery of digital filtering has developed rapidly: it permits useful transformations. Nevertheless, sound processing is often disappointing in that it is not easy to perform intimate transformations, affecting only certain aspects of the sound. For instance, one can alter the frequency of a given sound, by interpolating samples, -which has the same effect as changing the sampling rate; but this has the same effect as changing the speed of the tape recorder: the speed is changed in the same proportion as the frequency. The so-called sampling machines permit to digitally record sounds and to change their frequency in this fashion: for instance, one can record a "A" (440 Hz) on a violin and play a full chromatic scale from that tone - the frequency transposition is specified by the key hit on an organ-like keyboard. The change of duration is compensated by manipulations that make the sound overlap itself. Also a simple frequency transposition usually does not ensure a good quality throughout a wide frequency range: the spectrum as well as other parameters tend to change in certain ways that can be specific to a given instrument. More intimate transformations capable of divorcing frequency changes and speed require elaborate processes. Here synthesis remains superior, since one masters the parameters of the sounds at the source, and one is free to mold them at will. 115 Analysis-synthesis processes In the field of computer music, analysis is resorted to with a view toward synthesis. Even when one intends to process a given sound, it can be useful to analyze the sound in such a way that it can be resynthesized from the analysis: this can be the best way to perform intimate transformations upon this sound. Between analysis and synthesis, the user has some latitude to alter the parameters to accomplish intimate transformations. For instance, many instrumental tones can be imitated by additive synthesis. A tone of a given instrument can be described by line segment envelopes schematizing the evolution of the first harmonics: these envelopes permit to synthesize a good copy of the original tone (Moorer & Grey, 1977). Now one can interpolate between two such tones by interpolating between the line segment envelopes to obtain a continuous transition from one tone to another: this interpolation does not give the impression that the two tones are mixed with different weight, but rather that the sound undergoes a gradual metamorphosis. As was explained for the case of the trumpet, the parameters for synthesis are not always easy to extract from a given analysis. In fact, it is a worthwhile challenge to try to derive, from the analyses of a certain class of sounds, parameters which permit to simulate these sounds using a given model for synthesis (e. g. FM). This often requires skill, trial and error, and even research. However some processes build up the synthesis from basic functions that form a "complete set", in the sense that a reasonably regular sound signal can be expanded in term of these basic functions. In other words, the signal can be approximated as a sum of contributions of such functions. One then speaks of analysis-synthesis processes: the signal can be reconstructed automatically from the analysis data. The best known example is the Fourier expansion of periodic functions in term of sinusoidal (sine and cosine) components: the line spectrum determines the timbre for such waveforms. Such a process is valid only for certain classes of signals - periodic functions in that case. For non-periodic functions, the Fourier transform yields amplitude and phase spectra which in some sense generalize the expansion in terms of sinuso"ids. Such spectra are continuous, which makes resynthesis impractical. As was noticed by Gabor (1946), it is paradoxical to analyze a sound limited in time in terms of sounds which last forever: an infinity of sine waves are required so that they cancel each other in the regions where the signal is zero. When one arbitrarily limits the spectrum, the reconstruction suffers difficulties: oscillations appear, known as the Gibbs phenomenon. Also the Fourier transform does not display in a clear way the evolution of the sound in time: the information about this evolution is compressed in a very cryptic way in the phase spectrum. 116 One often uses Fourier-derived running analyses, using a gliding time window to limit the analysis to a short portion of the signal (Schroeder & Atal, 1962). Such an analysis is used in the sonagram. Several efforts have been made to resort to basic functions that are limited in time as well as in frequency. Gabor proposed an expansion in terms of sine waves of variable frequency "windowed" by a fixed length bell-like gaussian envelope. The idea was pursued for synthesis only by Xenakis (1971), Roads (1978) and Truax -the process is called granular synthesis. This expansion has been proven to be complete (Bacry & aI., 1975, Bastiaans 1980). It is now obvious that the wavelet transform is more powerful (Grossman & Morlet, 1984; Kronland & aI., 1987): the reconstruction calls for granular synthesis. Using a Morlet wavelet, Kronland has shown that a quasi-perfect reconstruction can also be done via additive synthesis, with both amplitude and frequency varying with time according to functions derived from the wavelet analysis. This has been implemented with MUSIC V by Daniel Arfib. In the meantime, Dolansky (1960) has expanded speech signals in terms of exponentially decaying sinuso'ids, a set of functions that had been proven to be complete by Szasz (Cf. Paley & Wiener, 1934). Some people have advocated the Walsh-Hadamard transform, which is easy to implement with digital circuits. However, this process is inadequate for sound, in that it cloche H"'" 't tl ,. ., • .. " )" p~r distoysion non line~iye 1-t1 2140' t ·.. c;1" t H· ... Fig. 2. A 3·dimensional display of a synthetic sound. This diagram represents 5.42 s of a bell·like synthesized by Daniel Arfib (1979) using non linear distorsion plus amplitude modulation. Time runs from left, bottom, to right, top; frequency runs from left, top, to right, bottom. The z axis shows amplitude. The analysis and display programs, using FFT, were implemented on a IBM PC compatible computer by Patrick Sanchez. 117 does not "deteriorates" gracefully for the ear: it tends to give too rich spectra, and the expansion must include a tremendous number of terms to get close to sounds such as those of a flute. Recently, the Karhunen-Loeve transform has been found useful for reducing the computation necessary to carry out additive synthesis (Stapleton & Bass, 1988). Two analysis-synthesis processes have been successfully exploited in computer music. The phase vocoder (Moorer, 1978; Dolson, 1986) permits to obtain information which leads to resynthesis for quasi-periodic tones, but also for tones made up of nonharmonic components. The linear prediction process (Moorer, 1977, Lansky, 1981) modelizes the signal as the result of passing an excitation through a time-variant recursive filter: the analysis consists mostly of estimating the variable characteristics of the filter. This process is efficient when the signals are reasonably well described by this model, as is the case for speech: the variable filtering here results of the variation of the vocal tract through articulation. Although these two processes are quasi-universal, they can lead to information explosion rather than information reduction for many types of signals which correspond poorly to the model they implement. Complete analysis-synthesis processes can in principle work for any type of signal: it does not require a priori knowledge on the nature of the sound to be replicated. (Flandrin, in his presentation at this meeting, uses the term of non-parametric representation). However, both the quality and the computing efficiency of a given process is often much better for certain types of sound. For instance, linear prediction works well for speech-like sounds. The wavelet transform seems to give results of excellent quality regardless of the nature of the sound, as Richard Kronland-Martinet demonstrated at this meeting: however it is likely that the ultimate information reduction can be greater if one takes in account the kind of signal and chooses the type of information reduction accordingly. Here the model is to some extent specific, adapted to the characteristics of the sound (Flandrin speaks in this case of parametric representation). Utility of analysis-synthesis models of sound The models drawn from analysis-synthesis provide insight and understanding on the correlates of sound identity. They lead to representations than can be reveali'ng. Looking at a sonagram gives some idea about the sound: it is much more informative than looking at the waveform (Leipp, 1971; Cogan, 1984). Analysis-synthesis may also lead to information reduction. But the most interesting aspect to the musician is the possibility of producing variants. 118 Sound modification here may be used with two different goals. One may want to preserve timbral identity while changing pitch and duration (also, possibly, articulation and loudness). For instance, analysis-synthesis processes may permit changing pitch and speed independently. Also, as Herman pointed out one century ago, it is often improper to keep the same spectrum when one changes pitch. The case of the trumpet indicates that it may also be necessary to change the spectrum as one changes loudness. Such changes are essential to really take full advantage of the processing of real sounds and not be limited to rather superficial transformations, which the listener will find too "mechanical" or "electronic". But one should realize that these transformations are delicate and demanding. So generality may be ocasionally abandoned to diminish the amount of computation, and also to simplify the specifications, so that the sonic transformations can be easily and efficiently controlled. As Barry Truax noticed, generality can be at the expense of strength. Another goal is attractive to the creative computer musician: expanding sonic resources. Here the goal is to change certain aspects of the sound so as to completely change the timbre while preserving the richness of the original model. Analysis-synthesis processes are musically exploited to this effect. They make possible intriguing transformations such as cross-synthesis - creation of a sound with certain features of sound A and other features of sound B; interpolation between timbres (Cf. Grey & Moorer, 1977, and the compact disk by Chowning); extrapolation beyond the register of an instrument; perversion of additive synthesis to produce sound paradoxes and illusions (Risset, 1978, 1985); metamorphosis of percussive sounds into fluid textures, while preserving their harmonic content (Cf. the INA compact disk by Risset). The expansion of digital music For many years, computer music could only be realized in scientific laboratories endowed with large computers. The development of the dedicated computer made it easier. IRCAM, which opened around 1977 in Paris, was the first large scale musical institution where the computer played a central role. There as well as in other centres like Bell Laboratories, where computer music was born, and at Stanford University where it grew a lot, special digital processors were developed to speed up the operations of sound synthesis and processing. Recently computers have become personal tools, and the development of microelectronics has made it possible to produce inexpensive chips implementing certain types of digital sound processing. So digital music has expanded a lot in the last few years, as witnessed by the success of digital synthesizers such as the DX7. The recent MIDI standard (Ct. Loy, 1985) has played a major role here, making it 119 easy to exchange control information between synthesizers, computers, and gesture-captering devices. More often than not, however, digital sound is used in a rather conservative fashion. Digital synthesizers are usually resorted to as instruments with preprogrammed tone qualities rather than as a creative medium. However the knowledge and knowhow developed since thirty years is accumulating and available, although not always easy to access, and several approaches have proven original and successful. The proof of the cake is in the eating, not in the cooking: the listener should not have to know a lot about the processes involved to appreciate the music. There is no room here for an evaluation of the most significant computer music works so far : the reader is referred to the discography. The field of computer music is gradually loosing its distinct identity as it is penetrating larger portions of the musical world. If he wants to take full advantage of the vast and novel possibilities open by the computer and the digital domain, the musician, unlike the listener, should widen his traditional solfege. It should be clear by now that the software - in a wide sense- the knowhow, the expertise, both in signal processing and in musical psychoacoustics, is more important and durable than the hardware. In this presentation, I have discussed some signal processing aspects in relation with certain specific features of perception. These should always be considered together. As I merely indicated in this article, our understanding of auditory perception has progressed a lot thanks to the flexibility gained in manufacturing sound digitally (Risset,1988). And our usage of digital processing for sound is valid only insofar as it can satisfy the perceptual demands. The best signal processing models are contingent upon a certain scientific and technical state of the art, and they physical or can only be justified as proper implementations of psychoacoustical models, which are of probably of more enduring significance. Musicians are still required to make music out of those possibilities. It looks like the introduction of the wavelet transform will considerably affect the field of computer music: it may be a big help the musician as well as the scientist and the engineer, but it will make none of them obsolete ! BIBLIOGRAPHY On computer composition and musicology L.A. Hiller, Jr., & L.M. Issacson (1959). Experimental Music. McGraw Hill, New York. H.B. Lincoln, editor (1970). The Computer and Music. Cornell University Press, Ithaca, N.Y. I. Xenakis (1971). Formalized Music. Indiana University Press, Bloomington. H. Charnasse & H. Ducasse (1973). Informatique musicale. Ed. C.N.R.S., Paris. Artificial Intelligence and music, by various authors (1980). Computer Music J. 4, n° 2 & 3. 120 On music printing L.Smith (1973). Editing and Printing Music by Computer. Journal of Music Theory, 17-2, pp. 292-309. On computer synthesis and processing of musical sound Textbooks M.V. Mathews (1969). The Technology of Computer Music. M.I.T. Press, Cambridge, Mass. Music and technology (1971). Unesco & La Revue musicale, Paris (disponible aussi en franc;ais). I. Xenakis (1971). Formalized music. Indiana Univ. Press, Bloomington, Indiana. E. Leipp (1971/1984). Acoustique et musique. Masson, Paris. H.S. Howe (1975). Electronic music synthesis. Dent, London and New York. Computer music (1980). UNESCO report, Canadian Commission for Unesco. Musica e Elaboratore (1980). L.I.M.B., Biennale di Venezia. H. Chamberlin (1980). Musical applications of microprocessors. Hayden Book Company, Rochelle Park, N.J. D. Deutsch, editor (1982). The Psychology of Music. Academic Press, New York. B. Schrader (1982). Introduction to electroacoustic music. Prentice Hall. J.R. Pierce (1983). The Science of Musical Sound. Scientific American Library, Freeman, San Francisco (with disks of sound examples). R. Cogan (1984). New images of musical sound. Harvard University Press. C. Roads, ed. (1985). Composers and the computer. W. Kaufman, Los Altos, Ca. J. Strawn, ed. (1985). Digital audio signal processing. W. Kaufman, Los Altos, Ca. P. Manning (1985). Electronic and computer music. Clarendon Press, Oxford. C. Dodge & T.A. Jerse (1985). Computer Music: Synthesis, Composition and Performance. Schirmer Books, MacMillan, New York. Nuova Atlantide: il continente della musica elettronica (1986), LI.M.B., Biennale di Venezia (with two cassettes of musical works). J. Chowning & D. Bristow. Frequency Modulation: Theory and Applications. Yamaha Foundation, 1986. D. Morrill (forthcoming). The little book of computer music instruments. W. Kaufman, Los Altos, Ca. Articles J.B. Allen & L.R. Rabiner (1977). A unified approach to short-time Fourier analysis and synthesis. Proc. of the IEEE 65, 1558-1564. D. Arfib (1979).Digital synthesis of complex spectra by means of multiplication of non-linear distorted sine waves. J. Audio Eng. Soc. 27, 757-768. H. Bacry, A. Grossman & J. Zak (1975). Proof of the completeness of lattice states in the kqrepresentation. Phys. Rev. B12 1975, p. 1118. M.J. Bastiaans (1980). Gabor's expansion of a signal into gaussian elementary signals. Proc. of the IEEE 68, 538-539. C. Cadoz, A. Luciani & J. Florens (1984). Responsive input devices and sound sybthesis by simulation of instrumental mechanisms: the Cordis system. Computer Music J. 8 N°3, 60-73. S.J. Campanella & G.S. Robinson (1971). A comparison of orthogonal transformations for digital speech processing. IEEE Trans. on Communication Technology COM-19, 1045-1050. G. Charbonneau (1981). Timbre and the perceptual effect of three types of data reduction. Computer Music J. 5 n02, 10-19. J.C. Chowning (1973). The synthesis of complex audio spectra by means of frequency modulation. J. Audio Eng. Soc. 211, 526-534. 121 L.O. Dolansky. A novel method of speech-sound analysis and synthesis. Ph. D. Dissertation, Harvard Univ., 1959. Ct. also IRE Trans. on Audio, AU-8 n06 (1960) p. 221. M. Dolson (1986). The phase vocoder: a tutorial. Computer Music J. 10 n° 4, 14-27. D. Gabor (1946). Theory of communication. J. Inst. Electr. Eng. 93, 429-457. J.J. Gibson (1966). The senses considered as perceptual systems. Houghton Mifflin Company, Boston. G. Di Giugno & J. Kott (1981). Presentation du systeme 4X. Rapport IRCAM n° 32, Paris. J.M. Grey & J.A. Moorer (1977). Perceptual evaluation of synthesized musical instrument tones. J. Acoust. Soc. Am. 62, 454-462. A. Grossman & J. Morlet (1984). Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15, 723-736. W.H. Huggins. Representation and analysis of signals. Part 1: the use of orthogonalized exponentials. John Hopkins Univ. Report n° AFCRC TR-57 357 (1957). Cf also IRE Trans. on Circuit Theory, CT 3 n° 4 (1956) p. 210. W.H. Hartmann (1985). The frequency-domain grating. J. Acoust. Soc. Am. 78, pp. 1421-1425. L. Hiller & P. Ruiz (1971). Synthesizing sound by solving the wave equation for vibrating objects. J. Audio Eng. Soc. 19, 463-470. R. Kronland-Martinet, J. Morlet & A. Grossman (1987). Analysis of sound patterns through wavelet transforms. Intern. J. of Pattern Recogn. & Artit. Intell. 1, 273-302. M. Kubovy & J.R. Pomerantz, ed. (1981). Perceptual organization. Erlbaum, Hillsdale, N.J. Cf in particular A. Bregman: Asking the "what for" question in auditory perception, and R.N. Shepard: Psychophysical complementarity. P. Lansky & K. Steiglitz (1981). Synthesis of timbral families by warped linear prediction. Computer Music J. 5 n03, 45-49. M. Le Brun (1979). Digital waveshaping synthesis. J. Audio Eng. Soc. 27, 250-266. G. Loy (1985). Musicians make a standard: the MIDI phenomenon. Computer Music J. 9 n° 4, 8-26. M.V. Mathews & J. Kohut (1973). Electronic simulation of violin resonances. J. Acoust. Soc. Am. 53, 1620-1626. J.A. Moorer (1977). Signal processing aspects of computer music : a survey. Proceedings of the IEEE 65, 1088-1137. J.A. Moorer (1978). The use of the phase vocoder in computer music applications. J. Audio Eng. Soc. 26, 42-45. J.A. Moorer (1979). About that reverberation business. Computer Music J., 3 n° 2, 13-28. J.A.Moorer & J.M. Grey (1977). Lexicon of analyzed tones. Part I: a violin tone. Computer Music J. 1 n° 2, 39-45. Part II: clarinet and oboe tones. Computer Music J. 1 n° 3, 12-29. D. Morrill (1981). Loudspeakers and performers: some problems and proposals. Computer Music J. 5 n04, 25-29. R.E. Paley & N. Wiener (1934). Fourier transforms in the complex domain. Amer. Math. Soc. Colloquium Publ. XIX, New York. J.R. Pierce (1983). The Science of musical sound. Scientific American Books, W.H. Freeman, New York & San Francisco (with disks or cassette of sound examples). G. De Poli (1983). A tutorial on sound synthesis techniques. Computer Music J. 7 n° 4 , 8-26. J.C.Risset (1969). An introductory catalog of computer synthesized sounds. Bell Laboratories Report, Murray Hill, N.J. J.C. Risset (1973). Sons. Encyclopedia Universalis, 13, 168-171. J.C. Risset (1978). Paradoxes de hauteur. IRCAM Report n° 10, Paris (with cassette of sound examples). Cf. also Pierce, 1983. J.C. Risset (1985). Computer music experiments 1964-.... , Computer Music J. 9 n° 1, 11-18. J.C. Risset (1986). Pitch and rhythm paradoxes: Comments on "Auditory paradox based on a fractal waveform". J. Acoust. Soc. Am. 80, 961-962. J.C. Risset (1988). Perception, environnement, musiques. Inharmoniques 3, 10-42. J.C.Risset & M.V. Mathews (1969). Analysis of instrument tones. Physics Today 22 n° 2, 23-30. J.C. Risset & D.L Wessel (1982). Exploration of timbre by analysis and synthesis. In D. Deutsch, ed., The psychology of music, Academic Press 1982, 25-58. 122 C. Roads (1978). Automated granular synthesis of sound. Computer Music J. 2 n° 2, 61-62. X. Rodet & Y. Cointe (1984). FORMES: composition and scheduling of processes. Computer Music J. 8 n03, 32-50. X. Rodet, Y. Potard & J.B. Barriere (1984). The CHANT Project: from the synthesis of the singing voice to synthesis in general. Computer Music J. 8 n° 3 (1984) 15-31. M.R. Schroeder & B.S. Atal (1962). Generalized short-time power spectra and autocorrelation functions. J. Acoust. Soc. Am. 34, 1679-1683. J.C. Stapleton & S.C. Bass (1988). Synthesis of musical tones based on the Karhunen-Loeve transform. IEEE Trans. on Acoustics, Speech, and Signal Processing 36, 305-319. G. Weinreich (1981). Synthesis of piano tones from first principles. J. Acoust. Soc. Am. 69, 588. DISCOGRAPHY L'IRCAM: un portrait. IRCAM 0001, Paris. MATHEWS, PIERCE, RISSET, SLAWSON, TENNEY. Decca DL710180. J. APPLETON. Mussem's song, Zoetrope, In deserto. Folkways FTS 3345. J. RANDALL. Mudgett. Nonesuch H 71245. M. SUBOTNICK. Silver apples of the moon. Nonesuch H 71174. W. USSACHEVSKY. Computer piece. CRI 2D 268. B. TRUAX. Sonic landscapes. IRC, Vancouver, SMLP 4033. HILLER, ISAACSON, BAKER. Computer Music from the University of Illinois. Heliodor H/HS 25053. I. XENAKIS. ST10. EMI VSM C061-10011. G.M. KOENIG. Function Blau. Philips 6740 002. J. CAGE with L.J. HILLER. HPSCHD. Nonesuch H71224. C. DODGE. Speech Songs, The story of our lives, In celebration. CRI SO 348. New Directions in Music (Ghent, Olive, Petersen, Risset, Wessel). Tulsa Studios, Tulsa, Okl. AUSTIN, CELONA, DODGE, HAYNES, PENNYCOOK. International Computer Music Conference Recording, 1981 (Denton, Texas). The Digital Domain: A Demonstration. Compact ElektraiAsylum & WEA 60303-2. LANSKY, ROADS, DASH OW, WAISVISZ, BARLOW, KASKE. New Computer Music. Compact disc: Computer Music Journal & WERGO. CHAFE, JAFFE, SCHOTTSTAEDT. Computer Music. Compact CCRMA. ALBRIGHT, BALABAN, BRODY,CHILDS, DASHOW, DODGE, LUNN, ROADS, VERCOE. Music for instruments and computer. Compact M.I.T. J. CHOWNING. Turenas, Stria, Phone, Sabelithe. Compact WERGO 2012. J.C. RISSET. Sud, Mutations, Dialogues, Inharmonique. Compact INA C003. J.C. RISSET. Songes, Passages, Little Boy, Sud. Compact WERGO 2013. 123 Part III Wavelets and Signal Processing Wavelets and Seismic Interpretation J L. Larsonneur 1 and J. Morlet 2 IDRGG SNEA(P) (ELF AQUITAINE), CSTCS, F-64018 Pau Cedex, France 20RIC 371 bis, Rue Napoleon Bonaparte, F-92500 Rueil-Malmaison, France The concept of the wavelet is not new in the petroleum industry, and particularly in geophysics , but this term is used here with a different meaning . For a geophysicist the source signal used during seismic acquisition is often approximated by a wavelet form. The mathematical approach of Wavelet Transform (W.T) permits to go further in the exploitation of their interesting properties. In this article, we explain some possible uses in Seismic Interpretation. INTRODUCTION. There are three major steps in a Seismic study: - acquisition : an acoustic signal is created and sent in the Earth and the signal that is returned contains information on discontinuities. - computation : data are processed to focalise the energy sent by a reflecting point at its supposed exact position. From this, data can be considered as a 2D or 3D image of the discontinuities in the Earth. - interpretation : the image is analysed to determine its homogeneous zones from the geological point of view and their content. We could use W.T to improve the first two steps. It would consist in splitting the signal received into different channels at the acquisition , one channel corresponding to one dilatation parameter in the W.T , and then use programs adapted to each channel, or frequency. 126 To improve step number three ( interpretation ) , the tendancy is now to propose different displays of the same data. Along with the black and white seismic paper section we can also obtain amplitude or phase colored section , either on paper or screen . With this in mind, the W.T can bring some new possibilities, which could be correlated to some types of geological event. The first two steps require much larger investments in man hours and equipmenUhan the third step. This is the reason for exploring this path first. 2D INTERPRETATION. The general problem of 20 interpretation is to follow horizons (lines) on a Seismic section and correlate it to a geological event. If the transmitted signal is good and the geological objectives simple, their contouring is very easy. On the other hand, if there is some noise or some incorrect hypotheses introduced in the processing , there can be some crucial choices to make . In this situation , the experience and the geological knowledge of the interpreter is a determining factor but it is not always sufficent . An error in the Petroleum industry can result in prohibitive cost, it therefore appears important to put the interpreter in the best conditions to make correct hypotheses. One possible help is the use of new displays containing phase information (analytic Signal) or amplitude values, or even instantaneous frequency information. In this direction , the W.T can naturally add some new attributes: amplitude and phase of the signal in each frequency channel. Figure 1 is one example of a seismic section and figure 2 shows the result of the transformation for one channel . The representation is very simple : positive values are 127 Figure 1. original image Figure 2. transformation for one channel . 128 white, negative are black. The second section contains very homogeneous information and is much more interpretable than the first. There are no more interferences and the horizons are very easy to follow. If the interpreter wants to keep the original section as a reference , he can point to some horizon on the second section and then report to the first one. Furthermore, any combination of channels can be computed very easily due to the simplicity of reconstruction in W.T . For example, if some noise is present in one channel , we can smooth it and then reconstruct the section. This filter will produce a minimum deformation in the shape of the object. The position of an object is no longer affected by the filter that is used to resolve its position more accurately. 3D INTERPRETAllON . In 20 interpretation, the problem is to follow one line (horizon) in an image (section) . In 3D , the object is to determine a level (surface) in a 3D volume, with a much greater amount of data. If the objects to determine are complicated, the errors occur frequently and the comprehension of their geometry can be very difficult. In this context, it is even more crucial than in 20 to propose different attributes of the possible solutions: amplitude along the surface, etc. Similar to 20 , W.T can provide the amplitude and phase attributes for each channel. In 3D , the job is very often made on a 3D Workstation (computer with large capacities for storage and visualisation) . This makes the choice for these displays easier because they can be processed locally very quickly. When surfaces have been determined, the interpretation job is not finished. We have the general geometry of an object, but we are also interested in other properties: micro129 faults, porosity, etc. The usual attributes are very limited in this research because they are dominated by the major frequency components of the signal . Very little change in the aspect of the data can indicate for example a good or bad porosity , and consequently the presence or absence of a hydrocarbon reservoir. In this domain , which is under development, the W.T is very promising. Figure 3 illustrates an example of its application for a thinning wedge problem . Synthetic models were computed for each basic wavelet for comparisons with a real case. This could be used to determine the exact position of the edge of the thinning wedge. ':. Ijr1AC,E Figure 3. 130 transformation for four channels of the impulse response of a thinning wedge . CONCLUSION. In these examples. results of W.T have been used essentially to improve the visual aspect of the data. This aspect of its application is not very different from the numerous linear filters used in the image processing techniques. The advantage obtained here comes from the theory that supports the Transform and allows us to more easily associate the results to a "physical" meaning. In addition. it is confortable to use a family of filters that provides at the same time a reversible decomposition of the signal. With the present tools. we can offer new types of information to the interpreter. As the number of displays offered must be limited. the next effort will be to develop new techniques by exploring them through intensive use. REFERENCES. P. Goupillaud • A. Grossmann and J. Morlet : Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23 (1984-1985) pp85-102 Y. Meyer. S. Jaffard and O. Rioul : "I'analyse par Ondelettes' . "Pour la Science" no 119 • Sept. 1987 R. Kronland-Martinet • J. Morlet • A. Grossmann : Analysis of sound patterns through Wavelet transforms. 131 Wavelet Transformations in Signal Detection FE. Tuteur Department of Electrical Engineering, Yale University, New Haven, CT06520, U.S.A. A new method for dealing with transient signals has recently appeared in the literature [2-11]. The basis functions are referred to as wavelets, and they employ time compression (or dilation) rather than a variation of frequency of the modulated sinusoid. Hence all the wavelets have the same number of cycles. The analyzing wavelets must satisfy a few simple conditions, but are not otherwise specified. There is therefore a wide latitude in the choice of these functions and they can be taylored to specific applications. We have applied them to detect ventricular delayed potentials (VLP) in the electrocardiogram. 1. INTRODUCTION The detection of weak signals embedded in a stronger stationary stochastic process is an old and well-studied problem. Probably the best-known example is the detection of radar or sonar signals in zeromean Gaussian white noise. It is well known that the optimum detector in this problem involves correlation of the observed signal with a replica of the desired signal. Correlation is most effective if the shape and time of occurrence of the expected signal is known. It is only marginally effective if this is not the case. Instead of correlation one can also use a matched filter, but the effectiveness of a matched filter detector also diminishes if the shape, or at least the bandwidth of the expected signal is not reasonably well known. Another approach to the detection of signals in noise is Fourier analysis or spectral estimation. This works best if the expected signal has spectral features that clearly distinguish it from the noise. Its advantage over correlation or matched filter methods is its insensitivity to the shape or time of occurrence of the desired signal. It is not well adapted to the detection of speech or biomedical signals. In these applications one is often confronted by short-time signals whose exact shape is unknown. Although these signals alter the Fourier spectrum of the observed signal the effect is generally proportional to the ratio of desired signal to noise energy, and for short, low-energy transients the change is very subtle and not easily detected. The fundamental problem with ordinary spectral analysis is that the basis functions of Fourier analysis 132 (sines, cosines, or complex exponentials) extend over infinite time and are therefore poorly matched to the short-time transients of interest. Signal representation in terms of basis functions that are in some sense localized in both time and frequency would appear, therefore, to be better suited to the detection of localized signals whose precise shape, location, time extent, and spectrum are not well known. An obvious method for dealing with short-time or nonstationary signals is Fourier analysis with a sliding time window. This is often used in speech analysis. The location of the sliding window adds a time dimension and one gets a time-varying frequency analysis. Thus instead of the usual Fourier integral expression S(f) = ~(t)eXp(-j2~ft) dt ( 1) one gets a time-frequency expansion of the sort S(a,b) = ~S(t)ha,b(t) dt (2) where a is the frequency and b is the location of the window in time. A possible form for h b(t) is simply the product of a complex exponential with a wind8w function; i.e ~(t-b)exp(j2~at). If the window function ~(t-b) is a gaussian function one has a so-called Gabor wavelet, after Denis Gabor who used them in the analysis of radar signals in the late 1940's [1]. A disadvantage of the Gabor wavelet is that the window function is fixed, and therefore as the frequency is increased there are more and more cycles included inside the window. Hence the Gabor wavelets do not treat all frequency components of the signal in the same way. This can result in lack of robustness or even instability in reconstructing the signal s(t). 2. WAVELET TRANSFORMS Wavelet expansions that avoid some of the problems of timewindowed Fourier expansions have been proposed in papers of Morlet, Meyer, Grossman, Mallat, and others [2-11]. The wavelets used in these expansions are complex functions that are concentrated in time and frequency, and that all have the same shape. They are functions of two parameters a and b just as the Gabor wavelets, but the parameter a is a dilation (or compression) of the time scale rather than a frequency change. Hence ha,b = l/j.a g[(t-b)/a] where g(t) is the basic (i.e. unshifted and function. With this definition, Eq. (2) becomes (3) undilated) wavelet 133 W(a,b) = 1/{c3~g*[(t-b)/a] s(t) dt (4) where the notation W(a,b) is used to signify "wavelet Equivalently the wavelet transform can be expressed as W(a,b) = ~*(af) S(f) exp(j2~bf) df transform". (5) where G(f) and S(f) are the Fourier transforms of g(t) and s(t) respectively. There is some evidence that the human vIsIon and hearing processes employ something like wavelet analysis [12]. Wavelets must satisfy a number of restrictions. The important are integrability and square integrability. Also the frequency behavior of g(t) must be such that ~(f) /Ifldf < 00 most Iow- (6) This conditions implies that if G(f) is a smooth function in the neighborhood of the frequency origin then G(O) = 0; i.e. g(t) has no dc component. Additional assumptions are often made about wavelet functions for convenience. One such requirement is that G(f)=O for f< O. It is also convenient to assume that G(f) is real for f > O. Another important property of wavelets is that they should be concentrated in the time and frequency domains as much as possible. This means that the time-bandwidth product for wavelets should be as small as possible. It is well known that the smallest time-bandwidth product is achieved by the gaussian function [13]. Hence a frequently used wavelet function has the form g(t) = exp(-t 2 /2 + jmt) (7) G(w) = f[;" [exp (w-m) 2/ 2] (8) for which See Fig. I. To satisfy the requirement G(O) = 0 one must add a correction term, but if m > 5 this correction term is negligibly small and can be omitted. Grossman and Kronland-Martinet [8] have shown that this wavelet function is well adapted to the problem of locating abrupt changes in a signal. We have used it to locate an abnormality in the clinical ECG (see below). However many other functions have been proposed that will satisfy the general requirements for wavelets. For example, Mallat [9] has described a wavelet that is quite different from the one in Eq. (7) and that is well adapted to image processing. We see from Eqs. (4) and (5) that the wavelet transform is essentially a filtered version of the signal s(t), with the filter 134 (al Fig. I (bl An analyzing wavelet at various values of the scale parameter. (a) real part, (b) imaginary part. chosen to satisfy certain admissibility conditions. In order to analyze the signal the dilation parameter a is given an initial large value (e.g. 1.0) and is then decreased in regular increments to examine the signal in more detail. Equivalently, the wavelet filter function considers successively narrow section of the signal spectrum S(f). Since spectral properties are frequently better displayed on a With logarithmic frequency scale, it is customary to use a = 2- u . this definition integral increments in u result in octave increments of a. Note that small a, and therefore large u, corresponds to high frequencies. Thus small u corresponds to an analysis of the largescale features of set), and as u is increased finer detail of the signal come into focus. 3. VENTRICULAR LATE POTENTIALS Ventricular late potentials (VLP) are an abnormality of the clinical electrocardiogram (ECG). A typical ECG is shown in Fig. 2. Its most distinctive feature is the sharply rising QRS peak that results from the rapid depolarization that occurs as the left ventricle contracts and forces blood into the aorta. This is followed by the much lower frequency T wave. Normally just after the T wave there is a short refractory period where there is no electrical activity. The magnitude of the QRS spike is typically between I and 2 millivolts, and its duration is on the order of 10 milliseconds. Many deviations from the regular sinus rhythm, or arrhythmias, arrhythmias are recognized. have been Two life-threatening and ventricular (VT fast heartbeat) ventricular tachycardia fibrillation (VF fluttering heart, no pumping action) Recent research [14-17) has demonstrated that it is possible to record from the surface ECG certain post-QRS waveforms during normal 135 sinus rhythm that are predictors of VT or VF. These are the ventricular late potentials (VLPs). Their amplitude is typically between 4 and 25 microvolts, and they are observable only by widening the ECG bandwidth to about 300 Hz and attenuating the background noise. The main difficulty with using the VLPs to identify patients at risk for certain life-threatening arrhythmias is that the waveform is usually swamped by external interference or muscle artifact noise. Thus various signal processing methods have been brought to bear on the problem of extracting VLPs from the background noise. A commonly advocated method is signal averaging, but this is only effective if the VLPs have the same shape at each heartbeat and come at the same point in the cycle. Neither of these conditions necessarily holds in practice; in fact one of the complicating characteristics of the VLPs is that they occur somewhat randomly [14]. We have investigated wavelet analysis as a way for more clearly displaying VLPs. The ECG waveform used in our experiments was taken from a child with a cardiac defect. To be sure that this signal would contain something like a VLP we added a small signal consisting of a 25. Hz sine wave modulated by a gaussian envelope. The duration of this added signal was about .1 seconds, or a little less than 10~ of the pulse period of 1.05 seconds. Its peak amplitude was about 51. of the QRS peak. The ECG with the artificial VLP is shown in Fig. 3. The added signal can be made out right after the second QRS peak, particularly if one compares the "abnormal" ECG with the "normal" one of Fig 2. In a clinical situation where one would not have a normal wave for comparison and where the location of the VLP would not be exactly known the VLP might not be quite so obvious. In the wavelet analysis we used as an analyzing inverse Fourier transform of the frequency function wavelet the G(f) = exp[-(af - m)2] 1.89 ... , .. ~.- 1.2 0 . . " ..;... .....:... , " " , -..... ; " ... , .; .. , ...:...... , .:.., ..... : ..... . i ...... ~ I •• -. :' . . . . . . . ~ _ . . . , •• : . ". " . : •• " ... ~ . .. . • . . 1.89 ..... ! ... , ... : ...... . :, .... . _:- " •••• • : _ . " •• : " _ . , ... ,', ...... ,'. . .. .,.!" .. . ,.: ' ... , .. ~' ... ... ,'. - , -"'~" '" -.:.. -., .. :..... .. , "-,, . j., .. ... ~ ••• " 1.20 .."., :, .. .. ,;, .......~, -,-, .. ' . -.".,: .. _ .m ,m , B99 ,999 " .;. .. ,; .. ,- . 2,BO mONOS Fig. 2 136 "Normal" ECG . - ~ . - ... :' , '" ...;... ,. . ., ~., .. 3,6~ SEeOHDS Fig. 3 ECG with artificially added VLP with m = 5.33. With this value of m G(O) = 6.5 x 10- 7 which is small enough to satisfy the requirement that G(O) = 0 and that G(f) = 0 for f<O. Wavelet transforms were calculated for. values of a-I equal to 2, 4, 8, II, 16, 22, 32, 64 after first clipping the QRS peaks. Results ~re shown in Fig. 4 which shows the magnitude of the transforms for a= II, 16, and 22. The peak after the second QRS spike observed for a-I = 16 is very noticeable and gives a clear indication of the added VLP. . ..........:.......... ... :......:...' ... :.. ~ ..; .. -;......: ; .. ~ .. Lt.i ; U ,.. .: .... . . "" ......... . ...."" ...... , '- : ., l .U Hems Fig. 4 Wavelet transforms of the "abnormal" ECG for dilation parameter a-I = II, 16, 22. Note the bulge to the right of the middle QRS peak indicating the presence of the VLP. 4. REFERENCES 1. D. Gabor, "Theory of Convnunications," pp 429-457. J. lEE, vol. 93(3), (1946), 2. A. Grossman and J. Morlet, "Decomposition of Functions into Wavelets of Constant Shape, and Related Transforms." Center for Interdisciplinary Research and Research Center Bielefeld-BochumStochastics, University of Bielefeld, Report No. 11, December 1984. Also in "Mathematics and Physics 2," L. Streit, editor, World Scientific Publishing Co., Singapore. 3. P. Goupi llaud, A. Grossman, and J. Morlet, "Cycle-Octave and Related Transforms in Seismic Signal Analysis," Geoexploration 23, 85 (1984/85) 4. A. Grossman, J. Mor let, T. Pau I , "Transforms assoc i ated to square-integrable group representations, I: General Results. J. Math. Phys 26 2473, (1985). 137 5. A. Grossman, J. Mor I et, T. Pau I, "Transforms assoc i ated to square-integrable group representations, II: Examples. Ann. Inst. H. Poincare 45 293 (1986). 6. I. Daubechies, A. Grossman, Y. Meyer, "Painless nonorthogonal expansions", J.Math. Phys. 27 p1271, (1986). 7. A. Grossman and J. Morlet, "Decomposition of Hardy functions into square integrable wavelets of constant shape", SIAM J. Math. Anal. 12, p.723, (1984). 8. R. Kronland-Martinet, J. Morlet, A. Grossman "Analysis of Sound Patterns through Wavelet Transforms." To appear in International Journal of Pattern Recognition and Artificial Intelligence, Special Issue on Expert Systems and Pattern Analysis. 9. S. Mallat, "A Theory for Multiscale Decomposition: The Scale Change Representation," Grasp Laboratory, Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 19104-6389 (1986). 10. R.R.Coifman and y, Meyer, "The discrete Wavelet Transform", Preprint, Yale University Department of Mathematics, 1987. 11. Y. Meyer, S. Jaffard, O. Rioul, "L'analyse par ondelettes," la Science, Sept. 1987, pp 28-37. 12. Pour D. Marr, Vision, W. H Freeman and Company, 1982. 13. A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill Book Co. Inc., 1962, p64. 14. E. J. Berbar i, B. J. Scher I ag, R. R. Hope, R. Lazzara," Record i ng from the body surface of arrhythmogenic ventricular activity during the S_T segment". Am. J. Cardiol. 1!i1l, p697-702, (April, 1978). G. Breithardt and M. Borggrefe, "Pathophysiological mechanisms 15. and clinical significance of ventricular late potentials". Eur. Heart J. li2l, pp 364-85, (May, 1986). D. L. Kuchar, C. W. Thoburn, N. L. Sanvnel, "Prediction of 16. Serious Arrhythmic Events after Myocardial Infarction: SignalAveraged Electrocardiogram, Holter Monitoring and Radionuclide Ventriculography," JACC vol 9., No.3, March 1987, pp 5318, (Austral ia). 17. E.S.Gang, T. Peter, M.E.Rosenthal, W.J.Mandel, and Y.Lass, "Detection of Late Potentials on the Surface Electrocardiogram in Unexplained Syncope," Am. J. Cardiol. vol. 58, Nov. 1986, pp 10141020. 138 Use of Wavelet Transforms in the Study of Propagation of Transient Acoustic Signals Across a Plane Interface Between Two Homogeneous Media s. Ginette 1, A. Grossmann 2 , and Ph. Tchamitchian 3 1Laboratoire de Mecamque et d' Acoustique, Equipe Ultrasons, C.N.R.S., 31, Chernin 1. Aiguier, F-I3402 Marseille Cedex 09, France 2Centre de Physique Theorique, Section n, C.N.R.S. Lurniny Case 907, F-13288 Marseille Cedex 09, France 3C.P.T. et Faculte des Sciences et Techniques de Saint-GerOme, F-13397 Marseille Cedex 13, France 1. Introduction. The problem we study can be defined as follows: In three-dimensional space, we consider two homogeneous media - "air" and "water" - separated by plane interface. There is a point source of sound in "air" at height h above the interface. Its emission is given by a function F(t) of time. We are interested in the behaviour of pressure in water, at time t, depth z, and distance r from a vertical line going through the source. This problem has been discussed innumerable times, both for monochromatic and transient sources [1-6]. The feature of interest to us is the existence, first recognized in the monochromatic regime, of contributions to the total solution called lateral waves. They tend to be concentrated near the interface, and have properties of propagation and attenuation different from waves in homogeneous media. They decrease exponentially with depth, and their attenuation length is frequency-dependent. Consequently, their penetration depth decreases with frequency. For a recent discussion, see e.g. [7]. It should be mentioned that this attenuation does not correspond to absorption of energy; a lateral wave is merely a contribution to an elastic propagation. In the case of time-limited sources, one can see the contributions of lateral waves arriving at times different from the "geometric" contributions. At sufficiently shallow depths, they can contribute significantly to the the total acoustic field. The classical methods of resolution, while well adapted to the case of monochromatic sources, are less suited to the description of transients. This is due to the fact that waves of different frequencies both follow different paths and undergo different attenuations. In such time-and-frequency dependent situations, it is natural to apply wavelet transform techniques. As a matter of fact, the main motivation for the introduction of wavelet methods in J. Morlet's work was the need, in geophysics, to study frequency-dependent propagation phenomena. In the first part of this paper we shall briefly describe the time behaviour of the wavelet transform of the acoustic field at a fixed point under water. (This is not the same as attempting to use wavelet techniques to solve the partial differential equations of the propagation problem). It is straightforward to write down an expression for the wavelet transform of the propagator for our problem. This expression is much smoother than the propagator itself, and allows selective reconstitutions with arbitrary precision. It has a natural decomposition into three contributions corresponding to branch-points of the integrand. 139 These contributions, which we study separately, are obtained without the help of stationary phase approximations which are used in the standard definition of lateral waves. Nevertheless, the main features of the geometric and lateral waves can be seen in our decomposition. In order to avoid possible confusions we enclose our names in quotation marks. The expressions, given below, have been numerically evaluated. The results are discussed and displayed graphically. In a second part, we use the different contributions to obtain a formula for reconstruction of the time-dependent source. 2. Expr ess ions for the pr opagat or. All quantities of interest can be calculated from the scalar acoustic potential <1>(r,z,t). In the case described above, its Fourier transform with respect to time is given by the integral (1) below, where the notations are as follows: C1: sound velocity in air; n: the refraction index for water: one has n <1. m: ratio of densities of the two media: one has m >1. A r(co) (1) f· 1 =~ Co Cj,)(r,z,co) e'ClltF(t)dt; . Jo: Bessel function. f _ gl' = i sgn(co) C} " " ; ; r(co) Ico I (ucor) 1 Jo ~ N(co,u,z) D(u) u du , o where N(co,u,z) and D(u) = exp{ . I lcol.~ sgn(co) [h\' 1-u 2 + Z~ C1 }; = m~ + ~ Here, for u > n, the expression "n 2 -u 2 is defined to mean i sgn(co)" I (n2 -u 2) I. Similarly, for u >1, we define" 1-u 2 as i sgn(co)" 1(1 -u 2) I. We shall divide the integral over u into three contributions, corresponding to the branch-points in the integrand: The first contribution, which we call the "geometric", is defined by (1) where the integral over u goes from 0 to n. The second, or "lateral" contribution corresponds to n < U < 1; the last one, "evanescent", corresponds to u >1. These names have been chosen by analogy with the terminology used in the case of the monochromatic source. 3. Wavelet transform of the propagator: formulas. In order to study the behaviour of the three components of <1>, we introduce their continuous wavelet transform with respect to an analyzing wavelet g(t), which we assume progressive (Le. without Fourier components for negative co; Le. g(t) is what is known as an analytic signal). With the conventions of the present paper, the wavelet transform of each of the three components (2) is given by: 140 (3) (L<1>)(b,a) = -va of<£j(r,z,oo) g(aoo) e-ibrodoo. (j = 1..3) A We have evaluated these expressions in the case where F(oo) = const, i.e. where the source emits a Ii-function pulse. The analyzing wavelet, in frequency space, was chosen as g(oo) = exp(- (00 - ~. 5) 2} this is the "standard Morlet wavelet" which has been extensively used (See e.g. [8]). For all practical purposes, it is admissible and progressive. In our analytic and numerical evaluation, we do not make any use of asymptotic estimates. 4. Wavelet transform of the propagator: numerical results. In the numerical evaluation of the propagator and of its wavelet transform, we have used algorithms of the Romberg type, adapted to the somewhat singular behaviour of the integrand at the boundary. In all the graphic representation, it is the level lines that are displayed. The modulus of the transform is shown in logaritmic scale, with a dynamic span of 40db. The phase representation is linear, between -1t and 1t. The color scale is in increasing order: mauve, blue, green, yellow, orange, red and black. For phases, mauve has been replaced by red, for the sake of better visualization. The values of m, n chosen for the computation are respectively 800 and 0.2266 The figures 1p (p for phase) and 1m (m for modulus) correspond to an observation point directly under the source (i.e r= 0), at a finite depth z under the interface (z/h = 0.2). The functions displayed are the phase and the modulus, respectively, of the wavelet transform of the contribution <1> 1, considered as a function of time. The moment of arrival of this contribution can be seen as the vertical line of constant phase on Fig 1p, corresponding to the abscissa a little over 3.0. The same abscissa appears as a line of maximum modulus on any horizontal line on Figure 1m. This result is a very clear indication of the arrival of a discontinuity. Quantitatively, the position of this discontinu ity corresponds exactly to the calculated time of arrival. In figures 2m and 3m the pOint of observation is at a finite radial distance from the source (namely r/h = 1), and at a shallow depth (z/h = 0.1). In figure 2m, one sees the modulus of the wavelet transform of the "geometric" contribution <1>1' The figure 3m, which uses exactly the same coordinate system and graphical conventions as 2m, displays the modulus of the wavelet transform of the "lateral" contribution <1>2' It is very instructive to compare the two pictures. One notices that the "geometric" contribution arrives before the "surface contribution" , in agreement with theory. The figure 4m describes the "lateral" contribution at r/h = 2. We have also calculated, independently, the contributions to the propagator. The results, for the modulus of "lateral" contribution, are shown in Figs. 5 and 6 for different positions of the observation point. A more systematic and quantitative discussion is in preparation [9]. 141 Fig.l.p Figure caption see page 144 6 r--------~--~----~------------------ o Fig.l.m Figure caption see page 144 142 2 3 4 1 2 3 5 6 7 Fig.2.m Figure caption see page 144 Fig.3.m Figure caption see page 144 143 Figure 1P : Phase of the wavelet transform of the "geometric" contribution. The observation point is under the source, r/h = 0 at a depth of z/h = 0.2. The smallest dilation parameter is equal to 0.3 . The number of voices per octave is 5 in a range of 5 octaves. Figure 1m : Modulus of the wavelet transform of the "geometric" contribution with the same parameters of the figure 1p. Figure 2m : Modulus of the wavelet transform of the "geometric" contribution at a radial distance from the source r/h = 1 and a shallow depth of z/h = 0.1. Here, the smallest dilation parameter has been taken equal to 0.15, with 5 voices per octave. The number of octaves is 6. Figure 3m: Modulus of the wavelet transform of the "lateral" contribution at the observation point of the"geometric" contribution seen in figure 2m. The analysis of the signal was performed with the same scale parameters and the same graphical conventions as in figure 2m. Figure 4m: Modulus of the wavelet transform of the lateral contribution where the radial distance is r/h = 2. The signal was analyzed with scale parameters equal to those of figure 2m. 144 Figure 5 Modulus Time la Modulus Figure 6 o Time 12 II la Figures 5, 6: Modulus of the contribution of lateral waves to the propagator (Green function) for r/h=1 and r/h=2 respectively. la = artefact due to the decomposition into geometric and lateral contribution seen [9] for a detailed explanation. 11 , 12 = delays appearing in formula (4). II = time of arrival of the "lateral" wave. 5. A reconstruction formula. We now return to the consideration of a point source with arbitrary time variation F(t). Our aim is to reconstruct F(t) from information obtained through "underwater listening". It turns out that there exists a simple reconstruction formula that involves the three contributions, evaluated at all depths for a fixed horizontal distance r. The notations being as above, define P(r,z,t) as the pressure obtained from the total acoustic potential <l>. The formula to be written is valid asympotically as r/h -->00. Consider I(r,t) = fP(r,z,t)dz. "r2+h 2 h~ r Define t2 = -C-l- and tl = - - - + - . Cl C2 We have then: (4) I(r,t) ~ C(r,h)F(t-t,) + K(r,h) F(t-t2) where C(r,h) and K(r,h) can be explicitly calculated. If t2-t 1 is larger than the duration of the emitted sound F(t), we obtain the exact signal emitted in air repeated twice but with different amplitudes. We have not yet performed any numerical evaluations of (4). 145 Acknowlegdements: One of the authors (S. G.) thanks Dr C. Gazanhes for helpful advice. This work has been partially supported by the company DIGILOG. References. [1] Gerjuoy, E., Refraction of Waves from a point source into a Medium of Higher Velocity, (1948) 1442-1449. Phys. Rev. :za [2] Brekhovskikh, L. N., Waves in layered Media (Wiley, New York) 1960, pp. 292-302. [3] Gottlieb, P., Sound Source near a Velocity Discontinuity, J. Acoust. Soc. Am. 1117-1122. ~ (1960) [4] Towne, D. H., Pulse shapes of Spherical Waves Reflected and Refracted at a Plane Interface separating Two homogeneous Fluids, J. Acoust. Soc. Am. M (1968) 65-76. [5] De Hoop, A. T. and Van Der Hijden, J., Generation of Acoustic Waves by an Impulse point Source in a fluid/solid configuration with a Plane boundary, J. Acoust. Soc. Am. I I (1984) 1709-1715. [6] Casserau, D., Nouvelles Methodes et Applications de la Propagation Transitoire dans les milieux Fluides et Solides, These de I'U. E. R. Paris VII (1988). [7] Saracco G., Transmission acoustique Ii travers Ie dioptre air-eau, J. Acoust. (1988) 79. 1 71- [8] Grossmann, A. and Morlet, J., Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM Journ. of Math. Analysis, ~(1984) 723-736. [9) Grossmann, A., Saracco, G., Tchamitchian, P., Study of propagation of transient acoustic signals across a plane interface with the help of the wavelet transform, in preparation. 146 Time-Frequency Analysis of Signals Related to Scattering Problems in Acoustics Part I: Wigner-Ville Analysis of Echoes Scattered by a Spherical Shell J.P. Sessarego 1, J. Sageloli 1, P. Flandrin2, and M. Zakharia 2 lCNRS Laboratoire de Mecanique et d' Acoustique, 31, Chemin Joseph Aiguier, F-13402 Marseille Cedex 09, France 2ICPI Laboratoire de Traitement du Signal (UA 346 CNRS), 25, Rue du Plat, F-69288 Lyon Cedex 02, France Numerous studies (both theoretical and experimental) have been devoted to the problem of acoustical scattering by targets of simple shapes. This paper will show some of the main results which have been obtained. It will point out the encountered problems and the classical signal analysis tools which are available. Many methods have been proposed to characterize acoustical scattering : some of them operate in the frequency domain (separation of resonances). the other ones operate in the time domain (observation of acoustical paths). In both cases. one cannot take into account all the physical phenomena encountered in the echo formation mechanisms. We propose here a new method which makes use of a joint time-frequency signal representation. The chosen representation is the Wigner- Ville distribution. The method has been applied to the analysis of the backscattering echo of a spherical shell (duralumin) immersed in water. INTRODUCTION Many papers dealing with acoustical scattering by elastic targets have been published during these last ten years. They provide us with many results in both theoretical. numerical and experimental domains. For instance. they emphasize the importance of resonance phenomena in acoustical scattering [1-21. Resonances are intrinsic parameters of the target. They are closely related to its physical properties and to its geometry. We can thus understand the great interest in identifying those resonances. both theoretically and experimentally. Various methods have been proposed to extract these target characteristics: they have been designed either in the frequency domain [2-31 or in the time domain [5-61. In the frequency domain. these methods lose completely the temporal structure of the scattered field and cannot take into account informations on the non-stationarities included in the echoes. On another hand. if we deal with the temporal response of a target. we lose the frequency content of the echoes and we can even find difficulties in separating events which are very close in the time domain but which occur at different frequencies. To avoid these difficulties. we propose to make use of a joint time-frequency representation in order to "spread" informations related to the scattered field in a timefrequency plane. The signal representation we chose is the Wigner- Ville distribution. Apart from a very suggestive graphical display, we will demonstrate the ability of this representation to provide quantitative parameters for echo classification. In this 147 paper we will show, as an example, results obtained with a spherical shell, but the proposed method is very general and can be applied to any type of target. ACOUSTICAL SCATTERING BY ELASTIC TARGETS: THEORETICAL ASPECT. Let us consider a incident plane wave reaching an elastic sphere immersed in a fluid (water). In the back scattering situation (emitter and receiver at the same point) the acoustical field can be written as : ~ Pd'CC(r) I -i(o)t '" n = Po e k i 0 n· D(I) n (I) (2n + I) h (kr) P (cos e) , Dn n n where: -PO is the amplitude of the incident wave; -k is the wave number in the fluid; -r is the distance from the observation point to the center of the sphere; -h n (I) are the spherical Hankel functions of first kind; -Dn and Dn (1) are determinants that take into account the limit conditions. In the case of a spherical shell Dn are 6X6 determinants 171. For given values of the frequency (or of the ka parameter), the denominator of the previous expression is equal to zero: this defines the target resonance frequencies. Recent works [2, 8-9) have shown that these resonances are closely related to surface waves running around the target. For a cylindrical target, for instance, we get a resonance when the Ith surface wave travels exactly n wavelengths around the cylinder (8). For a spherical target one can show that the same phenomenon happens for a travel of n+ 112 wavelengths around the large circle. These results provide us with experimental means to identify a target by putting into correspondence its resonances with the associated creeping waves. CLASSICAL METHODS USED FOR ECHO CHARACTERIZATION Classical methods used for echo characterization can be divided in two main groups: - temporal methods [4-5), used in impulsive mode: one tries to determine the arrival time of energy packets. Fig. 1 shows such time events that one could try to identify. The first component is due to the specular echo. It is followed by various surface waves. These waves can be rather dispersive and have different celerities. After some time, they begin to overlap in the time domain, which makes the time measurement very hard and even impossible. - spectral methods 11-3), where we observe F",,(ka), the farfield form function. This function is related to the scattered pressure by Pdiff a = Po 2r e i(kr-(o)t) Fjka) The form function is responsible for the "resonnant behaviour" in the spectral domain in the case of wideband echoes (superposition of optical echo and of surface waves). Fig. 2 shows the energy spectrum of the spherical shell echo displayed in Fig.l. Experimental methods have been developed in order to extract resonances from steady state echoes [11). Nevertheless, they find their limitation in the case of close resonances where the results can be wrong or incomplete. It is important to notice that 148 IQ* ,r - -r - - - - - -- 4 - - - - ----, 1.9 I .J1A..1 Q. 9 - .. ,lh, ~f]" 9.5 ·2,~Ix IQIE-4 s. I 2.9 <1916Hz 9. Q, ........-=..L-.l-J....L..""'-'--'-J.....J......1..--.l-............J..-I.--'-J.....J...-'-l--L..J U E.i&..YItl: Echo backscattered by a spherical shell of Duralumin, Ei&.uIT.2.: Power spectrum of the echo displayed in external radius a - 30 mm, b/a - 0.9 (b : internal radius). Experimental data: impulse response. Fig. I some very close resonances are due to very different surface waves which travel around the target at different celerities. For the considered example (Fig. I). Table I displays characteristic parameters of some of these waves. Table I Some characteristic parameters of surface waves (spherical shell) ([ characterizes the different surface waves [3]) 1=10 1=2 1=10 1= 1 ka Cl 6.53 6.83 5.37 5.37 1992 6762 1450 839 Table I shows that surface waves corresponding to very close resonance frequencies can possess very different celerities. This leads us to propose a joint time-frequency representation of the echoes in order to separate phenomena that cannot be separated either in the frequency domain or in the time domain. WIGNER-VILLE DISTRIBUTION This distribution. first proposed in Quantum Mechanics by E.P. Wigner. has been introduced in Signal Processing by J Ville [11-12]. Its general expression is given by: +" W (t x ' v) = f z( t t. + -) Z 2 t (t - -) e 2 -i2""" dt , -II where z(t) is the analytic signal associated to the temporal signal (with finite energy) x(t) (the echo) such that z(t)= x(t) + i qx(t), where qx(t) is the quadrature signal of x(t). 149 Several theoretical considerations show that this type of joint representation possesses a nu mber of nice properties [11-121. Especially. it allows a nice localization of the signal energy along both the time and the frequency axis. From a practical point of view. we will use a discrete version. which will also be smoothed [121: N- I SPW x(t.v)=2 with: I n - -N Ih(ndKM(t.n)e-i4ltVn + I M-I KM(t. n) = I g(m) z(t + m + n) ZO(t + m - n). m - -M+I where: -h(n) is a short-time (frequency smoothing) analysis window (2N-I points) ; -g(m) is a time smoothing window (2M-I points). This formulation is very convenient as it admits a simple numerical implementation using fast Fourier transform algorithms. It also allows a clear separation between the time resolution and the frequency resolution. Although such a transform seems easy to implement. the interpretation of results is rather intricate and it often needs a priori informations about the phenomena we want to analyze. In this paper. we have applied this analysis to a spherical shell echo. We can therefore predict exactly the acoustical behaviour of the target and compare the results obtained from the analysis to parameters already known theoretically [13-1 SI. EXPERIMENT AL RESULTS The analyzed echo is the one described in Fig. 1. The experiment has been done in water with a wideband transducer and the result of the analysis is given in Fig. 3. One can notice the arrival of energy packets (numbered from 1 to 7). The first packet corresponds to the optical echo; the other ones are due to the successive arrivals of the surface waves as they travel a given number of times around the target. From a qualitative point of view. one can say that the surface waves lose some of their energy each time they turn around the target. This loss increases with the frequency. We also can notice that the surface waves do not have the same propagation velocity at all frequencies (lower speed for higher frequencies). This velocity dispersion shows up clearly in the shape variation (tilt) of the packets. More quantitative informations can be extracted from the time-frequency plane. As an example. let us consider the time interval between two successive arrivals of comparable waves (packets n° 2. 4. S. 6. 7) at a given frequency. As we know the target shape and size. we can compute the velocity dispersion of the surface waves as a function of frequency. These results have been summarized in Fig. 4. where we also present theoretical results [71. The good agreement between experimental data (extracted from the Wigner-Ville analysis via centroids of each packet [14)) and theoretical data (obtained from acoustical derivations [7)) shows clearly the ability of a Wigner- Ville analysis to provide quantitative informations about physical phenomena involved in echo formation mechanisms. 150 - 9 h 1 7 T IM E ~: Wigner-ViUe analysis of an echo of a spherical shell (Fig.l) display parameters: 20 ~s/div. 200 kHz/div. 3 dB/gray level. 6000 r-- - - t -- +-_ + _ +--+ -.-.,-.-. e:-:-.~ -e-, ...:----t_ 200 0 .5 ~: 1.5 f.e (M hz . mm) Velocity dispersioll Theoretical values (+) and experimental values (el obtained from Fig. 3. 151 CONCLUSION We have showed, on a real case, the interest of using a time-frequency representation to study acoustical target echoes. While the classical methods fail (time separability or frequency resolution problems), the Wigner-Ville representation of the echoes can still give informations on the physical phenomena involved in echo formation. This is due to the fact that it "spreads", in the time-frequency plane, informations that we usually know only through their projections (in time or in frequency). It is important to notice that, while using a wideband signal (impulse), the whole analysis is done on a single echo. This explains obviously the interest of such an analysis for moving targets. The simplicity of obtaining the acoustical information (single echo) finds a counterpart in the complexity of the resulting images and in problems related to their interpretation. Nevertheless, given some training, it is possible to extract quantitative informations from theses images such as velocity, attenuation, resonance frequencies, ... The extension of such analysis could lead to new target recognition schemes based on time-frequency displays of echoes. ACKNOWLEDGEMENTS This work was supported by DRET (Direction des Recherches Etudes et Techniques). The authors wish to thank J.L. Rousselot (LCT AR) for his kind collaboration. REFERENCES [1) D. Brill, G.C. Gaunaurd, "Resonance theory of elastic waves ultrasonically scattered from an elastic sphere", } Acoust. Soc. Am. 81 (1), pp. 1-21, 1987. (2) D. Brill, G.C .Gaunaurd, H. Oberall, "Acoustic spectroscopy",} Acoust. Soc. Am. 72(3), pp. 1067-1069,1982. (3) R. Burvingt, J.L. Rousse10t, A. Derem, G. Maze, J. Ripoche, "Reponse acoustique de cylindres elastiques", Hev. CBTH£1J£C, 21 e annee (78), pp. 73-94, 1984. (4) M. Fekih, G. Quentin, "Presentation et interpretation des experiences de diffusion par des cylindres elastiques aux va1eurs elevees de ka", Hev. CBTH£1J£C, 1ge annee (72), pp. 91-102, 1982. (5) M. Ta1mant, "Retrodiffusion d'une impulsion u1trasonore breve par une coque cylindrique a paroi mince", These Univ. Paris 7, 1987. (6) G. Deprez, R. Hazebrouck, "Diffraction d'une onde impulsive spherique par une sphere creuse plongee dans l'air", Hev. CBTH£1J£C, 1g e annee (72), pp. 73-90, 1982. (7) J.L. Rousselot, A. Gerard, J.P. Sessarego, J. Sageloli, "Reponse basse frequence d'une coque spherique mince immergee", Acustic/I , to appear. (8) J.L. Rousse10t, "Relation entre les ondes de surface et les resonances dans un cylindre elastique creux", Hev. CBTH£1J£C, 1g e annee (72), pp. 47-60, 1982. (9) H. Oberall, L.R. Dr agonnette , L. Flax, "Relation between creeping waves and normal modes of vibration of a curved body", } Acousl Soc. Am. 61 (3), pp. 711- , 1977. (10) G. Maze, J. Ripoche, "Methode d'isolement et d'identification de resonances (M.I.I.R.) de cylindres et de tubes soumis a une onde acoustique plane dans l'eau", Hev. Phys. Appi 18, pp. 319-326,1983. [11) T.A.C.M. Claasen, W.F.G. Mecklenbrauker, "The Wigner distribution - A tool for timefrequency signal analysis", Philips} Hes. 35, pp.217-2S0, 276-300, 372-389,1980. 152 [12) P. F1andrin, B. Escudie, "Principe et mise en oeuvre de I'analyse temps-frequence par transformation de Wigner-Ville", Traitemeot dl/ 5'jgoa12(2), pp. 143-151. 1985. [13) N.C. Yen, "Time and frequency representation of acoustic signals by means of the Wigner distribution function: implementation and interpretation", J ACOl/st. Jac. Am. 81 (6), pp. 1841-1850, 1987. [14) P. F1andrin, J Sageloli, JP. Sessarego, M. Zakharia, "Application of time-frequency analysis to the characterization of surface waves on elastic targets", Acol/st. Lett. 10(2), pp. 23-28, 1986. [15) P.F1andrin, j. Sageloli, J.P. Sessarego, M. Zakharia, "Application de l'ana1yse tempsfrequence a la caracterisation de cibles", Onzieme Colloque GRETSI, Nice, pp. 329-332, 1987. 153 Coherence and Projectors in Acollstics J.G.Slama COPPElUFRJIPEM, Caixa Postal 68503, 21944 Rio de Janeiro, Brasil Present address: LMNCNRS, 31, Chemin Joseph Aiguier, F-13402 Marseille Cedex 09, France A general method to decompose a stationary random sound field in the frequency domain is presented. This method is based on a geometrical interpretation of the partial coherence theory proposed by Dodds[1] and can be applied to the fundamental study of sound sources and fields in the low frequency domain [2][3]. 1. The cross spectrum as a hermitian scalar product. Given two stationary random processes X(t) and Y(t), a heuristic notation is proposed where the cross spectrum Sxy(f) of the processes is written as a hermitian scalar product. Sxy(f) = {X(f)IY(f)} as in Dirac notations in quantum mechanics, IY(f)) is a ket and {X(f)1 is a bra. We can derive from the properties of the cross spectrum function, the properties of the bras and kets. For example, If X(t) is the input of a linear system and Y(t) is the output then we have: Y(t) = H(t)*X(t) IY(f)} = h(f)lX(f)} {Y(f)l = h'(f){X(f)1 Where h(f) is the transfer function of the system, the Fourier transform of H(t) * is the convolution operator. 2 coherence. 2 1 Coherence function. Consider two stationary random processes X(t) and Y(t); we can find a deterministic function H(t) as a solution of a minimization problem: Y(t) = H(t) * X(t) + N(t) = C(t) + N(t) where N(t) is a noise with minimum power EqN(t) 12} minimum. * is the convolution operator. EqN(t) 12} is minimum also signifie that the power spectrum SNN(f) of the noise is minimum for every frequency. The H(f) which minimises the spectra of the noise is given by: 154 Sxy(f) = Sxx(f) H(f) and the spectrum of the noise is: where G xy(f) is the classical coherence function between X(t) and Y(t). Crt) is the part of Y(t) which is coherent with X(t) N(t) is the part of Y(t) which is incoherent with X(t) Gxc GXN =1 = 0 GNC = 0 2.2 Coherent sets: Now we consider a field of stationary random processes Y(a,t) depending on the parameter a, we can find a new field C(a,t) by solving the minimisation problem: Y(a,t) = H(a,t) • X(t) + N(a,t) = C(a,t) + N(a,t) where N(a,t) is a noise with a minimum power spectra. The set of C(a,t) for a fixed X(t) is a coherent set which is a field coherent with X(t). We have decomposed the field in two parts:the part C(a,t) coherent with X(t) and a noise field incoherent with X(t). 2.3 Spectral properties of a coherent set A fundamental property of a Coherent set is that his cross spectrum function can be written as a product. it is possible to define an equivalent complex amplitude cp(a,f) cp(a,f) = h(a,f)-v {X(f)IX(f)) In the case of acoustic fields, a is a spatial position in the field and X(t) can be the sound pressure for a value of a.The selective spectrum can be used to perform studies of the contribution of a noise source in a factory noise. 155 3 projectors. We consider the expression: Yet) = H(t) • X(t) + N(t) = C(t) + N(t) where N(t) is a noise with a minimum power. Then C(t) can be considered as a projection of Yet) on the reference X(t). Considering the ket IC(f)} corresponding to C(t),we have: ~ IC(f)} =Sxx(f) IX(f)} and by using the scalar product for the cross spectrum: IC(f}} {X(f)lY(f)} IX(f}} {X(f)lX(f)} {X(f)IY(f)} IC(f)} = IX(f}}{X(f)lX(f)} IC(f)} - IX(f)}{X(f)l IY(f)} {X(f)lX(f)} Then we can write : IC(f)} = [X(f})IY(f}} where [X(f») is a projector. [X (f}) _ IX(f)}{X(f)l - {X(f)lX(f)} we have to assume the associative property (lX(f)}{X(f)l)IZ(f)} = ({X(f}IZ(f)})IX(f)} In the case of a continuous field of random functions,the part of the field which is coherent with a reference (a COSET) is found as a projection of the field on the reference (in the frequency domain). IC(Il,f)} = [X(f})IY(Il,f)} This interpretation is interesting when we apply the coherence theory in the case of a continuous field af stationary random functions. 156 4.Decomposition of the unity: We can write 1=[X(f)) +(1-[X(f))) If we consider a cross spectra between two stationary random function, {Y(f)IZ(f)} we can introduce the decomposition of the unity in this product and we find: {Y(OIZ(O} ={Y(OI1(Z(O)} ={Y(f) I[X(f) )+(1-[X(f) ))Z(f)} ={Y(f)l [X(f) IIZ(f)}+{X(f) I(1-[X(f)))Z(f)} {Y(f)I[X(f))IZ(f)} is a partial representation of the cross spectrum of the parts of Y(t) and Z(t) which are coherent with X(t). It can be considered as a selective spectrum or as an estimation of the cross spectra between Y(t) and Z(t). The same representation can be used in the case of multiple coherence theory where a projector can be proposed where the field is projected on a set of references. In this case it is interesting to use an orthogonal set of references. This work was carried out with the support of CNPq Brasil. [1] Dodds C.I. Robson 1.0. "Partial coherence in multivariate random process"Iournal of Sound and Vibration Vol 42, 243-249 (1975) [2] Slama I.G. "Methode pour la representation de champs sonores aleatoires" FASE 87 , Lisbonne (1987) [3] Glegg S.A.L. "The accuracy of source distribution measured by using polar correlation",Iournal of Sound and Vibration, Vol 80 N°l, P 30-40,(1982) 157 Wavelets and Granular Analysis of Speech I.S. Lienard and C. d' Alessandro LIMSI-CNRS, BP 30, F-91406 Orsay Cedex, France 1 - Very short time analysis of speech The speech signal comes from the convolution of a source signal - due to the vibration of the vocal cords or to the airflow through a narrowing of the vocal tract - with the impulse response of the vocal tract. Both constituents rapidly change over time, and one usually considers that the phonetic information in the signal is mainly related to the evolution of the two or three first resonances of the vocal tract, called "formants" F1, F2, F3. The vocal cords vibrating frequency, Fo, is closely related to a perceptive quality of sounds called "pitch". In order to extract the phonetic information the signal is considered to be stationary over a time interval long enough to include several pitch periods, and short enough to capture the evolution of the spectral envelope. The usual tradeoff yields some 25 ms for the width of the analysis window, and 10 ms for the interval between successive windows. So, despite the fact that everybody agrees about the relevance of classical spectrographic analysis - which uses bandpass filters 300 Hz wide, with impulse responses as short as a few ms - the information extracted from the signal for transmission or recognition is altered from the start in the time dimension. Some rapid transitions in consonants are smoothed or even erased, the sound structure disappears, the possible perceptual interaction between segmental and prosodic information is deliberately discarded. The "granular" analysis we present hereafter aims at decomposing the signal into a set of discrete elements associated with energy concentrations in the time-frequency coordinates, with some emphasis on the time resolution (in the 1 to 2 ms range). In the voiced segments (vowels, some consonants) those grains correspond to the resonance maxima of each proper mode of the vocal tract, at each pitch period. In the noise segments (consonants such as "s" or "ch") the grains are randomly distributed in some region of the time-frequency plane. Finally the bursts found at the onset of some sounds like "p" or "t" give birth to one or several grains precisely located at the same instant, following a silence. In our view of speech analysis the notions of voicing, pitch, formants, for which no method gives a completely satisfactory answer, cannot be directly extracted from the signal, but should result from a structural study of the grain distribution. For instance the signal will be declared as "voiced" when, locally, comparable grains appear at regular intervals. 158 This analysis is based on hypotheses about the temporal coding of the acoustical wave by the human auditory apparatus. Consequently it is tempting to implement an auditory model in order to check them. However, it is difficult to validate and interpret the results of such models, because they implicitly take into account some further processing by the brain, which cannot be modeled or understood as yet. Thus we choose to implement the granular analysis in a way such that objective or subjective verification is permitted through a reconstruction process (ref 1). The analysis process is composed of two steps. The first one aims at decomposing the signal into a series of narrowband signals, covering the frequency band of speech, i.e. from 70 to 5000 Hz. The second step consists in modeling each of them into a series of successive elementary waveforms. At the present time the first step only can be related to the theory of wavelet analysis. 2 - Decomposing the speech wave into a set of narrowband signals In order to reconstruct the signal by a mere addition of its narrowband components it is necessary that all of the filters respond with the same phase, have the same slopes, and have their gains properly adjusted with respect to the distribution of the center frequencies along the frequency scale. We implemented a recursive filter structure, used twice with time-reversal in order to cancel any delay or phase distortion. The result is a zero-phase filter, the order of which is twice the initial filter order. For the basic unidirectional filter we choose a simple resonator (second order), so that the resulting filter is of order four (slopes at infinity tend toward -12 dB/octave, slopes around the cutoff frequencies depend on the quality factor, Q). The distribution of the center frequencies along the frequency scale is one of the filterbank parameters. We used several tunings, ranging from linear to logarithmic, with an intermediary choice (Bark scale) close to what is known of ear physiology (tendency toward the linear scale in the low frequencies, toward the logarithmic scale in the high frequencies). The gains are automatically adjusted with respect to the number and distribution of the filters. Usually the number of filters is between 12 and 32, the bandwidths range from 100 to 600 Hz, the quality factor remains within the 1 to 10 range. The filtering process is illustrated on Fig 1, which shows the decomposition of a series of impulses into 16 linearly distributed channels, as well as their additive reconstruction. Except for some noise due to the poor bandpass limitation of the signal, it is clear that the reconstruction is satisfactory. Fig 2 shows the analysis of a speech signal, with some differences in the filterbank parameters, and a different representation of the output signals : only the positive parts of each signal are represented, after logarithmic compression of the amplitude. This repre159 sentation exemplifies the synchronization phenomena occurring among adjacent channels when several filters capture the same signal component. Here again, reconstructing the signal through summation of the outputs yields a signal very close to the original. When listening to both, no difference can be heard. Channels I a I b I Fig 1 - Decomposition of a series of pulses (a) with a zero-phase filterbank, and signal reconstructed (b) by summation of the 16 output signals. Time scale 10 ms between vertical dotted lines (valid for all the figures in the present paper). '., kHz : ! • I 4 ---' 3 2 Irl-b-=---+--: I . I I , I I I t I I I I I I I I , I Fig 2 - Vowel "a" analysed with a 32-channel linear filterbank; a) original, b) reconstructed signal. 160 Basically our analysis process consists of the convolution of the signal with a symmetrical "wavelet" made of the bidirectional impulse response of the filter. Comparison with the Morlet and Grossmann wavelet theory yields some remarks, which can be classified as analogies and differences. The first similarity is to be found in the need to a better mastery the tradeoff between time and frequency resolutions. In both cases a better time resolution is expected in the high frequency part of the spectrum, while the frequency precision is expected in the low frequency part. Another important similarity lies in the additive reconstruction possibility. As for the differences, the formal expressions given by the wavelet theory are obviously an advantage, thanks to the insights and guarantees they provide. On the other hand, the logarithmic frequency scale imposed by the wavelet shape conservation in its compressions and dilations may not be perfectly adapted to our psycho-physiological needs, and is not mandatory in our approach. In most of our experiments the equivalent "wavelets" composed of the bidirectional impulse responses of the filters were closer to the Gabor type than to the Morlet-Grossmann type. Finally the last noticeable difference lies in the implementation of a recursive IIR filter, which allows the computations to be achieved very quickly, several orders of magnitude faster than the wavelet analysis. The filter used has a low Q factor, and does not cause any stability problem. 3 - Modeling the output signals Into discrete elements If speech processing could be reduced to decomposing the signal into n narrowband signals, we would have gained nothing; the information rate of the signal would simply be multiplied by n. We are actually looking for a decomposition Into a set of discrete elements, or grains, which we call Elementary Waveform Models, or wfms (fig 3). This decomposition has been validated for singing voice synthesis (ref 2), but our problem is the opposite, I.e. how to go from the signal to the list of grains ? A first way to deal with this probleme is to decompose each narrowband signal into a string of wfms. 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III 1111, '11111111111 11111,1 11111,111111 I I I I I I I I I I I I I I I • , '.1I II I I · , I I I I I I I I I I I I I I I 1 I I , I I I I I I I I I I I ,. • I I I I I II 1' , 1 I I I I , I I , I I , I ' II I I I I I I rI I , , I I l,l " I' "n T: ~ "' ..••• {;,tI;'1~W'Y'T~~~'~ 11 ' 111111 1 " 11 ,11111 111 111 I I I ' I ' • I I I I I ~W" • I U, L I I •+ • I ' I I,' ~"',~ I. I I I I I ' '1111 t I I 1 1 1 I111 Fig 4 - Speech segment /atyvys;J/ displayed as a set of wfms in the time-frequency plane. (a) : channel by channel modeling. (b): same, after iterative grouping of adjacent channels. modeled. This process produces a set of wfms for each channel (fig 4a). Reconstruction of the signal by regenerating the wfms and summation of all of the channels furnishes a signal perceptually very close to the original. Even though this represents a considerable economy compared to the set of narrowband signals produced by the filterbank, the channel-by-channel modeling process still is highly redundant. As the filters we use are not extremely selective, one given constituent of the signal has been analysed by several filters, resulting in several wfms in adjacent channels. It is only when the sum of all of the wfms in those adjacent filters is calculated that that the original constituent - grain - can be regenerated. We have therefore created a local grouping procedure for the narrowband signals surrounding any local maximum of the envelope in the time-frequency space. This procedure greatly reduces the total 162 number of wfms obtained (fig 4b), and should ensure their invariance against different filterbank configurations, but some spotting problems appear in the low frequency range where the pitch period and the center frequency of the analysis channel are close to each other. Since the grouping procedure described above is costly and not perfect, a third approach has been elaborated which makes use of the specific structure of the speech signal (ref 3). It consists of filtering a short segment of the signal (some 50 ms, in order to avoid any boudary effect) in the regions of spectral prominence, as evaluated by a classical LPC analysis, rather than in fixed, permanently defined frequency bands. The new regions are frequently reestimated (every 6 ms in our experiments). Each filtered signal is then segmented and modeled by the same procedure, giving the desired grains without the need of a grouping procedure. This approach gives satisfactory results if it is adapted to the lower part of the spectrum (modeling the first harmonics) (ref 4). 4 - Conclusion Our short term analysis, as well as wavelet analysis, has the desire to dominate the compromise between time and frequency resolutions. Both processes may be seen as filtering, or as the convolution of the signal with a particular, symmetric, Gaussian shaped kernel. But beyond this common desire, we are trying to model the signal into a set of discrete elements, or grains, which are supposed to be perceptually pertinent. This aim contributes to defining an inverse problem for which, at present, wavelet theory has no answer. 5 - References 1 - J.S.Lienard : "Speech Analysis and Reconstruction Using Short-Time, Elementary Waveforms", IEEE-ICASSP, Dallas, 1987. 2 - X.Rodet : "Time-Domain Formant-Wave-Function Synthesis", Computer Music Journal, vol 8, 3, 1985. 3 - C. d'Alessandro and J.S.Lienard : "Decomposition of the Speech Signal into Short-Time Waveforms Using Spectral Segmentation", IEEE-ICASSP, New York, 1988. 4 - C. d'Alessandro : "Analyse-SyntMse de la bande de base par formes d'ondes elementaires", 17e Journees d'Etude sur la Parole de la SFA, Nancy, 1988. 163 Time-Frequency Representations of Broad-Band Signals 1. Bertrand l and P. Bertrand 2 lUniversity Paris VII, LPTM, F-75251 Paris, France 20NERA, BP 72, F-92322 Chatillon, France I - Introduction Joint time-frequency analyses were first devoted to the study quasi-monochromatic signals for which frequency translations exist. problem, In familiar the notions axiomatic of time and of the formulation this situation has led to the strong requirement of covariance of the representation by the group of time and frequency However, of if the resulting representations applied to quasi-monochromatic are signals, very they translations. interesting lose meaning general case due to flagrant time localization anomalies. when in This the latter point is illustrated in section II. To solve the localization problem, changes is introduced in section III theory. This leads to the as the affine the consideration group basic of of group general clock of signal affine joint distributions. A subclass is singled out by its interesting properties: it reduces to Wigner-Ville's signals, function it gives the spectrum by time when applied integration to and narrow-band it is time localized when applied to a time localized signal. II - Analytical aspect of the broad-band problem. Given a physical signal s(t) (i.e. a real function of time) it is usual to represent it by the so-called analytic signal Z(t) up to a constant factor, its positive frequency part. which The is, Fourier transform of Z(t) can be noted S(f) and is given by: I 00 2 Y(f) S(f) e-2infts(t) dt (1) -00 where Y(f) is the Heaviside step function. definition the real signal s(t) is just the As real a result part of of this Z(t). The substitution of Z(t) for s(t) introduces the following advantages: a) The effect of a non distording linear represented by multiplication by a complex number. 164 filter is merely b) In the decomposition Z(t) = IZ(t) lexp(U(t» the modulusIZ(t) 1 can be directly interpreted as the real signal and the quantity as (1/2n)(d~/dt) envelope its of the instantaneous frequency. Properties a) and b) are very distinct in nature. first is valid for any signal without restriction, Indeed, the second holds for quasi-monochromatic signals. This obvious remark is the if only at the starting point of our work. A classical time-frequency representation provides generalization of the above interpretation of the analytic is a real function P(t,f) which admits a signal. It quasi-probabilistic a interpretation through the marginalization formulas: 00 J P(t,f) df IZ(t) 12 (2) dt IS(f) 12 (3) -00 00 J P(t,f) -00 The most familiar form of P is J Wigner-Ville~s [1),[2) given by: 00 P(t,f) e-2inut S(f-u/2) S*(f+u/2) du (4) -00 It is an adaptation in signal theory of the Wigner function introduced in quantum mechanics [3). Representation satisfactory when applied to quasi-monochromatic signals. previously (4) is Its use the description of broad-band signals is also possible but can lead surprising images. For example, very for to it is easy to verify that the localized real signal: s(t) which can be represented (eq. (1» S<f) = = 6(t-t o ) by the analytic signal: 2 Y(f) e-2infto admits the representation: P(t,f) 4 Y<f) sin 4nf(t-t ) n(t t o 5 o which is not really localized in a neighborhood of in the low frequency region. In fact Wigner-Ville~s (5) especially t=t o ' function gives a 165 FL~.i - pulse si~l Wi~ner - VLL L e function o f a sharp 8 . 000 f..ot 0'-------- . 700 -. 000 .7000 I.~O O bad time-localization of a broad-band signal. where (5) is plotted in dimensionless units . In recent works [4], it has been shown that the [5] of the Wigner function can be entirely arguments. This is manifest in Fig . l founded on introduction group theoretical In signal theory the group to be considered is that of time and frequency translations whi c h is represented by : S(f) (6) Such transformations are physically well interpreted when applied to narrow-band (i . e . quasi-monochromatic) signals . From a mathematical point of view, the introduction of group (6) implies an imbedding of the subspace of analytic signals into the space of general complex signals in order that it can operate . This does not affect responsible for signals. the the In fact, description bad of pictures narrow-band associated signals with the definition of time localized procedure but is some broad-band signals consistent with (6) leads to functions of the type: St (f) o whose Wigner~s function is 6(t - t o )' The support of frequency region . not restricted to the positive 166 = e 2i1lft 0 (7) functions (7) Hence they is cannot describe any physical signal and the notion of time localization of a real signal vanishes. It results from the above arguments that the weaknesses of Wigner's function are related to the use of (6) as a fundamental group. group of transformations which do not introduce extra physical signals. Such This suggests a reformulation of the problem an approach is developped in the following paragraph by of the affine group of clock changes: t I founded on a consideration a(t+b). III. A class of satisfactory time-frequency representations The affine group G of elements (a,b), a>O, b E ~ acts in the time-frequency half-plane (f>O) by: (a,b) (t,f) It follows that the joint (a(t+b),a distribution -1 f) (8) P(t,f) must transform as: (a,b) P(t,f) (9) where a q , q E ~ is a possible scaling factor. The group G is represented in the space of analytic signals by: (a,b) S(f) with again a scaling factor a r + 1 e-2inabf S(af) r+1 arE~. (10) The relevant invariant scalar product is: J 00 (S,S') S(f) S'*(f) f 2r + 1 df (11) o For each K(v,v') such that K(v,v') P(t,f) f 2r - q +2 J K*(v',v), the distribution e2intf(v-v') K(v,v') S(fv) S*(fv') dv dv' (12) (~+)2 is real and invariant when transformations (9) and (10) simultaneously. Moreover it is the most general are performed sesquilinear form of the signal having this property. Now we put additional requirements on the kernel K in agreement with the kind of time-frequency description we want to realize. of particular interest emerges functions in the (v,v') plane. when kernels are equal In that case the time Fourier to A class delta transform of (12) defined by: 167 I P(y,f) becomes: e-2inty P(t,f) dt I -P(y,f)= f 2r-q 6(y-v+v~) K(v/f,v~/f) S(v) S * (v~) dv (14) dv~ (1R)2 It is connected to the reparametrization of the change of variables function (v,v~) through quadrant. (v,v~) I mere a It follows naturally that the (y,f) has to be one-to-one in that no information be lost in the process. For convenience, we order write the most general diagonal kernel compatible with the above requirements in the form: I 00 K(v,v~) 6(V-A(U» 6(V~-A(-U» ~(u) du (15) -00 where A is a real monotonous function ensuring a bijective correspondence between IR and IR+ and ~ is such that: ~*(-u) (16) ~(u) Further requirements that P has to satisfy include: - Correct time marginal I 00 (17) P(t,f) dt -00 - Good time localization of P(t,f). By this condition we demand that the analytic signal Y(f) f- r - 1 e -2inft 0 transforming as a localized signal, be represented by P(t,f) = 6(t-t o ) f- q - 1 The above conditions lead to the constraints A(O) ~(u) 1 = (18) d IdU (A(U)-A(-U»I Such a choice ensures that band (A(U) A(_u»r+l limited signals are functions limited to the same band in the t-f plane. The family of joint distributions is given by: 168 represented by corresponding J 00 f 2r +2 - q P(t,f) e2irrtf(~(u)-~(-u»S(f~(u» S*(f~(-u» (19) -00 d (~(u) ~(_u»r+1 du lau(~(u)-~(-u»1 where ~ is a function to be chosen satisfying the above conditions (Fig. 2). ~(u) { ~(u) one-to-one: ~(O) = Fie.2 - For q A sketch of admissib~e o [R+-+[R+ 1 ~-functions u 2r+1 representations (19) satisfy the relation J 00 P(t,f) dt -00 and can interpreted be distributions reduce to spectra. instantaneous as the classical Wigner-Ville All these function for estimating the narrow-band signals. Another narrow-band feature is recovered group delay as the conditional mean of time for when given frequency. A straightforward use of (19) leads to: [f t P(t,f) dt To further specify J ;I f P(t,f) ~, P(t,f) - dt] d (1/2rr) aT (arg S(f» we may impose unitarity: P~(t,f) f2q dt df = I (S,S~)12 (20) [Rx[R+ This is the counterpart of an analogous function and is sufficient to determine ~(u) = property ~ for Wigner-Ville~s explicitely. We find: (u e- u / 2 )/(2 sh(u/2» (21) and (19) becomes 169 J co P(t,f) f 2r +2 - q e-2irrtfu S(fA(u)) S*(fA(-u)) -co u 2r +2 (2 sh(u/2))-2r-2 du This was first obtained in [6]. (22) It has the remarkable property that it gives delta functions on a whole family of curves in the time frequency plane. Indeed, signals r>,~ E (23) !R are represented by = P(t,f) f- q - 1 6(t-~-r>/f) Other choices of A are possible = obvious one is A if we (24) give up unitarity. An e U which yields: J co P(t,f) f 2r + 2 - q e2irrtf(x-l/x) S(fx) S*(f/x) (1+1/x 2 ) dx (25) o This function was introduced by A.Unterberger [7] and studied in a mathematical context. practice. It leads to relatively tractable expressions in Again, there are signals (26) represented by delta functions P(t,f) (27) IV. Concluding remarks. We have obtained time-frequency physical signals and representations real anomalies. An application of (22) has been realized through a form in a radar imaging problem [4]. free from adapted arbitrary any to localization smoothed Representation (25) could also be interesting in such an analyois. Beside practical considerations, this study supports the the affine group analysis. to solve localization From this point of view, problems in of it emphasizes the potential interest of the affine wavelets analysis [8] as compared to Gabor's. 170 use time-frequency References [1] J.Ville, Cables et transmissions, n 0 1, pp. 60-74, [2] P.Flandrin, in this volume. [3] E.P.Wigner, Phys.Rev. 40, 749-759, [4] P.Bertrand and J.Bertrand, Rech. [5] J.Bertrand and P.Bertrand, Found. 1932. Aerosp. n05,277-283,1985. Phys. [6] J.Bertrand and P.Bertrand, C.R.Acad.Sc. [7] A.Unterberger, Comm. in Part. [8] A.Grossmann and J.Morlet, Diff. SIAM J. 1948. Eq., Math. 17, 397-405,1987. Paris, ~, 299, 635-638, 1179-1236, Anal., 1984. 1984. 15,723-736, 1984. 171 Operator Groups and Ambiguity Functions in Signal Processing A.Berthon Ste d'Etudes et Conseils, A.E.R.O., 3, Avenue de l'Opera, F-75001 Paris, France The standard ambiguity function in signal processing is well suited for studying signal properties under translations in time and frequency shifts, which belong to the Weyl-Heisenberg group of transformations. Another group, the affine group, is intimately connected with wavelet transforms. We attempt to generalize the connection between target estimation and the properties of group representations and to show the interest of considering still other transformations which arise naturally in detection/estimation problems for moving targets. INTRODUCTION The ambiguity function of a signal was introducted by Woodward [ 1 ] in the context of radar, where the echo from a transmitted waveform is used to measure the range and velocity of a target. It turns out to consist of the matrix elements of the operators of an irreducible representation of the Weyl-Heisenberg group. Here we attempt to investigate how the connection between detection/estimation and operator groups extends to more general situations and how signal waveforms might be synthetised from some desired (generalised) ambiguity function. In the first section we recall under which assumptions the echo may be viewed as the result of performing on the transmitted signal a linear transformation pertaining to a suitable group and why the detection/estimation process is closely related to the coefficients of a unitary representation. Since the properties of the signal shape as a measurement tool appear to be contained in this generalised ambiguity function, the question naturally arises whether it is possible, once a "best" ambiguity function has been chosen (for instance one which optimises a criterion under a set of operational constraints), to derive from it the right signal to transmit. This question is addressed in section 3, after the examples of the Weyl-Heisenberg group and the affine group have 172 been introduced in section 2. For more details on the former we refer to the monograph by Schempp [ 2 ] and for the latter to the article by Grossmann and Morlet [ 3 ]. In section 4, we show that more general transformations would be natural to consider, specifically by extending the time contractions, which act as diagonal matrices on the time-frequency couple, to the whole of SL (2, R). We conclude with some remarks. 1. DETECTION AND GROUPS 1.1 Operators Information is desired about the position and movement of a target. The model for this is ranging via an active system, radar or sonar, that is transmitting a signal, a delayed copy of which is returned through some propagation medium. Detection and delay estimation yield the wanted information, assuming the velocity of propagation is constant or known along the paths. Only a stationary pointlike target will induce a mere delay on the signal. More often the target appears pointlike (it might be small compared to the transmitted wavelengths, or big but with a single specular point...) but with varying range ret). It seems reasonable to model the transmitted waveform by a square-integrable, real f(t), alghough it is convenient to use complex waveforms as well, like the analytic signal introduced by Ville [ 4 ] which has only positive frequencies and the actual signal for its real part; it belongs to the Hardy space H2 of L2 functions f whose Fourier transform, denoted by f (0), vanishes for ro < O. Assuming in addition that the target is far enough for the propagation losses to be constant during measurement, the total energy received from the target is equal, up to a constant, to the transmitted energy: At time t the received waveform is f(t - 2g (t» where the delay get) obeys the equation: r (t - g (t» = c get) (1) which has a unique solution provided the (radial) velocity ret) is always less than c. Taking energy conservation into account, the linear unitary operator mapping the transmitted to the received signal reads: 173 q f(t) = f (t - 2g (t» (2) (1 - 2g (t) )112 Obviously these transfonnations cannot fonn a group if we impose r (t) > 0, hence g (t) > 0. However a displacement of the trajectory relative to the source is equivalent to translations in time ; specifically, let q (t) = r (t) - R, gl (t) be the corresponding delay; let Tube the operator which delays a signal by a constant u. One obtains: Thus we are allowed to take negative r (t)'s, at least over a bounded period of observation. Then the composition law: (g' . g) (t) = g (t) + g' (t - 2 g (t) ) (3) is associative and the inverse of U g is just U g' where g' is obtained from equation (1) with r(t) replaced by - r (t). Thus quite generally the operators U g defined by (2) fonn a group. The simplest examples of subgroups correspond to stationary targets (r (t) = ro, g (t) = rot c) and to constant radial velocities (r t) = ro + vt, g (t) = (rO+ vt) / (c + v». The first case yields the subgroup of translations T u , the second one a representation ofthe affine or (ax + b) group G2· 1.2 Estimation Now we consider the problem of estimating r (t) by means of the return from the target for a transmitted signal f (t). It is assumed that the output y (t) of the receptor is the sum of the echo u (t) and some white gaussian noise n (t). It can be shown [ 1 ] that the conditional probability of observing y (t) if the input to the receptor was x (t) is: where NO is the noise power per unit bandwidth and K a nonnalisation factor. If some a priori probability Po for x is known, according to Baye's theorem the posterior probability of x given the observation y is : 174 1 Log P (x) = - No fY (t) x (t) dt + Log Po (x) + cw (4) since x2 (t) has a unit integral. It was tacitly assumed that the delay g (t), hence the response x = U g f, belong to some sets with probability measures. As shown previously, the domain for g may be enlarged to a group. If the actual value of g is gO the measurement gives a posterior probability distribution for g which depends from the noise through y (t) in formula (4) but has an expectation value given by: X, (h) f = u'" f (t) U, f (t) dt = (f, Uh f) Lz (5) where h = gO-l g for the composition law (3) (from now on we label by g, h. directly the elements of a group G with unitary representation g --> Ug in L2). All properties of the posterior probability (4), now a function of g, may be derived from the function 'Xf (h). In particular it is expected to show a maximum, if the noise is low enough, in the vicinity of gO, the accuracy of the measurement is related to the stiffness of the function xr (h) at its absolute maximum (h = unit element). On the other hand any value of Xr (h) close to the maximum value I leads to ambiguity in the estimation, since g and gh have comparable probabilities. It is why Xr may be called the G-ambiguity function of the signal f. Actually the name was given by Woodward to a function of two variables, time delay and frequency shift; although there is no group of transformations of the signals labelled by exactly these variables the genuine ambiguity funclon fits into definition (5), as shown below. 2. AMBIGUITY IN RANGE AND VELOCITY 2.1 Doppler shifts True frequency shifts cannot be generated by the movement of a target but it is well known that, for signal of narrow bandwidth Q around and an angular frequency roo, the transformation induced by a target of constant velocity v is equivalent to a delay and a Doppler shift roD = 2 vic roo, in as much as the inequality 2 v/c.Q« roo holds. In that situation the group action on f includes the frequency shifts D ro, the translations T't and their products. In view of the identity : 175 it must also contain the multiplications by a constant phase a. Then an element g of the group G is a triplet (to, COO, a) and the multiplication law: gg'= (to' coo' a) (t' 0' co' 0' a') = (t 0 + t' 0' COo + co' 0' a + a' + ( LCO' "U 0 - t' 0 co0 ) 2 ) G is the Weyl-Heisenberg group Hl. The HI-ambiguity function above is easily rewritten as : (6) This is exactly the expression of the original ambiguity function [ 1 ] except for the phase factor in front of the integral. 2.2 Doppler contractions Turning to the case of general signals, a target moving with constant radial velocity, r (t) = ro + vt, induces a transformation of the form : By setting A. = exp (u) we express g as the pair (u, to) with the law: (u, to ) (u', t'o) = (u + u', to exp (u') + t'o) and G is the affine group G2. The G2-ambiguity function was studied, among others, by Jourdain [5]. Here its expression: f- u f (t) f (exp (u) t - to) dt Xf °2 (g) = exp ("2) 176 3. SIGNAL W ALVEFORMS OF GIVEN AMBIGUITY 3.1 Generalities In a situation where the response of the target may be described, to a good approximation, by the action of some operator U g, g being taken from a group G, it would be useful to adjust the G-ambiguity function to the operational conditions; for instance, to enhance the expected accuracy of the estimation of a particular parameter, and/or to push the unavoidable ambiguity into regions of G where the prior probability is low. There are two problems to solve (besides the practical ones) ; fIrst the G-ambiguity function cannot be chosen arbitrarily; second, how can the wanted signal shape f be derived from XGf ? We address this second problem first. To begin with, if the representation is not irreducible, there is no hope of getting a unique answer. For let f = fl + f2 belong to two orthogonal invariant subspaces ; then Xf (c) is the sum of Xfl (g) and Xf2 (g) , which means it is invariant when the relative phase of fl and f2 is changed arbitrarily. So let G be locally compact with a left-invariant Haar measure IJ. and U be irreducible. We proceed along the lines of the analysis of Grossmann and MorIet [ 2 ] for G2. A function fO is called admissible (as an analysing wavelet, it is understood) if its G-ambiguity function is square-integrable on G : f I XIii(g) r d~ (g) =k'b Ill(, II' < ~ (7) G This equivalent to saying that the formula: Alb = f u, I fo> <fo IV;! d~ (g) G dermes a bounded linear operator. The necessity of ineqUality (7) is obvious; conversely if it holds, any G-cross-ambiguity function (f, U g fO) is also square-integrable and vermes the equality : I, (f, U, 10 ) f d ~ (g) = k,o II f II' G 177 It holds namely for f = fo and, because of the left-invariance of J..L, for any G- translated UgfO; for [mite linear combinations of such functions the integral makes sense and verifies the equality; finally, because the representation is irreducible, the closure of this linear subspace is the whole Hilbert space. Consequently, products of such functions are integrable and the matrix element (ft. Af2) is bounded by k fO II fO 11211 fIll II f211. Therefore A is bounded ; but it commutes with all operators U g, again because the measure is left-invariant, hence, by Schur's lemma, it is a multiple of identity: (f,. A'o 12) ~ f (f,. U, fol (f2• U, fo ) dll (g) ~ k,o (f1• f2) G where the coefficient is found for fl = f2 = fO .In other words, the mapping which assigns to a function f(t) the complex function on the group G defined by (f, Ug fO) / (kfO )1/2 is an isometry from the signal space to L2 (G). f If both fl and f2 are admissible, by the same reasoning we can write: (f,. Ug f1 )(f2 • u;.1 f.) dll (g) ~ k (f1• f2) (f,. f,J (8) G where k (fI, f2) is linear in fl and antilinear in f2. Now we see how to retrieve fO from its ambiguity function; we substitute fO for f2 and f4 in formula (8), so for every function u the scalar product (u, fO) is obtained, up to the constant k(fl, fO) by integration of the cross-ambiguity of u and fl weighted by the ambiguity of fO. This procedure needs the help of some "analyzing function" fl , unless the group G is unimodular; in that case namely the Haar measure is also right invariant, g may be replaced by g-I in (8), or, equivalently, the indices 1,2 and 3,4 interchanged. Moreover in inequality (7) one may replace g by h- 1 gh, which means that Uh fO is admissible for every h, and then all functions are admissible by the previous density argument. An inversion formula is : 178 If the cross-ambiguity (f1, Ug f2) conver~es, in the sense of distributions, towards a kernel when f1 and f2 tend to Dirac measures, one gets an integral formula giving the product f (t1) f (t2) like (9) below. 3.2 The case of HI and G2 The Weyl-Heisenberg group is unimodular. It is an easy matter to verify the inversion formula: -- f f (t1) f (~) = 1/21t X (ro, t1 - ~) e i ro (t1 + t2) /2 dro (9) The affine group is not unimodular ; furthermore the representation is not irreducible on L2, but it is on H2 (and its complement [ 3 ] ; the admissibility condition reads: 4. MORE GENERAL OPERATORS The ambiguity function of ( 6 ) transforms very simply under the dilations of G2, becoming Xf ( AtO, roO fA ) and in general, if we denote by w the column vector of its arguments and by g a 2 x 2 real unimodular matrix, it is found that for any ambiguity function Xf (gw) is still an ambiguity function [ 6 ] and it belongs to the function U gf, where U g is a unitary operator [ 7 ]. U is the Weil representation of the group SL (2, R). For details see Lang [ 8 ]. The interesting fact is that among these transformations are the multiplications by exp (iat2 ) ; for narrowband signals, these are the linear frequency modulations impressed by a target when the delay has quadratic time variation; thus the parameter a conveys information about the transverse velocity and acceleration of the target. For this sake the subgroup of triangular matrices (combined with translations) will do. However, it is tempting to consider the full group, since it contains all popular and practical transformations of signal processing among them the Fourier transformation. Finally, since translations cannot be left out, we arrive at a five-parameter group (plus a phase a ), the semi-direct product of SL (2, R) and HI, whose elements consist of matrix g, vector w, phase a, with the multiplication: 179 (g, w, a) (g', w', a') = (gg', w + gw' , a + a' + 1/2 wTFgw') where F is the matrix G -1 ) o which yields the Fourier transformation. Of course G-ambiguity function should be considered for subgroups G "small" enough to allow for admissible functions, i.e. with at most two non-compact parameters. 5.- CONCLUSIONS To summarize, ambiguity functions are quite generally coefficient functions of a unitary representation of a group in a Hilbert space of functions. If the total ambiguity is finite and the representation irreducible, the synthesis of a signal of given ambiguity is possible. The first problem, to characterize valid ambiguity functions, remains open. Thus "the basic question of what to transmit remains substantially unanswered" (Woodward [ 1 )). Perhaps the mathematical characterization given by Schempp [2] for HI will lead to useful developments. REFERENCES [ 1] [2] [3] [4] [5] [6] [7 ] [8] 180 WOODWARD (P.) - Probability and Information Theory with applications to Radar - Artech house, 1980 SCHEMPP (W.) - Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory. Longman, 1986. GROSSMANN (A.) , MORLET (J.) - Decomposition of Hardy functions into Square Integrable Wavelets of Constant Shape. SIAM J. Math. Anal. volU, n04, July 1984. VILLE (J.) - Theorie et application de la notion de signal analytique Cable et transmission, 1948, vol. 2, nO!. JOURDAIN (G.) - Synthese de signaux certains dont on connait la fonction d'ambigu'ite de type Woodward ou de type en compression. Ann. Telecomm. 32, 1977 PAPOULIS (A.) - Signal Analysis Mc Graw Hill. 1977 BERTHON (A.) - Nouvelle application de la representation de BARGMANN ala theorie du signal. lOeme colloque GRETSI, 1985. LANG (S.) - SL2 (R). Springer-Verlag, 1985. Part IV Mathematics and Mathematical Physics Wavelet Transform Analysis of Invariant Measures of Some Dynamical Systems A. Arneodo 1,., G. Grasseau 1, and M. Holschneider 2,•• 1Centre de Recherche Paul Pascal. Domaine Universitaire. F-33405 Talence Cedex. France 2Centre de Physique Theorique. CNRS Luminy. Case 907. F-13288 Marseille Cedex. France * Permanent address: Laboratoire de Physique Theorique. Universite de Nice. Parc Valrose, F-06034 Nice Cedex. France **Pennanent address: Mathematisches Institut. Ruhr Universitiit. NA3 Postfach 102148. Universitiitsstra6e 150. D-5630 Bochurn 1. Fed. Rep. of Germany We present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal measures. We apply this technique, recently introduced in signal analysis, to probability measures on self-similar Cantor sets, to the 200 cycle of period-doubling and to the golden-mean trajectories on two-tori at the onset of chaos. We emphasize the wide range of application of the wavelet transform which turns out to be a natural tool for characterizing the structural properties of fractal objects arising in a variety of physical situations. 1. INTRODUCTION In the past decade, much effort has been devoted to the characterization of fractal objects [1-3]. In the context of dynamical systems theory, the Renyi dimensions Dq have been proposed to describe the geometric and probabilistic features of strange at tractors [4,5]. These generalized fractal dimensions are closely related (by means of a Legendre transform) to the spectrum of singularities f(~) of the corresponding invariant measures [6,7]. In general, strange sets can be viewed as interwoven sets of singularities of strength ~ and respective Hausdorff dimension f(<<) [8]. Although the measurement of the f(~)­ spectrum undoubtedly provides very interesting global information about the structural properties of strange sets, it is powerless to describe the spatial location of these singularities. An analogous situation has been faced in signal analysis, where power spectra extracted from recorded time series suffer from a similar deficiency. The power spectrum identifies the underlying frequencies and quantifies their relative contributions, but says nothing about their temporal locations. In recent analysis of seismic data [9] and acoustic signals [10], the Fourier transform has been supplanted by a transformation which gives a representation of the signal as a function of both time and frequency: the wavelet transform [11]. Since then, this transformation has been applied in many different fields [12,13]. The purpose of this communication is to introduce the natural tool for investigating the self similar properties different length scales [14]. Our aim is to illustrate technique in capturing the local scaling properties general. 182 wavelet transform as a of fractal measures at the efficiency of this of fractal objects in 2. WAVELET TRANSFORMS 2.1 Defini tions The wavelet transform consists of expanding functions over wavelets which are constructed from a single function g by means of dilations and translations [11,13]. Let us consider a fractal represented by a real function f; let g be a regular function that is localized around zero and some of whose moments are zero (g should be at least of zero mean: !g(x) dx=O). Then the wavelet transform of f with respect to the wavelet g is defined as T(a,b) = J g (-;-) x-b 1 ~ a>O,bEIR. f(x) dx (1) T is generally a complex-valued function over the position-scale half-plane. This transformation can be seen as a mathematical microscope whose position and magnification are band l/a respectively, and whose optics is given by the choice of the specific wavelet g. No information about f is lost since this transformation is invertible for a large class of functions f [11]. For a fractal measure dm(x), we define its transform as :.-- Jg T(a,b) aD x-b (-) dm(x) a a>O,bEIR, (2) where the renorma1ization factor l/a D may be chosen to best reveal the· scaling structure of the fractal measure under consideration. 2.2 Scalings and wavelets A typical property of fractals is that they are asymptotically self-similar at small length-scales [1]. Thus, looking near an arbitrary point Xo at different scales, we always find the same function up to a scaling factor. Defining f (x) = f(x o+ x) - f(xo),then we have Xo f Xo (hx) N?-. "( x ) 0 f Xo (x) For non integer local scaling exponent ~(xo) and sufficiently fast at infinity, this scaling behavior wavelet transform which scales like [14] : T Xo (?-.a,;\b) =?-. "( x ) 0 T Xo (a,b) , (3) a wavelet which decays of f is mirrored by the (4) i.e. with the same exponent ~(xo) as f; again Tx (a,b) = T(a,x o+ b). (For integer exponents, things are a little bit more involve? since generally wavelets operate modulo some polynomials). Therefore every local singularity of f produces a cone-like structure in the wavelet transform pointing towards the point (a = 0, b = xo ) at the border of the half-plane where this singularity of type ~ is located [13-16]. A fairly general result is that a high regularity of the 183 function is reflected in the wavelet transform by a rapid decay of the coefficients T(a,b) in the limit a ~ 0·. A formal partial integration shows that the local scaling exponent ~(xo) of a fractal measure ["16] turns into an exponent ~(x ) o = ~(x 0 ) - n (5) in the wavelet transform, where n is the normalization exponent in Eq. (2). Usually, relation (3) will not hold for all hER but rather for an infinite sequence hmN ~., mEl. Henceforth the exponent ~ is complex and the fractal shows oscillatory scaling behavior [17]. In the wavelet transform, oscillations of period In(~) exist around a straight line with slope ~ when In(IT(a,xo ) I) is plotted versus In(a). Similar periodic oscillations have been found in the loglog plot procedure used to determine the Renyi dimensions of highly self-similar Cantor sets [18,19]. Let us note that the oscillations obtained when measuring the fractal dimensions of well-known strange at tractors look much more complicated and do not display any convincing periodicity [20]. In this letter we apply the wavelet transform to some probability measures lying on higly self-similar Cantor sets and to some important examples of measure arising in dynamical systems. We use a real wavelet g of gaussian type: (6) For simplicity, we constant wavelet show also results obtained with the corresponding piecewise .(x) = { -l Ixl < 1 1 " Ix I " 3 Ixl > (7) 3 We choose the rescaling exponent in Eq. (2) so that in all examples we shall consider, the local singularities will correspond to a power law divergence of the wavelet transform T in the limit a ~ O· (& < 0). 3. PROBABILITY MEASURES ON CANTOR SETS 3.1 Uniform Cantor set A simple example is the standard triadic Cantor set. We initially divide the unit interval [0,1] in two intervals each of length l = l = l = 1/3. Each of these i ntervals ' receIves the same probabilityp = P1= Pz= 1/2.1 At Z the next stage of the construction of the measure, this same process is repeated on each of these two subintervals (fig. 1). For this measure a straightforward calculation leads to Dq= In(2)/ln(3) for all q. The Legendre transform of the spectrum of dimensions yields a single scaling index [7] ~ = In(2)/ln(3) with the density f(~ = In(2)/ln(3» =In(2)lln(3). In fig. 2a, we show an overview of the wavelet transform (n = 2 in Eqs (2) and (5» of the uniform triadic Cantor set. The successive pitchfork branchings observed when increasing the magnification (l/a) provide an instructive illustra184 tion of the construction process of the Cantor set. The positions of the local singularities of the measure are easily identified as spatial points b' where the P,=1/2 ',= 1/3 ',=1/3 the eacb At ~ s e 9 me n t has 0 = 1/2 tbe tbe ceotral from aod the scale sot . like tblrd of of each Each segment .e = 1/3 at the n. (C) (d) o ii = ln(2)/ ln(3)- 2 .<> II " .<> .; E: ']' p constructlon of 0 , :=: removed Is measure generation , stage standard cantor set, -5 , " ,, - 10 , P = ln(3) , - 15 -15' Tbe ~ (a) wavelet In the (a) and transform Pl= (b) uniform cantor nonuniform cantor set . . LLL . . . g(x) Is Pz are = -15 IT (a. (s go (T) . b) 1\ of tbe two distinct dlffereot). (c) 10jT(a,b=0)1 sct. 1/2j (d) b* In InIT(a,b=b*)1 corresponds Eq(6) aDd 0=2 to In o In(a) (b) wbere defloed o -5 In(a) uniform measure scales for - 10 EQs scale) kneading (2) PI= Cantor set 3 / 4, (arbitrary (arbitrary the triadic measures aDd scale) vs sequence (~) P2= lo(a) wltb: 1/4 vs (the lo(a) for the RRRRRRRRlLlL .. . 185 wavelet transform displays an oscillatory power law divergence. Each of these points is a point of the Cantor set. At each of these points the power law exponent is &(b*) = In(2)/ln(3) - 2, which corroborates the theoretical prediction that all the singularities have the strength ~ = In(2)/ln(3). The estimate of this exponent at b*= 0 is shown in fig. 2c where IT(a,b = 0) I is plotted versus a in a log-log scale representation. The period of the oscillations around the straight line is P = In(3) = In(t- l ) which attests that, up to a scaling factor, the fractal measure is invariant under dilation of the length scale by a factor ~ = t- l = 3 in the neighborhood of zero. The same result is obtained at every point of the triadic Cantor set. 3.2 Nonuniform Call tor set A somewhat less simple example is obtained when considering a probability measure which is concentrated on the same Cantor set with a unique scaling parameter t = t l = t 2 = 1/3 but with two nonsymmetric probabilities Pl> P2. The spectrum of singularities of this measure can be understood using the technique of "kneading sequences" [7]. Any point of the Cantor set can be addressed by an infinite sequence of symbols Land R where L (left) labels the interval with maximal probability Pl and R (right) labels the interval with minimal probability P2. Clearly, the sequence LLLL... is associated with the singularity ~.ln= - In(Pl)/ln(3), while the sequence RRRR ... is associated with the singularity ~.ax= - In(p2)/ln(3). Other more complicated kneading sequences yields ~-values which range between ~mln and ~mnx In fig. 2b, we show the wavelet transform of the fractal measure generated with Pl= 3/4 and P2= 1/4. Although the support of the measure is the same geometrical object as in fig. 2a, a simple visual inspection of the wavelet transform allows us to differentiate the nonuniform from the uniform Cantor set. While the power law divergence of T still clearly indicates the location of the singularities at the points of the Cantor set, the power law exponent is obviously no longer unique and is actually found to range between &mln -In(Pl)/ln(3) - 2 (e.g. for b = 0) to &max= -In(p2)/ln(3) - 2 (e.g. for b = 1). Again this numerical result confirms the theoretical predictions [7] of the existence of a finite range of s~aling indices ~ E [~mlo'~max]' with additional information concerning the spatial location of each singularity. In fig. 2d we have cut the half-plane along b = b*, where b* is the point which corresponds to the kneading sequence RRRRRRRRLLLL ... LLL ... ; when plotting, as in fig. 2c, InIT(a,b*) I versus In(a), we find a value of & E [ &mlo' &max]. Indeed a crossover is observed from the exponent &max at large scale to the exponent &mlo at smaller scale due to the infinite tail of L's. We note that the period of oscillations is P = In(3) in both cases, reflecting again the geometry of the support of the measure. Thus fig. 2d illustrates the power of the wavelet transform which can estimate the scaling factors t 1 ,t2 , ••• t k and the probabilities Pl,P2, ... ,Pk of a Cantor set through the measurement of &(b) at a finite number of points, provided that k is not too large. 4. THE 200 CYCLE OF PERIOD-DOUBLING Dissipative dynamical systems that exhibit the cascade of period-doubling bifurcations are in practice well modeled by one-dimensional maps with a single quadratic extremum [21] such as the map: 186 x0+1 = f R (X n ) = 1 - RX2. n (8 ) As we increase the parameter R which determines the height of the maximum of fR at X = Xc= 0, we observe an infinite sequence of subharmonic bifurcations at each stage of which the period of the limit cycle is doubled. This perioddoubling cascade accumulates at Roo= 1.4011518 ..• where the system possesses a 200 orbit. Beyond this critical value, the attractor becomes chaotic, eventhough there still exist parameter windows of periodic behavior. As originally emphasized by M. Feigenbaum [22], P. Coullet and C. Tresser [23], this transition to chaos presents a strong analogy with second-order phase transitions [24], and the renormalization group techniques [25] are well suited to account for the universality properties of this scenario. The envelop L(R) of the Lyapunov characteristic exponent (which provides a quantitative measure of chaos) displays a universal power-law behavior L(R) N (R - Roo)V reminiscent of an "order" parameter near criticality. The period of the bifurcating cycles is a characteristic time which diverges at the transition according to a power law behavior P(R) N (R oo- R)-v with universal exponent. At criticality, R = Roo ' the attractor exhibits a scale invariance: the adherence of the asymptotic orbit of almost all initial conditions in the invariant interval is a Cantor set. As sketched in fig. 3, the iterates of X = Xc= 0 form this Cantor set, with half the iterates falling between fR (0) and f~ (0), and the other half between f~ (0) and f: (0). At the next stages ~f the cons~ruction each subinterval is againoodivided in oo two subintervals with equal probability. Consequently the probability measure is symmetric with PI= P2= 1/2. But this Cantor set is a nonuniform Cantor set with two asymptotic rescaling factors: .e I = oc.-PDI (L for large) and .e 2 = oc.-PD2 (S for small), where oc.pD is the universal scaling factor [22,23] involved in the renormalization operator (The criteria defining universality classes in perioddoubling bifurcations is the order z of the local maximum of fR(x); oc.pD (z = 2) = 2.502907875 .... ). In early studies [26], most effort has been devoted to the calculation of the Hausdorff dimension Do= 0.537 ... of the period-doubling Cantor set. More recently, the whole spectrum of fractal dimensions Dq has been investigated [7]. The f(oc.)-spectrum associated to the nontrivial q-dependence of Dq has been shown to be universal [27] with scaling indices in a finite range In(2)/ln(oc.2PD ) ~ oc. ~ In(2)/ln(oc.pD ). The universality of the multifractal properties of this invariant measure has been confirmed experimentally in a Rayleigh-Benard experiment [28], and in driven diode resonator systems [29]. 4 2 2 6 L 8 3 -S- 4 3 7 5I LS Tho E~ doubling to -0.5 o the Dumber critical 0.5 1.0 construction cantor point set; D X of c of the the of the perlod- indices iterate the refer of mapping the (8). 187 Fig. 4a shows the wavelet transform (n = 1 in Eqs (2) and (5» of the 200 orbit of the mapping (8) at criticality. As compared to the wavelets of both the uniform and nonuniform Cantor sets in figs 2a and 2b, the pitchfork branchings observed when increasing the magnification (l/a) no longer occur by pairs but successively because of the unequal rescaling factors tl and t2 with t 1> t 2. Again the positions of the local singularities of the invariant measure are easily identified as spatial points b* where the wavelet transform displays an oscillatory power law divergence (4) in the limit a ~ 0'. The a-dependence of IT(a,b = 0) I is illustrated in fig. 4c in a log-log scale representation. The (a) 10 ( b) (e) 0.' ;=:: ',l) I ,l) a=ln(2)/ln(al'D) - ~. I E: -0.5 ..5 P=ln(apD) - IS -10 ~ The wavelet cascade calculated dlaenslonal obtained when defining (8)0); In Singularities explained versus pond In(a) to the correspond 188 In(a) with to limit In the for the can text. wavelet the " -1.6 In(a) T(a.b) of quadratic map (b) on each a~ O· b *= ,, -5 representation. the -I " trannform these as " , the be g(x) piecewise 1. (0) white 2 by an wavelet defined constont In a white to the while (7); D for in the = 1 (a) Tbree- of T(a.b) coding T b*= (8), = of 8 max [T(a.b)] symbols eacb of L Bnd S O. (d) Inlr(a.b")1 (c) and (d) wavelet In [qs 0 period-doubling singularities; sequence In(e) transforms wavelet the (T)Y) threshold infinite Eq. (6), of -2 R = 1 . 4011010.... point InIT(a.b*)1 versus The cycle and line regions 00 for (T(1) = constant addressed (c) flO) a A black the -< corres- transform (2) and (0). In (b) slope yields the exponent &max = ~max - 1 in remarkable agreement with the renormalization group predictions [22,23,27]: the singularity at zero corresponds to the "kneading" sequence LLLL.... and its strength is ~ = In(p1)/ln(£1) = In(2)/ln(~PD) = ~max. This local self-similarity of the most rarified region in the set shows up through small amplitude periodic oscillations around this slope with period P = In(~po). In fig. 4d, the same analysis is performed at the extremum b fR (0) = 1 which on the opposit corresponds to the most concentrated region in oo the set. The slope is now &min = ~min - 1 where ~min = In(p 2 )1 In(£) In(2)/ln(~2). This strength of the singularity with address SSSS .... 2 PD can be deduced directly from ~mBx= 2~m1n because of the quadratic nature of the extremum at X = Xc= O. The same argument can be evoked to explain the period P = In(£-l) = In(~2PO ) of the oscillations around this slope. 2 As illustrated in fig. 4a, the singularities of the invariant measure are located at the points of the Cantor set and thus can be addressed by kneading sequences of L's and S's. Each of these singularities is associated with a powerlaw efPonent in the wavelet transform which is found in the range ocm1n -1 ~ & ~ ocmax -1 according to the relative number of L's and S's in the sequence. In fig. 4b the wavelet transform of the 200 orbit is shown in a twocolor presentation: the black regions correspond to T < T, where the threshold T is defined in proportion to max T(a,b) (>0) on each a = constant line in such a way that the white regions point to the singularities in the limit a - 4 0+. This presentation enlightens the full complexity of the period-doubling Cantor set, especially the basic rule of the construction of this set: from one stage to the next stage of the construction, the relative position of the large and the small subintervals is kept unchanged when dividing a small interval but is exchanged when dividing a large interval (fig. 3). 5. QUASIPERIODIC TRAJECTORIES AT THE ONSET OF CHAOS Among the well-known scenarios to chaos [30,31], special attention has also been paid to the transition from quasiperiodicity with two incommensurate frequencies to "weak turbulence". This transition is commonly modeled by circle maps such as the sine map [32]: (9) where the parameter K provides the strength of the nonlinearities and the parameter n sets the rate of rotation. Let W*= (~-1)/2 be the golden mean. Then for every K < 1 there is a n*(K) such that the rotation number : W(K,n) lim [f=,n(S) - S]/n (10) n~+oo is strictly equal to W". The mapping (9) is the lift of a diffeomorphism of the circle, i.e. fK,n mod 1 is a diffeomorphism of the circle. Since W* belongs to the set of winding numbers defined by Herman [33], fK,n* is analytically conjugated to a pure rotation. At K = 1, the sine map fails to be a diffeomorphism: f-1~ is not differentiable everywhere because of the cubic inflection K, " point. This critical line is of interest since it marks the onset of chaos for quasiperiodic trajectories [32,34,35]. 189 5.1 Critical circle maps (K = 1) Shenker [32J was the first one to discover how the universal properties of this scenario are related to the nature of the inflection point of the criticle circle map. Since this pioneering work, renormalization group methods have been used to account for the local scaling behavior observed numerically near the inflection point [34,35]. More global universal properties were discovered when investigating the f(a) spectrum of singularities of the critical golden-mean trajectory [7,36]. The associated invariant measure develops a whole spectrum of singularities with scaling indices in a finite range 0.6326 .. , a ' 1.8980 •. giving rise to nontrivial Dq and f(a) spectrum. Figure 5a shows the wavelet transform (n = 2 in Eqs (2) and ~5» of the golden-mean trajectory which displays structure at all scales. The a-dependence of IT(a,b = 0) I is illustrated in fig. 6a in a log-log scale representation. The (sgn(T) IT(a,b) I~) of the golden-mean trajectory ~ The wavelet t.'ansform calculated with the sine map (9) for K = 1 (0) and K = 0.9 (b). vex) Is defined In Eq,(6) and n=2 In [qS (2)and (0). (b) (a) ;;;:; ii = - 2 30 • .I> ,. .I> 20 1:: "§ " j~l 2 -u -8 t.!.lI..:.-!. lDeau The -6 wavelet trajectory -2 In(ll) transform calculated -,0 In Ir(a,b=b*) I with the slne-.ap vs (9) In(a) - 10 -15 In(a) -10 of the critical for Q::;: .QA:, K;::; (b) b*= Q*. The golden-mean trajectory has been approximated by tbe W2 ,- cycle (W n = Fn/Fn>1) wbicb explains the trivial slope ~ =-0=-2 sIIa11 scales. The wavelet g(x) Is defined in Eq. (6). 190 1: (a) goldenb*= 0; superstable observed at slope yields the exponent &max= ~max- 2, where ~max= In(W*)/ln(~;~) matches the renormalization group predictions [7,34-36] based on Shenker's remark [32] that the distances around 9 = 0 scale down by a universal factor ~om= 1.2885 ... when the trajectory is truncated at two consecutive Fibonacci numbers Fn ,F n• 1 . This local self-similarity shows up through -small amplitude periodic oscillations around this slope with period P = In(ocgm ). In fig. 6b, we have reproduced this analysis at b = Q* (K = 1) (Q* (K) is such that the rotation number is strictly equal to W*), i.e. at the first image of zero. The slope is now &.In= oc. ln - 2 where oc811 D = In(W*)/ln(oc-gm3 ) is the local exponent. This can be deduced directly from ~.ax because of the cubic nature of the inflection point of f K =l,n* at zero [7]. In the insert of fig. 6b, the slope has been substracted to reveal the oscillations of period P = In(~om ) which reflect the scaling properties in the neighborhood of every iterates of zero. The other singularities are created from the inflection point by the action of f K =1.Q* and f~!1,Q* and their respective strengths range between ocmln and ocmax In fig. 7a, the wavelet transform of the critical golden-mean trajectory is shown in a two-color representation as already used in fig. 4b: the threshold T is defined in proportion to max T(a,b) (>0) on each a = constant line. In such a way the white cones point to the dominant singularities situated at the image of zero. The most prominent cones define the Fo-iterates of zero; they actually accumulate at zero in an alternating geometric progression governed by the exponent ocgm . Each one of these white cones is itself an accumulation limit of white cones (the F -iterates of the F -iterates of zero) but with a different 3 convergence rate ocgm because of the c~bic inflection point. This hierarchy of white cones continues at smaller and smaller scales. We mention that when defining the threshold T with respect to min T(a,b) «0), one can identify in the same way the weakest singularities located at zero and its inverse images. Then the main white cones corresponding to the Fn- inverse images of zero will converge in an alternating geometric way to zero at rate ocg . ' while secondary white cones (the F.- inverse images of these inverse images) will accumulate to the main white cones at the same rate ~gm and so on. l .10" ....rmr. .m:lfIl a b Tbe ~ slne .ap codlng ls Iterates (9) wavelet for K obtalned of tbe transform = 1 11ke (a) and 10 flg. of inflection point tbe K 4b. at = golden-mean trajectory 0. 9 (b). Tbe The white conea black 1n calculated (T(7) (a) and correspond wltb wblte to tbe (T~T) the F n zero. 191 5.2 Subcritical circle maps (K ( 1) For K ( 1, the sine map (9) f K,"~. (K ) is a diffeomorphism which is analytically conjugated to a pure rotation [32]. The generalized fractal dimensions are invariant under a smooth coordinate change [37]; this implies that Dq = 1. Indeed, the invariant measure has no singularities and so the scaling is trivial with a single index «= 1. In fig. 5b, we show the wavelet transform computed for K = 0.9; the large scale behavior is mostly unaffected by this deviation from criticality. The structures which emerge at large scales are progressively smoothed out at small scales and !T(a,b)! unescapably decreases to zero in the limit a ~ 0+. This observation is highlighted by the black and white coding of the wavelet transform in fig. 7b; the complex hierarchy of white cones observed at criticality (fig. 7a) disappears at small scales. This loss of structure occurs at larger scales as K decreases, i.e. as the departure from criticality increases. This cross-over behavior [38] from an invariant measure which retains memory of its critical properties at large scales, to a non singular invariant measure at small scales is analogous to cross-over effects observed in phase transition phenomena near to critical points [24]. This can be understood using a renormalization group approach as a cross-over from a strong-coupling (K = 1) to a weakcoupling (K = 0) fixed point [38]. These theoretical results have been confirmed experimentally in ref. [39]. 6. CONCLUSION Most previous characterizations of multifractals have brought a global description of the scaling properties through the determination of the continuous spectrum of scaling indices« and their densities f(<<). Additional information about the spatial location of these singularities can be obtained from the wavelet transform. The full complexity of the self-similar properties of a fractal measure can be captured by simple visual inspection of its wavelet transform. Since its implementation on a computer is not excessively time consuming and does not require enormous storage (the development of a Fast Wavelet Transform algorithm is in progress), the wavelet transform provides a very efficient tool for analyzing fractal objects resulting either from simulations or from experiments. Its application to a variety of physical situations [2,3], for example, critical phenomena, percolation, growth processes and fully developed turbulence, looks very promising. We refer the reader to some color pictures inserted at the end of this artic Ie and which represent the wavelet transforms of the above discussed fractal measures as obtained on a Pericolor 2000 with the wavelets g(x) defined in Eqs.(6) and (7). 192 Wavelet transform T(a,b) of some invariant measures in the (b,a) half-plane. The 256 colors of the Peri color 2001 are ordered according to the natural light spectrum from red (max T>O) to black (T=O). (a) Uniform triadic Cantor set with g(x) as defined in Eq. (6): be[O,l], ae[2-9 ,2°]; (b) Uniform triadic Cantor set with g(x) as defined in Eq. (7): be[O,l], ae[3-7 ,2 9 3- 7 ]; (c) Nonuniform triadic Cantor set (Pi =3/4, P21/4) with g(x) as defined in Eq.(7): be[O,l], ae[3- 7 ,2 9 3- 7 ]; (d) Period-doubling Cantor set with g(x) as defined in Eq.(7): be[-0.401 .. ,1), ae[10- 3 ,0.5]. 193 Wavelet transform T(a,b) of the golden-mean trajectory calculated with the sine map (9): (ae [3.10- 4 ,0.17], be[0,1]). The 256 colors of the Pericolor 2001 are ordered according to the natural order of the light spectrum from red (max T>O) to black (T=O) in (a) and (b). (a) Critical golden mean trajectory, K=1; (b) subcritical golden mean trajectory, K=0.9. The colors in (c) and (d) are ordered according to the natural order of the light spectrum from red (min T<O) to black (T=O). g(x) if defined in Eq.(7). 194 REFERENCES [1] B.B. HANDELBROT, The Fractal Geometry of Nature (Freeman, San francisco, 1983) • [2] On Growth E. STANLEY 100 [3] [4] [5] [6] and Form: and (Martlnus statphys P. N. Nijhoff, 16, edited J.P. U. ECKMANN FRISCH and and "Enrico T.C. HALSEY, P h y s. [8] [9] [10] [11] M.B. COL LET, J. L. P. GOUPILLAUD, 1141 R. KRONLAND-HARTINET, Recognition Systems and I. and edited DAUBECHIES and Physics (World P.G. bv and 617 E, by vol. 1986). (1985). Predictability R. GUlL, in School Geophysical of Physics and BENZI, G. PARISI 84. p. KADANOFF, A. and MORLET, L. T. Scientific by PO R 2 I 0, J. 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BOLSCBNEIDER, J. A. ARNEODO and H. HOLSCBNEIDER, Ph)'s. G. Ph),s. 1,L, J. ARNEODO GLAZIER, GUNARATNE SHENKER, E.D. Physica SIGGIA, Ph)'s. !Ul, Rev. 370 (1982). Lett. ~, 132 (1983). A. J.A. KADANOFF, ~, I.H.E.S. FEIGENBAUH, and A. (1986). Stat. Ph),s. Rev. !Ul, 99~ (1988). li., 2007 (1987). Lett. LIBCHABER, Ph),s. Rev. ~, ~23 (1988). Holomorphic Integral Representations for the Solutions of the Helmholtz Equation J.Bros Service de Physique TMorique de Saclay, Laboratoire de 1'Institut de Recherche Fondamentale du Commissariat al'Energie Atomique, F-91191 Gif-sur-Yvette Cedex, France Summary: Two types of integral representations for the solutions of the (n-dimensional) Helmholtz equation in certain classes of unbounded domains have been presented and proved in [BO] (see also [B] for some generalization of these results to other equations). We shall describe here these representations, and give an extension of the second one to a more general class of domains. The main results can be summarized as follows. i) Outgoing-wave and incoming-wave solutions~' defined in the complement of a convex bounded region are characterized as Laplace-type transforms of entire functions F of exponential type taken on appropriate cycles of the complex (n-l)-sphere in ~n. Moreover. the function F is the analytic continuation of the radiation (resp. absorption) amplitude associated with the outgoing (resp. incoming) wave ~'. ii) Integral representations involving universal kernels are obtained for classes of solutions which enjoy a property of revoLution symmetry. We shall denote by (R-C) these representations (called (R) in [BO] and [B]), which associate each solution q, in a given domain 6. with a "weight-function" f. holomorphic precisely in the meridian section ~ of 6. (i.e. f E C(~)). 6.0 being identified with a domain of ~ invariant under complex conjugation. The universal kernels of these representations are defined by an analytic continuation of elementary solutions of the Helmholtz equation performed in an appropriate complex variable; they reduce to functions of elementary type for odd values of n. For the case of outgoing-waves and incoming-waves in a simply-connected domain 6., defined as the complement of a bounded, revolution-symmetric, but not necessarily convex, nor even connected, region. the function f is determined as the Laplace-Borel transform of the entire function F considered in i) (the latter being. in view of revolution-symmetry. a function of a single complex variable). The idea of introducing the Laplace-Borel transform of the analytic continuation of the radiation amplitude is due to R.Omnes who proposed a suggestive (although incomplete) derivation of the representation (R-C) for outgoing-waves defined outside a given surface ~ in ~3. This unpublished work was at the origin of [BO] and [B]. For the case of solutions q, confined in a bounded domain 6.. we obtain similar integral representations (R-C) with a different type of connection between q, and f: in the case n=3. the restriction of f to the real axis purely coincides (up to a constant factor) with the restriction of ~, to the revolution axis. 197 I. Outgoing waves and incoming waves as Laplace transforms of entire functions on the complex sphere In the space ~k)' k=p+iq={k 1 ••••• k n }. we consider the {complex} sphere s(a) with equation s{k}=k 2 -a2 =O {k2=kf+ ... +k~}. and in the space ~x) of the the corresponding Fourier-conjugate variables x={x 1 ••••• x n } we consider Helmholtz equation {1} While general temperate solutions W of h~a) in ~(X) are Fourier-type transforms of distributions F on the real sphere s~a)=s(a)n ~n {i.e. of the form I w{x}= ~n eik.XF{k}&{s{k}}dk= J s~a) eik.XF{k} dk 1 A ••• Adkn ds{k} I 8(a)' with k.x= ~ l<;;i<;;n kix i } the outgoing and incoming-wave solutions of h~a) in "exterior domains" will be characterized similarly as Fourier-Laplace-type transforms of entire functions F on s(a). integrated on appropriate "complex cycles" 1 •• 1_ of s(a) (instead of s~a)}. In both cases. the identity {k 2-a 2 }F{k}=O (in s~a) or S(a)}. will imply that the equation {~+a2}W{x}=O holds in the relevant convergence domain of the Fourier-Laplace-type integral. If L is a closed bounded hypersurface surrounding the origin in ~(X)' we denote by Uz the exterior domain bordered by L. and by B{L} the polar set of L in ~(q). namely the convex closed set B{L}={q E ~n;vx E L. q.x <;; 1} {q.x=qlxl+"'+ qnxn}' which we also represent in polar coordinates as follows: B{L} = {q=on ; n E Sn_l' 0 < Q <;; T{n}-l} • with T continuous on Sn_l' By definition. an outgoing-wave {resp. incoming-wave} in Uz is a solution W. {resp. W_} of h~a) in Uz that satisfies Sommerfeld's radiation {resp. absorption} condition: lim r r(n-l)/2 o {2} --> 00 The following property has been proved in [BOJ: Theorem 1: If L is a convex hypersurface. there exists a bijection ~~ {resp. ~~ from: - the set of outgoing-{resp. incoming-}waves q'.{resp. '1'_} in Uz that belong to Coo{Uz }. onto: - the set of entire functions F on s(a) that admit B{L} as a gauge-set of exponential increase in i~(q). namely that satisfy a set of inequalities of the following form {3} a} F The mappings a± -n and their inverses are specified as follows: ~{W±} is defined by the "Fourier-Green type" formulae {4} 198 b) >It. and '1'_ appear formulae as "Laplace-type transforms" L:t(F) of F, dk 1 A.... Adkn defined by the I ds(k) (5) s(a) where t.= {1.(0);0 E $n-l} (resp t = {1_(0);0 E $n-l}) denotes a class of homologous (n-1)-cycles of s(a) (specified below in ii)) that give a sense to formula (5) in the following way. For each 0 in $n-l' the choice of the integration cycle 1±(0) in formula (5) provides the definition of >It±(x) in the corresponding half-space nn= {x follows from the fact that E ~n;O.X-T(O) ~ O}. The definition of >It± in UL UL = LJ nn. nE$n_l c) The restriction of F to s~a) yields the radiation (resp. absorption) amplitude of q,. (resp. q'_), namely one has (with Cn given by Eq. (25) of [BO]): r lim e±iar - - - F(± an) (1 tI'±( r 0) 1 + (6) 0(-)) --> 00 r Comments and remarks i) Formula (4) defines an entire function of exponential type with gauge-set (in i~(q») on the ambient space ~(k). However, it is only for k in s(a) that the integrand at the r.h.s. of (4) is a cLosed differential form; in the definition of F (on S(a»), the integration hypersurface L of (4) can then be replaced by any hypersurface L' homotopic to L in UL • ii) For each 0 in $n-l' one defines l±(O)=l'(O)+S±(O), where supp S.(O) (resp. supp. S_(O)) is the real hemisphere {k~n;k2=a2,k.n>o(resp.k.~)} and supp 1'(0)= {k=p+iQn;p.O=O, p2_Q2=a 2 , Q ~ O}. The convergence of the integral (5), B(L) taken on 1+(0), in the half-space nn, follows from this definition and from condition (3). We note that, by using the following parametrization of 1.(0): 1.(0) vn E $n-l ' {k=a[cos8 0+sin8 vn], with vn.O=O, 8 E [O,n/2] U [n/2,n/2-ioo] } , nn) the corresponding integral (5) coincides (for x in with the Lax-Feshbach representation (see [M.F], formula (11-4-49)). iii) As a special case of (5), the outgoing and incoming elementary solutions E~a)±(X)=cstC:I)'n-2)/2Hi;'~2)/2a( Ixl) transforms" of h( n a) appear (on 1±) of the entire function F=1(L={O} and as B(L)=~n r the "Laplace in this case). The Fourier representation of E(a)±, i.e. E(a)±(x)= ___ 1 __ eik.x[k2-a2+io]-ldk n n (2n)n J~n is then reobtained by using the residue theorem in the variable k.O (for each 0) . iv) For x in nn, one can compute t. (Q:~ (tI'+)) (x) via formulae (5),(4) as a 199 convergent integral on r+{n)xL. Integration on r+{n) then yields (in view of iii)) the Green representation of ~+(x) (in terms of values of ~+ and o~+/on on ~), which implies that t+o ~~ = n (see in this connection [M.F.] p.1538-39). v) Property c) follows from formula (5) by choosing the cycle r±{n) associated x with n = - (r=lxl) and by using an argument of stationary phase. r vi) If ~ is non-convex, let ~c be the boundary of the convex hull of ~; then and (according to i)), ~ can be replaced by ~ in formula (4). The ~ associated with U~ then appear as restrictions of the corresponding associated with U~c' and are still injective (but not surjective); B{~)= B{~), mappings mappings correspondingly, the formulae (5) only define ~± in the subset Uc ~ of U~. II. Integral representations (R.C) In the following, we shall only consider solutions of h~a) which are revolution-symmetric with respect to a given axis Ov, v being a unit vector in ~x). If ~ denotes the domain of definition of such a solution, supposed to be bordered by a regular (e.g. C~ manifold o~, we consider the meridian section ~ of ~ as imbedded in the complex plane of the variable Z=X+iY=x.v+i[x 2 -{x.v)2]1/2, and we call ~eg{~) the space of holomorphic functions in ~ which are COO in ~ and (if ~ is unbounded) which tend to zero at infinity as cstIZI- 1 . For various classes of domains ~, specified below, one proves that every (revolution-symmetric) solution ~ of h~a) in~, COO in ~,admits an integral representation of the following form w{x) = fa~ ~~~(x,w)f{w)dw, (R.C) where f is a holomorphic function in ~eg{~) and where the kernel ~~~ is an appropriate branch of the function E~a){[x2_2{x.v)w+w2]1/2), defined e.g. by An _ 2 denoting the (R.C) reduces to area of ~(x) the unit (n-2)-sphere. For n=3, the representation (8) For each w in C, E~a) ([x2-2{x.v)w+w2]1/2) is a (non-uniform) solution of whose singular set is the (n-2)-sphere ay{w) (with axis Ov) defined by the equations: X=x.v=Rew and Y={x 2 _{x.v)2)1/2=Imw. It follows that any function ~ defined by a representation (R.C) is a solution of h~a)in ~; according to this representation, ~ appears as a superposition (with the "weight-function" f) of the revolution-symmetric solutions ~~~{·,w) associated with all the "parallels" ay{w) that generate the boundary a~ of ~. Rigorously speaking, the study of the h~a) 200 representation (R.O) in a given domain 6 requires the following complements, treated differently according to whether 6 is unbounded or bounded : i) selecting a uniform branch of the kernel ~~! ii) proving that the corresponding representation is a bijection from §reg (6) onto a relevant subspace of solutions of h~a) in 6. - Unbounded domains. Two distinguished branches of the kernel ~~!, denoted by ~~~. and ~~~- can be defined, for each value of w in !C, in a "cut-domain" of the form !Rex)' Lv(W)' where Lv(w) denotes a bounded "floating hypersurface" bordered by uv(w) and homotopic to the ball {x;X=Rew, IYI< Imw}. These branches are distinguished by choosing the sign of the argument ~(x,w)=[x2-2(x.v)w+w2]1!2 in the definition of ~~~, i.e. ~~!(x,w)=E~a) (~(x,w));this choice corresponds to the following asymptotic behaviour of outgoing-wave (resp. incoming-wave) type for ~~!'("w) (resp. :y~~!-(·,w)), as shown in [BO] (Eq.(74)): 1 (1+0(-) ) r (9) Remarks: i) For w real, one has ~~~±(x,w)=E~a)±(x_wv). ii) For n odd, :Y~~!' and ~~!- are the only branches of the function :y~~~ since E~a) is a uniform function of the form eia(x ~-(n-2)pn(~) (P n being a polynomial). For n even, the Riemann surface of ~~! is infinite-sheeted, and the branches ""v, ~(n)± are specified by choosing (A2-2iAa~)(n-3)!2>o for ~=i~, ~ > 0 a in the integral (7), and considering the integration cycle of (7) as a floating curve from 0 to +00 (all the other branches of ~~! are linear combinations of the latter). The following property was proved in [BO] for the class of unbounded revolution-symmetric domains 6 of the form 6=Uz (with the notations of section I). As explained below, the validity of this property for the more general class of domains 6 considered here is also directly implied by the methods of [BO]. Theorem 2. Let 6 be any simpLy-connected domain with COO boundary whose complement in !Rex) is a (connected or not) bounded set with revolution-symmetry around the axis Ov. Then there exists a bijection !~~!' (resp. !~~!-) from the space §reg (~) onto the space of outgoing (resp. incoming) waves 'I'. (resp. 'l'_) in 6, which are revolution-symmetric bijections are defined by the formulae \I x E 6, 'l'+(x) = - with axis Ov, and belong to C=(6). These f a~ ~n)±(x,w) ,a f(w)dw (10) and the corresponding asymptotic amplitude F(k)=Fo(-ik.v) of'l'± (defined by formula (6)) admits the following, Polya-type representation Fo= n(f): (11) of the proof. The representation (11) of Fo follows directly from (10) using formulae (9). The bijective character of -n.a i(V)± is first proved for Summary by 201 domains ~ of the form UL , with L convex. In this case, the mapping n is a bijection of §reg(~} onto a relevant space of entire functions Fo(z) of exponential type whose gauge-set Bo (in~) is the polar set of LO= o~; this space is then shown to be the bijective image of the space of revolution-symmetric entire functions F(k} satisfying the conditions of theorem 1; if ~~V) denotes this bijection (defined by Fo(-ik.v)=F(k}}, it follows from theorem 1 that ~~~~±=[~]-1~~V)-10n is itself a bijection. Moreover, the inverse mapping [~~~~±]-1 is computed explicitly by using formula (4) and the inverse of (11), i.e. the Laplace-Borel integral (12) f(w} where the integration cycle is the floating half-line result the following inversion formula is obtained : \JwE~, f(w}= J yn-2 o~ L~={z=pei~, p~}. As a o~+ oG(n) ] ---a---(X,y;w}~+(X,y} dY {[G(n) (X,Y;w}--=(X,Y)AX AX a (13) o~+ oG(n) ] } - [ G~n) (X,Y;w}Oy-(X,y)- ~(X,y,w}~±(X,Y) dX , in which ~±(X,Y}~±(x) and G~n)(x,y;w} is the following kernel (holomorphic in w) : (14) Formula (13) can now be used for proving that ~~~~± is a bijection in the case of more generaL domains ~, whose boundary is (as in [BO]) a non-convex hypersurface L, or even a union of disjoint hypersurfaces. In fact, the basic property of the representation (13) is that the integrand is a cLosed 1-form which, for each (X,Y) fixed, is holomorphic and uniform with respect to w in a "cut-domain" of the form ~"'Y(X,Y}, where 'Y(X,Y} denotes a floating path with fixed end-points Z=X+iY and Z. Starting from a general domain ~ in which the solution ~+ (or ~_) is given, let ~c= U C be the subdomain of ~ whose boundary L LC is obtained by taking the convex hull of the complement of ~. Then formula (13) yields a well-defined function f in §reg(~}, which (in view of Stokes' theorem) can be as well represented in ~c by a similar formula, with integration cycle Lc: since the latter is the inverse formula of (10) in ~c, this property still holds (via contour distortion and by uniqueness of the analytic continuation) in the given domain ~. Remark: If ~ is not simpLy-connected (i.e. if ~n {(X,Y};Y > O} is not simply-connected} formulae (10) and (13) do not define uniform functions (in view of the ramified character of ~~~ and G~n)}, respectively for x in ~ and for w in ~. However, uniform solutions in ~ can still be represented by a 202 formula of the (~~~+-~~~-), type (R.O) in which the relevant kernel is (~~~+ + ~~~-)or resp. for n odd or even: these solutions are not pure outgoing or incoming waves. domains. We consider for simplicity the case of domains ~ homeomorphic to a ball, whose boundary is a COO-hypersurface L. We note that when the point w varies on the meridian Lo of L, the point ~(x,w)=[(Z-w)(Z-w)Jl/2 describes a loop around the origin. Let us then restrict ourselves to the case when the dimension n is odd; since E~a) is uniform in ~'{O} in this case, either branch E~a) (± ~(x,w)) can be chosen as an admissible kernel ~~~ for writing a representation (R.O) in~. Besides, the weight-function f of this representation is related to the corresponding solution 'l'(x)?I«X,Y) by an equation of the form: Bounded \I w = X in L\ 'hx,o) n IR, = [Qn (d)fJ(X) (15) where Qn(d) is a certain differential operator. This follows from the fact that, for Y=O, the representation (R.O) yields (for a certain choice of the branch Y,,;, ~ ~): 'hx,o) f eia(X-w) - - - - Pn (X-w) f(w)dw Lo (X_w)n-2 (16) which reduces to the form (15) by application of the residue formula. In the case n=3 the situation is simple since one has (by contour distortion): f eia[(Z-w) (Z-w)J 1/ 2 1 J+1 cos[aY(1-t 2 )1/2J - - - - - - - - f(w)dw=--.- - - - - - - - f(X+itY)dt 41T L 1/ 2 [(Z-w)(Z-w)J 211T -1 (1_t 2 )1/2 o (17) The other choice for ~~~ gives the same representation up to a sign. Moreover, formula (16) becomes in this case : 1 'l'(X, Y)=-A >It(X,O) 1 r 41T ~ i eia(X-w) X-w f(w)dw = 2 f(X) (18) By using an integral of the form (13) which, for w real, is equal (in the case n=3) to cst.>lt(w,O), one obtains the inversion of the representation (17) and therefore, the following Proposition. For every bounded domain ~ in 1R3 (of the class specified above), the representation (17) is a bijection from Qreg(L\) onto the space of revolution-symmetric solutions'l' of hja) in ~ (COO in ~). Moreover 'l' is related to the weight-function f of (17) by Eq.(18). References [BJ J.Bros in Seminaire E.D.P., Ecole Poly technique 1985-86, Expose n° 22. [B.OJ J.Bros and R.Omnes "Holomorphic integral representations for outgoing-waves in IRn " , Saclay preprint, june 87, to be published in J.Math.pures et appl. [M.FJ P.M.Morse and H.Feshbach "Methods of Theoretical Physics", Mc Graw Hill (1953) . 203 Wavelets and Path Integral T. Paul* Courant Institute of Mathematical Sciences, New York University, New York, U.S.A. *00 leave from Centre de Physique Theorique, CNRS, Luminy Case 907, F-13288 Marseille Cedex 09, France The matrix elements between wavelets of the quantum propagator for a large class of Hamiltonians on the half-line are given in terms of path integral. It is a sum over path defined on the upper half plane with a Wiener measure associated to the hyperbolic Laplacian in the limit where the diffusion constant divcrges. The construction in the case of the circle is sketched. Talk given at the Conference "Ondclettes, Methodes Temps-Frequences et Espace de Phases" Marseilles, France, Dccembre 14-18 1987. 1. INTRODUCTION Solving the time dependent SchrWinger equation in term of path-integral has a long history in mathematical physics, since its formal definition by Feynman in 1949 [1] (see [2] for numerous references). Since path-integral produces a beautiful link between quantum and classical theories, a formulation of quantum propagator in terms of a sum over path defined on the phase - space seems natural. In [3] was proposed a procedure to compute quantum propagator by well defined path - integrals involving Wiener measure on phase - space in the limit of diverging diffusion constant. Not only does this formulation give a rigorous computation of the solution of the SchrOOinger equation, but it allows a natural geometrical formulation of the problem for very different natures of the phase-space, in terms of symplectic form and Wiener measures. In [3] was treated the case where the phase-space is the Euclidian N- dimensional space and the 2-dimensional sphere. In [4] the case of the Lobatchevski half plane was studied. In this paper we report on this last work by emphasising the "wavelet aspect" and give a last example where the phase-space is the cylinder (phase space of a rotator). T~e main result of this note can be stated as follows: let us consider on L 2(R+ ,dx) the SchrOOinger equation i ~t CP(t) = H CP(t) (1.1) where 11 is a self adjoint operator. Let us consider on L 2(R+) a family of continuous wavelets of the form (a > 0, b E R): (1.2) (they are obtained by translation dilation on the other side of Fourier transform). Let us suppose H of the form: (1.3) Let us consider the Wiener measure (Wiener bridge) dl1~ (a (s), b(s» assiciated to the Laplacian in the Lobatehevski half plane, with diffusion constant v (see [4]) defined on path with fixed extremities at (a',b') and (a • b). Then the propagator of (1.1), namely e- irH, is given by the formula: 204 =c~ lim eV'~ v -> - (a ('),b (I)) =(D, b) I (a (O),b (0)) =(a',b') - e , J j~ .1!!.!ll. a j J h(D(S),b(s))dr 0 dll~(a(s),b(s» (1.4) where c~ is a constant (c~ = (~- 1I2r1), 2. Coherent states, wavelet and the phase-space nature of quantum mechanics. The idea of localization in phase space for quantum systems comes back to the very beginning of quantum mechanics. The so called "coherent states" defined as the best localized on both sides of Fourier states of L2 (R) were introduced by SchrOOinger himself [5] and have been proved be be very useful in many areas of physics [6]. Generalized coherent states can be seen as a family of vectors 'Vpq(x) of L 2 (configuration space) indexed by points of phase space which we note (p,q) and "localized" in phase space around (p,q). This means thal: (2.1) (2.2) Since they usually satisfy a decomposition of identity: 7I (2.3) ('Vp,q -) 'Vp,q d Il(p,q) = 1 for some measure d Il(p,q), they naturally carry an isometric transformation U between L 2 (configuration space) and L 2 (phase space, dll) via the formula: <Xx) U / (p,q) '" 1 ..Jc ('Vp,q,<I». (2.4) This transformation is natural between L2 (configuration space) and its range, which is a closed subspace of L 2 (phase space, dll) and provides the national setting of the quantum theory. It is important to notice that in certain cases where the phase space is not a cotangent bundle (example sphere), there is no L 2 (configuration space) and quantum mechanics is directly defined on (which is, for the sphere, finite dimentional). The relation between this construction and the wavelet transform goes as follows: suppose we consider the quantum mechanics of a particle moving on the half line. The corresponding phase space is the half plane with its natural Lobatchevski structure. Then the family of "continuous wavelets" obtained from the vector (2.5) by dilation by a and translation in Fourier space by b, namely (2.6) are the coherent states on the half line. The decomposition of identity (2.6) results from orthogonality relations for the affine group [7] and the space is just the space of "wavelet coefficients" (reproducing kernel space). The reader is referred to [8] for their use in quantum mechanics and to [9] for their relation with the experiment of microwave ionisation of hydrogen atoms. It is important at this point to notice that the space is the subspace of L (C+, da~b) (a > 0) of functions of the a form a~1/2 x/(b+ia), with/analytic on the upper half plane. We will see in the next section how can be recovered in terms of geometry on the Lobatchevski half plane. 205 3. Path integral without Hamiltonian and reproducing kernel Hilbert space. Taking the fonnula (1.4) in the case where h (a,b) = 0 gives (3.1) Both sides have a very simple interpretation: the r.h.s. before taking the limit, is just the integral with a Wiener measure, of the complex exponential of the integral of the primitive of the symplectic hyperbolic fonn the .h.s. is nothing but the scalar product of two wavelets which is also the kernel (in the sense of integral operators) of the projector from L 2 (phase space) into the space of wavelet coefficients (see [7]). The fonnula (3.1) gives a way of computing (which works also for other geometric situation such as R2n , the 2. sphere, the cylinder) which plays a central role in geometric quantization [10]. The proof can be sketched as follows: - First of all it turns out that e vt~ Je a"b" -i~f~ a dl1~ (a (s),b (s» a'b' is the integral kernel of a scmigroup e-vtA on L 2(R X R+) - A is a second order differential operator: (3.2) - A factorises in A =a 2 (i 1.. +1..+i ~/a) (i 1.. -1..-i ~/a) aa ab aa ab (3.3) - A has positive spectrum with infinitely degenerated isolated eigenvalue zero. The corresponding eigenspace in the set of functions of the fonn a Pf (b + ia) with f analytic on the upper half plane as shown by (3.3). - Finally the limit over v gives the projection on . It is pointed out that fonnula (3.1) gives a link between a quantum object and classical geometrical object such as symplectic fonn and Laplacian. One can also remark that although the symplectic fonn is canonical on phase space, the Laplacian is not and can be chosen in different ways. 4. Path integral with Hamiltonian and the quantum propagator: Going back to fonnula (1.4) for non zero h(a,b) we have to compute the following quantity: J.1!!Jtl. -i !h(a(.),b(.»ds I cpe vtP (a (t),b (t)) = (a",b'1 J (a (O),b (0» =(d,b1 -iP e a d~(a(s),b(s». (4.1) Omitting here technical conditions on h which are given in [4], one can easily show as in section 3 that (4.1) is the integral kernel ofa semi group of contractions of L2(R xR+) of the fonn cp 206 e-(vA+ih)t (4.2) where A was defined in (3.2) and h is the operator of multiplication by the function h (a.b). The problem of the limit of (4.1) when v -+ 00 still remains. In [4] was proved the convergence of (4.1) in the strong operator sense and in the pointwise sense. Both give the formula (1.4) in the following sense: by defining Pp the projector from L2(R xR+) on the zero eigenspace of A (namely functions of the form a P /(b+ia). with/analytic). (4.1) tends when v -+ 00 to the integral kernel of the operator CII pp[ exp (- i PphPpt~ Pp (4.2) 'l'a"bj (4.4) Since the integral kernel of P II is [7] P lI(a'.b'. a".b") = C11l [ 'l'ab'. we get first that PllhPII= H (4.5) where was defined by (1.3) and then the formula (1.4). 5. The case of the cylinder. Coherent states on the circle: The preceding construction suggests to compute the same kind of path integral in the different situation where the phase space is the cylinder. cotangent bundle of circle [11]. Since models of particle on the circle have been proved to be useful in the study of stability and instability in both classical and quantum theories [12]. (pulsed and kicked rotators) path integral and coherent states on the circle may present some interest. The construction can be sketched as follows: The phase space is the cylinder SIX R with its natural Laplacian The path integral (without Hamiltonian) is p(vt;~'.P';~".P")= (P(1).~I» = (1'•.-> (p (0) •• J (0» = (p'••') J e Vl e i P4+dJl!(P(s).~(s» (5.1) where dl1::' is the Wiener measure associated to t.. P is still the integral kernel of a semi group e-vtA' with A' = (_...£.. + i ...£.. - ip) (...£.. + i ...£.. + ip) a~ ap ~ ap The null space of A consists offunction of the form e- p 'JJ2 /(~ + ip) with/analytic on the cylinder. An orthonormal basis of it is the set of functions .. 2 'I'm (~.p) = e -- 2 2 eim(++ip) e-P 12 (5.2) This space is unitary equivalent to L 2(.'1 1) via the integral transform of kernel: (5.3) 207 We remark that I +,p (9) - coherent states on the circle - is obtained from 1•• (9) - which is the Jacobi e -function - by translating it on both sides of Fourier. We finally remark that I •• is nothing but a "Gaussian" for Fourier series, which makes it analogue to the usual canonical coherent states [11]. References [1] R. P. Feynman; Rev. Mod. Phys. 20 367 (1948). [2] D. G. Babbitt; J. Math. Phys. 4 36 (1963, E. Nelson; J. Math. Phys. 5332 (1964), J. Tarski; Ann. Inst. Poincare 17 313 (1972), K. Gawedzki; Rep. Math. Physics 6 327 (1974), S. A. Albeverio and R. J. Hoegh-Krohn; "Mathematical Theory of Feynman Path Integrals" (Springer, Berlin, 1976); C. De Witt Morette, A. Mabeshwari and B. Nelson, Phys. Rep. 50 255 (1979); P. Combe, R. Hoegh Krohn, R. Rodriguez and M. Sirugue, Comm. Math. Physics 77269 (1980); I. Daubechies and J. R. Klauder, J. Math. Physics 23 806 (1982), J. R. Klauder "Quantization is geometry, after all" (preprint AT&T Bell Laboratories, Murry Hill, NJ. 07974 USA). [3] I. Daubechies and J. R. Klauder; J. Math Physics 25 2239 (1985). [4] I. Daubechies, J. R. Klauder and T. Paul; J. Math. Physics 28 (1987) .• [5] E. SchrOOinger, Sitzungsher Pruss, Akad - Wiss. Phys. Math. Klasse 906 (1930). [6] J. R. Klauder and B. S. Skagerstam, "Coherent States, Applictions in Physics and Mathematical Physics (World Scientific, Singapore (1985). [7) A. Grossman, J. Morlet and P. Paul; J. Math. Physics 26 2473 (1985) and Ann. Inst. H. Poincare 65 293 (1986). [8) T. Paul, Thesis [9] S. Graffi, T. Paul; Resonnance overlapping, quasi-energy avoided crossing and microwave ionization of hydrogen atom - preprint CPT, CNRS Luminy Case 907 13288 Marseille Cedex 9, France. [10] J. M. Souriau "Structure des Systemes Dynarniques" Dunod, Paris [11] In preparation (12) J. Bellissard; "Stability and Instability in Quantum Mechanics" in Trends and Developments in the Eighties, S. Albeveiro and P. Blanchard cds., World Scientific, 1985, Singapore. 208 Mean Value Theorems and Concentration Operators in Bargmann and Bergman Space K. Seip The University of Trondheim, The Norwegian Institute of Technology, Division of Mathematical Sciences, N-7034 Trondheim NTH, Norway 1. Introduction. A frame of wavelets corresponds to a discrete set of points in the plane or in the upper half plane. The density of a frame, if such a nuniber exists, will !lEan the density of this set with respect to the appropriate gecrretry (in the "ax+b" case this is the hyperbolic geanetry.) Assuming no regularity condition we ask the following question: Must the density in any part of the plane exceed sate lower bolmd, depending only on the analyzing wavelet (the "dual" question is for interpolation, then we ask for the existence of an upper bound)? Assuming regularity, we knCM that is a critical density in the ,"leyl-Heisenberg case [2, pp.37-47]. In the "ax+b" ; case it is kn= that such a density corresponding to the Nyquist rate does not exist [2, pp. 69-71]. The follCMing discussion could be applied to yield Sate nore infor- mation about this topic. The reason for the relevance of the Bargmann and Bergman spaces is the follCMing. The -1/2 -1/4 -x2/202 choice of analyzing wavelet go (x) = 0 7T e (0) 0) takes us in the Neyl2 -1/2 Heisenberg case into the Bargmann space B while the choice h8 (1;) = [8r (28)] A "ax+b" (21;)8e -I;(8> 0) inthe [4, II]. #, ~,q, Here case (a>8) A1,q (q= 8 +~) takes us into the Bergrran space denote the LP ~,q spaces of analytic func- tions respectively over C with !lEasure e- 1zI2 dxdY, over the upper half plane U q - 2dxdy and aver the unit disk /', with !lEasure (l_lzI 2 )2 q -2dxdy . with !lEasure i W: remark that in these spaces sufficient density bounds are kn= to exist [1], [5], [9]. It should also be remarked that part of the problems to be considered here have been solved previously [3], though in a different manner with the problems formulated in a different way. 2. Concentration operators in general. W: describe concentration operators analogous to the one leading to the farnilar pro- late spheroidal wave functions [6], [7], [10]. In stead of the Paley-Wiener space we shall be concerned with Bargmann and Bergman spaces. Let [2, I be any carpact subset of the darain aver which the space in question is defined. W: then seek the function f(z) in this space for which the concentration 209 A fllf(z) 12dw{z) = -=----=--- (1) frllf(z) 1 2dw(z) = f rll f (z) I2dw (z) ) ( II f 112 achieves its largest value. This leads to the following eigenvalue problem in L2 (rl,dw(z» Af(z) where P = (2) (Pxlpf) (z) denotes projection onto the space at hand and by the characteristic function of I. XI denotes multiplication By the same argurcent as in the Paley-Wiener case eigenfunctions and eigenvalues do exist. I f we next assume the situation to be as described in [4,1], we can make a similar construction with respect to the fonn of the Hilbert space :/C. Ig trans- Letting I be any CCII'pact subset of the locally oarq:act group G, the corresponding concentration problem can again be stated as an eigenvalue problem, nt:M in L2 (G,dx) (3) denotes projection onto the range of L. Also in this case we find that g -1 Taking the inverse transfonn L we get the g equivalent fornulation in :/C Here PL eigenfunctions and eigenvalues do exist. Af = c1 fI(f,U(x)g)U(x)g g (4) dx This is the localization operator used by I. Daubechie [ 2], [3, I] and I. Daubechie and T. Paul [3,11]. It is easily seen that the eigenvalues of PLXrPL and XIPLX I are identical. The kernel of the latter is k(x,y) = XI(x)XI(Y)~ g (5) (g,U(y-1 x )g) and we find (6) In the Paley-Wiener case we have [7] I A. = "Nyquist rate" ~ This suggests the significance of the number IIg1l 2 /Cg as a "critical 210 (7) • m(I) density". It is knONI1 that this munber has such a significance in the W:!yl-Heisenberg case (it which correspcnds to the Nyquist rate) [2, pp.37-47]. In the "ax+b" case Ugll 2/c both has and has not such a significance. In the case of an orthononnal basis of we must have a density equal to IIgIl 2/cg =l/2V2" [2, p. 36]. But 1. Daubechie and P. Tcharnitchian have shONI1 that it is possible to consequals 2l1T it is knONI1 that wav~lets truct wavelet bases at densities both lcwer and higher than this nurrber [2, pp. 69-71]. 3• The BarcylJailll case. 1. Theorem. For any f c BP (1 £ P £ (0) , we have the following reproducing for!tlUlas (e; =E, + in) (8) for any R, 0 < R £ 00, ;iZW) = {e;: i<:;-zj where (9) < n} (10) Proof. For 0 < R < 2 _ I z I -21T -e n! fR 0 00 we calculate the integral 2 n 2n+1e -pp d [d p - dwfl f (we ) -zw] w=z by Cauchy's formula after a change of variables. as R -r The formula obviously remains valid co. We single out two special cases as separate statements. 2. Corollary. For any ff f(z) = ____ 1 -- :\0 (R) !::'z (R) f' (z)-z f(z) 1 -TZ = f E BP we have f(<:;)K(<:;,z)e-I<:;1 ~() ff(R,f(L;) "1 R !::'z ) 2 dE,dn (11) (2-~)K(<:;,z)e-Ie;1 2 dE,dn (12) 2 (K (<:; , z) = TI e " , the reproducing kernel of B .) W:! also note the :i1lIrediate ccnsequences 211 Corollary. 3. For any 1;EC, the functions fn(z) = (n!1«1;,t:» -1/2 n (z-1;), n = 1,2, ... , fom an ortho- nonnal basis for B2, and they are orthogonal over any disk /:"1; (It) • 4. Theorem. Suppose I = /:"1; (R) (0 < n < 00). Then the o~rator PB2 XI PB2 has ei<]envalues An(R) as given by (10) and eigenfunctions fn(z) as given in Corollary 3. Proof. An For any fEB a~plication Re.mark 1. 2 we have by Theorem of Corollary 3 cOffi?letes the proof. Ne have here obtained the solution to the eigenvalue problem considered by I. Daubechie [3,1] avoiding the use of a commuting second order differential operator. Fernark 2. In addition to the fact that the eigenfunctions are independent of R it is interesting to note that they are orthogonal over any disk /:"1; (E) (Corollary 3). This is in analogy with the Paley-Wiener case, the prolate spheroidal wave functions are orthogonal both over R and the ooncentration interval. In fact, it is easily seen that this will hold for all ooncentration operators. 4. The Bergman ~ I1ere \'ie define Vl-Z Tw = - z 5. TheorETI. dn - ON? - z zw-1 f E A~,q (q > ~, For any [f (T w) (1-zw - -2 q]w=a 1';; p';; 00) we have the reproducing fonnulas (13) - where c n {l 2q(2q+l) •.. (2q-l+n) , n=O else (14) (15) 212 6.z (r) Proof. = Tz6.(r) 6.(r) = {s:lsl < o<r r} The fonnulas are easily proved for z =0 (16) ~ in the same way as Theorem 1. rest follows frOlll this special case by the method used in [8, pp. 90-9JJ. The (I am grateful to H.H. Hartens for showing me the relevance of this reference. We have follcwed the terminology of [1], [9]. In view of our rrethod of proof it may be rrore appropriate to use the tenn Bers spaces.) We note sane consequences analogous to the ones following Theorem 1. 6. Corollary . For any f E AP,q we have 6. (17) , ,2 - (18) (izl -l)f' (z)+2q z f(z) 7. Corollary. For any t; c 6., the functions h (z) = (T z)~(z,L;), n = 0,1,2, ... , furn 2 n t; an orthogonal basis for A6.,q, and thev are orthogonal over any hyperbolic disk 6.L; (r) . 3. Theorem. Suppose I 6.[, (r) (0 < r < 1). Then the operator P A 2 ,q XI P 2 A ,q has 6. 6. eigenvalues An (r) as given by (15) and eigenfunctions hn (z) as given in Corollarv 7. Proof. As the proof of Theorem 4. Remark 1'2 a Under2 the surjective iSOllletric correspondence f(z) -> {(t;~i)}2q f(~:~) between Af1' - and A(j,q this solves the corresponding problem for hyperbolic disks in the upper half plane. sidered by I. Hence \Ie have obtained the solution to the eigenvalue problem con- Daubechie and T. P'3.u~_ [J, II] avoiding the use of a CClllIllUting second order differential operator. Remark 2. As Remark 2 following Theorem 4. Re:nark 3. Transfonning and using (12) and (18) one finds both in the Bargmann and Bergman case that the commuting differential operator essentially takes the 5. fo~ zf' (z) . Application to sampling (frames) and interpolation. In this section, unless otherwise specified, we shall not distinguish between the Weyl- Heisenberg and the "ax+b" case. dx will denote the left Haar measure in both cases, dw(2;) the measure in the appropriate Bargmann or Bergman space, and "distance" will mean 213 either Ellcledian or hyperbolic distance. rated by a least distance 00 > 0. All discrete sets {si} will be assumed sepa- We will in both cases assume the special choices of analyzing wavelets described in the introduction. By (11) or (17) we have for any ¢(x) in the range of the transfoDII ¢(x) = _1_ "0 (c) f f d(x,y)<c (19) ¢(y)K(x,y)dy where K(x,y) is the reproducing kernel of this space. An application of Schwarz' inequality and the property of the kernel leads to Fran this we are able to prove exact parallels to Lemna 1 and lerma 2 in [6]. Using the asyrrptotic behaviour of the eigenvalues in the Weyl-Heisenberg case [3,1] we get (in the tenninology of [6]) 9. Theorem. With a gaussian analyzing wavelet in the Weyl-Heisenberg case the density of the set {Xi} must asymptotically be at least k to be a set of interIx>lation. k to be a set of sanpling and at lOClst In an attempt to prove a similar result in the "ax+b" case we prove a sharpening of one of Iandau' s lamias, fonnulated in the appropriate Bargmann or Bergman space. 10. Iemna. For °< r < 00 and {zk} let n o = nCr) be the number such that 0< d(z1'0) ~ ••• ~ d(zn'O) < •.• < r + 2 ~ d(zn+1'0) ~ .••• For the set {zk} to be one of sanpling there must exist a constant y < 1, independent of r, such that (21) where , ~ ! (k! = 2q .•• )2o-Hk - (k!)!..2 Proof. have let ¢n(z) ) !..2 = ~ n II (z-~). k=1 ~ , BerqIran case , Bargmann Assume {~} to be a set of sampling. Then by (19) we f f d(z,O)<r 214 case l¢n(z) 12 dw(z» By Corollary 3 or 7 and Theorem 4 or 8 the result follows with y An interesting observation is the following. are of the sane size if IZkl2 Heisenberg case. = (~1+k) <5 = 1-Acg AO(2) • The first and last ooefficient in (21) in the "ax+b" case and IZkl2 In either cases this corresponds to the density =k in the Wey1- ugn2/Cg. We may ask whether our larrna in conjunction with the known asymptotic behaviour of the eigenvalues [3,II] could be used to derive a necessary density condition in the "ax+b" case. We may also ask i f the density in the Weyl-Heisenberg case has to be strictly 1 greater than 271. References. [1] R.R. Coifman and R. lbchberg: Representation theorems for ho1CIOClrphic and hanronic functions in r..P. Astfu"ique 77 (1980), 11-66. [2] I. Daubechie: The wavelet transfonn, time-frequency localization and signal analysis. Preprint, AT&T Bell Lab. 1987. [3] Daubechie: Time-frequency localization operators - a gearetric phase space approach, 1. To appear in IEEE Trans. Inf. Theory. 1. Daubechie and T. Paul: - - , II. Preprint, AT&T. Bell Lab. and Luminy 1987. [4] A. Grossmann, J. Mor1et and T. Paul: Transfonns associated to square integrable group representations. 1. Journ. Math. Phys. 26 (1985), 2473-2479. II. Ann. Inst. Henri Poincare 45 (1986), 293-309. [5] S. Janson, J. Peetre and R. ~g: Hankel fonns and the Pock space. sala University, Math. report, 1986:6. [6] H.J. Landau: Sanp1ing, data transmission, and the Nyquist rate. 55 (1967) 1701-1706. [7] H.J. Landau: Necessary density conditions for sanp1ing and interpolation of certain entire functions. Acta Mathsnatica.!.!l (1967) 37-52. [8] J. Lehner. AutCIOC>rphic Fonns, Discrete Groups and AutCIOC>rphic Functions, ed. W.J. Harvey. Academic Press (1977) 73-120. [9] R. lbchberg. Operators and Function Theory, ed. s.c. Power. NA'ID ASI Series (1985) 225-277. [10] D. Slepian and H.O. Pollak: Prolate spheroidal wave functions, Fourier analysis and uncertainty, 1. Bell Syst. Techn. J. 40 (1961) 43-64. 1. Upp- Proc. IEEE H.J. Landau and H.O. Pollak: - - , II and III. Bell Syst. Techn. J. 40 (1961) 65-84 and 41 (1962) 1295-1336. 215 Besov Sobolev Algebras of Symbols G.Bohnke Universit6 de Nancy I, oepartement de MalMmatiques, B.P. 239, F-54506 Vandreuvre les Nancy Cedex, France 1. INTRODUCTION 1.1. We are studying Banach algebras for *-products associated to quantization on Hermitian symmetric spaces 1),2),7) and 8). We are giving here an example of such algebras on the usual flat symplectic manifold x = (x' ,x") and y (R n,a) = (y' ,y") where a(x,y) Rn x R n • = x"y' - x'y" for all E Other results will be published elsewhere. 1.2. The Tools Let for all W be the (projective) Weyl representation of W(x)(qI)(n) = ei<X"~ qI E L2(Rn) and all R 2n +n>qI(n+Y') (1) x E R 2n ; and let denote by f the symplectic Fourier transform f(x) = f R 2n e ia(x,y) f(x,y) dy • (2) Then, the integral represem.ation given by rr(f) = for all R ( 3) 2n f(x) W(x) dx f E <d(R 2n ) , extends to an isometry (under suitable normaliza- ti on) between tors on f L2( R2n) and the space ~ ~ 0 f Hil bert-Schmi dt opera- L2(Rn) 5). In fact, in the Weyl pseudo-differential calculus, the function f in (3) is the symbol of the operator rr(f) and we have rr(f) 216 0 rr(g) = n(f,jj. g) (4) for all f and g E L2(JR2n ) , where # bols (Moyal product). Moreover, we have '" g = f ~ f = *0 '" g (5) where, in the right hand set of (5), tion on R2n : f *0 g(x) = is the composition law of sym- J R *0 denote the twisted convolu- 2n e- io (x,y) f(y)g(x-y) dy (6) which is a bilinear product from L2 (JR2n) to L2 (JR2n) There exist explicit formulas for f ~ g in terms of asymptotic expansion, but we will not use them here. 2. AN EXAMPLE We have the following result 2.1. Proposition Let s ~ 0 and 1 ~ P ~ 2 • The ordinary Besov is a Banach algebra for the twisted product ~. This result has been announced in 3). 2.1.1. Proof. We use wavelets which give more information than HardyLittlewood decomposition used in 3) . We recall that the frequencies space is quasi-partitionned by a frame of overlapping paving-blocks which are the supports of the wavelets Fourier transforms (euclidean or symplectic as one can see easily by symmetries), and that there exist two functions, quite explicit in 6), ~ = ~ (1), ~ = ~ (0) (the "mother" and the "father" of the wavelets) with supp~c [-R.,-4R.]U[R.,4R.], supp ~c[-2R.,2R.] for R.= ¥- The family of wavelets 2nj /2 ~(E) (2 j x-k) , j E 71.. , k E 71.. , (e) (e1) (en) , e i = 0 or 1 (e i 1 0 at least for one i) is an orthonormal inconditionnal basis of L2(R2n) which will be denoted further by {ew}wEn ' w = (j,k,E). ~ ~ ... ~ 217 For the sake of simplicity, we sketch the proof in the case of Sobolev space HS BS2,2 • The, we have the following caracterization of HS : f <=:> < E wEn (7) co <=:> We know already that HO L2(R2n) is a =-algebra ; so, we have only to show that f = E y(w)e w E HS and 9 = E S(w')ew' E HS implies wi th (8) We write this latter sum in three terms (9) R(f,g) 2.1.2. Estimation of the "paraproducts" Parseval, we have Tf • 9 and f * Tg • By (10) and, on account of the tesselation of the frequencies space, one can take an integer NI > 0 such that there exi st subsets I(w') , w' En, uniformly finite (ll(w')1 ~ NI ) for whom the two subsets supp(~w' *w iw) and supp i w" are disjoints when w" ¢ l(w') and jew) ~ jew') - 2 Put (for w' fixed) then, since {ewl wEn 218 E( w'Elw' )o(w")e" = "( ) "( ') a(w)ew # ew' , w JW~Jw-2 is an inconditionnal basis, II) (w") I for all w' Ell (11) and w" E l(w') . Because of the uniform bound in (11), it remains to show that the sum l: a( ')e w" is in HS ', but this is clear if one w, Hl,w "l E ( w') "w remarks that the two integers jew') and j(w") are of the "same size" (more precisely, there exists an integer N2 > 0 such that !j(w") - j(w') I :;; N2 ) . 2.2.2. Estimation of the "error" R(f,g). We"can pick up an integer No > 0 such that, for jew) = jew') ± 2 and j(w") ~ jew) + No ,the two supports of w" and of w' *w w are disjoints. Consequently, if we put y(w") = <R(f,g),e w"> , we have e then, for e e jew) = k fixed, 11.( )_~( ') 2 J w -J w ± a (w)s(w')e w' #e)12 C' :;; 4ks then !y(w") :;; which implies l: j(w»j(w")=j-N o R(f,g) E H2S - E ,0 < E < S . 2.3. Remarks 2.3.1. "Microlocalization". We can restric the sommation to any subset C II (such that there exists a certain amount of "holes" in the frequencies space), we have always a subalgebra of HS for the twisted product ~ when s ~ 0 . II' 219 2.3.2. Ordinary multiplication. For the ordinary multiplication of functions, we have the well known result: HS(R n ) is an algebra if (and only if) s > ~ ; moreover, the function of one variable f E C""( R) , f(O) = 0 ,operatein HS : f(tp) E HS for all tp E HS if s > ~ • The proof of this fact, using wavelets, works as well as the one using Littlewood-Paley decomposition in 4) Bibliography [1] Bayen, F., Flato, M., Fronsdal, C., Lichnerowlcz, A. and Sternheimer, D., "Deformation theory and quantization", I, 111, 61-110 (1978). [2] Berezin, F.A., "Quantization", Math. USSR Izvest., 8,11091165 (1974). [3] G., "Sur les alg~bres de B~sov pour le produit C.R. Acad. Sc., Paris, t. 303,S~rie I, n° 15,729732 (1986). Bohnk~, crois~", [4] Bony, J.M., "Calcul symbolique et propagation des singularit~s pour les ~quations aux d~riv~es partielles non lin~­ aires", Ann. Sc. E.N.S., 14, 209-246 (1981). [5] Grossmann, A., Loupias, G., Stein, E.M., "Ana algebra of pseudo-differential operators and Quantum Mechanics in phase space", Ann. Inst. Fourier, 18, 343-368 (1968). [61 Meyer, Y., "Principe d'incertitude, Bases hilbertiennes et alg~bres d'op~rateurs", S~minaire N. Bourbaki, n° 662, 1-15 (1986) . [7] Moreno, C., "Geodesic symmetries and invariant star products on Kahler symmetric spaces", Letters in Math. Physics 13, 245-257 (1987). [8] Unterberger, A., 220 Ast~risque 131, 255-275 (1985). Poincare Coherent States and Relativistic Phase Space Analysis I.-P. Antoine Institut de Physique Tbeorique, Universite Catholique de Louvain, B-1348 Louvain-Ia-Neuve, Belgium 1. INTRODUCTION Group theory is one of the cornerstones of wavelet analysis. Indeed, at a very general level, one may say that the following three concepts are equivalent: (i) a square integrable representation U of a group G ; (ii) coherent states over G ; (iii) the wavelet transform associated to U .This analysis is familiar in the two standard cases [I], which have been thoroughly discussed during this colloquium: (i) the affine (a.x+b) group, which yields the usual wavelet analysis; (ii) the Weyl-Heisenberg group, which leads to various phase space or timefrequency representations. Our purpose in this work, done in collaboration with S.T.Ali [2], is to try to apply the same method to the Poincare group P (first in 1+ 1 dimensions, then 1+3), thus extending to the relativistic domain the usual phase space analysis. However, in doing this we encounter a major stumbling block: the natural (Wigner) representation of P is not square integrable! However it does become so when the integration is performed, not over the entire group P, but over a suitable quotient space prr (in 1+ 1 dimensions, T is the subgroup of time translations), which may be interpreted as phase space. Then the whole procedure goes through, with only minor modifications. As a by-product, we obtain a relativistic Weyl transfom1 .On the other hand, the results have an intrinsic mathematical interest, since this situation is more general than the one usually treated in the literature; in particular, we obtain in that way a generalized notion of coherent states [3], which may prove useful in other contexts as well, such as geometric quantization or functional integration. 2 . THE PROBLEM AND ITS GENERALIZATION First we review quickly the usual approach, following essentially [1], where the original references may be found. 221 Let G be a locally compact group, with Haar measure dg, U a continuous, unitary irreducible representation of G on a Hilbert space 9f. A vector S e 9f is said to be admissible if the following integral converges: c(s) = IG I< U(g) sis >12 dg < (2.1) 00. If the set .9l. of admissible vectors is not empty, it is dense in 9f and, in particular, .9l. =!Jl whenever G is unimodular (Le. if the left and right invariant measures are the same). If .9l. *0 , the representation U is called square integrable , since for each S e .9l., the relation: (W~<I»(g) = [c(s)r 1/2 < U(g)s I <I> >, (2.2) defines an isometry W ~ of 9f into L2(G,dg), called the wavelet transform associated to the vector S (the vector S is the analyzing wavelet of [ID.Furthermore, W ~ intertwines U with the left regular representation of G on L2(G,dg), thus showing that the square integrable representations are precisely those that belong to the discrete series. We remark also that the vectors 11g == [c(sWl/2 U(g)s form a family of coherent states corresponding to the representation U . The square integrability of U has the following interesting consequences (which are crucial in the development of wavelet analysis, in particular for the possibility of reconstructing a signal from its wavelet coefficients) . Denote by 9f ~ the range of W ~' which is a closed subspace of 9f. Then: (i) The family of coherent states {11g , ge G) is an overcomplete set of vectors in 9f, with resolution of the identity (2.3) (ii) The vectors <I>~ e 9f ~ are continuous functions on G, and 9f ~ is a reproducing kernel Hilbert space, with kernel K(g,g') =< 11g l11 g'> . (iii) The following orthogonality relations hold: There exists a unique positive invertible operator e on 9f with domain equal to .9l. (the set of all admissible vectors) such that: \I SI' S2 e.9l. and \I <1>1' <1>2 e 9f . In the special case where G is a unimodular group, e = I,?{ 222 . The next step is to lift the analysis from J{ to 'B2 (J{), the Hilbert space of all Hilbert-Schmidt operators on J{, with scalar product <P 1IP2>HS = tr[Pl *P2]. This is obtained by noting that: < U(g)C 1;;1<1> > = tr[U(g)*p C-l] == (Wp)(g) , (2.5) where p = 1<1» <CI;;I . Observing that operators p of this form are dense in 'B2 (J{), and that the orthogonality relations (2.4) precisely mean that W is an isometry, one obtains in this way, by continuity, an isometry W: 'B2 (J{) ~ L2(G,dg), called theWigner transform map. Using this map W, one may then obtain a decomposition of 'B2(J{) into mutually orthogonal sectors, each of which consists of a Hilbert space of coherent states, like J{1; above. As stressed above, the crucial condition in the analysis is the existence of admissible vectors. Now it may happen that no such vectors exist, but that one has an analogous situation over a transitive homogeneous space X of the group. In other words, there exists a closed subgroup H of G, for which X == G/H, and for which there exists a Borel section ~: X ---) G such that the following integral converges, for some I;; E J{: f c~(I;;) = x I< U(~(x)) I;; II;; >12 dV(x) < 00. (2.6) The measure dv on X is assumed to be invariant under the action of the group G . On the other hand, for the same representation U, there may not exist any vectors I;; which are admissible in the sense of (2.1). Nevertheless it turns out that the existence of vectors I;; which satisfy (2.6) (and possibly some additional conditions) is often enough to give rise to properties of the representation which parallel those described above. Besides, as mentioned there, square integrable representations are representations of the discrete series of G, so that studying a generalized notion of square integrability might enable one to catch some of the other representations of the group as well. We will work out these ideas explicitly in Section 3 below, in the specific case of the Poincare group in 1+1 dimensions. Before that, it is worth mentioning that similar cases have been treated in the literature, but only when H is the center of G [4] or, more generally, when the restriction of U to H is a unitary character [5]. Then the analysis is independerlt of the choice of the section ~ . This will not be the case, however, for the Poincare group. 223 3. RELATIVISTIC COHERENT STATES We denote by P/(1,l) the Poincare group in 1+1 dimensions. Its elements are written as (a,A), where a = (ao,a) is a space-time translation and A is a Lorentz boost. The matrix A may be parametrized by a vector p = (po,p) : p Po 1' P E (3.1) o/m +, where o/m+ denotes the forward mass hyperbola: o/m+ = {(po'p) E = m2 }. 1R2 I Po > 0, Po2 - p2 (3.2) The elements ~ of the Lorentz group act on o/m+ in the natural manner, k ~ k' = Apk, k E o/m + . (3.3) This action is transitive and the corresponding invariant measure on o/m+ is easily seen to be dk/k o ' (1,1) . The Consider next the following unitary irreducible representation of Hilbert space is !If.w = L2(o/m +, dk/ko) , whose elements are functions of the single p/ variable k E IR, square integrable with respect to dk/ko . The unitary operators constituting the representation will be denoted by Uw(a,A), (a,A) E P+ i (1,1) and their action is: (3.4) where k.a = ko ao - k.a . We shall call Uw the Wigner representation of P+i (1,1) formass m. It is easy to see that the Wigner representation is not square integrable in the sense of (2.1). Indeed, for any <l>w E f i !If.w, k Uw(a,A)<I>w l <l>w>12 dao da dp/po P+ (1,1) = (3.5) 00. However, we shall now show that in a certain sense U w is square integrable over a particular homogeneous space. Consider for this purpose the subgroup T of time translations of P+ i(1,l) and denote by rl and rr the corresponding left and right coset spaces, (3.6) It is easy to see that points in both that the map 224 13 : rl,r rl and rr can be parametrized by (q,p) ~ P+i (1,1) defined by E 1R2, and (3.7) is a Borel section for both spaces r l ' rr rl and rr. Since P+ i (1,1) and T are unimodular, both coset have a unique left, resp. right, invariant measure. A direct computation = dq dp is left-invariant on r l, while the measure dll/q,P) =dq dp/po is right-invariant on rr . It is important to notice here that although we may take the two coset spaces P/(l,l){f and 1\P/(l,I) to be equal, i.e. r l = rr = r, the two invariant measures dill and dllr are different. This feature is reminiscent shows that the measure dill (q,p) of non-unimodularity when the whole group rather than a coset space is considered. For the section 13 in (3.7), let us write, (3.8) where U w«O,q),1\) is defined as in (3.4). We now show that there exist vectors S E J{ w ' for which the functions f<l>.1;: fM(q,p) = < Uw(q,p) sl<1> >. r ~ ([; , defined by <1> E J{ w . (3.9) are square integrable. Let Ho be the free Hamiltonian operator on J{w : (3.10) Clearly, Ho is a positive operator with spectrum [m, 00) • Let 1J(Ho 1/2) denote the dense domain of its square-root Ho 1/2 . Then a straightforward computation shows that, for an arbitrary <l> E J{, the integral I(<l>,S) exists iff SE = fr 1fM(q,p) 12 dll(q,P) (3.11) 1J(Ho 1/2) . In (3.11), dll denotes either dill or dllr . Using this result, we may now define the wavelet transform associated to U w . Theorem 1 . - Let S E J{ w satisfy the two conditions : (i) >E 1J(Ho1/2), (ii) J k Is(k)12 dk/ko (3.12a) = 0. (3.12b) Then the relation : (3.13) where 225 c~(~) = 21tm- 1 fIR 1~(k)12 dk , (3.14) • defines an isometry W~~ : ~ ~ L2(r, dq dp). From this theorem follows that the analysis outlined in Section 2 goes through. First we adopt the following defmition for the admissibility of a vector ~ representation U w . (i) A vector ~ E !J{ w is said to be admissible mod(T,/3) conditions (i) and (ii) of Theorem 1 above. Note again that the set ~ which are admissible mod(T,~) is dense in !J{w . E !J{w for the if it satisfies the ~,~) of all vectors in (ii) Since the representation U w admits such vectors, we shall say that it is square integrable mod(T,b) . ~E The next step is to define coherent states for U w . Given an admissible vector ~T,~), we consider its orbit under U w : @5~(~) = {ll q,p = [c~(~)rl/2 ~q,p I ~q,p = Uw(q,p)~, (q,p) Er } (3.15) Then it can be seen that e~ is overcomplete in !J{ wand, moreover, f 11l q,p><llq,pldqdp = IJ{;'" (3.16) For this reason we shall call the family of vectors e~(T,~) = U ~ E (3.17) e~(~) jJ(T,~) the set of relativistic coherent states on the phase space r. For each fixed ~,the set @5~(~) will be called a coherent section. Let IP ~ = W ~~ W ~~* be the projection operator onto the closed subspace !J{ ~ of L2(G,dq dp), which is the image of !J{ wunder W ~~. Then there exists a reproducing kernel K~: r x r ~ a:: (i) K~(q,p; q',p') (ii) (IP ~<I»(q,p) (iii) = such that : < llq,plllq"p' >; = fr K~ (q,p; q',p') <I>(q',p') dq'dp' , V <I> E (3.18a) L2(G, dqdp); (3.18b) fr K~(q,p ; q",p") K~(q",p" ; q',p') dq"dp" = K~(q,p; q',p'). (3.18c) Thus everything parallels the general situation described in Section 2, and we try next to obtain orthogonality relations in the manner of (2.4) . The result reads as follows: 226 Theorem 2 . - There exists a positive unbounded invertible operator C and a bounded self-adjoint operator A , such that for all SI' S2 E 'D(Ho 1/2) and all <PI ,<P2 j{w' E fr < Uw(q, p) SI I<P 1 > < Uw(q, p) S21<P2 > dq dp = <CS2ICS 1> <<P 11<P2> - <ACS2ICSl> <<P 1IA<P2> , the second tenn on the RHS of (3.19) vanishing whenever SI = S2 E (3.19) • ~T,~) Explicitly, the operators C and A are given by (C<Pw)(k) = (2rr/m)1/2 ko 1/2 <Pw(k) , V <Pw (A<Pw)(k) = k/ko <Pw(k) , V <Pw E 'D(Ho 1/2), E (3.20) (3.21) ~. The fonn of the orthogonality relations in (3.19) ought to be compared to that in (2.4) . The appearance of the extra term, involving the operator A, has some very interesting implications. To understand this tenn better, let us begin by working out the Wigner transfonn for the representation U w ' Since C-l is a bounded operator, we may rewrite (3.19) as: fr < Uw(q, p) C-l S1 1<P 1 > < Uw(q, p) C- 1S21<P2 > dq dp (3.22) a relation which is now valid for all SI' S2 and <PI' <P2 we can now define a Wigner transfonn E j{ w . As in the last section, W, initially on all Hilbert-Schmidt operators p on j{w' of the fonn p = I<P> < Sl, <P, S E ~ .Thus, for all such p, (Wp) (q, p) = tr [Uw(q, p)* pC-I] and we then use a continuity argument to extend W to a linear map W (3.23) ~(~) ~ L2(G, dq dp) . Thus, V P E 'B2(~) , (3.22) assumes the fonn < WPl l WP2 >L2(G) = < Pl 1P2 >HS - < PI 1D p2 >HS (3.24) where D is the bounded linear positive operator on ~(~) Dp = ApA Thus the operator , V P E 'B2(~) . (3.25) W is bounded, but by no means isometric ! 227 1\.w of W is dense in Using the methods of [6], one can prove that the range L2(G, dq dp) and coincides with the domain of the operator cHo<1» (q, p) = Ho: (p2 + m2)1/2 <1>(q, p) . (3.26) The inverse operator W-l is unbounded, with dense domain introduce the new scalar product 1\.w' On that domain, we (3.27) = W- 1* D W-l. Clearly D w is an unbounded self-adjoint operator in L 2(G,dq dp), with domain 1\.w = 1)(W-l), and the scalar product (3.27) corresponds to where D w its graph norm. Thus the latter turns 1\.w into a Hilbert space, that we denote by J{(D J, and we get the rigged Hilbert space: (3.28) where Jf(D JX == Jf(D w- 1) is the dual of Jf(D J with respect to the inner product of L2(G, dq dp). The interest of the triplet (3.28) lies in the fact that the orthogonality relation in (3.24) becomes now: < WPl l WP2 >w = < Pl 1P2 >HS' \j PI' P2 E tJ32(~) Thus the Wigner map is an isometry from ~(~) into !Jf{D J (3.29) . A complete decomposition of the representation IT generated by U w on ~(~) can now be undertaken, as indicated in Section 2. The results are similar to those obtained in [6], except that now, with the new scalar product (3.27), the phase space representation turns out to be globally unitary. This result exhibits the fundamental difference between the nonrelativistic situation and the relativistic one, already noticed in [6]. In the former case, the operator W-l is bounded, in fact unitary. In the present context, however, W-l is unbounded, so that J{ (D w)"# L2(G). Now the difficulty encountered in [6] becomes clear. The U, via W, is not unitary, because the norm of L2(G) is not the correct one. Now, since 1\.w is invariant under U, as can be seen easily, we may consider the restriction Dr 1\.w . Then, if one replaces the L2(G) norm by representation U induced in L2(G) by the graph norm of D w ' everything falls in place and the phase space representation becomes globally unitary, as it should, but in J{(DJ instead ofL2(G). 228 4. A RELATIVISTIC WEYL TRANSFORM The Wigner transform, defined by the extension of (Wp) (q, p) = ..Jm/2n (4.1) tr [Uw(q, p)* p Ho-l/2] may be used to construct a relativistic Weyl transform in analogy with a similar transform which is used in non-relativistic statistical mechanics [7]. We start by defining a relativistic symplectic Fourier transform. Consider the set offunctions f: M(1,l) x ']Jm+ f(q,p) where q -7 Ie defined by = (e-iHot cD) (q, p) , <P E L2(r, dqdp), = (t, q), p = (II p2 + m2, p (4.2) ), M(l,I) is the Minkowski space and Ho is the operator on L2( G, dqdp) given in (3.26). Since, for fixed qo' f(p,q) defines a function on L2(r, dqdp), we shall denote the set of functions (4.2) again by L2(r,dqdp), for we shall only ~e concerned with fixed values of qo . Then the relativistic symplectic Fourier transform f of a function f is defined by the relations : f(q,p) = (2ntl f e-i(q.p'- p.q') f(q',p') dq' dp' (4.3) <lo' =const f(q,p) = (2ntl f e-i(q.p'- p.q') f(q',p') dq' dp' (4.4) <lo' = const For fixed qo' q'o ' the norms of f and satisfy II f 112 = f, considered as functions in II [II 2 , i.e.the symplectic Fourier transform L2( r, dqdp), :F: f H 1 is a unitary operator on L2(r,dqdp) . Since the representation operators Uw(a,A p) leave the domain of Ho in (3.10) invariant, it follows that the function f : 9.1(1,1) x o/m + -7 Ie defmed by (4.5) is an element of 1(w for all rank one operators p. Moreover, it is easy to see that f(q,p) = (e-iHo t Wp) (q, p) . (4.6) = :F W . It is easily verified that the range 1(9 of 8 w is dense in L2(r,dqdp), and indeed 1\.e = :F'R.w. Thus it remains simply to transport the triplet (3.28) through the unitary map :F. In other We define now the relativistic Weyl transform 8 w as the operator 8 w 229 words, one introduces on 1(e the scalar product: <'I'l' 'I'2 >e where De = <'I'l' (4.8) 'I'2> L2(1) + <'I'll De 'I'2 >L2(1) = 1'-1* D w 1'-l. With this scalar product, ~ becomes a Hilbert space 9-I(8,J , and from (3.29) we get <8 w Pl' 8 w P2>e = <PI' P2>HS 'V'Pl,P2 E 'B2(!J{w)' (4.9) We collect the properties of the relativistic Weyl transform 8 w in a theorem. Theorem 3. - The relativistic Weyl transform 8 w : ~(!J{w) ~ !J{(8,J, given by (8u,P)(q,p) = --Jm/2n I e-i(q.p'- p.q') tr [Uw(q, p)* P Ho-l/2] dq' dp', (4.10) CJo' = const (defined initially on rank one operators in ~(!l-\v) and then extended by continuity) is a Hilbert space isometry. The inverse map 8 w-1 : !J{(8,J ~ ~(!J{w) is given by 8 w-1 'I' = --Jm/2n I Uw(q, p) Ho-l/2 (X'¥) (q,p) dq dp , (4.11) CJo=const where ~ is the symplectic Fourier transform of 'I' , and X = m-2 Po Po-l (PoPo +P.p) is a self-adjoint operator on L2(r, dqdp) with domain 1(w' defined in terms of P = -iOloq , Po = --J p2+m2. • Notice that the the vector \¥ in (4.12) belongs to 1(w' Thus, 8 w- l associates, to any vector 'liE L2(r, dqdp) which satisfies f p 2 1",(q, p)12 dq dp < 00, (4.12) a Hilbert-Schmidt operator on!l-\v . Moreover, looking at it in this way, 8 w- l : 'I' H P {'¥E $W>, P E ~(!l-\v)) is also the inverse of the Wigner transform (4.1). Secondly, if we make the (very heuristic) non-relativistic approximation, Po - Po - Ho - m » p, P, then (4.1), (4.10) and (4.11) collapse to their non-relativistic counterparts [7]. 5. FINAL COMMENTS We end this paper with two remarks. First, our definition of a coherent section (3.15)-(3.17) is a generalization of the notion of a family of coherent states, introduced by Perelomov (see[l]) . In the latter case, H would simply be the stability subgroup, up to a 230 multiplicative phase factor, of a fixed vector ~ E 1{, satisfying (2.6) . However, the subgroup T of P+ t(l,I) , which we use in (3.6), is definitely not such a stability subgroup for any of the admissible vectors ~ in (3.12) that we consider. Second, we have restricted our considerations here to the Poincare group in I-space + I-time dimensions. However, in view of the analysis carried out in [6] for the standard Wigner representations, with mass * 0 and spin j = 0, 1,2, 3, ... , of the usual Poincare group in 3-space + I-time dimensions, it is clear that an exactly analogous theory of square integrability could be built in those cases as well . In that case, H would be the subgroup T ® SO(3) of all time translations and space rotations. The admissible vectors would have to satisfy the additional condition of being invariant under SO(3) . However, the p+ t(l,l) case already brings out all the interesting features related to square integrability without the formalism getting too involved due to the presence of the additional rotation variables. REFERENCES [1] A. Grossmann, J. Morlet and T. Paul, J. Math. Phys. 26 (1985) 2473; Ann. lnst. H.Poincare 45 (1986) 293 [2] S. T.Ali and J.-P.Antoine, Coherent states of the 1+1 dimensional Poincare group: square integrability and a relativistic Weyl transform, preprint UCL-IPT-87-39 [3] J.R. Klauder and B.S. Skagerstam, Coherent States - Applications in Physics and Mathematical Physics, World Scientific, Singapore 1985 [4] A. Borel, Representations des Groupes Localement Compacts, Lect. Notes in Mathematics, vol. 276, Springer, Berlin et al. 1972 [5] H. Moscovici, Commun. Math. Phys. 54 (1977) 63 H. Moscovici and A. Verona, Ann. lnst. H. Poincare 29 (1978) 139 [6] S.T. Ali, Rivista del Nuovo Cim. 8, #11 (1985) 1 S.T. Ali and E. Prugovecki, Acta Appl. Math. 6 (1986) 1, 19,47 [7] A. Grossmann, G. Loupias and E. M. Stein, Ann. lnst. Fourier, Grenoble, 18 (1968) 343 I. Daubechies and A. Grossmann, J. Math. Phys. 21 (1980) 2080 I. Daubechies, A. Grossmann and J. Reignier, J. Math. Phys. 24 (1983) 239 231 A Relativistic Wigner Function Affiliated with the Weyl-Poincare Group J. Bertrand 1 and P. Bertrand 2 lUniversity Paris VII, LPTM, F-75251 Paris, France 20NERA, BP 72, F-92322 Chatillon, France I. Introduction In recent years, wavelets analysis associated with the "ax+b" group [1] has emerged in signal theory as a substitute to the Gabor one which is basically related to the Heisenberg way, .affine time-frequency representations were solve time localization difficulties group. In the devised in order connected to the use Wigner-Ville function [2]. The purpose of the present work is that a similar situation may be found in the area of formalism of non relativistic same to of the to show relativistic quantum mechanics. In the phase space quantum mechanics, states are characterized by functions f(x,p) and expectation values of operator observables A are written as: op = <A> J f(x,p) A(x,p) dxdp where A(x,p) is the symbol of A op A widely used formula for the quantum states is where ~(x) space representation of [3]: Wigner~s f(x,p) phase =J e-2inu.p ~(x+n~u)~*(x-n~u)du is the wave function. (1) The relativistic generalization of the above formula is usually performed by adding an extra time variable a Lorentz-covariant way, = in thus leading to [4]: Je 2in(uo o_u. ) * 4 p p ~(x+n~u)~ (x-n~u)d u (2) where x stands for (x,xo) and p for (p,po). However, the physical interpretation of such problems due to spurious interferences between frequency parts. a function positive In particular, states localized in x leads and and to negative t involve necessarily both signs of the energy and cannot therefore be associated with truly elementary particles. These considerations representations founded on show a representations do exist and can 232 the need physically be derived for more admissible from the significant group. Such Weyl-Poincare group which is the semi-direct Poincare transformations. product of positive dilations and Among typical features of this group, we note that the inner positive and negative energy cones are distinct orbits. In the following, the explicit construction of function is performed through a tomographic method for the affine group [5]. In section II, we the distribution previously recall how allows to recover the usual Wigner function when applied group [61. In section III, we Weyl-Poincare group relative to apply one the space same devised the to technique dimension method Galilei~s and to the obtain a relativistic phase space representation. II. Tomographic method applied to the Galilei group. Given a time t, Galilei~s group G reduces to translations in phase space according to: (a, In the following, v)e G: (x, p) -----+. (x+vt+a, p+mv) we consider simultaneously the usual of this group on rays in Hilbert space and its ~ phase space functions f(x,p). for the correspondence ~ ~ The requirement representations classical of galilean action on covariance f is thus expressed by the commutativity of the diagram: ____-+. e -imv. x+i4> ~(x-vt-a, t) 1 f(x,p) (3) f(x-vt-a,p-mv) where 4> is a phase depending only on v and t. To introduce the tomographic construction space dimension, Go = «a,v)e G;a=ov), For a given 0, in the case of one we now study observables invariant by subgroups 0 e ~. such observables are simultaneously diagonalizable in the basis defined as: -1/2 1 2rr (t+o) 1 exp(i(t+o) -1 2 «m/2)x -rx» The expectation values are then given by (4) _ 1 (~, zo (r» 12 is a true density. In phase space, the functions A(x,p) representing such observables are constant therefore be written as: on the G -orbits. o The invariant expectation can 233 <A> where I(~,a) J = (5) f(x,p) dx dp 6(~+(t+a)p/m-x) (6) plays the role of a one-dimensional density. At this point, we can express the so-called constraint by identifying the marginalized distribution tomographic with the I(~,a) density pa(~). The matching of parameters in (4) and (5) is performed through the use of an arbitrary monotonous function A • We thus impose: (7) The function A is determined by requiring covariance with respect to Galilei~s group, as shown in (3), and space inversions. The result is: = A(~,a) m ~ Thus the Radon transformation expressed by (7) is readily inverted, yielding the standard Wigner function. dimensions by considering translations in x and p. Among the new features arising The method can be generalized to three space in this extension, we mention: Only a restricted family of subgroups is needed. - The families of diagonalizable observables are all associated with subgroups whose phase space orbits are lagrangian hyperplanes. The procedure leads again to the Wigner function in In one space dimension, the Weyl-Poincare group W is the set a non-ambiguous way. III.Relativistic phase space representation. 1. Defini t ions. elements w = ch €I A '.0 (a,A,y) where a = AOi. = sh €I I > 0 is the dilation, A = is the boost and y = (y~) = (A~ ), v (y~,yo) A of A U 00 are the space and time translations. The group law is w where A.y = (A~ w~ (a a~,A A~,y+aA.y~) yV). v The action on Minkowski space is given by: w.x Let ~ =a be the Hilbert space of functions momentum space with scalar product: 234 A.x+y ~ on the forward light cone in (8) where p = (p1,pO), dp = dp1dpo,re ~. We shall consider the following unitary irreducible representation U of W in 1Je: (9) To get a more manageable description of the group action on phase space, we use the light cone variables: X X P 1 P 2 P° - P1 1 2 Then setting e =a 1 oe -e = a oe 2 (1/2) (yo+ y1) b (1/2) y1) b (yO_ 1 2 we remark that W is the direct product of two affine groups, = where H is the set of elements h H X W- H (a,b) with multiplication law: = (a,b) (a',b') (aa',b+ab') The action of W on phase space becomes: _ _ _W _ _ _+I (a.X. + b., l. Representation (9) can now be realized \. \. in a~lp.), L ;.=1,2 (10) L the space of functions S(P1,P2) such that: by U(w)S(P ,P ) 1 (11) 2 We follow the same steps as in section II. 2.Two-parameter suberoups and correspondin8 densities. The Lie algebra of H X H is generated by A., (dilations) and B (translations) with commutation relations: [ A. , B. 1 " = B., , ;. = 1, 2 The two-parameter subgroups are either abelian or solvable. systematically discard isolated cases as they would have in the Radon transform. The abelian case is considering direct products of H-subgroups. The labelled by two real numbers e ,e 2 : completely resulting We zero shall measure solved groups by are 1 235 H = {(h ,h)e W; t;~t;2 1 h. = (a,C<1-a.))} 2 \. \. \. The solvable subgroups labelled by (ry,a)e 1. ~2 and (e,~)e ~2 are given by: Hrya = HI' .. ~ ~ h {(h ,h ) 2 ~ Of course, many of these conjugacy by W. h =(a,b), {(h~ ,h 2 ) ~ h =(a,a b+ ry(l-a»} 2 = (a,e(l-a», h subgroups are 2 = (a~,b)} equivalent with But we actually need all representatives in respect the to above families to be able to perform a Radon transformation along the orbits in phase space. Now, into we decompose the restriction of U to the irreducible diagonalization of the abelian case, This components. corresponding will lead invariant various subgroups ultimately to observables. the decomposition is performed as in [5] by a In the introducing the Mellin transform according to: (12) where and the functions O(~~'~2)/O(~~'~2) , ~.(~,e),i=l,2, are such that the jacobian J(~,e) is different from zero. The inverse formula is The restricted representation U1He e 1 then acts upon ~(~,e) in the 2 following way: U(a ,a )~(~,e) ~ 2 and invariant observables are diagonal in the basis Ze (~). In the sovable case, invariant with respect it can be shown that to the subgroups H diagonalizable in general. arbitrary rya or observables are not 3.Determinins the relativistic Wisner function. In phase space, He e - invariant observables must be described ~ 2 constant functions on the subgroup orbits. The latter are given by: 236 by Going back to variables (x,p), space representation ~(x,p) we define relativistic phase by: = ~(x,p) F(X,P) where F is the function characterized by relative to subgroups which reads: He e 1 a the tomographic constraint 2 (13) To fix the arbitrary functions AL' we shall impose a number of constraints: WeyL-Poincare covariance This is expressed by the commutativity of the following diagram: a ~(x,p) 2q ~(o The result is that A can depend only on Space inversion covariance. This implies the relation : -1 A -1 (x-y),oA -1 p) ~1 and ~2' A2(~1'~2) Uni tari ty condi t ion. (14) where ~1'~2 This gives are the representations respectively associated with = A/~1'~2) ~1+ ~1'~2' cst. Time reversaL invariance. This eliminates the arbitrary constant. It is then straightforward to invert the Radon transform in (13) and find: f (u u )2r+2 (4 sh(u /2) sh(u /2»-2r-2 1 2 1 2 (15) where m m m m 00 01 If q 11 10 -1 (u ) (e u 1-1) -1 + (u ) (e u 2-1) 1 2 (-u ) (e 1 u 1-1) -1 + (u 2 ) (e u -1 2-1) 2r + 1,this representation has the additional property: f ~(x,p) dx 0 dx 1 = I~(p 0 ,p) 12 237 4.Coherent states. As is easily seen from the (12)-(14), represented by 6-functions in phase space. basis elements In particular, Ze are if the positive energy state is represented in (x,p) space by a state localized in xO,x~: Beside these states truly localized on phase space lines, one can find minimal spread recalling that V = states in the neighborhood of any point by H X H. Thus the results obtained in the study of the affine group coherent states (wavelets) can fiducial state to be transported by group readily V, we be applied. choose a As tensorial product of minimal states related to H [7] which reads: o ~ ~ 0 0 ~ exp[-2in(J.I~(p-p)+J.l2(P+p»](p-p) -2inv -r-1 ~ 0 ~ (p+p) -2inv -r-1 2 IV. Concluding remarks Ve have built a phase space representation with respect to the Veyl-Poincare group and (or negative) energy elementary particles. interferences arising are between which which More positive is covariant localizes generally, energy positive the only components or negative energy components of the field .. When the representation is applied to a field sharply localized in Fourier space (po,p~), the result is very close to the standard (2). In particular any progressive wave is represented by a one delta funcion in the momentum subspace. References [1] I. Daubechies, in this volume. [2] J.Bertrand and P.Bertrand, in this volume. [3] E.P.Vigner, Phys.Rev. 40,749(1932). For a review, see for example V. I. Tatarskii, Sov.Phys.Usp. 26,311(1983). [4] S.R. de Groot, V.A. van Leeuwen and Ch.G. van Veert, Relativistic kinetic theory (North-Holland,1980) and references therein. [5] J.Bertrand and P.Bertrand, C.R.Acad.Sc.Paris 299,635(1984). [6]J.Bertrand and P.Bertrand, [7] J.R.Klauder, Found.Phys.~,397(1987). in "Functional Integration: Theory and Applications",J.P.Antoine and E.Tirapegui Ed. 238 (Plenum, New York 1980). Wavelet Transforms Associated to the n-Dimensional Euclidean Group with Dilations: Signal in More Than One Dimension R.Murenzi* Institut de Physique Th6orique, Universite Catbolique de Louvain, B-1348 Louvain-Ia-Neuve, Belgium 1. INTRODUCTION When one wants to extend to more than one dimension, the whole wavelet machinery developped for the one dimensional ax+b group, while keeping the group language, it is natural to consider the n-dimensional Euclidean group with dilations, to be denoted by IG(n). It is a non-unimodular locally compact group and its most natural unitary representation of in L(lR n, dn x), is irreducible and square integrable. We believe that this representation can be used to analyze signals in more than one dimension; this applies, in particular, to the analysis of images, which involves the group IG(2) . 2. THE GROUP Let n be an integer, n ;::2 . Consider the set G(n) of real n x n matrices v of the form v= ar (2.1) where a> 0 and where r belongs to SO(n),the special orthogonal group in n dimensions. The set G(n) has a natural structure of direct product ( dilations and rotations commute). An element v of G(n) depends on 1+ n(n-I)/2 parameters . Consider next the inhomogeneous group IG(n) : the Euclidean n- dimensional group with dilations. Elements ofIG(n) are pairs {v, b} with v e: G(n) and b e: IR n .Here b is the displacement vector in IR n. The element {v ,b} of IG(n) acts on IR n in the obvious way: x -+ {v,b}x=vx+b, xe: IRn. (2.2) *Boumer du Conseil du Tiers -Monde, VeL, Louvain-la-Neuve 239 The group law of IG(n) is given by {v, b}.{y, b'} = {YV, yb'+b}, (2.3) hence the inverse of {Y, b } is {v, b}-l = { v-I, - v-I b}. (2.4) Finally, one has the following decomposition: IG(n) = [IR + * x SO(n)] ® IRn . (2.5) The group IG(n) is a non-unimodular Lie group with : where: . left Haar measure a-(n+ l)da dr dnb (2.6) . right Haar measure a-Ida dr dnb (2.7) . a-Ida the invariant measure on the dilation group . dr is the Haar measure on SO(n) (see [1] for the explicit formula of dr) . In the case n denoted by (a, = 2, we have IG(2) = (IR+ * e, b) with a > 0, b E IR 2 ,e x 50(2) ® E [0,21£[ ; 1R2. An element of IG(2) is e is the rotation angle in the plane. This group contains the following discrete subsets, that will be important in the construction of frames below: G<Io, ~,y, L ={(~I , eo j (2.8) ,w)} where: w =(~I ~ m,~1 y q) 1, m, q, L E Z, j E ZL+l ={ 0,1, ... , L}, 13, y > 0. eo = 21£/Land 3 . THE REPRESENTATION We are going to defme a natural unitary representation of IG(n) in O(lRn,dnx) which turns out to be both irreducible and square integrable. We start by considering the operators corresponding to the three types of operations that constitute IG(n) . Given a function 4> E: L2(lRn, dnx), we defme : ~Xx) = 4>(x-b) , (r::>B4>Xx) = x, b E IRn (unitaryshiftoperator) (3.1) a-n/2¢(a- 1x), a>O (unitary dilation operator) (3.2) (Rr4>Xx) =4>(r 1x) , 240 x E IRn, (unitaryrotationoperator) (3.3) The commutation relations for these operators are : (3.4) RrOa= oaRr, (3.5) Introducing the family of unitary operators Eb defmed by : (E%Xx) = ei b·X<jl(x), (3.7) and the usual Fourier operator F: (F<I>)(k) = «k) = (211")-012 f dnx e-ik.x <l>(x) , (3.8) we obtain the following commutation relations : F Oa = Ol/ap , (3.9) FTb=E-bF, (3.10) FRr=RrF, (3.11) In terms of the operators (2.1) - (2.3), we construct now the operator ,Q(a,r,b) = T b oa Rr , (3.12) (0 (a, r,b) <1» (x) = a-llI2«a- 1r -l( x -b» (3.13) with action Using the commutation relations (3.4) - (3.6), it is then easy to check that 0 defmes a representation of IG(n) in L2( JRn, dnx) . In the frequency variables we obtain : ,., 0=FOF-1, n(a, r, b ) = (3.14) E -b oa R r , (3.15) (!l(a, r, b) $) (k) = an/2 e-ib .k «ar-1k) , (3.16) Proposition 3.1. - 0 is an irreducible representation of IG(n) in L2(JRn, dnx) . This representation is square integrable in the sense of [2], Le. there exists a dense set 1) c O(lRn, dnx) of vectors g such that the matrix element < 0 (a, r, b) gig> is square integrable over IG(n) with respect to either one of the measures (2.6), (2.7) . A vector g belongs to 1) if and only if its Fourier transform satisfies the condition: 241 f dnk 1k 1- n 1g(k) 12 < (3.17) 00 where g is the Fourier transfonn of g , and 1k I = (k 12+ ... +kn 2) 112 . D A proof of the proposition will be found in [5]. Remarks : (i) Vectors in1' are called admissible vectors or analyzing wavelets [2] , and the condition (3.17) , admissibility condition. (ii) As in the one dimensional case the admissibility condition is a restriction of g around the frequency k = O. When g is admissible, we will denote by cg the constant: Cg = (2rr)nr dnk Ikr n ~(k)12 (3.18) Corollary 3.2. (Decomposition of the identity) - For any fl' f2 in O(lRn, dnx) and any analyzing wavelet g, we have the following decomposition of the identity [2]: < fll f2 > = cg-l.or a-(n+l\iadrdnb < fll a, r, b where I a, r, b> = > < a, r, b I f2 > n (a, r, b) g . (3.19) D Let us give now some examples of analyzing wavelets . (ii) The 2-dimensional Mexican hat: The 2-dimensional Mexican hat is defmed, in Fourier coordinates, by : gco(k) = <k I D k > e- 1I2 <k I C k >, (3.20) where < kiCk> >0, < k I Dk > > 0 . (ii) The n-dimensional Morletwavelet : We define the general n-dimensional MorIet wavelet[2] [6] by gc(x) = ~(k) ei ko·x e- 1I2< x 1ex> _ e- 1I2 <koIDko> e- 1I2< x IC x > = (det D) 112 (e- 1I2<(k-ko) ID (k-ko) > - e -112< 1\0 D1\o>e- 1I2< kiD k » (3.21) (3.22) where C is positive defmite and C = D-1 , the substraction tenn (counterterm) in Ik and gc guarantees that OC(O) = 0 and ko is chosen in such a way that the counterterID is negligible in practice, for example : lkel = rr(2/ln2)112 242 . (3.23) 4. WAVELET TRANSFORM Lg , REPRODUCING KERNEL, RECONSTITUTION FORMULA. One can now apply to this representation the whole machinery of square integrable representations [2] [3]: (i) Lg transform (wavelet transform), (ii) the characterization of the range of Lg by a reproducing kernel, (iii) the inversion formula for Lg. Let us consider an analyzing wavelet g and f E O(lR n, dnx) ; then the wavelet transform associated to g is the map Lg : L2(lRn, dnx) (Lgt)(a, r, b) = a-n/2 (cg)-1I2 = a rv2 (c gt1l2 -+ O(IG(n) , a-(n+ 1ljadrdnb) f dnx g(a- 1 rl(x-b» f(x) f dnk ei k.b g(ar1k) f (k) (4.1) (4.2) The wavelet transform has the following properties: (i) Isometry andreconstitution formula: Lg is an isometry, i.e. : < Lg f I Lgf > = <f If> , (4.3) and it is invertible on its range by : (Lg-ILgf )(x) =f(x)= (cg)-l ill a-(n+lljadrdnb <D(a, r, b)g I f> !1(a, r, b)g(x) (4.4) (reconstitution formula) where f E O(lR n, dnx) . (ii) Reproducing Kernel: The range Lg L2(lR n, dnx) c O(IG(n), a-(n+ Iljadr dnb) of Lg is a Hilbert space with reproducing kernel [2] [3]: Pg«a',r',b'),( a,r,b» = cg -1I2( Lgg)( ( a',r',b' t1(a, r, b». (4.5) 5. FRAMES OF L2(1R2,d 2x) DEFINED BY IG(2) Let us consider an analyzing wavelet g for the representation n and the discrete subsets CJ<loV B L of IG(2) dermed in (2.8) . We are going to formulate a proposition that gives conditions on <\" ~, y, L for which the family: 243 (5.1) constitutes a frame in O(1R2, d 2x) , in the sense of [6] , i.e. for any there exist two constants A, B, with 0 < A < B< 00 , f E L2(1R2,d 2x) such that (5.2) This proposition can be generalized for IG(n), n 2: 2, and generalizes the corresponding one given by I.Daubechies for the one dimensional "ax+b" group [6] . Further details about frames maybe found in the papers [6] [7]. Proposition 5.1. - Assume that the following conditions hold: (1) s(ao' L, g) = ess infkE 1R2 Lj Lli g(801 r ·6oj k) 12 = ess inf (lkl.cp)Lj Ll 1 gp(801 1k I.cp+ 6J) 12 (5.3) is strictly positive ; (2) S(ao• L, g) = ess sUPkE 1R2 Lj Lli g(801r .6ojk) 12 = ess SUP(1k1 ,cp)Lj Ll 1 gp(801 ik 1,cp+ 6 0 j) 12 IkI ( coscp, is fmite, where k = sin cp); IkI (5.4) E ]O,ao [ , cpE [0, 21£ [ , gp is the Fourier transform of g in polar coordinates; (3) sUPuE 1R2 (l + u 2)1+e a (u) where £> = CE , (5.5) 0 and a(u) = sUPkE 1R2 L 1Ig(801k +u)11 g(801k) 1; (5.6) then there exists constants Ve, ~e > 0 such that : (i) for any (V,~) E ]0. L is a frame; (ii) for any ~ vel x ]0, ~d . the family {glmqj} associated to a 0' V. ~. > 0 • there exists (V. ~)E lYe, Vc+~] family {glmqj} associated to a o , v, (3, L is not a frame. See [5] for the proof. 244 x [f3eo f3c+~] such that the a Corollary S.2. - Let us denote by E(3o, y, t3, L, g) the expression: L (m,q) EZ*2 { x i( r -80j(301k+ v » II g(301r -80*)11 [suPkE 1R2 Lj Lli g( r -80j(3oIk- v) 11&(301 r -80jk)IJ} 112, [suPkE 1R2 Lj Ll (5.7) where v= (21r m Iy, 21r q/~) . Then lim (y.~)-(O.O) E(3o, y, 13, j, g) = 0 and the frame bounds can be estimated by where 0< A ~ B :S (41r/yl3) (s(ao ' L, g) - E(a, y, 13, L, g)} (4;r/yl3) {S(ao ' L, g) + E(a, y, Y < Yc ,0 < 13 (S(aa, L.g) - E(ao ' y, < I3 c and y c = t3, (5.8) (5.9) L, g)}, {inf y} , I3 c = {inf 13} such that 13, L,g)} :s O. a We refer the reader to [5] for numerical estimates of those frame bounds and more details on applications of the above results to signals in more than one dimension. ACKNOWLEDGEMENTS We want to thank Prof.Alex Grossmann for suggesting this problem, and also I.Daubechies and M.Ho1schneider for useful discussions .We acknowledge gratefully the hospitality of the Centre de Physique Theorique II, CNRS-Marseille, where this work was done. Our thanks go also to the Unite FYMA (VCL, Louvain-la-Neuve), the Conseil du Tiers-Monde (Louvain-1a-Neuve) and the R.C.P. "Ondelettes" (Marseille), for their [mancial support. REFERENCES [ 1] N .la. Vilenkin, Special Functions and theory ofgroup representations, (A.M.S .,Providence,RI 1968). [2] A. Grossmann, J. Morlet, T. Paul, Integral transforms associated to square integrable representations.I, J.Math.Phys.26 (1985) 2473-2479 245 [3] A. Grossmann, J. Moriet, T. Paul, Integral transforms associated to square integrable representations.II, Ann. Inst. Henri Poincare. 45 (1986) 293-309 [4] A. Grossmann, R. Murenzi, Integral transforms associated to square integrable representations. III. The Euclidean group with scale changes (in preparation) [5] R. Murenzi, Doctoral thesis in preparation (U.C.L,Louvain -la-Neuve) [6] I. Daubechies, The wavelet transforms, time-frequency localization and signal analysis (Preprint). Bell Labs, 600 Mountain Avenue Bell Labs .Murray Hill , NJ07974. [7] I. Daubechies, A. Grossmann, Y. Meyer. Painless non-orthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283. 246 Construction of Wavelets on Open Sets S.Jaffard CMAP (Ecole Poly technique), F-91128 Palaiseau Cedex, France and CERMA (ENPC) We shall describe a way to construct wavelets on an open set 0 of R n (this construction is a joint work with Y.MEYER and can be found in (1) ;the reader should look there for precisions),then we shall give a more explicit description of the two following points that are important for possible applications: The asymptotic behavior (wavelets that are localized around very small cubes which are far from the boundary of 0 are numerically identical to the" corresponding" wavelet on R n ) and the fast decomposition algorithms (which aJ;e of a similar kind as in R n except that the storage of more filters is needed). 1 Construction and properties of wavelets This construction is related to the multiresolution algorithms that were invented by S.MALLAT and Y.MEYER and are described in other papers of this book. We shall construct an orthonormal basis of wavelets of L2(0) that are C 2m (m E N).For that we define a new form of multiresolution analysis as follows. Let Qj,k be the cube defined by 2 j x - k E [0,1] and Vj the subspace of L2(0) composed with functions C 2 m, such that supp j C 0 and j, restricted to Qj,k is a polynomial of degree 2m + 1 at most in each variable. The Vj are then an increasing sequence of closed subspaces of L2(0) whose reunion is dense in L2(0) . We shall at first construct an orthonormal basis of each Vj. We define the B-spline 0' by &(~) = ( )2m+2 . Sl;:"l .:. ~ ( . ~ SIn <"n )2m+2 ~n A basis of Vj is then obtained by taking the functions such that supp O'j,k en. These functions form a Riesz basis of vj, i.e. (1) 247 The constants that appear in the equivalence being independant of j. The set of n-uples k/2 j such that supp O"j,k is caracterised by en d(~ (2) 2J' an) >- m 2J~ 1 (if d(x, y) = sup IXi - Yil). A function of Vj is determined by its values on Aj .More precisely , if F E Vj (3) where 0 < Cl ::; C2 < +00. We construct two new bases of Vi, The first one is orthonormal and thus obtained: Let G be the operator defined over Vj by 2:= G(F) = < F,O"j,k > O"j,k k/2 j EAj by (1) G is positive definite. Let then A... k 'l'J, the rPj,k be = G- 1 / 2 (0".J, k) , are the required basis. The second basis is composed with cardinal splines, i.e. functions Lj,k of such that Lj,k(1/2 j ) = 8k,/. If rPj,k Vi then the matrix : is the inverse of the matrix (The existence of this inverse is assured by (3) ). LetWj be the orthogonal complementary of Vj in Vj+!.We shall construct an orthonormal basis of Wj. Such a basis is obtained by projecting over Wj the functions Lj+l,k such that 2J~1 E Aj+! \Aj and orthonormalizing the basis thus obtained by the "Gram matrix device" we already used. we thus obtain the wavelets 1jJj+!,k we were looking for. the union of tllese bases yields an orthonormal basis of L2(n). If this construction was made with n = Rn , we would obtain the "usual" wavelets, i.e.,in this case,there are functions rP and 1jJ(i) such that 248 (4) (i is a n-uple belonging to (0, 1)n and different from 0). These wavelets are also unconditional bases of the functional spaces (with the usual substitutes when m is an integer, for example CJ (n) will denote the Zygmund class) and of their duals ,of course as long as the smoothness of the wavelets is compatible with the space we analyze, and if the boundary of the domain has some smoothness (for example if the domain is bounded, with a lipschitz boundary). We shall now show that, if 2i d(k/2 i ,an) is big enough, the wavelets are close to the" asymptotic wavelets" of (4). 2 Asymptotic estimates We note ¢i,k and tP i,k the wavelets we constructed and: ¢j,k' tPj,k the corresponding wavelets on R n .Then the following estimates hold Proposition 2.1 IIa (¢j,k IIa (tPj,k - ¢i,k) 1100 ~ c2(n/2+a)j exp (-'Y2id(k/2i, an)) tPi,k) II 00 ~ c2(n/2+a)i exp (-'Y2i d( k /2 i , an)) for all a such that 1011 ~ 2m + 1. This proposition shows that, though the wavelets will theoretically be all different, we do not need to calculate those that are centered far enough from the boundary.This property is very clearly illustrated by the figures, obtained by R.SEBILLE (Ecole Poly technique) , that we give at the end .In this case, the wavelets are piecewise linear and the construction is made on the interval [0,1]. We need for proving the proposition the following lemma Let T be a discrete metric space such that Lemma. 2.2 V€ 3c such that sup L exp( -€d(s, t)) ~ c sET tET Let M be the set of matrixes M = (m(s, t))(s,t)ETXT defined on [2(T) such that 249 Im(s, t)1 :::; cexp( -,des, t) let A be a subset ofT and MI and M2 two matrixes of M sucb tbat aId:::; MI :::; {3Id aId:::; M2 :::; {31d and Iml (s, t) - m2(s, t)1 :::; cexp( -,Id(s, A) IMI- I / 2 (s, + d(t, A)I). tben v t) - M:;I/2(S, t)1 :::; c' exp( -,'Id(s, A) + d(t, A)I) wbere c',,' only depend of c", a, (3. and tbe same estimates bold for tbe coefficients of M I- I and M 2- 1 We now sketch the proof of this lemma.let Rand S be so that MI = I-R a M2 = I - S a then MIl = a-I L: Rk and M:;l Then,on one side, = a-I L: Sk. with p ::; l, and,by induction, one can obtain one thus gets IMI-I(s, t) - M 2- I (S, t)1 ::; ko 00 k=O k=ko I: c,k exp( -,'Id(s, A) + d(t, A)I) + I: l the optimal choice for ko yields the required estimate. The same proof yields the same type of estimate for IM;I/2(s, t) _M:;I/2(s, t)l. Prop 1 is an easy consequence of lemmal 3 Fast decomposition algorithms The simplicity and efficiency of the algorithms that MALLAT introduced in [2] comes from the two following facts: First that Vi and Wj are both included in Vi+I,SO that pyramidal algorithms can be achieved,and second that the set of spaces 250 251 Vi on one side and Wj on the otp.er are deduced from one another by dilations and are invariant by translations so that the discrete filters used in the different levels of decomposition are the same. The first of these two properties still holds for the wavelets we constructed but the second doesn't,so that pyramidal algorithms can be written, but the values of the filters will be different for the computation of every coefficient. Though, it is not such an important drawback since the asymptotic estimates show that the values of the filters will be very close to the " asymptotic' values" as soon as one of the wavelets will be localized far from the boundary of n .This propriety holds because, as in the Rn case ,the values of the filters are given by scalar products of wavelets; Proposition 1 gives then the desired result. We can go a bit farther with certain peculiar geommetrical settings, since then , it can happen that the filters to be used near the boundary will be the same for different values of j (for example,if the open set we consider is a half-space,or a cube (when j is big enough). 4 Bibliography 1. S.Jaffard et Y.Meyer , Bases d' ondelettes dans des ouverts de R n to appear in Journal des Mathematiques pures et appliquees 2.S.Mallat, A theory for multiresolution signal decomposition: The wavelet representation. Dept of computer science, University of Pennsylvania, PA 191046389,USA 252 Wavelets on Chord-Arc Curves P.Auscher UER de Math-info, Universite de Bordeaux I, 351, Course de la Liberation, F-33405 Talence Cedex, France Abstract. We give a new proof of a theorem of G. David which says that the Cauchy integral on a chord-arc curve r is a bounded operator on L2(~). The main tool we use is the multiresolution analysis to get wavelets adapted to r. Introduction. Let r be an unbounded Jordan curve in the complex plane and Q be the region on one side of r. To solve the Dirichlet or the Neuman problem in Q with data f in LP(r), we need some geometric constraint on r. We suppose then r chord-arc that is, r is oriented and locally rectifiable and its arc-length parametrization x ~ z(x) satisfy (1) 30 0 > 0 \;1(x,y) E ~2 Iz(x)-z(y)I" 0 0 Ix-yl. For example, graphs of Lipschitz functions or logarithmic spirals enjoy this property, a parabolic curve does not. A.P. Calderon and A. Zygmund [CZ] showed that these problems are related to the study of the Cauchy integral which is (2) 1 f (y) z' (y) 271i z(y)-z(x) lim t:~O dy a.e. The real variable methods tell us that the L2(~) estimate (3) implies all LP(~) estimates for 1 < p < 00 [CM]. Calderon [C] proved (3) if r is the graph of a Lipschitz function A, with IIA'II. < 1)0 for some unknown 1)0. R. Coifman, A. McIntosh and Y. Meyer [CMM] proved moreover that one could take 1)0 = +00 and, eventually, G.David [D] concluded this study by characterizing the set of curves for which Tr is bounded on L2(~). Chord-arc curves are in that set. Our aim in this work is the following. As it is explained in Y. Meyer's paper [M], the wavelet representation formula is nothing else than the Calderon reproducing formula which is the main tool to get the boundedness of Tr on L2(~). Here, we show that the advantage of the concept of multiresolution analysis, especially orthogonality, is 253 to simplify the needed quadratic estimates. P.Tchamitchian [T] had already succeeded in the same direction in the lipschitzian case. All results exposed here (obtained in collaboration with P. Tchamitchian, see [A] or [AT] for more details) extend to higher dimensions and can be applied, for example, to the double layer potential related to a surface in ~ •. We shall use freely most results about multiresolution analysis and refer the reader to [Ma] or [M]. I - Wavelets on chord-arc curves r-wavelets must have three basic properties : localization, smoothness and cancellation. They are indexed by (j,k) E 2x2 and will be denoted by 9 j ,k(X). Take w.(x) = (l+lxj)-l-. for 11 E ]0,1]. By definition (9 j ,k) are r-wavelets if there exist two constants C and 11 with C ~ 0 and lIE]O,l] such that for all (j,k) E 2x2,(x,x') E ~x~ and (6) f o. aj,k(x) z'(x)dx IR = Note that, if r ~, 9 j ,k(X) could be an ordinary wavelet on 2 j / 2 1P(2 j x-k) (with small regularity). ~ II - Results We get first a theorem concerning r-wavelets. Write b(x) for z' (x) and if f,g E L2(~), B(f,g) = (7) f~f(X)b(X)g(X)dX Theorem 1 : There exists two families of r-wavelets (a in L2(~) such that if (j,k) B(aj'llJej~Jk~) = (8) (a J, (9) k) o( J , k ) , ( j' , k') forms a Riesz basis in L2 = {: (~) if (j J ,k) = ,k) i and (6 J ,k) (j' ,k') (j' (the same for ,k') (6j,k». In other words (9 J, k) and (6 j , k) are two Riesz basis in L2 (~), biorthogonal with respect to the bilinear form B and each function f in 254 L'(~) has the representation (10) (L f LIB ( f , ej , 1 / 2 k) I2 are two equivalent norms. ) From this follows immediatly The Cauchy integral given by(2)is a bounded operator on Theorem 2 L2 (IR) . We outline the proof to show how our basis are a powerful tool. Write formally T r (e j , k) LL = Y( j , k) , ( j , , k') e j , , k' then j ' k' ¥ ( j , k) ( j J , k J , k' ) 1 im c-t 0 - _1_ 4n: 2 ff z' (x) z' e j' , k' (y) e j, k (y)dydx (x) Ix-yl>e z (y) -z (x) Some technical estimates show that the matrix M with entries Y(j,k),(j',k') is almost diagonal so that M is bounded on ~2(22). Because (ej,k) and (ej,k) are Riesz basis, this is equivalent to (3). III - Scheme of the proof of theorem 1. The main idea in this proof is the construction of a special multiresolution analysis in L2(1R). On this basis, we imitate algorithms to get ordinary wavelets but for the bilinear form B instead of the usual scalar product on L2(1R). This is possible because of geometrical assumption (1). Now we sketch the proof in a few lemmas. Lemma 1. There exist a function g in L2(1R), lution analysis Vj,j E 2, such that for all (12) Ig(x)1 ~ Cw,(x), Ig(x)-g(x')1 ~ E ]0,1] and a multireso(x,x') in 1R2 ~ Clx-x'I'lw,(x)+w.(x')I, for all E 2, {gj,k(X) = 2 j / 2 g(2 j x-k),k E 2} is a Riesz basis of Vj and the matrices Mj with entries B(gj ,k,gj,,),k E 2, ~ E 2, are bounded and invertible on ~2(2) with the following norm estimate UM~IUop ~ 28~1. This multiresolution analysis comes from the simplest we known the multiresolution analysis of the Haar system. Take ~ = X[O,I[' In that case the matrices ~j are very simple : they are diagonal and each diagonal entry has a modul us larger than 8 0 by (1), but we cannot conclude yet because of lack of regularity. So we use a perturbation of this case. Let ~(~) denote the Fourier transform o f ~. We ha v e, if mo(O = e- i U 2 cos ~/2 then iP(O = e- i U 2 (~/2)-lsin~/2 = IT mo(~2-j). Define q(O = exp [-10g2(cos2~/2)],m(0 = j~l mo(O(q(O)' and g by 255 = IT m(e2-;)" e > 0 will be fixed at the end of computations. j;>l The choice of q insures that, whatever e will be, m is a C· function, 2n-periodic. Then, generates the multiresolution analysis we are looking for with ~ ae and a is a strictly positive constant depending only on q. Now consider Mi the matrices introduced in the lemma. There exists a constant C independant of e such that, for all j E 2, HMj-~j".P ~ Ceo This ends the proof. g(e) = We recall that Wi is the orthogonal space of Vi in Vjtl and that there exists a real valued function IJJ in Wo ' satisfying (12), such that {lJJ i ,k(X) = 2;/' 1JJ(2 j x-k),kE2} is an orthonormal basis of W; .Next define a subspace of V;tl by : (13) B(f,IJ) = O} For the sake of simplicity we take now j = 0, everything working the same uniformly in j E 2. We write gk for 'O,k""" and V,W,X,H for Vo,WO'XO'V l " (i) H = V &I X (direct sum, not orthogonal in general) . (ii) Define ITx and ITv the two bounded projection operators related to this direct sum, then ITx:W ~ X is an isomorphism onto X Lemma 2 (i) comes from the invertibility of the matrix Mo on ~'(2) and is formally expressed by the following diagramm which explains the action of ITx on W. (ii) X W H (the plane here) In order to calculate IJ k and 9k we use the following general result. Lemma 3 : let V be a separable Hilbert space, {e k ,kE2} a Riesz basis of V and B a sesquilinear bounded form on V such that the matrix M with entries B(ek,e,),k E 2,~ E 2 is bounded and invertible on ~2(2). Then there exists a Riesz basis {ek,k E 2} of V such that for all k E 2, ~ E 2, B(ek,e t ) = cSk,t. Moreover, if for all m E ~ there exists a constant C m such that for all k E 2, ~ E 2 IIl(ek,e,)1 tants C~ such that ( 14) 256 ~ Cm(l+lk-~I)-m then there exist cons- The last part of lemma 3 is due to S. Jaffard and Y. Meyer. We thank them to let us use this result still unpublished. B is not a valued, we can if v = Akg k ~ek) So we sesquilinear form on V but, because of g and ~ are real make use of the following involution on V (resp. on X) E V then v'= ~ gk E V (resp. if e = gkek then e' = define a sesquilinear form on V (resp. on X, by taking L L 8(v,v') L = B(v,v'*), v,v' L E V (resp. 8(9,9') = B(B,B'*),B,fP EX). Applying then lemma 3 to {gk,kEZ} in V o ' we can deduce a representation formula for il x . As we get e k = rrX(~k)' we obtain e k = ~k (15 ) - L B(~k,g,)g, ~ A consequence of the invertibility of M, on ~2(Z) (remember that X C H = V, and M, is given in lemma 1) is the invertibility of the matrix (B(ek,e,)) on ~2(Z). Then we apply the same trick to exhibit 9 k . So we have built two families (ej,k) and (9 j ,k)' These satisfy (8) by construction because B(9 j ,k,e j , , ) = Ok" and if j 'I j',eEXj,e'EX j , then B(e,e') = 0 by (13), a.nd verify (4) and (5) by (12) and (14). To get (6) we fix jEZ,kEZ. If j' < j and f E Vj then B(f,e j ,k) = O. It is easily checked that g(x-k') = 1 for all x E ~, so take fn(x) = k' gj' , k ' (x) E Vj' and let n go to infinity. By the Lebesgue domi- L L I k' I ~n nated convergence theorem we obtain B(l,e j ,k) = O. Next result concludes the proof of theorem 1. That is Theorem 3. Define a linear operator T on the basis (~j (16) T (~ j , k) = e j ,k) (6). by , k Then T is bounded and invertible on L2(~). The fact that T is invertible on L2(~) is easy once its boundedness is established. Last point is not an evident a priori and follows from the concrete situation in which we work. To see this, we introduce a distribution kernel K(x,y) of the operator T. Because of the orthogonality of the ~j ,k with respect to the scalar product we get, if (X,y)E~2 and x 'I y, K(x,y) = ej,k(x) ~j,k(Y)' This kernel is said of Calderon-Zygmund type and the boundedness on L2(~) of operators associated to such kernels has been characterized by G. David and J.L. Journe [DJ]. In our case, to apply their theorem we must show the following estimate (see [LM]) denote by D the collection of all dyadic intervals 1= Ij,k = [k2- j ,(k+l)2- j [ kEZ,jEZ, there exists a positive constant C such that for all lED, LL (17) L ~ CI I I where I I I is the measure of I JED,JCI 257 (17) is known as Carleson's condition. To calculate c J we use formula (15) and remember that B(f,g)=<fb,g> for f,gEL 2 (m). Then there exists a family (W j , .. ) in L2 (m) satisfying (4), (5) and (18) fm wj, .. (x)dx = 0 such that, if we write wJ = Wj ,k' we get c J = <b,w J >. Next bEL- (m) and the three properties (4), (5) and (18) of the functions wJ give us the desired Carleson' s condition. Now, to inverse T, first remark that (9 j ,o.) is a total family in L 2 (m), second take K' (x,y) = !/Jj,,,(x) iij,k(y) b(y) for x "I y and T' the operator associated to that distribution kernel. Because of biorthogonality of (9 j ,0.) and (ii j ,k) with respect to B, we get T'T = I = TT'. This ends the proof. LL References [A] These de doctorat, P. Auscher. Universite de Paris Dauphine. To appear. [AT] "Ondelettes, pseudoaccretivite, noyau de Cauchy et espaces de Hardy", P. Auscher, P. Tchamitchian. To appear. [C] "Cauchy integral on Lipschitz curves and related operators", A.P. Calderon. Proc. Nat. Ac. of Sciences 74, tome 4, (1977), 1324-1327. [ C Z] "Singular integral operators and differential equations", A.P. Calderon, A. Zygmund. Am. J. of Math. 79 (1957), 901-92l. [CM] "Au-delA des operateurs pseudo differentiels" R. Y. Meyer. Asterisque n057. [CMM] "L'integrale de Cauchy definit un operateur borne sur L2 (m) pour les courbes lipschitziennes", R. Coifman, A. Mc Intosh, Y. Meyer. Ann. of Math. 116 (1982), 361-387. "Operateurs integraux singuliers sur certaines courbes du plan complexe", G. David. Ann Sc. de l'ENS 17 (1984) 157-189. [D] [ DJ] [LM] [Ma] [ M] [ T] 258 Coifman, "A boundedness criterion for generalized Calderon-Zygmund operators", G. David, J.L Journe. Ann. of Math. 120 (1984), 371-389. "Ondelettes et bases hilbertiennes" P.G. Lemarie et Y. Meyer. Rev. Mat. Iberoamericana, vol. 2, n° 1, (1986). "Multiresolution approximation and wavelets", S. Mallat (1987) Dept of C.I.S.S.E.A.S., Univ. Of Pennsylvannia, Philadelphia, PA 19104-6389. "Wavelets wieved by a mathematician "Y.Meyer. This proceedings. "Ondelettes et integrale de Cauchy sur une courbe lipschitzienne", P. Tchamitchian. To appear. Multiresolution Analysis in Non-Homogeneous Media RR.Coifman Department of Mathematics, Yale University, New Haven, CT06520, U.S.A. We would like to describe various versions of "wavelet analysis" valid in a non translation invariant setting. Here the scale is allowed to change at various points in space, as well as the analyzing wavelets. This theory has been developed previously [1] in order to carry over various aspects of Fourier Analysis, such as Littlewood-Paley theory and singular integral operators to various settings, where a group structure is not available. It would seem natural that such a development could find uses in a varie~ of signal and image processing contexts, as well as in the study of partial differential operators with variable coefficients arising in a non homogeneous medium. One can easily imagine a picture viewed through an imperfect lens or through a fog. The light intensity seen at various points of the picture depending on its geometric location. Any multiresolution analysis, say for an edge detection problem, should take into account the variable geometry and sensitivity of the camera. A simple mathematical description of such analysis can be given in the context of spaces of homogeneous type: We are given a space X equipped with a metric (or quasimetric) d(x,y) and a measure dx. For example, we can consider a surface in R3 equipped with surface measure and Euclidean distance or, more generally, a Riemannian manifold with it's natural metric and volume. A different example is provided by imposing different scaling rules for different variables say, time and space, leading to non isotropic "metrics", such as d«x,t),(XI,t ' » x'I + It - = Ix- tit. This distance is natural for the study ofthe heat operator on R 2 which exhibits different homogeneity in space and time. 259 Discrete spaces such as the integers or graphs can also be considered provided they respect some mild regularity and growth conditions on the balls defined by d. l To be specific, the quasimetric d( x, y) has the following properties: a) d(x,y) > 0 {:} x b) d(x, y) =1= y = dey, x)3C > 0 such that for all x, y, z inequality d(x, z) ::; C[d(x, y) If Bx(r) = {y we have a weak version ofthe triangle + dey, z)]. EX: d(x, y) < r} denotes the open ball ofradius r centered a x, IB(x, r)1 its volume (or measure) we assume that there exists a constant C such that for all x IB(x,2r)1 ::; CIB(x, r)l· It is useful to renormalize the metric so that a ball of radius r will have volume ~ r. This can be achieved by defining a new quasi distance called the measure distance m(x,y) as the measure of the smallest ball containing x and y. We can also assume without loss of generality that we have some Holder "smoothness" for the balls, i.e. there exists a > 0, C such that Im(x,y) - m(x',y)1 ::; Cm(x,x,)a[m(x,y) + m(x',y)]-l-a. A wavelet or multiresolution analysis can be achieved whenever we dispose of an approximate identity Pr(f) ( x) corresponding roughly to a "smooth" weighted average of f on a ball of radius r around x. To be precise, we assume that Pr(f)(x) = J Pr(X,y)f(y)dy where Pr(x,y) =0 , m(x,y,)a IPr(x,y) - Pr(x,y)l::; r1+a and m(x,y) > cr for C (Pr(X, y) ::; -) r J Pr(x,y)dy = 1. As an example, we can take Pr(x, y) to 1 for x E [0, 1 For 260 t] and 0 x> t = 7P( m(~,y)) and c;:-l = ~ more examples see the survey [2]. where P : ~ Jp(m(~'Y))dy. -+ R+ is smooth equal Clearly, lim Pr(f) r--+O = f. We can now consider the variation in P r defining Qr measures the change in obtained by taking r = f as we move from one scale to the next. A discrete version is 2- k . (One can also pick a discrete set of points xj). Reconstruction formulas for f given Qk(f) can be obtained easily, the simplest is ob- tained by taking -00 in operator notation. This formula, although simple, does not provide us with a representation of f in terms of a fixed set of functions varying at different scales. Such a realization is obtained by writing where the Qj or Qj can be calculated as follows: We let EN = "L Qj+kQj Ikl>N j and Qj = (J - EN )-1 "L Qj+k Ikl::S;N It can be checked that by taking N sufficiently large Qj and Qj can behave roughly like Qj. In fact, they form a simple weighted average of Qj leading us to the realization of any function as f = ~QjQd =~ J qj(x, y)iii(y, t)f(t)dt or 261 The functions qi(x, yj) are wavelets centered at yj on the resolution 2- k • The advantage of this formula is that it involves average of "coefficients" (f, iii) permitting the over sampling and linear dependence of the functions qi, iii, and enabling transformations and error correction on the coefficients. (Unlike the case of orthonormal basis, the functions qj are not linearly independent, implying many linear relations among the coefficients.) These formulas have been used effectively as a substitute for the Fourier transform in a variety of nontranslation invariant contexts (1),[2],[3). It would be interesting to explore their applicability to situations where measurements are distorted with time or location in analysis and reconstruction. REFERENCES 1. R.R. Coifman and G. Weiss, Analyse harmonique non.commutative sur certains espaces homogenes, Springer-Verlag 242 (1971). 2. R.R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. 3. G. David, J.L. Journe, and S. Semmes, Operateurs de Calder6n-Zygmund fonctions para acretives et interpolation, Revista Math Ibero Americana 1 (1985). 262 About Wavelets and Elliptic Operators Ph. Tchamitchian C.P.T. et FacuIte des Sciences et Techniques de Saint-Gerome, F-13397 Marseille Cedex 13, France This paper is intended to show some possible applications of the wavelet transfonn to the study of elliptic operators, and then to pose an open problem. The proofs of the announced results will appear elsewhere. I - THE INVERSE OF SOME NON SELF-ADJOINT SCHRODINGER OPERATORS We begin with a SchrOdinger operator on ]Rn, with a complex-valued potential, which we assume to be bounded and strictly accretive (i.e. 1V(x) 1:s; C and Re V(x) ~ 8 > 0). Hence, the bilinear fonn associated to L, B(f,g) = f Vf. Vg + f V fg, defined on HI (lRn) x HI (lR,n) (where HI (lRn) is the first Sobolev space), is bounded and strictly accretive: Re B(f, f) ~ f 1V f 12 + 8 f 1f 12. Now, we choose a multi scale analysis on L2 (lRn), of the type defined in [2]. Our notations are the same as those of Meyer's paper in these proceedings. The important point is that the wavelets 'l'jk are compactly supported in frequency . 21t· space, and even that A'l'jk ( co) = 0 if 1co 1:S;T 21. We fix an index jo, which will be chosen later, and we call1t and 1t1. the orthogonal projections onto Vjo and V!-. Jo 263 The general organization of the construction of L-l is the following: we construct an approximate inverse on Vj ,and another one on V-:-. Finally, we use orthogonality o Jo relations to obtain a global approximate inverse, then the exact inverse. 1) The approximate inverse on Vjo : by hypothesis, the operator 1t L 1t : Vjo ~ Vjo is invertible. In fact, its matrix in the basis (<J>jok)kEZn is given by the elements and thanks to our assumptions on the potential, it is continuous and invertible on 12 (zn). We call A its inverse, and the correponding operator defined on Vjo. Then, the operator 1t A 1t is a perfectly well-defined operator from L2 to L2. 2) The approximate inverse on V-:- : here we will use the localization properties of the Jo wavelets 'Vjk, for j ~ jo. First, we consider the wavelet 9 defined by - /). (9) = 'V, and then 9jk = 4-j ~n/2 9(~ x - k), so that - /). (9jk) = 'Vjk (notice the different normalizations for 9jk and for 'Vjk). Finally, we define P by Pf= L L j~jo k < f, 'Vjk > 9jk. P is bounded from L2 to If2, and acts only on the high-frequency components, because Pf = P 1t.l f. Hence, we have II Pf IIL2 :S; C 4-jo II f II L2. In fact, - /). (Pf) = 1t.l f, and this implies that LP = 1t.l + VP. The operator VP, from L2 to L2, has a small norm, dominated by C 4 -jo II V 11 00 • 3) The global inverse: it follows from the definitions of A and P that L (P + 1t A 1t) = 1t.l + VP + L 1t A 1t = 1t.l+VP+1tL1tA1t+ 1t.lL1tA1t = 1t.l + VP + 1t + 1t.l L 1t A 1t = I+U, where U is the operator VP + 1t.l L 1t A 1t. 264 Now, the key remark is that, although U is not small in norm, I + U is invertible, because U2 is small. This comes from the orthogonality relation 1t.L 1t = 0, which implies (1t.L L 1t A 1t)2 = O. So, we find that U2 = (Vp)2 + VP 1t.L L 1t A 1t + 1t.L L 1t A 1t VP. With the help of the following estimate: 111t.L L 1t A 1t 110 0 ~ C (1 + II V II 00), , which can be proved independently of jo, we get that Finally, if jo is large enough, we obtain that L-1 = (p + 1t A 1t) (I + Uy1. The construction of the operator A is nothing but a version of the usual Galerkin method. The specificity of the wavelet transform appears in the construction of the operator P and of the global inverse L -1. We show now that this can be generalized in order to obtain parametrices of general elliptic operators. II - PARAMETRICES OF ELLIPTIC OPERATORS We turn our attention to general elliptic operators on Rn, with weakly regular coefficients. Namely, we consider operators 2 L = - apq (x) dIXI. + bp (x) dp + c (x), The construction of the operator A is nothing but a version of the usual Galerkin method. The specificity of the wavelet transform appears in the construction of the operator P and of the global inverse L -1. We show now that this can be generalized in order to obtain parametrices of general elliptic operators. 265 II - PARAMETRICES OF ELLIPTIC OPERATORS We tum our attention to general elliptic operators on Rn, with weakly regular coefficients. Namely, we consider operators L =- apq (x) aPl2 + bp (x) ap + c (x), where apq (x), bp (x), c (x) are bounded, and the matrix (apq (x)) is uniformly definite positive. We assume that the coefficients are of regularity ex> 0, i.e. I apq (x) - apq (y) I : : ; C I x - y la, and the same for b p (x), c (x). Of course, we have ex : : ; 1. ° U sing the same multiscale analysis as in I, we fix jo so that there exists ~ > with I apq (x) COp COq + i bp (x) COp + c (x) I ~ ~ I co 12 21t . for every co of modulus ~ T 2Jo. Then, if j ~ jo, we define 'tjk by and the operator Q by Qf = L L j~jo Theorem k < f, 'l'jk > 'tjk. Q is a parametrix of L, in the sense that LQ=I+ S, where S is a smoothing operator, continuous from L2 to Ha-£, for every e > 0, if ex < 1, and continous from L2 to HI if ex = 1. III - AN OPEN PROBLEM We do not know how to use more deeply the bilinear form associated to an elliptic operator. Let us describe which kind of result we would like to achieve, for the operator L = - div A (x) grad, 266 where A (x) = (aij (X)h~ij~o is a complex-valued, bounded and accretive matrix, which means that I < A (x)~, ~ > I ::; M I ~ 12 and Re < A (x)~, ~ > ~ 0 I ~ 12, for every ~ E RD. To such an operator one associates the bilinear symmetric accretive form B (f,g) = A (x) V f (x). V g (x) dx, bounded on the homogeneous Sobolev space HI (RO). f Our problem is the following : prove that there exist a family of functions Ojk , j E Z ,k ZO, such that E (i) I Ojk (x) I ::; C 2in/2 2-j co (2i x-k), (ii) I Ojk (x) - Ojk (y) I ::; C 2in/2 2-j 2i£ I x-y 1£ [co(2i x-k) + co (2iy-k)], (iii) B (Ojk, O"pq) (iv) f =L = O(jk),(pq), B (f, O"jk) O"jk for every f HI (Rn). E j,k -n-E Here, CO (x) = (1 + Ix12) - 0 2 , and E depends on M . This would imply that the Green kernel associated to L could be written as G (x,y) =L O"jk (x) O"jk (y). j,k This would be a decomposition of G into pieces localized in phase space, in the spirit of C. Fefferman ([1]). We do not know whether the O"jk exist or not, if n ~ 2. But, if n = 1, it is easy to construct them, with E = 1. In that case, A is an accretive function a (x), and so is b (x) Consider the bilinear form b (f, g) = =atx)" f b f g, bounded and accretive on L2 (R), and the associated basis of wavelets ([3]) : (v) (vi) I Pjk (x) I ::; C 2i!2 co (2i x-k), I Pjk (x) I ::; C 2 3j!2 co (2i x-k), (vii) fb (Pjk, ppq) (viii) Pjk b =0 =OUk), (pq), 267 (ix) f = Here, CJ) L b (f, pj0 Pjk for every f E L2 (R). jk (x) = (1 + x2)-1. Then, the desired O"jk are defmed by x O"jk (x) = I Pjk (t) b (t) dt -00 We leave the details to the reader. REFERENCES [1] C.Fefferman, The uncertainty principle, Bull. AMS, Volume 9, Number 2, September 1983. [2] P.G. Lemarie et Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2, 1-18, 1986. [3] Ph. Tchamitchian, Ondelettes et integrale de Cauchy sur les courbes lipschitziennes, to appear. 268 Towards a Method for Solving Partial Differential Equations Using Wavelet Bases V.Perrier ONERA, BP 72, F-92322 Chlitillon Cedex, France Wavelets frequency present good localization) and properties of global approximation their spatial localization allows (good precise approximation of discontinuities, without producing spurious fluctuations all over the domain. Wavelets so provide the combined assets of finite difference schemes and spectral methods in approximating functions, solutions of partial differential equations, governing flows of compressible viscous fluids. Interpolation and differentiation of periodic Heaveside functions using wavelets bases are performed. Much more accurate results are obtained with wavelets than with classical trigonometric Fourier functions. 1. INTRODUCTION 1.1 Motivations Numerical experimentation in fluid mechanics and turbulence has been much developed during last years. It involves the simulation of non-linear phenomena: highly inhomogeneous flows which can present discontinuities for some variables, speciaI.ly in the compressible case. Let us consider, for example, a typical 2-D compressible problem. A flow develops in the vicinity of a flat plate a "boundary layer", within which takes place a rapid variation of the velocity, (due to the friction on the boundary), superimposed with small fluctuations induced by the vortex generation along the plate. When a shock wave reflects on the plate, outside the boundary layer pressure, velocity and density will exhibit discontinuities across the shock ; inside the boundary layer, small fluctuations are superimposed to these discontinuities. 269 The problem is then, in numerical experimentation, to develop methods accurate enough to represent these behaviours and design "filters" able to distinguish between fluctuations. 1.2 turbulent (physical) and numerical (spurious) Classical Methods Three different classes of methods are mainly used for numerical solution of partial differential equations (P.D.E.) : finite differences, finite elements and spectral methods. Roughly speaking, the finite difference method consists in defining the different unknowns by their values on a discrete (finite) grid, and in replacing in the equations any differential operators by a difference operator, using neighbouring points. In finite element method, the equation is integrated against a set of test functions, with small compact support (chosen in a adequate functional space), and the solution is considered as a linear combinaison of this (finite) set of test functions. These two methods can adapt easily to complex geometries. They also correctly represent irregular functions (due to the localized character of the calculation ). In spectral methods, the unknown functions are developed along a basis of functions with global support (in general eigenfunctions of the system). This development is truncated to a finite number of terms which satisfy a system of coupled ordinary differential equations in time (solved by standard techniques). The main advantage of spectral methods is their greater accuracy. However, this accuracy is partially lost when the function to be approximated is not regular : the globality of basis functions induces the well known Gibbs phenomenon. To take advantage of both kinds of methods (accuracy of spectral methods in "smooth" regions and robustness of finite difference or finite element methods in "shock" regions), mixed methods have been developed, two of which are briefly described in the following. 270 1.3 Mixed Methods For compressible fluid flows with "shocks", mixed methods can be combined with both "shock capturing" and "shock fitting" techniques. In the "shock capturing" technique, at each time level, the whole flow domain is treated using a spectral method ; then a postprocessing using difference schemes is performed localizing the shock and smoothing it [1]. This technique greatly improves the results. However its computational cost is high and the spectral accuracy is lost. In the "shock fitting" technique, the computational domain is divided into two subdomains, separated by the shock. In each subdomain, the regular solution is computed by a classical method. The two solutions are connected along the shock using analytical Hankine-Hugoniot relations for the non VlSCOUS case. Subdomains overlapping over one mesh may be used for the viscous case. In both cases, the two solutions are obtained without loosing spectral accuracy [2]. When this connection is used with high order finite difference methods, non centered schemes are to be used near the overlapping mesh and without special care, the accuracy can decrease in this region. In this sense, one can speak of mixed method too in this case. Unfortunatly this method is difficult to adapt to complex shock geometries. Both technique have disadvantages : the first is numerically expensive, the other does not adapt to complex geometries. A single global method, which would present none of these defaults is still to be found. The wavelets are expected to provide a satisfying answer to this problem. 1.4 Wavelets Interest During last years, new orthonormal bases of L 2(H) have been constructed which have been called "wavelets" ([3], [4]). Houghly speaking the aim of the construction is to obtain basis functions as localized as possible both in time (or space) and frequency (spectral space). These functions are generated from a single "generating wavelet" by translations and dilatations. 271 A wavelet analysis of a 1-D signal leads to its representation in a 2-D coefficient space of time (or space) and frequency. Several generating wavelets have been proposed ([4], [5], [6], [7]) ; depending on their regularity the associated wavelet basis is also an unconditional basis of higher order functional spaces (Lebesgue, Sobolev, Besov, ... spaces). Wavelets have been originally derived to analyse seismic signals in the field of petroleum research [8]. At present they are used in image processing and analysis [9], and in sound (speech or music) analysis [10]. No results have been yet published concerning the use of wavelets in the numerical solution of partial differential equations. However wavelet bases seem to combine the advantadges of both spectral (good frequency localisation) and finite difference (good time -or spaCEr localisation) bases. One can expect that numerical methods using them should be able to attain good accuracy while resolving properly shocks. Four this point of view we compare in this paper some properties of spectral and wavelet bases. 2. INTERPOLATION AND COLLOCATION METHODS 2.1 DIFFERENTIATION IN Problem Statement For sake of clarity, the one dimensional periodic case is chosen. To highlight the problems involved in numerical simulation of P.D.E., let us consider the simple transport equation : (1) where af -(x, t) at af + u-(x, t)=O ax t> 0, ufO, 1]. f is the unknown function, and where the transport-velocity u (x, t) is given, 1-periodic in space. The simplest semi-discretization in time of equation (1) reads, where f n (x) denotes an approximation of f (x, n tlt) : 272 (2) In+l(x) _ In(x) - - - - - - + u(x, .6.t al n n.6.t)--(x) = 0 ax Now consider a set of N collocation points (xl' ... , xN ) in [O,I[ , where I n is supposed to be known. Solution of the discretized P.D.E. (2) requires al n to compute at the same points an approximation of - - , from which the ax values at the collocation points for time level (n +1).6.t can be obtained. Let us choose N linearly independent functions (¢l , ... , ¢N) and consider the finite dimensional vector space generated by the ¢ i. In this space, we calculate the interpolating function: which verifies </J(xi )= I(x j ) at each collocation point. Then </J(x) provides an approximation for I (x) , and : . . f or -al () an approxImatIOn x· . ax I Clearly, in this method, the choice of the basis functions ¢i is crucial, especially if I presents "pseudosingularities". In the following, we compare two cases: the first one uses Fourier trigonometric functions, the second one wavelets as basis functions. From the function </J defined above, one can obtain approximate informations on the function I : the derivative of I at collocation points, the interpolation of I between collocation points. In fact these two problems are nearly equivalent: one imagines easily that if </J has a strange behaviour between collocation points, a</J will not be accurate at these points, and vice ax 273 versa. AB the interpolation problem is easier to express graphically, our examples will be given in this framework. 2.2 Fourier Interpolation The natural basis in the periodic case is the Fourier trigonometric basis: 1/J m (x)= exp(21rmx), mf-Z. With the Fast Fourier Transform algorithm (F.F.T.) , we can easily compute the coefficients em from the f (xm) . The function: N/2 l/J(x) = em exp(21rmx) E m=-N/2 will approximate f and verifies l/J( xm )= f (xm) at each collocation point. In this case, the derivative problem is easy and cheap to solve, because the Fourier coefficients of al/J are obtained simply from the em ' and the ax resynthesis at collocation points can also be done by means of the F.F.T .. On the other hand, the interpolation problem cannot use a fast algorithm and is consequently expensive, each evaluation requiring the sommation of N trigonometric functions. Figure 1 shows a periodic Heaveside function f , characteristic of our "shock" problem. Figure 2 shows the approximation obtained by Fourier interpolation with N =32 and N =128 collocation points. Due to the 1.33 0.77 y 0.22 -0.33 274 FIGURE 1 : Periodic Heaveaide lunction 1(%) 0.00 0.25 0.50 0.15 1.00 FIGURE t : "Fourier" interpolated HeatJe.ide function 4J 1.33 0.77 y 0.22 -0.33 0.00 0.25 O.SO 0.75 1.00 1.33 4J{z) •••1 ~. .~ 0.77 N=128 Y 0.22 ..1 -0.33 0.00 0.25 O.SO 0.75 1.00 disc.ontinuity, outside collocation points, Gibbs oscillations appear all over the domain; these oscillations do not decrease when increasing the number of collocation points (figure 2). 2.3 Interpolation Using Wavelets 2.3.1 Choice of a periodic wavelets family. Y. Meyer describes in [l1J how to construct an orthonormal basis tP m of the space L 2[0,1J periodic, from any generating wavelet tP of L 2(R) . The method is the following: 1 em = J f (x )tPm(x)dx o which can be approximated by the Riemann sum: 275 However, the result is not satisfying (figure 4a) , the function cP does not correspond with f at collocation points. This is due to the fact that the family (tPo' .... , tPN-l) is not an orthonormal family in 12(Z j[NJ) , indeed I ... 1.0 o. - 0 -1.0 -2.0 ... 1.0 2.0 1.0 •• 0 '.0 x FIGURE 9a : Generating wavelet .p -1.0 FIGURE 96: l-periodic 6a8ia/unctioft8 276 tP;,j the N /2 wavelets of smallest scale have a significant variation on a grid mesh. The collocation approach is more satisfying, it consists in solving the Nth order linear system obtained by considering the function N-l <fJ(x) = L:; m=O cm1/;m(x) ,which verifies <fJ(xJ= !(xj ) at collocation points. One can see (figures 4, 5, 6) the approximation <fJ , computed with the two methods, for a sampled periodic Heaveside function ... 1.1 ~(:r) Wavelets are ¢(:r) ... f - -- •• t !. ----, .... .... .... ... (b) Collocation method (a) Galerkin method ... ~(:r) .p(:r) , ..·.y...--- -'\I1 ... ,.. -t-----....., . t-~--~+---~---4 .... .... (a) (b) .... FIGURE,, : 24 order "wavelet interpolated Heavuide function ¢ 277 ... I •• ... .. ~. +---~------~----------~~ •.• ..... ., .... +------~~--+-----~--.." .. ... N=32 v N=32 (a) Galerkin method ... ~ (b) Collocation method ••• ... . ,.. -w /" +------------i--------~___j ... - .... y N=64 fa) FIGURE 5 : 4" order "wavelet" interpolated Heave,ide function 41 278 (b) ... I .. ~(%) ~(%) \0' .. .. ., .... ., N=32 N=32 (a) Galerkin method ., (b) Colloca~ion method ..I ~(%) \0' ~. , .. I.' y +---~---~~~------i .A .... ... N=64 N=64 (a) FIGURE 6 : 6 14 order "wavelet interpolated Heave&ide function (b) cf> 279 respectively of order 2 (i.e. CO), of order 4 (i.e. C 2 ), of order 6 (i.e. C 4 ). The number N of collocation points is here equal to 32. According to the wavelet regularity, a small Gibbs phenomenon appears, however localized in the vicinity of the discontinuity. This differs strongly from the Fourier method, where the Gibbs phenomenon extends all over the domain. Although the collocation method is a priori more expensive than the Galerkin method, it is more accurate ... ... ... ... it gives the correct values at WoO . .. ..... .... .... ... ... . N=32 _ ... (a) Galerkin method .... ~z) "' <1% .. ..... (b) Collocation method .. .. . ... . - ...... N=64 FIGURE 7.' "Wavelet" den"vative 280 N=32 ... ... .... ... ( a ) .... N=64 ~ 01 HealJe,ide I"nction ~ lJz (b) collocation points, and the L 2-error between f and <P is much smaller (of order liN % of the L 2-norm of f). Using this last method, we have drawn (figure 6) the derivative of the interpolated function <p , obtained in derivating the wavelets basjs. It is clear on this figure, that the wavelet approximation allows to numerically differentiate the Heaveside function. On the opposite, the Galerkin method as the Fourier interpolation (see figure 2) does not allow to satisfactorily approach the derivative of the Heaveside function. 3. CONCLUSION AND FUTURE PROSPECTS Comparing Fourier and wavelets interpolation in the periodic case, we have pointed out that the collocation method using wavelets is best suited for the approximation of discontinuous phenomena. However, what we call "wavelets interpolation" is nothing else that "spline interpolation" : the wavelets we used being spline functions, their linear space is exactly the vector space of spline-functions defined with the same nodes. Furthermore, the method we use to obtain the interpolating function is not competitive at all compared to the well-known spline decomposition algorithms. Now, is there any interest in using wavelets? If one IS merely interested in interpolation or differentiation procedures, the response is no : use rather spline techniques, they are simpler and less expensive. The interest of wavelets comes from the interpretation of the wavelet coefficients. It can be proved [13] that the wavelet transform detects singularities. Indeed, in the time-frequency space, the asymptotic decrease of wavelet coefficients, as frequency increases, depends on the local regularity of the analysed function. Thus for example the largest coefficients will concentrate near discontinuities. For P.D.E. simulation the interest will lie in the possibility of regularization or smoothing of given functions ; for example, as mentionned in the introduction, in numerical codes one has to filter out spurious numerical fluctuations, without filtering physical ones and particulary 281 without smoothing shocks. In that respect, the wavelets give us an alternative basis of the spline-function vector space, which is a priori more suited for filtering purposes than the classical B-spline basis. Our aim is now to derive appropriate filters. ACKNOWLEDGEMENTS The author wishes to thank C. Basdevant and K. Dang for their contributions. REFERENCES [1] Loisel, P., " Resolution des equations de Navier-Stokes compressibles instationnaires par methode spectrale de Tchebycheff ", These Universite Paris VI (1986). [2] Pernaud-Thomas, B., " Methodes numeriques d'ordre eIeve appliquees au calcul d'ecoulements compressibles ", These Universite Paris VI, to be published (April 1988). [3] Grossmann, A., Morlet, J., " Decomposition of Hardy functions into square integrable wavelets of constant shape ", SIAM J. Math. Anal. 15 , 723-736 (1884). [4] Meyer, Y., " Principe d'incertitude, bases hilbertiennes et algebres d'operateurs ", Seminaire Bourbaki, nr. 662 (Feb. 1986). [5] Daubechies, 1., " Orthonormal bases of supported wavelets ", Bell. lab. (1987). [6] Battle, G., " A block spin construction of ondelettes, Part. 1 : Lemarie functions ", Comm. Math. Phys. (1987). [7] Lemarie, P.G., " Ondelettes a localisation exponentielle " Journ. de Math. Pures et Appl., to be published. [8] Goupillaud, P., Grossmann, A. and Morlet, J., " Cyclo-octave and related transforms in seismic signal analysis ", Geoexploration 23, 85-102 (1984). 282 [9] Mallat, S., " A theory for multiresolution signal decomposition ", preprint GRASP Lab, Dept., of computer and Information Science, Univ. of Pennsylvania (May 1987). [10] Kronland-Martinet, R., Morlet, J. and Grossmann A., " Analysis of sound patterns through wavelet transforms ", International Journal on Pattern Analysis and Artificiel Intelligence, voU (Jan. H)87). [11] Meyer, Y., Wavelets and operators ", Ceremade, Cours de l'Universite Paris Dauphine (H)87). II [12] Meyer, Y., "Ondelettes, fonctions sp.lines et analyses graduees ", Univ. of Torino (1986). [13] Grossmann, A., Holschneider, M., Kronland-Martinet, R. and Morlet, J., " Detection of abrupt changes in sound signals with the help of wavelet transforms ", preprint, Centre de Physique Theorique, CNRS, Marseille, (1987). 283 Part V Implementations A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform M. Holschneider 1;*, R. Kronland-Martinet 2 , 1. Morlet 3 , and Ph. Tchamitchian 4 1Centre de Physique Theorique, CNRS Luminy, Case 907, F-13288 Marseille Cedex, France 2Faculte des Sciences de Luminy and Laboratoire de Mecanique et d'Acoustique, C.N.R.S., 31, Chemin J. Aiguier, F-13402 Marseille Cedex 09, France 3TRAVIS, c/o O.R.I.C. 371 bis, Rue Napoleon Bonaparte, F-92500 Rueil-Malmaison, France 4c.P.T. et Faculte des Sciences et Techniques de Saint-Gerome, F-13397 Marseille Cedex 13, France *Permanent address: Mathematisches Institut, Ruhr Universitat, NA3 Postfach 102148, UniversitatsstraBe 150, D-5630 Bochum 1, Fed. Rep. of Germany I. Introduction The purpose of this paper is to present a real-time algorithm for the analysis of time-varying signals with the help of the wavelet transform. We shall briefly describe this transformation in the following. For more details, we refer to the literature [1). The main goal of the wavelet transform is to decompose an arbitrary signal into elementary contributions which are labeled by a scale parameter a. Consider a fairly arbitrary function g(t), which is localized both in the time and the frequency domain, and look at all its translated and dilated versions g( (t-b)/a ). Then the wavelet transform S(b,a) of a signal s(t) with respect to the wavelet g(t) is given by: (1.1) S(b,a) 1 ={i J-(t -b) g -a- s(t) dt (the bar denotes the complex conjugate). Expressing equation (1.1) in terms of Fourier transform we obtain the following: (1.2) S(b,a) ={i J g(aw) e ibro s(w) ow where the Fourier transform of a function f(t) is defined by: f(w) = (21t)-i/2 f f(t) e-iOl dt. So for the simplicity of notation we shall distinguish a function [(t) from its Fourier transform f(w) only by its argument. Formulae (1.1) and (1.2) allow us to interpret the wavelet transform as a time-frequency analysis of s(t) with filters g (aw) of constant relative frequency resolution (Ao¥w=Cte). 286 For mathematical reasons [l], the wavelet get) should satisfy the admissibility condition, which reads in Fourier space: (1.3) c g = 21t J Ig(w)1 2 dw < Iwl 00 This condition essentially means that get) is of zero mean f get) dt = O. In this case, the wavelet transform is invertible: (1.4) 1 set) = Cg Jf -;,r-' 1 g (t-a- b) da db S(b,a) {ii Here, we have supposed that the signal set) was of fmite energy, f Is(t)12 dt < 00. There exist many other reconstruction formulae. Some of them use only the values of S on a suitable grid [3]. The main properties of the transformation are : - the correspondence s --> S is linear, - the transformation preserves energy : f Is(t)12 dt = l/cg If IS(b,a)12 db da / a2 In practice however, one works with sampled signals obtained from set) by measurements at the instants ti = i.Ts (i E Z), where 1ITs is the sampling frequency. Therefore, formula (1.1) should be replaced by its discrete version: (1.6) S(iTs,a) =Ts a- 1/2 L. s(n.Ts) g (n -!) Ts) n Now, suppose that the wavelet get) has fmite support. In this case, the number of sampling points of get) at the scale a growths linearly with a. So the calculation of S with an algorithm based on the formula (1.6) cannot in general be satisfying on today's machines, especially in audio acoustic where the dilation parameter a ranges typically from 1 to 210, which corresponds to frequency analysis of the signal set) over 10 octaves. So, the need for a more elaborated algorithm is imperious. 287 2. A real time algorithm. 2.1 Notations and definitions As a general notation we use the arguments of the functions to distinguish the different spaces. We define the following operators: Let r,h E L2 Dilations: (XE Convolution: CK.hr)(x) R, a>O) = f h(x-y) r(y) dy (1. r)(x) = r(-x) Inversion: Then the wavelet transform of a signal s E L2 with respect to the wavelet g(t) is expressed as a set of convolutions, each of them labeled by the scale parameter a: (2.1.1) S(. ,a) = Kqas, with 9a =Da 1. g . In the following, we shall work with sequences s E 12, that is the space of sequences of complex numbers s(n) (nE Z) of fmite energy: II s II = Ln Is(n)12 < 00. It is sometimes more convenient to use the z-transform of s which we denote s(z): z-transfonn: s(z) = Ln s(n). z-n The following operators acting on sequences will be used constantly: let f,s E }2, and pEN Translations: 288 (Ts)(n) = s(n-l) (Ts)(z) =z-l. s(z) Dilations: p-l/2 s(n/p) for n = 0 mod p o (Dps)(z) = Convolutions: (KfS)(n) = elsewhere p-l/2 s(zP) Lm f(n-m) s(m) (KfS)(Z) = s(z) f(z). We shall denote by 1 the sequence which is zero everywhere except in 0 where it 1. So the identity can be written as: Identity: The length of a sequence s (the number of non zero elements) will be denoted by lsi: lsi = .2,s(n)Al 1. The most time consuming operations that we shall encounter are actually convolutions. In order to compare different algorithms we introduce the notion of complexity. For a convolution KfS it is quite reasonable to measure the complexity by the length 1ft of the filter f we convolute with: Complexity: I Kf I = 1ft The reason for this is that this is exactly the number of operations - multiplication of two numbers and addition of the result to an accumulator - to realize this convolution. The complexity of the product of two convolutions is given by the sum of the respective complexities: (2.1.2) The passage from an everywhere defmed function r(x) to a sequence is done by the perfect sampling operator P. For the sake of simplicity, let us suppose that the sampling time is unity, Ts=1. Sampling: (Pr)(n) =r(n) (ne Z) 289 In view of fonnula (1.6), we derme the discrete wavelet transfonn of a sampled signal se 12 with respect to the wavelet g(t) as a set of convolutions with filters ga labeled by the scale parameter a: (2.1.3) 2.1 The need of an efficient algorithm We now want to calculate the discrete wavelet transform for N octaves; that is the scale parameter a takes the values a=I,2,4, .... ,2N . Typically N is of the order of magnitude of 10 e.g as in audioacoustic applications. Obviously, there is a direct method of computing Sa just by using the definition (2.1.3), and evaluating the convolutions. So suppose now that g = ga=lhas finite length, Igl < 00. In practical applications this will always be the case. Then the complexity of this algorithm to calculate the n-th octave, Sa=2n, can be estimated as follows: (2.1.1) We see that the amount of calculation grows exponentially with the number of octaves, which is a serious problem when n-lO. 2.3 A class of wavelets. The algorithm we shall establish now will reduce the complexity of the convolution with the dilated wavelet, by factorizing it into convolutions with smaller filters (compare with fonnula 2.1.2). This will be possible under certain hypotheses on the wavelet. In a fIrst step, we shall construct an operator acting on sequences which shall be the analog of the dilation operator D2 acting on functions. To be more precise, we are looking for an operator 0: 12__>12 satisfying (2.3.1) on P g =P (Dvn g , (ne N) for a sufficiently large class of functions g. In particular this class should contain some interesting wavelets. Equation (2.3.1) means that sampling the dilated versions of g can be replaced by the action of 0 on the original sampled sequence. Additionally we should require that 0 is numerically simple. The a priori choice 0 = 02 is not satisfying since there are too few functions satisfying (2.3.1): the only continuous function satisfying (2.3.1) is g = O. This is due to the fact that (02&)(n) = 0 whenever n is odd, independently of its neighbouring values. A better choice might be to obtain 290 the values at the odd position by means of an interpolation procedure. Let us suppose that there is a filter F E 12 doing this job for us. We then define (2.3.2) To illustrate the action of 0, let us give two examples. a) Piecewise affine functions Let F be given by: F( -1) = F(D) = 112 , all other elements are zero, then 0 is doing a dilation by means of linear interpolation: 2- 1/2 g(n!2) for n even (Og)(n) = 2- 1/2 (112) [g( (n-l)/2) + g( (n+1)12)] for n odd The class of continuous functions for which (2.3.1) holds are exactly the functions which are affine on each interval [n,n+ 1[ . b) Piecewise constant functions Let F be defIned by: F(D) = 1, all other elements are zero, then the action of 0 is : (0 g)(n) = (0 g)(n+ 1) = g(n!2) for n even There is actually no continuous function satisfying 2.3.1 apart from the trivial one, g=D. However, the piecewise contmuous function satisfying (2.3.1) are the functions that are constant on any interval [n,n+l[ . From these two examples, one might be tempted to guess that ftIters corresponding to higher order Lagrangian interpolation (quadratic, cubic, ... ) might give rise to the corresponding spline functions. But this is not true. However for higher order interpolations, the functions satisfying 2.3.1 become more and more regular.[2] However, in view of numerical applications, condition (2.3.1) is much too strong. Instead it should be sufficient to require that the difference, e.g. in norm, of the right and the left hand side are smaller than some given precision E for all N octaves in consideration: (2.3.3) II on P g - P (D~n g II < € ; D::; n ::; N This condition can easily be checked numerically for a given function g. 291 We now want to show, that convolutions Kong, with dilated versions of a filter g, can be factorized into convolutions with smaller filters: Lemma: let g E 12 , and let P E 12 be a fIlter defming the pseudo-dilation operator O. Then the convolution operator Kong factorizes into simpler convolutions: (n = 11.../2) (2.3.4) with So with the help of this lemma we can realize the calculation of the n-th octave of the wavelet transform, which is a convolution with a fIlter of length IOngl = 2n Igl, with the help of smaller convolutions, which correspond to an algorithm of complexity (2.3.5) I Kgn KFI ... KFn I = Igl + n ( 1 + IPI ). So, for wavelets satisfying (2.3.3), we have reduced the exponential growth in n of (2.1.1) to a linear one. More than that, as we shall see, the calculations for N consecutive octaves can be organized in a hierarchic way, yielding an additional gain of calculation time. Proof of the lemma: Let us write 0 in the z - representation: (Og)(z) = 2- 1/2 g(z2) [ 1 + z-l p(z2) ]. n Iterating this identity yields:z 2 (on g)(z) = 2- 1/2 (on-l g )(z2) [1 + z-lp(z2) ] n n-l n = 2-n/2 g (z2 ) [1 + z-lp(z2)] [1 + z-2p(z4) ]... [ 1 + z-2 P( z2 ) ]. So using the z representation of the dilation D2 and the convolution, we have proven the lemma. 3. The implementatjon of the aleorjthm We now shall give two possible implementations using the algorithm presented above to calculate the wavelet transform for N octaves of the signal s with respect to the wavelet g. The hierarchic structure is clarified if one rewrites the necessary operations in the following way: (n = 11.../2) 292 (3.1) Sa=l = Kg s; F1 = 1 + T gl = (X. «x. -I D2) F -lD 2 g Xl = (X. KPI S; Sa=2 = Kgl Xl; F2 = 0.- 1 D2 Fl X2 = g2 = Sa=22 = Kg2 X2; (X. -ID 2 gl (0 octave) (1 octave) o.K P2 Xl; FN = 0.- 1 D2 FN-l XN = gN=o.-lD2 g N_1 Sa=2 N = KgN XN; (X. (2 octaves) KFN XN-l ; (N octaves) In the following we shall present two possible implementations of this algorithm. First we define some symbols that we shall encounter throughout this section. (3.2) £~ ---r---t~~ g I-----~.~ O. octave 1. octave ~ ·GJ ~ =~ cp ~ .~ • 2. octave • N octave 293 The convolutions with filters Fi ' which are all the dilated versions of one fIxed fIlter can be realized by an "algorithme a trous". We suppose that the non-zero elements ofh are h(n), h(n+l) ... h(m). A delay shall be denoted by: The convolution with a fIlter h shall be denoted by: The myltiplication by a (complex) number a shall be symbolized by: The addition of two numbers shall be symbolized by: Then a fIrst implementation of the algorithm is merely a direct translation of formula (3.1). It is given by the following diagram: (3.3 ) II 294 Another possible implementation makes use of a multiplexer: It separates a sequence sen) into an even (s(2m» and an odd (s(2m+l» sequence. The following multiplexer identity is obvious: (3.4) Then the convolution with a dilated filter a -iDz h is realized as: (a=l/'>I2) (3.5) The convolution with FI = 11. + T a-IDz F is obtained by the following butterfly diagram: n~~ I U--.t~ F ----:--.t F Here we have used the following symbol: 295 If we now replace all these identities in diagram (3.2) and do some graph algebra, then we see that the calculation of the wavelet transform on N octaves can be realized as follows: (N=3) ooctave 1. octave 2. octave 3. octave Here we have used the following abbreviation for the elementary cell: ~ •• e.c. ~ = 4. The wavelet transform on N voices. Up to now we only showed how to realize a real time'algorithm to compute the wavelet transform on N octaves, which corresponds to a geometric progression in the scale variable a. It sometimes may be necessary to calculate the wavelet transform for dilation parameters which progress arithmetically: a =1,2, ... N. In a first step we replace the dilation by 2 encountered in section 2 by any dilation by an integer number p. In complete analogy with (2.3.2) we defme a dilation on sequences with the help of p-I interpolation fIlters FIll. Fp-l: 296 The following lemma is a generalization of the lemma of section 2. It shows how to decompose the convolution with a dilated fllter hp,q =Op Oq h, by smaller convolutions: Lemma: Let Pi, Qi be the interpolation filters for Op, Oq respectively. Then the convolution with the dilated fllter Op Oq g factorizes as follows: Kop,qg=K~ -"'1 with ~ Kp P= 9 = (p.q)-1/2 Dp.q g. The proof of this lemma is as straight forward as for the lemma in the previous section. Let us now suppose that for any prime number p we have chosen the interpolation fllters. Then we can simulate the dilation by any integer N of the sampled wavelet g in the following way: we first factorize N into prime numbers, N =PI ... Pm ,and then we define the dilated version gN of gas: Then the calculation of the voice corresponding to the convolution with gN can be factorized into smaller convolutions if N itself is not prime. The complexity of this algorithm depends on some number theoretic properties of N. There is an order problem in equation (4.2), since the co~tinuous dilations commute whereas its discrete analogs do not in general. But for convenient wavelets g we may expect that the energies of the commutators applied to g are small. In particular for pseudo-dilation operators corresponding to linear interpolations, the commutators of these operators vanish on the affme wavelets. References: [1] Proceedings of the meeting "Ondelettes, methodes temps-frequence et espace des phases" and references in there. C.I.R.M Luminy Marseille France 14 -18 dec. 1987 to appear. [2] M.Holschneider, R. Kronland-Martinet, I. Morlet, Ph. Tchamitchian, The "algorithme a trous", in preparation. 297 An Implementation of the "algorithme it trous" to Compute the Wavelet Transform P. Dutilleux Laboratoire de Mecanique et d'Acoustique, CNRS, 31, Chemin J. Aiguier, F-13402 Marseille Cedex 09, France and DIGILOG, 21, Rue F. Joliot, Z.I., F-13763 Les Milles Cedex, France 1 lotroductioo The computation of the wavelet transform involves the computation of the convolution product of the signal to be analysed by the analysing wavelet. It will be shown that the computation load grows with the scale factor of the analysis. We are interested in musical sounds lasting a few seconds. Using a straightforward algorithm leads to a prohibitive computation time, so we need a more effective computation procedure. trous" is introduced and an example The basic algorithm is shown first, then the "algorithme of implementation on a general purpose sound processor is given. a 2 Basic algorithm The basic algorithm is presented in ref (2). The wavelet transform S(b,a) of a signal s(t) with respect to the wavelet g(t) is given by : 1 S(b,a) =..ra f -(I . b) g - a - s(t) dt (the bar denotes the complex conjugate) (1.1) In practice however, one rather works with sampled signals, therefore, formula (1.1) should be replaced by a discrete version : . 1 "£..J _g(n'i)T S(ITs,a) = Ts..ra a s ) s(nTs) (1 ITs is the sampling frequency) (1.2) n Let 9a(iTs) = 9(iT s/a). So, for each value of a, the analysing wavelet g is sampled, yielding the sequence 9a(iTs). Then the convolution product between s(nTs) and 9a(iTs) is computed. Now, suppose that the wavelet g(t) has a finite support. In this case, the number of sampling points of 9a(t) is finite and grows linearly with a. Here is a wavelet that we often use (2) 9 1(t) ~(-t2 ) = exp(jwot) eX"'\.""2 with Wo between 5 and 6 (1.3) All 9a (t) wavelets are derived from 91 (t) through the dilation operation, that is the resampling with a narrower interval. The computation of (1.2) can be performed by the structure shown in Fig 1. This structure was first implemented and produced interesting results but it yields a heavy computation load. Where is the problem? We are interested in analysing not only short synthetic sounds but also real sounds. With speech signals as well as with musical sequences, the duration is at least a few seconds. As far as we are concerned with audio signals, the bandwith of the analysis should encompass the 298 x(n) x(n-1 ) Fig 1. Transversal filter y(n) bandwith of the hearing system, this implies that the wavelet analyser should cover at least 10 octaves. The analysing wavelet must be sampled with enough precIsion. The mathematical criteria to choose the sampling rate of the wavelet are not y~t settled but, with the wavelet chosen in (1.3), experimentation shows that 50 pOints can be sufficient. In practice we have used between 81 and 121 points for the analysing wavelet. Let us denote lsi the length of the sequence s(n). This length is a measure for the number of operations (multiply-adds) to be performed. At the scale a, we have to compute convolutions with 9a(n) =91(i} at the 10th octave, a = 2 10 so if Igll = 101, then Igal = 103424. Now, let us think about the implementation on an existing computer. We use SYTER, a comprehensive digital sound processing station. The structure presented in Fig 1 is called a Finite Impulse Response Filter. It is readily implemented on digital signal processors such as the SYTER processor. A key figure in signal processing algorithms is the number of multiplyadds they require. At the full audio bandwidth, the system can compute 81 point convolutions, that means that it can compute the wavelet transform in real time on a single voice, at the scale a = 1. At the scale a = 1024, the convolution is a times longer so the duration of the analysis is a times the duration of the signal. Another key figure with signal processors is the size of the data and program memories. Even if we accept a non real time computation, at the scale a = 1024 the FIR structure of length 103424 cannot be implemented in a single run of the processor. We have to use a sectionnend convolution scheme. With 431 runs of 240 multiply-adds the task can be done. It is clear that we have to look for a more effective algorithm. We have implemented a simple and effective algorithm that makes use of the scaling properties of the wavelet transform. 3. •Algorithme a trous· A formal presentation of the algorithm is found in (1). Our goal here is to show how it works and how it can be implemented, in the case of piecewise affine wavelets. We are looking for an algorithm that keeps constant the number of non zero coefficients as the scale parameter grows. A preintegrator and other functionnal operators are introduced and then the algorithm is presented. 299 Instead of using a continuous wavelet. dilating it before sampling it. we wish to use a wavelet sampled at a fixed number of points. The dilation operates on the sequence of samples. Dilating by a factor of 2 means inserting a zero every other sample in the sequence. ~ 0, Fig 2. Dilation operator The interest of this operation is that but the dilated version of the wavelet. analysing wavelet. We should find a way to perform a linear interpolation between the number with all its to fill in the the original of non zero coefficients is held constant zeros. is a poor representation of the holes. A simple and effective solution is samples. Let us define some basic operators Linear interpolator F Unit sample delay (Tg)(n) = g(n-1) Convolution of sequence sin) by wavelet gin) (Kgs)(n) = g(m-n)s(n) m L Preintearator Fl F1 .. 1 + TD2F F 0, ~ It can be further dilated D,~ With these notations it can be shown that the convolution with the wavelet at the scale 2" is equivalent to the product of the preintegrating filters with the dilated version of the wavelet. 300 Amplitude correction In (1.2) an amplitude factor Ja appears, it has been omitted in the previous formulas. It can be shown that to take it into account, the preintegrator F has to be 1 1'-1-1-4"'1-,-1---r12 1-/4-'1 instead of 1,/2 1 ' 1,/2 1 Complexity Originally the number of multiply-adds for a Single convolution was Ig2 n l = Ig,1 2 n , the "algorithme a trous· yields IKO ng I = Igl + n(1 +IFI). where KO ng is the convolution with the piecewise affine wavelet and n is the number of the octave. The exponential growth in n is reduced to a linear one. Inaccuracy The substitution of a piecewise affine wavelet for the original analysing wavelet leads to some errors. It can be numerically checked that these errors are minor in most cases. The point is that the piecewise affine wavelet has all the properties of a wavelet, so the wavelet transform is actually a time scale representation of the signal. We would like to think of this time scale representation as of a time frequency representation. It is only from this point of view that the difference between the original wavelet and the piecewise affine wavelet is relevant. Implementation Using the "algorithme a trous", the processing can be organised as shown in Fig 3. Each bloc is in fact an "a trous" convolver, i.e. a convolver with a given number of non zero coefficients that forgets all the signal samples but every 2ith. The size of the convolver must be longer or equal to the longest wavelet used in the analysis. The number of non zero coefficients is a tradoff between accuracy and computation time. Fig 3. Parallele structure Table memory In §2, we spoke about the limitations of digital signal processors. SYTER has the additionnal feature of a large table memory. This table memory can be used to store a long sequence of signal, say the 4096· last samples, in a circle memory. Retrieving the useful signal samples is only a matter of computing as many address pointers as there are non zero coefficients in the convolver. 301 Synchronisation We usually use non causal wavelets. The delay introduced by the convolution increases as the wavelet is dilated : Delay = Ig,l.a/2. To compensate for the differential delays between voices and the delay introduced by the processor pipe-line, an offset (SYNC) is applied to each voice. To check the synchronisation, a very good test signal is the delta function. Many voices per octaye When we analyse a signal, we wish that the transform contains all the original information, we are not always concerned by the fact that, in the transform, the information can be redundant. Interpreting a transform is done by eye observation, so the picture should be as easy to read as possible. A simple mean to do this is to display highly redundant information. This makes easier the spoting of a maximum in the module or the following of an isophase line. What we call a voice is the output of the convolver for a single version of the wavelet. In this paper, the pictures shown were made with 8 voices per octave, whereas all the information trous" is only would have been retained with only one voice per octave. The "algorithme suitable for wavelets that are one octave apart each other. To implement an n voices per octave analysis, we can define a set of n analysing wavelets, separated by 1/n1l1 octave, and use an "algorithme a trous" for each wavelet. In order not to go n times through the F1 preintegrator, the n algorithms can be interleaved. a Fig 4. "a trous· convolver Refining the preintegrator The wavelet transform is basically a time-scale representation. In many instances we wish to have a time-frequency representation of the Signal, the wavelet transform doesn't always give this representation. Remind that whith the 3 point preintegrator, we get piecewise constant functions. In the Fourier plane, these functions have a main lobe and side lobes. That means that at a large scale, the wavelet can capture a part of the small scale features. In order to reduce the side lobes we can use smoother wavelets, this implies the use of longer preintegrators. We give an example in Figures 5 and 6 where a clarinet tone is first analysed using a 3 point pre integrator then with a 7 point preintegrator, the coefficients of which are : 1-1/321 0 19/3211/219/3210 1-1/321 With the 3 point preintegrator, in the lower octaves, the phase rotates at the speed of the higher octaves. The module, not shown here, is very small in the low octaves. The effect of the side lobes is not hidden by the energy of the signal, which is very low in this area. With the 7 point preintegrator, the side lobes are much smaller so that the phase behaves as expected : the rotating speed is related to the center frequency of the octave. 302 Fig 5. Clarinet tone 3 point preintegrator. Fig 6. Clarinet tone 7 pOint preintegrator. Dyadic operation The material presented above is concerned by the wavelet transform on a fine grid, i.e. a wavelet transform coefficient is computed at every signal sample. This leads to an increase of the information volume. If the analysis is carried out with a complexe wavelet and on n octaves, the resulting information volume is 2n times as large as the signal volume. The scaling properties of the wavelet transform enables us to compute the wavelet transform coefficients only on a dyadic grid. On this grid the number of coefficients is devided by 2 when trous". the dilation parameter is doubled. This feature can be exploited in the "algorithme After the processing of the octave n, one sample every 2 n th is discarded. To compute this octave, the "a trous" wavelet has ignored 2n-1 points every 2n th, so the decimation process can be done immediately after the preintegration and the convolver no longer forgets any of the remaining samples. With this implementation, the growth of the information volume is no longer proportionnal to the number of octaves. a Fig 7. Dyadic implementation, 1 voice per octave 303 4. Conclusion In the first experiments with the wavelet transform, the computation of the transform needed a few hours per second of sound. With the dyadic implementation, on the same machine, the computation time is reduced to a few minutes per second of sound. The SYTER system was demonstrated during the conference. Many wavelet transforms were made, on request, on synthetic or natural sounds. Real time operation will be aChieved through parallel processing, for the fine grid, and multirate processing for the dyadic grid. On the fine grid the frequency range of the analysis is limited by the size of the buffer memory. With a 4096 sample memory and a 59 points analysing wavelet, the frequency range covers 6 octaves, with a 65536 sample memory, the 10 octave range could be reached. It must be pointed out, that after the transform is computed, much time is still needed to compute a module or a phase picture. 5. References (1) M. HOLSCHNEIDER, R. KRONLAND-MARTINET, J. MORLET and Ph. TCHAMITCHIAN : A real time algorithm for signal analysis with the help of the wavelet transform. Preprint, CPT, CNRS-Luminy, Marseille, France. (2) R. KRONLAND-MARTINET, J. MORLET and A. GROSSMANN: Analysis of sound patterns through wavelet transforms. International Journal of Pattern Recognition and Artificial Intelligence, Special issue on expert systems and pattern analysis, Vol 1 n02, World Scientific Publishing Company, 97-126. (3) R. KRONLAND-MARTINET: The use of the wavelet transform for the analysis, synthesis and processing of speech and music sounds. Preprint, LMA CNRS, 31 Chemin Joseph Aiguier, 13402 Marseille CEDEX 9, France. (4) J. F. ALLOUIS : Use of High Speed Microprocessors for Digital Synthesis, in Foundations of Computer Music, Article 18, Edited by C. ROADS and J. STRAWN, The MIT Press, 1985. 304 An Algorithm for Fast Imaging of Wavelet Transforms P.Hanusse Centre de Recherche Paul Pascal, CNRS, Dornaine Universitaire, F-33405 Talence Cedex, France 1. INTRODUCTION We consider the use of wavelet transforms as a tool to analyze the structure of complex signals through a two dimensional representation of the transform rather than through its capabilities of coding, decomposition and reconstruction. Indeed, the remarkable properties of this transform can be used with great profit to obtain a very natural and visual access to some of the structural properties of a signal, which can be typically viewed as a time series(1). It turns out that, for most practical purposes, the two dimensional imaging of such signals can be achieved using a simplified view of the wavelet transformation. The use of the underlying interpretation of the wavelet transform does not cover, of course, all its- possible properties, nor does it solve or take into account all the questions associated with recontruction, that is, inverse transform. We have focused our attention on the multi-scale representation of a one dimensional signal as such. In fact, our starting point originates in two dimensions, namely digital pictures, for which multi-scale resolution, provides a good example of discrete wavelet transformation. A large convergence has been recently achieved in this field, which is greatly reflected by various contributions in the present issue~,~. 2. WAVELET TRANSFORM AND DIGITAL FILTERING First, as it is now well recognized, many convolution operators used in picture processing, as well as for other types of digital signals, are indeed equivalent to wavelet transform or related to it. This is the case, for instance, for the Sobel or Laplacian operators [~. However, it may be worth pointing out that the way one is usually lead to use them in this field is partly related to some aspects of their behavior, sometimes considered as resulting in spurious effects or limitations, which are in fact connected to their unrecognized wavelet transform nature. Similarly, the existence of various length scales in a picture, that may correspond to objects of different sizes, calls for various methods to decouple the contributions of these length scales. Thus, the idea of multi-scale analysis is rather natural in this field, although it is seldom expressed as such. Many of the methods, if not tricks, used in various circumstances are clearly related to the hidden or unexplicited properties of the wavelet transformation they implement. To make this point clearer, let us consider an example. A digital picture being given as a 512x512 pixel matrix, one shall perform on it a convolution using a 3x3 mask such as 305 o -4 o which 1 o 1 is of laplacian type. The transformation can be symbolically expressed as J = : P ~ I * M+ T where I is the initial picture, J the resulting picture, P and T integer constants. The summation stands for the weighted average permformed on a 3x3 neigborhood using mask M. Note that in this context all operations are performed on 8 bits integer pixel values. This transformation is very short ranged, and as such, very sensitive to high frequency spatial noise or transitions. To avoid this effect, one often must perform first a somewhat long range smoothing or averaging, or, alternatively, reduce the picture to 256x256 or 128x128 format, but summing up 2x2 or 4x4 neigborhoods, then applying the convolution operator. This is precisely a way to perform a scaling of the transformation. Clearly, all the ingredients of wavelet transformation are used. In this case they represent discrete compactly supported wavelets [2]. 3. WAVELET TRANSFORM AND FINITE RANGE AVERAGING The view of wavelet transforms originating from digital filtering can be used to emphasis an aspect of this transformation that is helpful when one is essentially interested in forming a two dimensional multi-scale representation of a one dimensional signal. In that case, the detailed shape of the wave let may not be as relevant as some global structural features, the most important concept being finite range averaging. Consider the general expression of a wavelet transform h(a,b) = 1 - a J+OO -00 x-b dx f (x) g (-) a b E R a E R' where g(x) has the usual properties of a wavelet 1, essentially J+OO -00 g(x) dx = 0 g(± 00) = 0 Typically, the shape of the wavelet function is either symetric (g even) or antisymetric (g odd), as, for instance, the Korlet wavelets 1 exp(-x 2 ) cos(kx) , exp(-x 2 ) sin(kx) or the following simple wavelets exp(-x 2 ) x 306 (a) The essential morphological features of using the following symbolic notations [-1,1] these two functions can be described [-l,2,-lJ which denotes the relative weight of each piece of the surface area between the the curve of the graph and the horizontal axis. For function (a) there are tho such pieces, the weight of which is measured by the integral of g(x) from -00 to o and from 0 to 00, respectively. For function (b) there are three pieces corresponding to the integral over the three segments]-oo,-l/f!J[-l/{2" 1/f!J , I and [1/f! ,oo(,The property that the integral of both functions should vanish is simply reflected in the fact that the weights sump up to zero. This discretized notation can be viewed as the analog of a discrete convolution filters as those introduced earlier in two dimension for digital pictures. In other words, a coarse view of these functions can be depicted as a one dimensional discrete convolution filter. Clearly, the first one describes a first order finite difference differentiation, whereas the second calculates a finite difference second derivative. More generally, let us consider a filter of size 2n+1 defined as g =[x _II x -od ••• , x -I x 6 x { ••• , x o-I X] 0 Given a set of values fi representing a fixed time step sampling of a signal, the convolution with filter g is defined as +n h1 = L j=-n f l+j gj where gj is the j-th component of filter g. This is the discrete analog of the convolution involved in the wavelet transform definition given above. It is not our purpose to discuss here the relevance or limitations of such discrete wavelets, in particular the consequences of their poor localization in Fourier space. It is worth noting that any filter like g can be expressed, for instance, in terms of only three basic filters such as [-1,0,1] l1,-2,l] [0,1,0] For symetry reasons we choose here to use only "centred" filters, i.e. with an odd number of terms, which somewhat increases the size of this "basis". The operations that we need to decompose any filter g are the multiplication by a scalar a [x J= [a xJ and the convolution product of filters 307 For sake of space we do not give a formal presentation; examples should suffice: [l,l,lJ =[1,-2,lJ+ 3 [O,l,OJ [ 1 , 2 , 3] = [ -1 , 0 , 11 + 2 [1 , - 2 , 1J + 6 [0, 1 , 0] [-1,0,2,0, -1] = (-1,0, 1J * [-1,0, 1J r-1,2,0,-2,lJ=(-1.0,lJ * [1,-2,1) Apart from the "constant" filter [••• 0 ••. ,1, ••• 0 •••], or simply[l), the two other basic filters produce the first and second order finite difference derivatives when applied to a sample of discrete measurements, which amounts to saying that any filter g of half-size n is a linear combination of derivatives up to order n. The first pratical consequence is that a filter g could be applied using only these three short three-element filters which can be very efficiently implemented in hardware in highly vectorized and parallel operations. It is also significant that a complex shape filter can be viewed as resulting from the differentiation of the constant filter which performs a finite range averaging when applied at different scales. We essentially want to stress here that the main ingredient of such an intrepretation of the wavelet transformation is the averaging process. In a number of appplications this is more relevant than the details of the shape of the wavelet. For instance, let us consider the follwing gaussian averaging function g(x) - 1 a.r;t e- x 2 and the multi-scale averaging of function f J:: h(a,b) dx f(x) g(X:b) g(x) is not a wavelet, and the this transformation is not a wavelet transform. But let us expand h(a,b) in b about b=O h(a,b) h(a,O) - -b a 1 +- 2 J+OO f(x) -00 ~r x-b a g(1) ( - ) I:: dx x-b f (x) g(2) ( - ) a dx + (9(3) where g(D) is the n-th derivative of g(x). Clearly, these functions are wavelets, and the previous expression shows that the wavelet tranform of f(x) using g(D) (x) is simply the n-th derivative of the "averaging transform" with respect to b. In other words all wavelet transforms based on this family of wavelets are embedded in a simple way in the variable scale average. 308 As an example, consider the following function f (x) = x2 (1 + d sign (x» where sign(x) discontinuity to be a good function g(x), is +1 or -1 depending on the sign of x. Function f presents a in its second derivative at x=o. Wavelet transform has been shown tool to detect such discontinuity [~. Averaging with gaussian we obtain h(a,b) + .' + d {(~ .'+ "J b ba erf(-) + -- exp(- a .[1t with special values h(O,b) = b2 (1+ d sign(b» 1 h(a,O) = - a 2 2 The taylor expansion in b about b=O reads h(a,b) 1 2ad 2d - a2 + b + - - b3 - --- 2 15 .[1t 3a.[1t d a3 .[1t b'l + C)(6) All derivatives of h with respect to b, at b=O, of diverge as a goes to zero, which is the signature discontinuity. Equivalently, performing the wavelet would have reveal the. same behavior, since it would lead of h : d b2 a . a2 order greater than two of the above mentionned transform using g(3) (x) to the third derivative - exp(- --) It is in that precise sense that we consider scaled averaging as the fundamental process, independant of the shape of averaging function, provided it possesses minimal properties. Similar qualitative results would be obtained using a different averaging function, for instance g(x) which is precisely about b=O 1/a x E [-a/2, a/2] a "constant" filter as ° otherwise introduced previously. One obtains 1 1 2d h(a,b) = -- a 2 + - a d b + b2 + -- b3 12 2 3a which presents the same divergence. 4. IMPLEMENTATION OF A FAST DISCRETE WAVELET TRANSFORM The previous discussion shows that in various circumstances, certainly not in all, the use of very simple wavelets or more generally of discrete convolution 309 filters of type g defined above, can be very helpful to obtain very effeciently a two dimensional picture of a multi-scale view of a signal. Let us consider the N point fixed time step sampling of a signal X, Xl' i=1, •. N and a convolution filter g of half-width n, Le with 2n+1 elements, noted gk' k=-n, .•• ,no Let us define S1 , j as the average, or more simply, the integral of signal X at time i, over range 2j+1 j L Sl.J k=-j X1 + k , i+j ~N, i-j ~1 The transform will be defined by n T1, j L k=-n gk SH (2 J + 1) k, J Index j represents the scale factor a of the wavelet transform; index i is the position b. The previous expression is therefore a discrete convolution with filter g applied on the signal averaged over range 2j+1. Obviously, for a finite sample of size N, the transform is only defined for values of i and j satisfying the following relations i + (2j+1) n + j i - (2j+1) n - j ~ ~ N 1 which leads to 1 + (2n+1) j + n o ~ j ~ Therefore T is defined over height (N-1-2n1/2(2n+1). The efficiency of definition of Sl,J i ~ N-1-2n 2(2n+1) a triangle the implementation Sl,j+l N - (2n+1) j - n of base T1 , 0 , i E 1+n,N-n and of is increased by using the recursive Sl,J + Xl+J+l + XI-j-l so that from one scale j to the next the cost of the averaging process is negligeable as compared to the convolution process. Furthermore, the convolution could be performed concurrently with that at a previous scale, so that both vectorization and parallelism could be effectively used to implement this algorithm. In practice, on a scalar processor like a VAX 8600, a 512x512 picture can be generated within a few seconds. It could be much faster in a dedicated environment. Finally, an important practical aspect of wavelet transform imaging that we have not considered so far, concerns the renormalization factor used at each scale j. Indeed, when calculating a picture of the transform, the intensity 310 values of each pixel are usually coded using 8 bits integers providing a 0 to 255 range. In many applications, the dynamic range of the transform can be quite large. This is for instance the case in the example treated above where a divergence or at least a large local enhancement is expected. For this reason a fixed renormalization factor is often inadequate to reveal the full structure of the transform, which is, in fact, the main goal of such a tool, in particular as far as our picture interpretation ability is concerned. To allow an effective visual understanding of the transform information content, whatever the intensity dynamical range ,when forming the transform picture it is necessary to renormalize the amplitude of each line of the transform, which corresponds to each scale factor j. The picture intensity I 1,j will be defined by where ocj and ~j are evaluated for each line j for so that I 1 , j varies between 0 and 255. The observation of the variation of the amplitude renormalization factor oc j as a function of scale j reveals a qualitative as well as quantitative important information which should be used to interpret the transform behavior along with its picture representation. 5. CONCLUSION We have performed the imaging of transforms of various sample signals on a color bitmap display. A selection of several pictures have been presented, which we cannot reproduce nor discuss here. We have found that, on a qualitative basis such an algorithm was able to quickly produce a view of the structural features of a signal. It was applied to periodic signals, with or without white noise, and to aperiodic or fractal signals. We have found the same behaviors as already described by various authors [5-8]. In particular we have confirmed that the multi-scale averaging by itself is able to extract much information without having to use refined wavelets. In other words, looking at the multiscale average picture itself provides much qualitative insights on the signal structure. This is particular true in the case of fractal signals which have been also studied using a more conventional wavelet transform appraoch by other authors [7,8]. To conclude, we think that the idea of finite range averaging could be pushed further. Work is presently under progress along these lines. REFERENCES 1. A. Grossmann and J. Morlet, "Mathematics and Physics, Lectures on recent results ", edited by L.Streit (World Scientific,Singapore ,1987). 2. I. Daubechies, "Orthogonal bases of compactly supported wavelets", (preprint ATT Bell Labs 1987). 3. S. Mallat, "Multiresolution approximation and wavelets", (preprint GRASP Lab, University of Pennsylvania, 1987. 4. W.K. Pratt, Digital Image Processing, (Wiley, N.Y. 1978). 5. R. Kronland-Martinet, J. Morlet and A. Grossmann, in Int. J. Pattern Recognition and Artificial Intelligence, (special issue on "Expert systems and Pattern Analysis" 1987). 311 6. A. Grossmann, M. Ho1schneider, R. Kron1and-Martinet ans J.Mor1et, in "Advances in electronics en electron physics", P.C. Sabatier ed., supplement 19, "Inverse Problem", (Academic Press, 1987). 7. A. Arneodo, G Grasseau and M. Ho1schneider, "Wavelet transform of invariant measures of some dynamical systems", (preprint CRPP, CNRS Bordeaux 1987). 8. A. Arneodo, G Grasseau and M. Ho1schneider, "On the wavelet transform of mu1tifracta1s" (preprint CRPP, CNRS Bordeaux 1988). 312 Subject Index Reference is made to the first page of relevant articles Abrupt changes 99 Acoustical scattering 139,147 Adaptative (Methods, Models) 68 Admissibility 239 Affine group 2,21,38,164,172,232 Algorithm 305 - "a trous" 298 - real-time 286,298 - Romberg 139 tree 38, 286, 298 Ambiguity function 68, 172 Amplitude Radiation 197 Amplitude Scattering 147 Analysis - electrocardiogram 132 - seismic 126 - sound 102 - speech 158 Analytic signal 2,164 Approximation 2,21 Asymptotic estimates 247 Audition 2, 102, 158 Balian-low theorem 38 Banach algebras 216 Bargmann space 209 Bergman space 209 Broad-band 164 Calderon-Zygmund operators 21,253 Cauchy integral 21,253 Chord-arc curves 253 Coherence (in acoustics) 154 Coherent states 221,232 - on the circle 204 - Poincare 221 Complex cycles 197 Complexity 286,298 Composition by computers 102 Concentration operator 209 Contouring 126 Detection 132 - discontinuity 2, 305 - estimation 68 - failure 99 - target 172 Diagnosis 99 Dyadic 21,38,253,269,286,298 Echo characterization 147 Electrocardiogram 132 EQR model 158 Evolutive (methods, models) 68 Filter(ing) 21 - digital 305 - interpolation 286 - quadrature 38 Fluid mechanics 269 Form function 147 Fourier analysis - Gabor transform 38,68, 132 - local 21 - short-time 132, 158 - time-frequency 68,164 Fractal - measures 182 - properties 38,182 Frames 38,209,239 Gabor 38,68, 132 Galerkin's method Gaussian wavelet 2,38, 126, 132, 139,298 Geophysics 126 Gram matrix 247 Green kernel 263 Haar measures 239 Helmholtz equation 147,197 Hyperbolic geometry 2,204,209 313 Image 126 - processing 259,305 Implementation 286,298 Interpolation 2, 21, 209, 286 Interpretation 126 Laplace-Borel transform 197 Littlewood-Paley theory 21 Matrix - elements between wavelets 204 - Gram 247 Mean value theorem 209 Multiresolution analysis 21,38,247,253,259 Noise 132 Non-self-adjoint operators 263 Non-stationarity 68,99 Numerical methods 247,269 Operators - elliptic 263 - non-self-adjoint 263 - of Calderon-Zygmund 21,253 - SchrOdinger 204,263 - self-adjoint 204 Parametrix (of elliptic operators) 263 Paraproduct 216,263 Path integral 204 Periodic functions 269 Phase space 204,221,232 Piecewise affine wavelet 286 Piecewise constant wavelet 286 Preintegrator 298 Progressivity 2,132,139,164 Propagator 139,204 Quantum propagator 204 Real time 286,298 Reconstitution formula 38,139,239 Renormalisation group 182 Renormalisation 305 Representations - holomorphic integral 197,204 - square integrable 2, 172, 239 - Weyl, Poincare 216,221,232 Scaled averaging 305 Scaling exponents 182 Scenarios to chaos 182 SchrOdinger operators 204,263 314 Segmentation 99 Seismic 126 Self-adjoint operators 204 Singularities 2,99,132,172,182,305 SL (2, R) 172 Sound analysis 2, 102 Sound synthesis 102 Spaces of homogeneous type 259 Spectral analysis 68,147,154,158 Spectral method 269 Spectrum of singularities 182 Speech analysis 158 Splines 21,38,247,269,286 Stationarity 68,99,132, 154 Stationary phase 139 Surface 126 - contribution 139 - wave 147 Symplectic form 204,216 Syter 298 Timbre 102 Time localisation 164 Time-frequency 68, 164 Time-varying signals and systems 68,99 Tomographic 232 Transient 68, 132, 139 Twisted product 216 Velocity dispersion 147 Ventricular late potentials 132 Visualisation 2 Wave - evanescent 139 - geometric 139 - lateral 1"39 - outgoing 147,197 - surface 147 Wavelets - bases 21,247,253,263,269 - continuous w. transform 2,164,182,204, 221,239,259 - finitely supported 21,38 - frames 38,209,239 - Gaussian 2, 38, 126, 132, 139,298 - orthonormal 21,269 - piecewise polynomial, spline 21,38,247,269,286 Weyl 216,221,232 Wiener measure 204 Wigner map 221 Wigner-Will 68,147,164 Index of Contributors Antoine, J.-P. 221 Arneodo, A. 182 Auscher, P. 253 Ginette, S. 139 Grasseau, G. 182 Grossmann, A. 2, 139 Basseville, M. 99 Berthon, A. 172 Bertrand, J. 164,232 Bertrand, P. 164,232 Bohnke, G. 216 Bros, J. 197 Hanusse, P. 305 Holschneider, M. 182,286 Coifman, R.R. 259 d'Alessandro, C. 158 Daubechies, I. 38 Dutilleux, P. 298 Flandrin, P. 68,147 Jaffard, S. 247 Kronland-Martinet, R. 2,286 Larsonneur, J.L. 126 Lienard, J.S. 158 Meyer, Y. 21 MorIet, J. 2,126,286 Murenzi, R. 239 Paul, T. 204 Perrier, V. 269 Risset, J. -c. 102 Sageloli, J. 147 Seip, K. 209 Sessarego, J.P. 147 Slama, J.G. 154 Tchamitchian, Ph. 139,263,286 Tuteur, F.B. 132 Zakharia, M. 147 315