Preface
What is a signal? What information ca,n be drawn from a signa|? How caa~ we exm~gt, th}s information? ttow is a signal denoised? At whidl point, does a prior/knowledge about a signal enter and how cmi we utilize it? These are essentiN questions m which no answers coutd be given until goday, h ib.ct, they have been subject to considerable change during the past fifty or more years, Between 1940 and I.q60 signals were anal(g, and signal processing wa,s primarily a part of physics, tto~,,,'ever, with the onset of microprocessors the analog signa.1 lost~ its rank to the digital signd, and information theory irrupted. Fast computational algorithms (such. as fast Fom'ier ~ransform) Nlowed most filtering operat.ions to be performed Nmost instantly, and this enlarged treme~tdousty the scope of possible manipulations. Nirth:ermore, signal processing also gained an advantage from all the new achievements in statistics. T N s occurred to such a degree that one could sometimes look upon signal processing as a, part of statistics. A third revolution (1970-1980) in this field occurrod with the Mvent of methods and techniques of mathematicnl physics and quantum mecha.nics. ]'his cleared the way for mathematicians to pa.rticipat;e in scientific activities reIa.ted to signal proces;sing. T?oday signN processing ofN,> a forum to tmmerous disciplin~ and requires manifold knovAedge, which can be combined only rarely by a single scientist~ P. Flandrin takes up this challenge with so much competence and a truly multid~sciplinal\y vision. During the study of his book devoted to time-frequency signal a.nalysis l once more ~dmired the brigh.i,ness of hN spirit, the depth of h £ insight, the wealth of his knowledge, and an erudition that is never p<mderous. He devotes most of this book to the Hme-ti'equc~cy amd)~Js of signals. This am~lysis involves unfolding the signal in the time-frequency plane in the way b ~ t suited to comprehension. This operation can be compaxed to writing the score of a symphony as one listens to the music. There exists no unique ~iution, of course. However, P. Flandrin exposes a chues of methods xi
xii
Timc-FrequencL/i[Sme-Scale AnaIysis
(Cohen's class) and provides an exhaustive di~:ussion of the q~mlities and defects of each of them. Onee of the most celebrated members of t:his class is the Wig'neJ~Viltetr~'m,s'/})rm. It lays the foundation fbr the field in which qumltum IImehanics, the thex~ryof pseudo-differentia! operators, and signal statistics meet. This work by P. FIandrin ira;ires us to become (or con~imxe m be) r~earchers in this field; ~ft,er having read and rereM this hook, one is wellwepared to participa:te in the exploration of the unacces~sible instantaneous freque~my . . . .
Prench Academy of Sciences
Yves Meyer
Foreword "There must be, in the re,)resented ~hing:s, the ir~sistent murmuring'of the resernblance; there must. be, in the xv~p~sentation, the ever pose'ible recourse of the imaginationF~ Michel Foucault
A signal ks the physical carrier of some information. It carl originate f~om a multitude of different sourc~ (acotLsties, radioelectronics, optics, mechanics, etc.). Beyond this diversitE however, the main object of interest is the observation of a time-varying quantity, which is collected at one or more ~nsors. This constitutes the basis on which a "signM processor" can perform operations for ex~,rax:ting some useful information. Tile facility of gaining and pwcessing this information certahfly depends on its readaMlity. This is our motivat.ion fbr leaving the immediate represemational space, in which plain data are given. We pass, instead, to a transformed space containing the same information, in order to obtain a clearer picture of specific chara.'teristies d the signal [\y "looking at" it fl'om a particular
aitlgle, The choice d a representation is crucial for the ultimate t ~ k of processing dat~a, which often comprN~ several consecutive steps for solving a statistical deeMon problem (detection, estimation, ckussification, recognb tkm, etc.). The tmrtinence of a representatrion N rooted in its cat)ability to provide well-suited descriptors tot this task. Viewed from a perspective (d signal analysis, t~he representation should "tell" the user something about the structure of the signal. As long as one knows very little about the constitution of the signal a priori, the representation should require as few external specifications about this structure ~ possible. This situation falls in the category of nonparametric methods, which often empk~, a much larger representational space than used for the initial data. A better structuring of the transformed in%rmation should counterbalance the onset of redundancy. This is our preferred point of view in this book, and we shall make use of it in many nonstationaL~y situations.
2
Tim~,-l?equenc3?"Tim<.-Scale Amdysis
Tile necessity of dealing wilh no~ist~:tt:i,:marysignals is root(d in a para~ dox. On one side, a huge amount of work has already been devoted to tim stationary c~se. "lt.~day one can say tha~ this e~,~e is well-analyzed f?om a theoretical perspective and enjoys a muhi~ude of performing ~;ools that. are often linked r.o ii'equentiaI descriptions. On tile other side, it is Mso lrue that the real signaN in their vast majori V a.re nons~ationary; expre~ed e~,n more drastically, an integra/I part of the inIbrmation (if not the dolrfinant one) can he found in the nonstationary property itself (e.g., start aI~d termination of events, drift.s, rup~,ure~, modulations). Th~se two counterparts have led to the &'~elopment of t.ools that are d~igned specifically to match noImta~iong~y situations. Among them emerge ~,he ~.ime-frequenW me,hods, which explicitly consider the time-dependerme of the frequentiM content of the signal. This book concentrates primarily oT~ such methods. The following describes the organizm.ion of the book: ,, The first chapter ("The Time-Frequency Problem") addresses the description of some general concepts associated with the notions of ~ime and frequency. ~,.~)?dwell on the relations between these two variables and encompass the pro~lems resulting from their combined use. The limits of ~he physicN interpretation of the Dmrier ~rm~sfbrm and the e~xNtence of cet~ i n n mathematffcN obstructions wiI1 be discusxsed. ThN forces ~t~sto search for possible substitutes that better match nonstationary situations. A first a.ttempt to fl.mfish a ch~z*;sifi(atkm of the conceivable solutions is eoatained at the end of this chapter. Q After emphasizing in {he first chapter a mrmber of problems tha*~ arise when we search for a rime-frequency description of signMs, the secmld chapter ("Clazsses of Solutions") is dew,ted to t.he discussion of some concei> able approa,:hes, inchding an extensive inventor3' of t,heir properties. In a terra.in seI~se the presentation can be regardc
resm~tation should be che~+en?) and facilitate their interpreeation (whidt information can be extracted?). Because no{ aI} approaches can be covered
in detail, we focus oI~ the bitim~,r representations (Cohen's ctass and aft[the class) in this chapter. * The tim¢~-frequency representations do more t~tmn offi?r an arsenal of adaptive methods tor no~lstatiom~ry signals: they mar~ifest a new paradigm. It is from t.his perspectiv~ that the last cha.pter ("Time-Frequency a~s a ParMigm") attempts to illustrate, by some Vpical examples, how an explicitly joint description (:an amount to a new vision of several classical problems in signal anMysis and signN processing, and how it em~. le~M to solutio~ts thae h a ~ "natural" interpretations. As we do not aim go write an encyclopedic t,reat;ise on the very generM theme of %ime-frequency," we had to make our choices as to which subjects to include and which to leave aside. They result, from two different motiva.tions: o~le is the concern fbr competence (describing facts that one believes to know), and the other one is the desire to ofi?r a large panorama. of subjects that were mmvenly covered in tl~c few existing surveys (which means writing a book that one would have liked eo exist: already). These wen-~ the main reasons fi~r selecting the maeerial fbund in this book. t~rthermore, we emph~size mostly generM principles (omitting most of the algorithms and applications), and concentrate on the nonparametric b5 linear apNoaches (to the detriment of linear methods a n d / o r paranmtrie modeling). We have described here what the reader will be Mile to find in this book. It might, also be helpful to point out, those subjects not treated and which other sources can be consulted instead. Only little or nothing is said about Mgorithms (in any case there are no "codes" included in a "ready to use" form~). However, after the first edition of this book was published in French in 1993, the situation concerning algorithms that put the time-flequency methods to work changed quite a bit. ~ e would like to re%r to a public %oolbox" written in Matlab and fl'eely accessible via t,he im;ernet under the address http ://www •physique, ens-lyon, fr/tslti~b, html This collection of rout,ines became available thanks to the efforts of O. Lemoine, F. Auger, and P. Gon~:alv~s, and was supported by CNRS through its successive GdRs TdSI and ISIS. The interested reMer can th~s access over a hundred functions that allow one to create or manipulate nonstationary signals. These include efficient algorithms for th~ computation of the time-frequency representations that are presented from a mostly theoretical point of view in this book. Moreover, they offer different postproeessing methods. The collection of these functions is documented in due form in a reference guide as well as in a tutorial that has been sorted by examples.
V~i~ wouM like to mention that t;his toolbox w,u, used to g~,::erate most of ~he figures in Chapter 3 of this t~;:lglish ediVio~:, We also do not develop at~y '*concrete" applications, These issuers alone would necesshate tile writil~g of a::other book. While awaiting someone tarundertaki~:g such an effort one should look at the article by Grenier (1987). The subjects included here receive a different degree of attention. As already meneio~ed, bitinear time-fre~q~lency met, hods constitute the central pare of the book. Nevert, tmless, we gra:~t; a significant portion to ~he Bnear time-seMe methods (wave:eg theory}, without trying to be as exhaustive a;s t.he description in Daubechies (1992), for instance. The importance of wavelet theory (which is Nr ti'om being coafined go the ti:ne~frequeney problem) justifie:s our decision t.o give a short account of its basic principles. On the other hand, we hope that the coexistence of the time-freque~cy and time-scale approadles in one book witl help improve comparativo understanding a~d facilitate complementary use. C(mverse:y, some important issues in tim ~heory of bill:mar representations have been h~ft aside; such as the signal s3:I~thesis problem. Here we re%r the reader to the generM presentations by Hla,watseh (1985: 1986) and HIawat.sch and Kragtenthaler (1992; 1998), which were written by the investigal:ors themselves. The same is dolle with the bivariate extensions {more specifically directed at image methods (see Zh:: et aL, t992)) and the multilinea.r (or 'q':ighep order") methods that have flourished recm~.tly (see, e,g,, Amblard a::d La~'oume: 1992, or, Nikiaas and Petropulu, 1993). Let us finally remark that many of the proof~ are uot given in a. mathematicatly rigorotLs form. We p:'eti:rred rather to stress a "sensible" understanding of the results and their physical in~erpretat.ion. Although it h ~ become almost trivial (at least well :mderstood and accepted) to tNk about signals in a time-frequency stetting today; it is worthwhile to remember that this was not Nwag,,s obvious. Not very tong ago these descrip~hms ranged from bei::g either a theoretk:al exception or ~ heuristical mishmash. It is, therefk)re, my great, pleasure to salute the memory of B. Eseudi#3, who we~sone d the believers during the early days. Remembr~uo? of him st,ays alive in NI wt~o knew him. I am thankfl:l to him tbr introducing me to a research area tha.t was ~*ot marked out clearly at that time; and which in due course proved its importance.
bbreword
5
Some of the work preser~ted in this book, and whiclL ,ax)rk I perik)rnied later, has greatly profited from odier collaborations that~ I wish to nlention here. In particular, Imiiding the theory of the Wigner-Vilte spectrum a~ld its estimaiion was (toiie widi W. Martin, tim "geouietry of the interferences" w~s explored together with F. Hlawatsch, and the introduction and investigation of time-scale distributions obt.ained by affine smoothing w~s work (totie together with O. Rioul, and later with P. Gon(:atvbs. Thanks to all of them, e~s well ~s to P. Abry, F. Auger, Y. Biraud, J. Gr(~a, J. Sageloli, and J. P. Sessarego fk)r other fruithtl (ooperalions. I also include in my thanks my tormer colleagues from the Laborai~oire de Traitemeng du Signal de t'ICP! Lyon, M. Chiollaz, J. P. Corgiat{i, T. Doligez, N. Gache, and M. Zakharia. There were other influences, which I do not underestimate; these are feK in a more diffuse way. In particular~ I am refbrril~g to the workgroups "Adapta~.it]/Evolutig' and "Non stationnNre]' which I eondueted together with O. Maeehi al~d M, Bassevilte within the GdI{ 134 CNRS 5Ii~dtemeId, du Signal et Images, and to the work on comparing lhe time-frequency methods peri)~rmed there. I also think of several encotmters with the "wavelet community" (with A. Grossmami and Y. Meyer in the tirst place), where we could discuss our more complementary rather than differing points of view. P. Abry, R. G. Bara:muk, G. F. Boudreaux-Bade}s, and O. Michel were kind enough to reread and criticize prelimiliary versions of the manuscript (especially the tirst two chapters). I protit.ed from numerous remarks that they made, and I wish to express my thanks to them. 1 am also gratehfl to Y. Meyer, who granted me the honor of writing the prefa,:e. Finally, ~ wish t;o express n)y gratitude to Po Duvaut. and D. Garreau, who asked me to contribute to their book series 'I)'aitement d~l Signal, and whose friendly pressure led to the exis{ence of tchis book. Thanks also to J. Delkm, who went without his bek)ved computer for severM months, in order to let. me accomplish {;his project. While almost all materiM in this tratlslation is taken from the original "Temps-frdqueltce" from 1993, several smM1 corrections and additions have been incorporated. Concert~ing the eorreetio~:~s, I especially wish to thank the transla.tor, J. StSdder, whose numero~ls remarks during the translation process helped clarifying some details that were still missing in the first t~¥ench editiom These corrections are also incorporated in the second French edition, There ix oMy a small number of additions, which are intended mainly to give aceotmt of some more recent developme~ts in the field sil~ce 1993, This makes i{ easier to situate the book in the present literature on this subject. T h o ~ additions can be found mainly in the con-
6
A~a]vsis
cIuding remarks at the end of each chapter~ and they ref;~r to ar~ updated and extended bibliography. Most fig~res ha~e been re,Jraw~ a~M slightly e~ha~med. L~n July t993 and M~y 1998
Pa,trick Fbmdrin
Further introductory remarks, 1. Ea~ch (%apter starts with a short rdsumd to ~ r v e ~,s both an outline and a guidelirm for the reader. At tile end of each cha.pter a significant number of notes and remarlcs are to be found, rhey are concerned mai~ly with ,;~)mI~mnts o~ the subject theft car~ be omitted during a firs~ re~uJing, precise bibliographic re%fences, and brief sketches of fllrther issues thai; cannot be covered in this book. In order to keep the text uni~brm (it is illtended to be seiLcontained apart from a few exceptions) and to avoid overloading it, we include references to the or~ginat work i~a these l~o~,es. They contmn pointers to cumuluS;lye Re%rences that have been placed at the end of the book. 2. T h e time-i?equeney distributions are denoted mostly by the initiNs of those authors who defined them (example: B for Bertrand, C W for Cho'£-Williams, etc.). Other frequently used notations are listed in the Nomenclature. 3. In m ~ t illustrations of the timeq~-requency reweseutations the x-axis denotes time, the y-axis denotes frequency. The values of the representati(m are eiti~er drawn as a contourpiot or a densi V plot, whichever way w~.~s deemed r~tore appropria,te under the (ircumst~ances. I~ order to improve the clarity of these plots, we drew only the positive amplitudes in every ease.
I;breword
7
NOMENCLATUI~E time variables frequency variables scale variable : continuous (resp. discrete) time signal : frequency represen~at, ion of x(t) or x[n 1 : space of finite energy signals scalar product of L2(]F~) energy of signM x ( = (x, x})
.(t) 0esp, x[,4) X(.) g2(~)
i
u{.} pv
:
: :
E{.}
:
r~=(t,s)
:
~',,:(r)
:
r:Jl,) .<~(t, ~.) or .~:..(t, .; h) T , : ( t , a ) o r 7i,:(t,a;h) &(~, r)
: : : : :
c,.(t, ~; f)
:
f~;~,(La;f)
:
t4-';,(~,,.)
p~[t, .) U(.)
:
e(.)
:
{St~ ry ~
1~(.)
: :
:
V/~I;] -
complex conjugate reM part, imaginary part, modulus tlilbert transform Cauchy principal value operator associated with a Nn(.'tion A identity operator expectat, ion value of X ( = (~ilx, x}) stoch~tie expectation vahm autocovariance hmction ( = E {x(t)x" (s) }) autoeorrelation fllnction (stationary covariance) power spectrmn delzsity short-time Dourier transibrm (with window h) wavelet transform (with wavelet It) Wigne> Ville distribution (naxrowbaad) ambiguity ['unction Cohen's class, zparamet~er function f agine class, parameter function f time-frequency distribution with discrete time Heaviside unit step function Dirac distribution Kronecker symbol ( = 1 if n = m, 0 othm~Mse) characteristic flmetion of the interval I
Chapter 1 Problem
The Time
This chapter describes some of the generM concepts ~sociated with notions of time and frequency. We dwell oil the relations between these two variables and encompass the problems resulting from their combined use. In the first part (Subsection 1.I.1) we explain why purely freque~tiat representations (based on the h~mrier transtbrm, which ertLses all time dependence) prove to be insut~cient from a physical point of view, Ks one cannot dispense with the t.ime for describiItg a signal. Then we dis~ cuss two other consequences, which result fl'om employing the Fourier transfi~rm. They are both related to mathematical impossibilities. The first is expressed by the Heisenberg-Gabor uncertainV principle (Subsection 1,1.2), which stipulates that a signN cannot be concentrated on arbitrarily small time-frequency regions. The second result (theory of SlepianPollak-Landau, Subsection 1.1.3) shows thai, a signal cannot confine its total energy to finite intervals in the time and frequency domMn, no mat.. ter how large these intervals might be. Without being able to overcome the inherent limitations of the ~burier transform, Section 1.2 hoMs out a prospect of possible substitutes fbr the Fburier frequency, which ,~re better suited for nonstationary situations. In Subsection 1.2.1 we,. introduce some local concepts such as the instant:aneol~s frequency, thus giving a meaning to the intuitive notion of a temporal evolution of a (determi~.fistic) spectral properV. Then we investigate the problem in Subsection 1,2.2 of how locality can be introduced to the representation of a nonstationary signal. Here we work ill a probabilistic setting. However, the given solutions provide only partial answers to the posed problem (or a,nswc.rs that. are diNcult to interpret). This motivates the study of more general appro~u:hes in an explieit!y joint time-frequency fl:amework. An imroduction to such approa,~hes is the subject of Section 1.3. V~re sketch briefly a classification d possible solutions, focusing on those that witl be studied in more detail in Chapter 2. 9
t~ime-li}'c~quonc_tg/Time-Scah -, Anal;~=~;is
10
!il.1. The Time-Frequency Duality and Its Bars 1.1,1, Fourier Analysis The Fourier anatysis ~ is one of the major accomplishments of physics and mathematics, It is indispensable to signal ~heory and signM processing ff)r severn reasons, Certainly, the first among diem is ~he universal conc,ept of [}~fl~oncy in which it is rooted: a frequentiat description can oAen be the basis of a bet;ter comprehension of the underlying phenomena, because it supplies an essential complement to the excI,,Lsive/y temporal description (r~:eiver ou~pu~ or ~quem'e of eva,his) t.hat is usuNly taken first tbr mlalysis. This occurs in several areas of imerest, t)e it in physical waves (acoustics, vibrations, g(<@O~sies, optics~ etc.) or in some other frmne~wk where periodic processes play an importan~ role (economy; biology, a;stronomy, et,e.), The second reason comes from the mathematicM structure of the Fourier transtbrm it~,tf, aus it is naturMly suited to ~x)mmon transfbrm methods (sud:~ as tim linear filtering) by its abihty ~,o render them in a p a y ticutarty simple form, Finally; we mention a~s a third and more pragmatic reason, that the collection of these a, tvantages has led to the development of a la,rge nmnber of a,lgorithms, programs, processors, and machines for frequem:y analysis, atI of which contribute to its good reput~tion fbr practical use. Limitations. Import, ant though i~ is from a mathematicM point of view, the Fourier analysis possesses severM wstrietions concerning its physical interpretation and its ravage of applicability, tn order to explMn tills, let us look at the usual definition of ~:he Fourier transform
(L1) x
Evidently, the computation of one frequency value X ( @ necessitates the knowledge of ~.he complete histoo, of the signal ranging from - . ~ to 4 ~ , Com~rsely, the inverse Fourier transibrm is given by f - K,
du
(~,2)
9~
Hence, any va|ne :r(t) of the signM a~. one instant t can be regarded as an infinite superposition of comptex exponentials, or everY:sting and completely nonlocal waves, Even if this mathematical poi~t of view may reve.al t M true properties of a signal in e,ertain cases (~ktu~@monoehromatic" situa,tions, steady statG etc.), i~. can also distort the physical reality, This happens, for exa.mple, with transient signals, which vanish outside a certain time
Chapter 1 'lthe Time-.ti)'(~)ue~iqr P r o b l e m
it
interval (e.g., by switching a maclfine on and off). Although the 1mllity of its values is reflected t9' tile [~burier analysis, this occurs only in an artificial manner: it results from an infinite s~perposition of virtual waves that interfere sucii that t,hey annitfitate ca,oh other. Hence l;he situation on the domNn, where the signN vanishes, can1 be described ;.~s a ~'dynamic" zero {there exist waves whose resulting eor~tribution is zero by interference). This contradicts any proper understanding of the reM physical situation as a %tatic" zero (tile signal does not. exist). Citations. In this regard we wfstl to give promi~telme to three particularly appropriate citations. (These are translations of the original texts in I*¥eneh.) The first, by Vilte, is from 1948. In his flmdamenta/ article, which appeared in C;ibhas et 2)~msmis,si~n~s, he defined both the notion of analytic signals and the t,ime~frequeney representation that is nmned aft~er him today. We can reM the following in the Introduction: Indeed, if we consider a piece [of music 1 which contains several bars (which is the least we can demand), arid if one note, N for example, appears once in the Neee, the harmonic [Fourier] aualysis will present the corresponding frequency with a certNn m~plitude and a certain phase, without locating the ta in time. It is evident, however, that there are InomeI~ts in the course of the piece where the la cannot be heard. Nevertheless, tlm repi~sentation is mathematically correct, because the pha~ses of the notes near N a,re arra.nged in such a way, that they destroy tNs note by interference when it is not hea.rd, and reinforce it, a,gain by interference, when it is heaM; but even if there is a versatility in this eo~lcept honoring the mathematieM anMysis, one should not coneea,t that there is also a distortion of l.he reMit:f: indeed, when one does I-~ot hear the la, the true reason is ~hat the N is not emitted. The second citation, though more recent ;&s it dates h:om 1966, is taken from all equa]ly illustrious source. It is due to de Broglie, who ~ s e r t s in his ;'Certitudes et Incertitude's de la Science": If we consider a quantity which can be represented in a Fourier manner, i.e., b.y a superposition of monochromatic components, then it is the superposition which has a physical meaning, but not the isolated Fourier components. If we deal, for example, with a swinging c.hord whose motion can be described by a series of harmonies, a movie of this motion would reveal that the chord h~ks a very complicated form at e~u:h momei~t, and that it varies incessantly according to a complex rule. Nothing in this motioi~ allows us to distinguish the -carious monochromatic components: these
[2
Ti,n~-f')'eq~tc,nqy?/75me-Sc¢dc A~mlysis components exist only in vim minds of theorists who endeavor an abstract analysis of this motion. They wouId only come into physicM existence if one couM achieve {.heir isola{;ion by an operation, which, in turn, would break up the superposition. Besides, the whole theory of intert>rences would be inexact if this were not true. The idea. that the monochromatic components ha,,,~' a real existence in the physicM woeess, which comes from their superp(>ition, seems false to me, as it vitiates parts of the th(x)retieM reasoning which is actuMly common in Quantum Physics.
FinMly, the third quotation is found in ~he work of the ineffable Bou~sse, who makes a strNghtforward statement in his Acoustique GdndxMe in 1926, saying that "[...] unless one has lost tim most elementary common sense, it is impossiNe to attribute an objective existence ~o the harmonic oscillations which emerge in the Fourier series." Even though the Fourier analysis has such restrictions regarding availability of inteIlaretations o r ad~xtuacy tor certain Vpes of signals, it still remNns tr~m that it p~ssesses an immense utility, be it ~-floneor ~s a cmnputational tool. Moreover, we will see that numerous time-frequency methods, though deviating from the spiri~ of a Fourier analysis strict:o sel~u, stay elc~e t.o it in their definition and expkfit its weNth of mat.hematical structures.
1.1.2. Heisenberg-Gabor Uncertainty Principle Let us first cot~sider the case of a bandihnited signal. As its support in the frequency domain is compact, it.s (im,erse) Fourier transiorm must be aa~alytic. Therefore, the signal cannot, vanish on a set of positive measure or be strictly confined to a finite duration: indeed, its ana.tyticity would impIy thug it va~fistms everywhere by anNytic continuation. Another vc~ to prove this Net is to start from the converse hypothesis of bounded supports in t,ime (with dm'ation T) and frequency (with bandwidth B). Any nonzero signal with these properties would s a t i @ the relation
a t~,e:
=
'
du
=
0,
}tt
> 7'/2.
J -B/2
As x(t) is of bounded duratim~, the same is true for its ,~th derivatives. Fu> thermore, as X (u) is supp(,sed to vanish outside t,he interval I-B~2, +B/2}, we would also have
(;hapter 1 Tim Timc-tCrcquency Problem
;13
for all ,n > 0. The value of the signal in ~ point s, which belongs to the support of z(t) (so }sl < T / 2 ) , can now be writte~ as , t:3/2 J
I.q < : r M ,
1~/2
Itt > r / 2 .
By replacing tim first complex exponential wit;h ills power series =
'"
'u! we arrive at our final relation
x s()=
.......................... rz! ~,~/
(i2rcu) '~ X (u) e '2'~'t du = 0 ,
where Itl > 7:/2. This holds for all Is I < T / 2 and thus contrMicts our initial assumption that the signal is nonzero in the interval I - T / 2 , .FT/2]. Even if we relax the strict constraint of finite supports, it is well known that l;he (essentiM) support of a signal cammt be arbitrarily small both in time and frequency: our experience prow?s, for example, t.hat a short impulse extends over a large fl'equency range. Vice versa, the narrower the band of a filter, the lounger is its response time. This type of cor~straint. is imposed by the Fourier duality (which exists between the time and the frequency representations of signals). It is dearly illustrated by the pair "Dirae distribution const.ant flmction," and in a somewhat smoother way by the Gaussian flmctions which satis(y
If we regard the width of a Gaussian as proportionaJ to a '-112 in time, tile preceding relation shows that the equivalent width of its Fourier transform, which is also a Gaussiam is proportionM to ct 1/2 These magnitudes vary in opposite directions depending on the pm:ameter c~. One of them increases when the other decrea~ses, and vice vema. The product of the two mlmbers remMns constant. (Let us fllrther remark that the forementioned pair "Dirac distribution constant function" turns up when g tends to infinity.)
14
Time~hi>quen( K/T ime-'Scalc Ana!ysis
The time-tYequency inequality. This behavior of dualib" is a direct coI> sequence of the definition of the D~urier transform. It finds its simplest. mathematical tbrmutation in the so-called tfeisenberg~Gabor uncertainty p r i n @ l e . 2 tt is nam~l after the 'uncertainty principles" dis~:overed ~Heise~berg in the 1920s in the context of an arising quantum mechanics, a.nd aher Gabor, who performed analogpus s~udies directly after World x3~:~r II in the field of comlnunication theor> 'I~ est,ablish this inequality, let us consider a signal re(t) with finite m~ergy , @~X: &. =
/
7X~
,(~)t
2 dt < +oc.
\ ~ assume tbr the sake, of simplicity that the signal and its Dmrier {.ran> form X0e) have a vanishing center of gravky, that is //i "+:>< t [z(t){e dt = 0 .
and
/ti" :~ ,, i X (,,}I~~,d~ = 0 ,
x)
x
(This can ahv~\vs be obtained I~" a, suitable shift of the axes.) As measures of the time and frequency supports of the signal we introduce the respecth~ moments of inertia At2-
E1. . . . /. ~ t ,~, ' ! x { t } [ ~ d t
_
''
A~ 2
=
1 E,~
/_....+>" :~ ~
x(,,)i
~ a,.
(:t.3)
Let, us finally define the auxiliary quantity I -
f
~' ~~'* (t) Zdx (t) dt "
By using Parseyat's identity, we can immedia,tely deduce that ~ t 2 tx(t)K~dt. / + ~~ (Re{I}) 2 ~: [Ii 2 ~ / + .:~
dx t} i~ dt = 4rr ~ E'~ A t ~ A t , ~', ~-,17[,
where the first inequality holds for ar\v complex number and the second fbllows frorn dm Cauchy-Schwarz inequality. Integration by tmrts shows thai, I equals
t hence
=
[ t :ix(t)l• 2 j_~. ~+:~
• Er~ - J -l~x.i/ ~ . t r r ( t } -d:r~ ~ ( • d "0 t=-E~,-F,
Chapter I T h e Thne-Fr~.que,
15
this compmadoti is jusdfied, supp(xsing |,hat tile squared al)~lut( va,tue of x(t) decays fast enough such t}mt tix(t)l 2 va.lfishes a{ infi~fity. (This hypothes~ is clearly satisfied if x(t} has compact support; it also results from tile fi~fiteness of At.) Under this assumption the IIeisenberg~Gabor uucertain~ ~p r i n ( i p b follows, which is expressed by the inequality
I Employing the preceding definitions for the duration At and bandwidth Au of a signal, we ha~,,e thus seen that the "duration-bandwidth product" At - Au of any signal is bounded f}om below. The lower bound is at:rained if we meet the conditkms of equality in the Cauchy-Sehwarz inequaIity (on which the derivation of the uncertainV principle w~Lsbased), Viewed fi'om a general geome{rica.l point of view, this means that the two vectors in d~e inner product are coliinear. In other words, the two signals t x ( t ) and (d:c/dt)(t) must be proportional. For elements of the w~cmr space of real-vNued signals, riffs implies the differential equathm d.r
....
kt dt ,
;r
k ~. lR .
Its finhe mm~gy solul.i(ms a(tmi~ the genera.1 form x(t)
(~ e -~t"
(_,, ~) ~ ~ x g{.+
Hence dm Gaussian flmctions are the only solutions that minimize the duration-bandwidth product in die Heis~e~berg-Gabor sense. This relates to our wevious statement about the Gaussians and their Fourier transforms. Rems~rk 1, Let us note, h'om al~ angb of physica,1 interpretation, that the given definitions of the durmion and bandwidth of a sigrml lead to perceptibly smaller values thm~ other measures, such as an "equi~dent width" or "half-wideh." If we compute the quantikv All~=, at a size of exp(-lr/4) of the Ga,ussian peak value (which is rather cl(me to the half= width, as exp(- ~/4) = 0A56) dmn we obtain Ate!2
=
2,/~At.
This leads to a minimal duration-bandwidth product of I. Remark 2. One could be willing to accept that the results concerning the minimization of the duration-bandwiddi product ((mid be extended to eomNex-valued signals, whose absolut.e values x(*)i and IX(u) l are both GatLssians. A Vpieal instance of such signals is a "chirp" (linear frequency modulation and G a ~ s i a n envelope) =Ce
'~+
t
((.,a,~3)~=IR×N~xlR.
I{s Fourier transform is g[v(,l~ by
E
'
'
'
,
Some direct comput;ations show, however, that in this ca.se 1 < ..................... =, 1 -.-- vZl + ~" > --.
with
~,' ~
9
(1,5)
In fact, for a given duration (fixed by ct) the bandwidth is enlarged by an amount relative to the range of fi'equencies, which is scanned by the freo quency modulation due to the quadratic phase, Consequently, Mthough the absolute valuers of the signal ~nd its Fourier transform are Gmissians, the duration-bandwidth product may exceed the lower bound in tim HeisenbergGabor uncertMnty principle. An obvious excep{:ion is the case ~ = 0, which reduces to the situation of a reN signal without modulation. This simple example illustrates the following fact: If a signal a.ttNns the minimal value of the duration-bandwidth product, (:hen its absolute values in time and frequency must be Gaussians. On tim other hand, the converse is not true; that is, if a signal has Gaussian absolute values in time and freq~mncy, then it imed not attNn the minimal value of the duration-kvmdwidth product. Finally, olin should recall that this discussion is also valid tbr signals that are shifted ii1 time al:~d/or fl:equency. The only change to be made is a proper eompuga.tion of the moments of inertia with respect to the cenge~.-s of grea.'it.y of the shifted signals, it is important to note, however, tiiat an extension of the same results to the case of reM signMs, for which only the half-spectrum of positive frequencies k~ considered, is not allowed, 3 Interpretations. The Hei~nberg-.Gabor un(:ertainW principle admits differm~t intelqpretations. As we ha:re doom so Nr, we can consider At and At* e.s extensions of the supports on which a well-defined signal exists in thne and in frequency. B e c a u ~ ly(t)i 2 a.~d iX(,~)t 2 are nonnegat, ive flmctions, we can also ascribe the rank of a probN)ility density Nnetion to them. Then the tbrementioned quantities m~,turalty become standard deviatiot~s, This is, in fact, the predominant a4)proa~(:h in quantum mecha~ics, where the actual equivaIent of the inequality (1.4) refers to the variables of position and momentum, and not to [h~se of time and energy as in "'Heisenberg's fourth uncertainty prineiple." 4 This latter principle can easily be derived, a,t; le~-~st formally, from the time.frequency inequMity via the relation ~
h U
(where h is t.he Planck constant). Then a simple mu|tiplication of the two members in Eq. (1.4) by h shows that, 5 t . ,'~E ~ h/4rc.
Cb~p~er 1 The Time-t%'equency Problem
17
When we attempt, to use a common formMism tbr quantum mechani(s a.nd signal theory, we have to pay attention to the fact that the notions of "thne" in both disciplines are rather different by nature. The "time" in quantum mechanics is an evolutionary parameter relative to observable quantities that depend on the variables of the description of the system, such a.s the position or the momentum. Wid~ these observables we a:ssociaee certain operators. (Recall that the measurements are expectation values of these operator functions over the possible states of the system. 5) In this context, the %ime" itself does not correspond to an operator, which would endow it with the status of an observable. In contrast to this situation, the "time" in the operational formalism of signal theory (a ptays a role dmt is mmlogous to the position variable in qum~tum mechanics. Hence it has no dynamical na.ture in the foregoing sense. Rather, it; can be a~sociated with an operator
(i:,.)(t) ~ t:r(t). As is welt known, the Fburier transform maps a multiplication by the x-,~riM)le in one domain into a derivative in the other domain. Hence, by duality of the Fourier transfbrm, the frequency ~e (which is the analogue of the momentum) can be t~ssociated with the operator
(i)x)(t) ~ I dx (t. ' i27r ~ ,0. One can easily see that the product of these two operators depends on their order. In fiw~t, they satisfy the commutation relation
where I denotes the identity operator. This is traditionally expressed by saying that the time and frequency variables are canonically conjugate. This operational fbrmalism allows us to prove the Heisenberg-Gabor uncertainty principle in a different was2 Let us first employ the notation of the expectation x~4Iue of an operator A, which is defined by
{;i),: - <Xx, x} where (., .} denotes the usual inner product of L2(N); that is,
(z, y) ~
z(t) y*(t) at =
X ( , ) Y*(,.) d . .
The equations
~xt~= ~:L~
E
t ,8
"t'ir, ~. F>eq ue n qv/Z~m e- & :ale A , a ]3"sis
hold by definitiom :if we introduce t.he operator [ + {AP, with A being an arbit.rary real number: titan the positivity of die inner pr(Muet yields
o < ( (i + .ia~>).c (i +-,:;wp: )
( (i
i),o) (i
iax>)
Here the equality on the right-hand side holds, t~s both operators [ and P are self-adjoint. The commutation reIation for i and b leads further to
As El,: is positive, this last inequality is satisiied for all A. if and only if the discrim}nant of the quadrat.ic polynomial (in A) is alw<¢s r~egatiwe. This anew furnishes the relation 1
At, Av > -- . - 47r
1.1.3. Slepian-Pollak-Landau Theory As flmdamel~ta,1 ~s it is, the tteisenberg-Gabor u~merta,in W principle is not. the only possible approax:h aiming at a mathematical des(ription of the Fourier duality, which implies that the confinenmnt of a signal in o~e domain (time or fr~ttmncy) causes the loss in its confinement in the canonically eot\fugate domain. The Heisenber2Gabor inequality p a r t i c u l a d y accentuates the impossibility that a signal have a,rbbrarily smM1 supports both in time and frequency; But it tells noghing about the impossibility of restricting its total energy to eompac~ supports in time and fl:equency, if arbitrarily large (but finite) intervals are aliow(,.d. Concentrations. This impossibility is well knowm however, and it calls for a quantitative explanatiom instead of characterizing the equivalence of supports in terms of mea.sures of dispersion, it is preferaMe for the given tt~sk to u ~ measures of energy concentration. In view" of Pars~?val's idengi~y we know that, for a signal x ( t ) , ~:~ =
!, + ;.a
, :jl~+i/
Ix(t) ~ dt =
Ix(,dt 2,>
Hence, the following definitions of the energy concentrations in a time inte~v,,l [-~172, +~r/2] o~ a f,'eq.en,,y interval -...;~/2, .+-B/21 seem to ~,., appropriat, e:
E,,('7')
-
J-T~2
I~,(~.)l~zt: and
~::'x(B) '
a ,.,,,B/2
ix(~,)t2d~,.
(1.6)
C/mpt~er I T h e 't'ime-t;¥eq~zem3," P l ~ b l e m
[9
'Fhese quantities rely on two trm~catiox~ Ol)Cr,igors, which are defined by tile restriction to the respective intmval, namely (~:':)(~)
= { 0:':(:) ,' I*IItt>~r/2,T/2,
and
(l.7)
x(,.,) , i,:l <- B/2, (&X)(~,)
=
0
,
Iz:b > B / 2
(1.s) .
Equation (1.6) can now be written in the equivMent form <;~ >,: = e , ( r )
,,.nd
<~%>~,,:= E x ( B )
.
(t.9)
The preceding operators are projections (which mealLs that I~¢.. i}4 = ~ and t~I~. _fi~ = ];]i), Withil! this formalism, a perfect concentration both on the time i,~te,wal [.....-'1"/2, +T/2} a,~d the ii'equeney interval [ - B / 2 , +/3/2] \ = E~., would result in (~tr).~ = {Pf ~ ~:~/:~ , with finite "~alues tbr T and 1~, Sampling. ~v\~ealreMy mentioned the impossibility of such concentrations owing t;o tile anNyticity of the Rmrier transform of a function with compact support. From another point of view, we encounter one of the most common manifestations of this impossibility when we t~%"to sample a signal. It is known as a general fact that the sampling in the time domain ca.uses a periodiza:tion in the frequency domNn. Hence, it can only be used without any loss in infbrmation (which means without changing the spectral contents of the ~ontinuous signM), if the original signal is bandlim~. ited. Unfortunately, this forces tim signal to Iast infinitely, which obviously never h~ppens in practical situations. All this means that: a perfect sampiing is impossible in practice. However, we know from our experience that a. re~soimble approximation is often achievable, which is compat4ble with the fir~ite duration of observations and, at the same time, with the finite frequency band of the receivers. Hence, the newly set problem of the t.ime-frequency duality is to establish a precise mathematical framework for the empirical not,ion of signMs that are "practically" finite in time and frequency. A first approach consists of a qualitative analysis of the approximation error, which is induced by imperfhct sampling. Let us suppose momenta> ily that a signal x(t) is strictly limited to a frequency band [ - t ? / 2 , + / 3 / 2 ] . Then it can be sampled with a sampling period G < 1 / B without intr~ ducing all error. Denoting by x[~] - x ( ~ / B )
,
~ ~ ~ ,
2{1
"I imc-h)~equ~>ncy/'*l'ime-Scale :1 ualysis
a seque~ce ~f samplhJg values at~ this minimal sampling rate~ all xalues :~:(t) of the sigmd can be ~ecovered ~,,ia the interpolat[o!~ fbrmula r
r~(Bt ~:::
(1 .I0)
n)
>¢
As far as the summation actually extends over the infinite sequence-, this interpolation [s totally exact, as one can veri(v that
j a,(t)
:4-
sin rc(I3t - ;'t) :'~
7
dt ~ 0 .
In order to work with a finite sum only, we must. drop a countable nmnber of sampling vatu(r.s fi'om the sequence, Intuitively, we obtNn an approximate reconstruction by re,~trieI:ing the summation to all sampling ~vdu{ s inside au interval [ - T7 °f . , ~7'/21, if .r(t) has neg~igibl~, values oni;side this interval. This means that all values for n, which are used i~ the s u m m a t i o n must satisf~y ~t 5': ( T / 2 ) / ( 1 / 7 ; ) , or equivalently tu[ < B T / 2 , The resulting error of the reconstruction ha.s the form
,
f
+~I
F/T/2
si~I v(Bt
.... n )
dt .
This error can never be zero, yet its size,;, iv giveu by
B
BT/2
where T is fixed and [7 is sufficiemly large, The attticipat, ed approxima:tion of finite duration T of a, band[trotted siguM becomes better, if the temporal energy conce,ntratioJ~ is large. The problem still remains of how to flied the exact bound for the largest; possiMe energy concem;ra,tion in the given time-interval. A precise a~sw'er to this question, among other rela~.ed ones, was given by the gheoW of Slepian, Pollak, and Landau. s It was developed around the beginning of ~he 1960s and is based on the study of the eigemadues and eigen%netions of the forementioned projection operators (eft Eqs. (1.7) and (1.8)).
(Giapter
t The :l:iuic-l;?'equenqyProbh~m
21
eigenvalue equation. Given an arbitrary (nonzero) signal x(t) in L2(IR), let. us first+ apply ~he truncation in time and then the trurmation in frequency. The new signal must be diflbrent fl'om the originN, as its energy was reduced by at ten,st one of the operations. The question that arises here is how the minimal deeiine of the energy by this double t.runea~ion can be characterized, and for which signal it is attai~ed~ A solution to this problem wouId enable us to define a precise notion of simultan~ums concentration in time and frequency. The
The operator of double trunea,tion corresponds to the transfcmnation
[
+2'/'2
+t3/2
hence, sin ~ B ( t
s)
-
Became the trurication operators are ,.Cf-a.([lomt, the ratio ft(B, T) of the energy of the twice-truncated signal and the original signal ca,n be written aN t
t
,
B t;lX]
.........
(1.11)
bhirthermore, tim second factor in Eq. (1.11) depends only on the values of x(t) inside the tinie interval [--1/,, +T/2i, hence "
f
r
#(tlt l ) =
/+v72 I /'+'V/2 sinrc[3(l,- s ) .-r/2
L,L~?2 -~2:~~
.r(.,.)ds x * ( 0 d t .
Minimizing the eKix:t of the double trmlcation is hereby equivalent to maximizing p(B, T). This is obtained if x(t) is an eigenflmction of the integral equation
~.+:~ '/2
~'/2
~) ~r(t - s)
x(s)ds = ~z(t)
'
!tI < T/2
'
(1J2)
77me-t ?'equet~qv/'7:ime-Sca.le At~al3".sis
22
relative t.o the larges{ eigenvahm k ........ Consequently, an upper bound [br the rat, io of the energi~> (after and before {he double truncation) is
Durat.ion-bandwid~h product At first, sight t.he eigenva.lue equation (1.12) (and its solutions) seems to depend on B and T independently. Ilmvever, it can e ~ i l y be rewritten by mea.ns of ,~ subst.itution of the ~rriabte s = T'u., and by incroducing the auxiliary signal ,t(u) ~ .~:(Tu), thus rendering it in t,he simpler form f+v~
sin rcBT(.u
[[,)
:!ffw)&~
=
.,\
~j(,,~:'t
bl < 172
This shows that the dependence on B and 7' occum only via the intermm diary of the duration-bandwidth product BT.
Eigenvahms. Because of its strtmture the eigenvMue equatio~ has a discrete spectrum of posith.'e eigenvatues ),.,~, which lie between 0 and 1 (as they provide a measure of the relative energy concentration). We can thus arrange them a~ccording to 0 < ,.. < A,~ < ... < )q < A0 = ,\~-~,.× < 1 . I£ach eigenvahle is to be considered as a flmction of the product BT.
Eig'enth~(tirms. There is one and only one eigenfunetion (~= (t) g~soeiated with each dgenvMue k~. Properly normalized~ this collection of eigenf)nctions (which are called 'protate spheroidal wave functions" and "fonc~ions sph4ro'ida,les aplatie<,") forms an orchonormat system on IR, We thus ha,re that Likewise, it is a. simple exercise to show that
f Fhfite bandwidth. lf+~
+.,./2 ~:,,.,(t) .,:~,.~(t) Jt~ = .~,,, ~ ...... . :* 7 2
If we rewrite the eigenvalue equation (1,t2) in the form sin roB(! - s)
it. is of convoIu~ion ty'pe. We c a l iminediateiy see by" an applieatim, of the Fourier transform that alt eigenflmction.s have finite bandvAdth/3.
Chaptm 1 The [Tm<-FT'eque~sqy Problem
23
A p p r o x i m a t i o n of b a n d l i m i t e d signals. Due to t;lie fa*~t that the eigenflmct.ions %rm a.n orlhonormal system, every signal with I:)andwi(tt,h B has a series expansio~l
x(t) = E x,,. ,,;,,:(t)
where
x; = /
r(t) :%(t) dr. "x
This enables us to give a definite answer ~o one of the previously posed problems: the bmMlimited signal with bandwidth B, which maximizes {.he energy concentraUon in the time interval [ 7 t 2, { T/2], is the eigenflmction ~%(t) relative to the ina.xilnal eigenvalue k0. In&~,d, this result, is a direc{, {onsequence of the relations , } ,~<
:2*:>
<<
t'~; 0
'
aaM ,~>: ~
\* T I *
/,.','i2 = .LT/2
x2(t) de = '
>E k,, n~.{>
z.
2 <_ Aoi],
(]t.14)
,
wit.h equality iI~ the last st.ep for re(t) = g'o(t). Mor~}ver, we can quam, ify the approximation error, whidi is commitI,ed when representing a. signal of bandwidt.h 117 by only iinitety many eigenfunctions relative to the support T. Measuring it in terms of a mean square error a~ before, we obta.in
/
.....E
','
E
This error is bomMed by 8
t
1-),x+l n:=N
1
~
"
:r,i
(I
-
,\,,)
,
r~::: U
hence Eq. (1.14} yields 2 e,; _< l - .X~,..~ I
(I,15)
Time-b)*equenc3;/lime-Scale Am@sis
24
A p p r o x i m a t i v e dimension of a s i g n a l The approximat.ion error in gq, (1.15) obviously decrea~5~s when the size of the first, omitted eigemv~lue in the expansion gvts smaller. This leads to the notion of the appfoximative dimensioi~ of a signal, which is lwgarded as the number of eigenfunctions that are associated with non,-negligibte eigenvalues. In ~l~e ease of the sampling operation, ~he following empirical argmnent for a signal with a fl'equency band [--B/2, + B / 2 ] w ~ raised: because it requires a minimal sampling rate of 1/B, it should be repre~mta, bfe hy roughly T/(1/B) = BT sampling values, if its (essential) ~emporal support is confined to [-.T/2, +T/2]. This point of vie'," can now be specified more accurately by the new approach using the truncation operators. Indeed, we can conclude from Mercer's thc~)rem 9 and the value for the norm of the eigenflmctions on I - T / 2 , +T/2], that the kernel of the integral equation has an expansion in terms of the wolat, e spheroidal wave t~anctions given
by
From this relation ~a~>can deduce the fa*gt that
This result is relaged to the idea that approxim~tely B T of the dominant eigenvaiues are close to 1, while the others are cloe~ to 0. An actuM computation of the eigenvatues ~s functions of ,~ for a fixed value of BT' reveals such a behavior: one observers a fast decay of the x~flues as soon as n > BT. This knowledge can be usod in order to find better estimates for the error, when we want to approximate a signM with baz~dwidth B by a finite collection of prolate spheroidal wa~e fuim~,ions relative to the duration T. B a ~ d on the fact that 3,wf~ (BT) < 0.916 one obtaiIxs the frequently used estima.te
e~ ~ 1 2 [ E . -
E~(T)]
if
N > BT".
I n e q u a l i t y of t h e concentrations. If we consider the time-frequency duality from an angle of the energ2¢ conceni, rations in l,ime and frequency, the adopted point of view allows us to state another interesting inequMity. It is closdy associated with the following question: given a concentration in ~;ime (or in fl-equency), what is the best possible concentra.tion in fl'equeney
(.ha~ ter I The ~l~me-Frequency Problem
25
(in time, ~spectively), which can be expected, mid for vchich sigmd is it obtained? We should note, after all, that this issue arises only if the imposed concentration E,(T)/E~. (or FJx(B)/E~.), for a give~ value of BT, is strictly larg~-;r than the largest eigenvalue Ao(BT). Otherwise, in the case of equMity, we showed before that a unique solution exists and is equal to the eigenfunction "J/o(t). Even more can be said, when we suppose that E~,(T) < Ao(BT) F;:,.; then there exist, infinitely many bandlimited signals with E x ( B ) = E~. Hence, this ease implies no restriction of the fl'equency concentration whatsoever. The situation is totaIIy different for the remaining cases when H;r(T) > A0(BT)E~. or Ex(B) > A~(BT')I~2:,:. One can show that the individmd concentrations satisfy the time-fl'equeney inequality •
~
~/:~
a,rcos
(3,~ (BT) i/2) ,
and that equality is attained fbr th,e signal
)
_
As an illustration of this result, we depict several curves in Fig. t.1 that are associated with different values of BT. Each curve forms the boundary of the domain of jointly admissible values of the energy fractions on the supports B and T. ~br increasing values of BT, the upper borderline of these domains stretches out towards the point (1, 1), which corresponds to the extreme cause of total energy concentrations in time and fl'equeney, Rema.rk. It is also interesting to comment on the other extreme case ~J.ss~ elated with B T = 0, which corresponds to the antidiagonal in the diagram. If we suppose, in fact,, that the energy fractions i~.r(7') and E x (B) are such that Ej,(T) + Ex(B) <; E:~, then we hax~e
-2 This guarantees that the inequMity of the concentrations is verified for any value of BT, including 0, [mcause, the relation 0 < Ao(BT) < 1 is Mw®~s true. Hence, there is no constraint at M1 in this case, and the given concentrations can be achieved on arbit, rarily small supports in the time and frequency domains.
Time-t'i'e(frm~cy/Timc-Scale A~a/2wis
26
(k8
,.\ N \
0{~
.....,,, ...."",
4,t BT = 0 04'
}
%, % 5i
""\ ".. .......,
{
i
'i;?!
", \
\
",.% "% i
"\ i
0
02
0.4
06
0,8
1
E (TI/E
F i g u r e 1.1.
E~mrgy con(entrati(ms in thne and freque~cy.
Borderli~es of the admissible domains of eaergy coime,~tra.tions in time grad frequency, fc,r dii%rent values of the duratiot>bandwidth product B;t
(of. E q . ( 1 . 1 7 } ) .
{il.2. Leaving Fourier? A}I fbremention~M |imitations center around the difficulty caused by the use of global descriptions '% ta talmrier '' (and thus having a time or frequency nature) for the appreh.ension of a, reMity that exists jointly in time and frequency, Indeed, most time-fre(pmncy problems can be specified in t e r m s of locM quaI~tities, which can either be .joint (in terms of time and frequem:y~ or not. Such specifications arise through the ~toption of deftnitions that incorporate certain nonstationary properties; or ~tm ~ a r c h for possible interpretations of these definitions. First of all they require some deeper insight into both the motivation a~d toots t,ha~, exist tbr s~.mh }ocat objects.
1.2.1. L o c a l Q u a n t i t i e s When we want to describe a signal b o t h i~ time and fi~equency, the most desirable alld most natm'M Iocal qua~tity is one t h a t gives a meaning to an "instantaneous" spectral content. This terminology aeems to be based on an inner contradiction; as a t;%urier frequency in ~.he mathematicM sense
Chapter 1 The Timc-F~('q~el~(\v Problem
27
is associated with a global behavior. Yet our experience (a.nd esp~x:iMty that of our auditory system) suggests that~ one ca~) imbue such a local quantity with a physical meaning. Only the "frequency" brought into pta;,; in this context should be defined in a different; way than the usuM Fburier frequency. I n s t a n t a n e o u s frequency. 1,:) In order to define ghe notion of an "ir~stantano:ms f}equency," it is appropria,te to revisit the protoWpe of a signN associated with the concept of steady state and stability in time: the monochromatic ~v~v< It can be unambiguously represented (apart from a pure phrase) by
x(t) = a c e s 2rra¢ , where the constants a ~ l d vo are to be read as the amplitude ai~d the ~equ(mQy, respectively. The latter measures the rate of change of the argm~mnt of the co,sine, or its derivative with respect to the time variaMe (except for a factor 2~r). It is quite tempting to extend this point of view to evolutionary situation,s, simply by letting the constant a vary in time and by i1~troducing al~ argument of the c(~sine with a time.varying derivative. Fhis would lead to definitions of the fbrm . ( t ) = ..(tt c o s Unfortunately, this exwession is not unique. In contrast to the ideal monochromatic case, there are infinitely many pairs (a(t),~{t)) [br the representation of a given signal z(t). This can be see~ lb~ chorusing m V NImtion b(t) with 0 < b(t) < 1, Then the equation
z(t) = a(t)cos~(t) = i~i:~-b(t)cosp(t) shows that z(t) can be written in a~tot.h.er tbnn s~, =
with a'(t) =
a(t)/b(t) and ~*(t) = arcos(b(t} cos p(t)).
A proper sotution to this problem can be found, when we first reconsider ~tm monochromatic ca.~e prior to its generalization A real monoctm~ matic signal call cer£ainiy be regarded as the real part, of a complex exponentiM (1.18) acos2~r~,o! = Re r.... -/.~ 2~-,.¢.;"I . The amplitude and frequency of the monochromatic signal are the modulus and p h ~ e (except tbr the factor 27r) of this exponentiN, respectively. Its
28
'l lime-D'eq~,~et~cF?"Tim+.-ScMe ,4 natStsis
imaginary part, which is (~sin 2~r~.,~)t,is derived from tile real part by ~ pha:~+e shift of ~/2: ~¥~ say that the real a~KI im~).ginary parts are in quadr;,~tu'e. The matlmmatical operation behind this transformation (an be described in the frequency domain mosg easily. It, maps the Dmrier {,ransform of a cosine Nnction 2
to the transform of tJ:~e ccwresponding sine function
2i This operation is a linear filtering whose frequency response is --i sgn u (so its impulse response is pv(I/Tct) where "pv" denotes the Cauchy principal value). It is called the HilSert, t~ar~sform. ConsequerMy, if we axssoc}ate with each real signM :r(~) a complex signal L~:(S) ,
==
--
pv
ds
where H denotes the Hilbert trm~sform, we ob~ai~l a "modulus-phae<~ pair" in an equivalent (ram unambiguous) w~ky, So we are able to (tefiim an i~lstantaneo~Js amplitude a~.(t) and an fltst,an~aneous f}'equency u~(t) (by reli~rence to the mono&romatic case of Eq, (tA8) and as well uniqtm) t4y
t
dargz~.(t).
(1.21.}
Tile "complexified" signal z; (t) is called an analytic si~na,L Using polar coordinates the anaiytic signal describes a turning vector wh(~e length and ~mgular velocity are time-dependent. This differe~ftiat(~s it from the mono(hromatie ca~e, which is associated with the distinctive picture of a vector that spins round the origin at a cons~,ant angular velocity and w h o ~ trNectory is a circle (eL Fig. 1.2). The a,nalytic signal ~xtmits a simple interpretation in ti~e iYequency domain. By definition, we know that = x ( u ) +- i ( - i s g , ,
x(u) = 2 u(,) x
(1.22)
where U(u) denotes the normalized Hea:viside step function. Fhis shows that, tile anMytic signal is obtai~md t~om the real signM })3"removing tile nega,tive frequencie~ from its spectrum. This does not change the inibrmation
Chapter I 77m Thne-f~equerlqy Prohlem
'29
Real Signal
Analytic Signal
'
i
] . . . . . . . . . . . I
o o
~ o
!
0.5 ' o
i
o
o o
o
o o
o
o
o
i i
oI
0 1
-0.5
°
°
°
o
o
o
ot
o
o
o
I ;o o. . . o .o I
l
gO
gg
[
1
ol
E
o
"" ~.5
ol
o o
of
o
gt
o °
o
0,5
time
oo
051 0o
0.5
[
0 (},5 real pa~l
oo
!
o
i
g
i
oo
I
o o
°
0
I \/ !
o
E
-- -0.5
, time
-|
! -~0.5
0 0.5 real pm-t
F i g u r e 1.:2, Representation of a signM by' a turning vector. Top: a pure monochromatic signM; the representation depicts a vector of const,ant modulus (associated with the amplitude) and ~onstam angular velocity (associated with the frequency), Bottom: a signal with modulated amplitude and frequency; the modulus and ~mgular x~;lo(::ity vary in time mid define the quantities of instantaneou:~ amplitude and
instantaneous ~'equency. contents of the signal~ because x(t) wa;s assumed to be real, and, therefore, X ( - u ) = X~(u). The truncation of the negative frequencies h~.s the effect of "complexifying" tile original signM, which can be interpreted as a realiocagion of the redundancy; dividing the fi'equency band of a signal by two aiIows us to sample the signal at half t~he sampling rate in the time domain. l~br a fixed period the anMytic signN requires only half the intoner of sam-. piing values as compared to the real signal; however, we have to compute two values at each sampling point, namely the real a,nd imaginar}, part of the complex signal ttence, the "dimension" of the signat, reN or complex, giobNly remNns tim same.
l ' i m ~ - lo?eq ue~lq~'/Time.~S( ale A ~la/ysis
3{}
t~'.:mark. If we consider (rely the positive fl'equem:v axis. the mea~ frequency ~ ~-lmit, s two equivalellf; det:irfiti(ms "-
,~
)
,
z, .Xl (~,) ~ &, =
.....
.3,
(t .23}
Indeed, by using the fact that Z,.(*') = 2U(,.,)X(~/), and. therefore, E~ = 2E,:, we can write
An applica~.ion of Parseva,l's identity b~.u:ls to
'i27r E~
"
"
The announced result Mlows, if we assume that iz:,:(-,:x;.)t ~ = Iz~(+~.::)I 2, and t;his is true both for signMs of finite or quasi-finite duration (whose. envelope va~fishes at infinity) a~ld for signals with a constaI~t envelope. Therdbre, we can t,hiIlk of the mean frequency ~ a weighted mean value of the iasta,ntaneous frequency, with the weight given by the squared instat~ar~eous amplitude. The ream pm:t of an alm.lytic signal
has the form
Re {zAt)}
=
a~{t)cos~:~(t)
.
It is important to keep in mind, however, that we cannot draw an.}: conclusion in the opposite direction. This means that the ana~yeic signal, which is associa:ted with tim real signal a.r (t) cos p~. (t), n(>~t not haa,e the same expressions a:,,(t) and p~.(t) as its modulus aald pha~se, respectively. Motiva~,ed by a simple physical interpretation, one is at least close to such a situation, if t,he eft)ors of the modulations t~r(~ smNl, In pa.rt,icu[ar, if a~a(t) has the form of a lowpass with its spectrum restricted to the inte,~,~l [....~B,+ B i, and if cos ~ . ( t ) h ~ the form of a ban& pass with its spectrmn in ] -- ~,-...oB'} U [+B', ~crc[, atut if B ~ > B, then ( T h e o r e m o f Bedrosia.~ i 1)
H {a~:(t)cos~a,(t)}
= a:~(t) H { c o s ~ : ~ ( t ) }
.
(1.24)
Chapt:er 1 The Time-/~)'equen(T Problem
31
These considerations still do not justiI~' the exponential expression fbr the analytic sigmfi, unless the additional requirement H {cos
= sin
is satisfied. This is not valid in general; but it is approximately verified in the quasi-monochromatic cause, where the width of tim frequency band is small as compared t:o the cetltral frequency of the spectrum. Group delay. The instantaneous frequen(y characterizes a local frequency behavior ~ a funct;ion of time. In a dual manner, o/~e can also investigate the locM time behavior a;s a flanetion of the frequency. The correspond~ ing magnitude, which measures the instant of the appearance of a certain fi'equel~cy, is called the group delay: it is defined by
_ -
I n t e r p r e t a t i v e remarks. for several remarks:
1 dar,,~Z 2~r
du
(1.2z) "
"
The loom approa~q~ tha* we just ~lopted calls
(i) In the ease of deterministic signals our intuition usuMty t~sociates the notion of a stationary signal with the idea of a steady ~state governing both amplitude and frequency. The previously introduced concepts admit a more formalized point: of view: it is appropriate to call a signal a stati(ma5y" del,erministic sigma.l, if it is the sum of several components each having a constant instantaneous amplitude a.r~d fi'equency.
(ii) Although tile instantaneous frequency and the group delay are loom quantities with respect to the time or the frequency variable, they are stiIl averages relative to the dual v~riable. In fact, ~ simple i~tterpretation can only be given fbr the case of single-component signals. This denotes all sit~ uations where at a fixed instant (or frequency) the signal exists only neat' one frequency (or one instant, respectively). In tile oN~osite cruse (multicomponent signal) the local quantifies defined in Eqs. (1.20), (1.21), and (I.25) account for the local characteristics of M1 single components taken individually, together with their interferences. For example, the simple situation of the superposition of two monochromatic wa:ves with the same (xmstant amplitude a and different fYequeneies zq and t.,2 leads to
= 2a I ( os Tr(u~ - u2 )tI exp { i27r ( u~ + u2 ) t + irr sgn(cos s( u~ - u.z)t ) } 2
Time- F?'oquel~cy/75me-b'catc A~mtj,csis
32
In this special case the physical interpretation of x(t) a.s ~he sum of two pure sinusoidM waxes disappears where there is an a~,ra.ge frequency with a modulated ampfitude. This corresponds nicely to the notion of a bea.t frequency when u~ and u~ are very close. However, it strays ii'om the m(~st natural int.erp:retat[on for a large ffequential gap, although the cha.racterization offered by the a n d y t i c signN remains formNly valid, It is clear that the same restrieth:ms apply to the group delay, mutatis mutandis, when it is ~ e d for a signal whose components lie in the same frequency band, are Iocalized in time, but have a time offs,~t.
(iii) Let l~s finally remark that ~;he notions of instantaneous amplitude and frequency, as given in Eqs, (1.20) and (1.2 t), are ceaMnly toea.l with respect to time. But they are obtained by the intermediate gJobal knowledge of the signal that enters through the Hi, bert tramsfbrm, which itself has an infinite impulse response (cf. Eq. (1.19)). An example. ~ t us consider the common idealization of a linear chirp; hereby we denote a signal with a linear frequency modulation and a Gaussian envelope. Its fr(NuentIy adopted exponential tbrm is given by t2
x(t) = ,.;-,~l :,-~J~' c,2~-~ . Note
that
(I.26)
[
2
/
f
However, this model does not define an analytic signal in the strict sense, as its l:ourier trm~sform
.}
(
X(u) = (2' - i,3)" L.'2exp ~ -r~ ~;,.55. (v .,- vo) 2
(1.27)
is nonzero for u < 0. Nevertheless, we can regard its spectrum as aNlost equivalent to the spectrum of an ap.@tic signat, if the contributkm at the zero tYequemey (and a fortiori at N1 negative frequencies) is negligible (eft Fig. 1.3). If we measure the width of a Gatkssian at a value of exp(-rc/4) = 0.456 of its peak (which Mmost coincides with the hMhwidth), we find that the duration T and the bandwidth B of the signal x(t) are given }:V 7'=
i v-~;
B=
,/( ?
""
i+~7~]
=+
BT=
i1 + 7 2" ,
according to Eqs. (1.26) and (I.27). The r<~ult of Eq. (I.27) implies that the condition of quasi-analytici V CaM be expressed as the narrowband condition B --
vo
1 <<
-
2
.
(1.28)
C h a p t 31" 1 T h e Time~lT}'equency P r o b l e m
33
Spectrum
Signal
L 4
4
0,5
F
[......................
, ...............
.-.0,5
.a i _
_2
o
0.5
0 frequency
0.5
II 2, J
0,5I
0 qk5
time F i g u r e 1.3.
Linear chirps and quasi-ana]yticity.
The signals in the time domain (obtained as real parts of the exponential model i,l Eq. (I.26)) are shown on the left, and their spectra are shown on the right. For a given bandwidt, h, o~le verifies thai, a higher central frequency ( i.e, a smaller relative bandwidth) leads to a better satisfaction of the condition of qua~si-anMytieity (negligible contribution of negative freqummies). Under this hypothesis the instanta;neous frequency arid the group delay turn out to be u,:(t) = uo + ;3t , (i.29)
t..(,.)
= ~.2 + ;32 (~" - . 0 ) •
(1.a0)
This result enlightens the fax't that the instantaneous frequency and the group delay define two curves in the tim~frequency plane, which are usually distinct. One could be willing to accept, intuitively, t h a t these two quantities are il~verses of each other for frequency-modulated signNs, a~s they provide two different readings of the same physical reMity (fi'equency a %nction of time and time as a Nnction of frequency). However, this
34
l'i~.:,.b~-eque~cy/Time-Scale
Ana/ysis
oniy }>ecomes true if ~ui asymptotic co~tdition of ~he type "large durationbandwidth product" holds: indeed, we obt;Nn ~ha~,
t~,(~.,;,.(t))=
.,~_t /32 t
.... ]
(BT) 2 ~,
which yields
Remark. One can show that this a~sympl,otie condition is stronger t.han the simple assumption of fast ~cilla~ions compared with the evolution of tim envelope. This latter situation can persist, ior example, ibr a lmrrowband signal withou! any frequency modulation, in our previous ~ota~ions this corresponds to 3 = 0 and BT = 1. In this extreum c~se the instantaneous frequency is given by 14~.{t) = v~ fbr N1 t, while the group delay is t;,.(v) = 0 for ali ~/. The corresponding curves in the time-frequency pla~m are certMnly distinct. The asymptotic reciwoeity of the instantaneous frequency and the group dela5~can be a~alyzed by means of certain arguments that alg}eal to the method of stathma~N phase..v~ This cm~ be carried out for sit,Ms with a mo~otone frequency modulation, a~ we wilt see in wha,t fl~i~c;ws~ Note that we can actually think of the F~mrier transform of an anMytic signal
y~
o~s an oscilla~.o~3, integral
Let us further assume that the variat.io~s of the phase are f~st in comparison with the variation of ~he envelope, which meails that
;da': I //;t:
Then the essential contribution to the bburier integral is suppIied by the l~eighborhood of the point t~ (cMl~t stationary point) for which the deriv~tire of the pha.se vanishes, that is, wtmre we have
Let us highlight this claim by giving the following qualitative interpretation: The im;egral over ~;he fa.,t oscilIatio~s yields a negligible comribution to t~he total integral, because the posit.ire and negative a r c h ~ of the oscillatory function are sufficiently symmetric, ag least tocMly, in order to ca.heel each other. The only spot with a retewmt col?tribution occurs where the ~ l o c i t y of the oscillations Mows down sufficiently in order to make these
C/2a:pter
I
"l h(~
Tim~.F}'equencv
35
l-'robh:,m
(a) 05
(b)
....................................................................~ .............
0.5
r~
i
(41
J . ....- [
I
i
o.I ~ "
.//
0 ............................................
[
4
0.1
J
o
time
I"
I
.......... ..z
.. j ' ' j
7
time
{c)
(d)
time
time
F i g u r e 1.4.
I;)mrier transform and stationary phase.
(a) aad (b) are symbolic representations of the analysis of a signM with linear frequency modulation {solid lines) by two different reference frequencies (dashed Iines). (c) and (d) are real parts of the corresponding l~)urier integrals, tt cm~ dearly be seen that an essential contribution to the integral is only provided by the neighborhoods of the point (called stationary) whose trine abscissa corresponds to the intersection of the fi'equencies.
positive and nega,tive arches asymmetrical, In other words, the contribution to a fixed frequency of a nmdulated signal is a low-fi'equeImy signature of the hita'Awence between the signal and the reference frequency. This intertbrence is only effective in the neighborhood of the point where the instanta,neous frequency and the Fourier frequency "meet" (cf. Fig, 1,4). Subject to the actaM conditiol?& the approximation of Z,~,(u) by the method of stationary phase can be written as d~J~'
"I'Ij2
, s
36
f'imc'-tq'eque~O'/Thn~,-&:~de~ Azml3.'sis
where
This impti~ that the group delay has the approximal;e vahle (again confined to the validity of this method)
This clearly shows t.hat, the group delay is the inverse flmction of the inst.antanc~:ms frequency. Not,e that ~,he a.ppr(.~ximation by the method of st.ationary phase only makes seI~se, if the derivative of the instantaneous frequency does not vaxfish at the stationary point, thus exetudil~g NI eases wtmre constant ins~,antatleo~ls frequencies occur.
1.2.2. Nonstationary Signals As in t,he c~;~e of deterministic sig~als, the most naturat local quant.i~ies: which intrude into the description of nons~at.ionary far, dora !4 signals, are related to the time. The %gationary" property, in its usual definition, means the ~'fiMependence of statistical properties relative to an absolut.e time." The incorporation/ of a possibly nonst.ationary behavior forces ~_us to reintroduce the time as a parm~mt, er needed for the description. We therefore wish to cow,sider properties of a signal, wMch evew~ually change: f r o m o n e moment: t;o the next and thus become time-varying. Definitiom Viewed from a physical point of view, the most important properties of a stationtu'y sig~M are those of second order, This nmaI~s that only the statisticai properties of degrees one and twe, m'~st be inva.riant under a translation in time. (With the exception of Gaussian distributions, {br which the stagistieaI properties of the first two degrees are ki~own to determine those of a~V degree, a random signal which is stationary in this broader sense, need not be stationary in the strict sense.) By definit&m, a random signal is cMled stationao" in the wide sense or weak[)" stnt:io~laD', if (i) its expectation vahm is indepe~dent of the time; that is, i~: has the tbrm E {:~:(t)).~, .... (1.as) where p:~ is a (xmstam (which we assume t;o be zero without k~s of t e n e t ality}, and
Chapr¢:~r I Iih¢~ Tim¢.4%'~'quoncy Problem
37
(ii) i~s autocovariauce f\mction~ which is the same as its m~toeorreta~io~ f}li~(t:Jon ii~ (:~4e of a zer(>mean signa.l ( i.e., its expectation value/,:;., vanishes), depends only mi the diflbrence of tim two (onsidered instants; it dins has d'm form e {:,:(t) s i s ) } .~ :,.re - , s). {Laa} In the weakly stationary (and zero-mean.) cause, the w~.rianee must be constant ms well, as we have vat {a:(t)} = E {l*(t): e'} = ~'~ (0) . If the varia:me is finite, the signal is called of serond ordeL (One should note that the usual idealization of white noise in continuous time does not fall in this category, as it verifies
E {:,:(t) a" (s)} = ~:,oa(t
s)
(1.,34)
with a constant %, a,nd consequently E {!;r(t)l:' } is not defined.} It will be useful t.o include a brief sketch of some important properti(~ of the autocorrehtion function or the auto(o,carianee function of a stationary signa,1. First of all, iv possesses a Hermitian symme~w
->, (-- r}
:
<(r),
(1.as)
,tnd its maxima~ modulus is attained at the origin h,.(r)t >~ :,,(0) .
(1.a~)
Moreover, it. is posi*:i,<~ defini*.< this means that for all. pairs of instants t,, and t,,~ and all collections of complex nunff)ers k,~ tile inequalily
E
E
>(e,,-
>
o
holds. This implies that t.he llburier transform of the mJtocorrelation function of a stadormry signal {s nonmcgative (Th(~rem of Wiene>Khinchin),
F,:(,,) z
,.+:xe
~ >.(r,e~
-
2rr*,,r dr > O.
(1.37)
The funcd,_m Fy (~) is calMt the power spectrum d{msiq>"due Vo the i&ntity
]/'~q"7,..
s {f:~:(,Ot ~ } =
r, (~,) d,,.
(1.as)
TilLs e~ll iU~"is ir~dependem: of the time, a,m:l this suggests that a stat}onary random sig~al has some frequency com,en~s that are (xmstm~ in ~im( (a,~smuing we know bow to make sense of this statement). The characterization of a sigt~aI by ~he mere knowledgt" of its power spectrum relie,~ on a Spectra] Decomp(~sitio~ Theorem. It tells that every signN x(t), which is statio~lary in the wide sense, admits a harmo~fie decomposition (named after Cram6r) in the form =
f
'i- : ~
c ........ dA Iv') ,
(1,39)
The [brement.ion~l integral is of Fourier-Stiettjes type, and the equality holds iri the sense of a quMratic meam The main interest in this de(oreposition stems from its w o p e r t y of double orthogol~ality. T N s means that: (i) the complex exponentiats, serving as (deterministic) flmctions tbr the decomposit, ion, are ord~ogonal with re~pect, to ~he usuM inner product,
{~:
) dt :::: .(.~ -.,,,,u) ;
(1.40)
(i 0 the spectral in(rements, being tantamount to the statistical weights a~sociated wit& these flmctio~s, are orthogona| with respect to i:he inner product defined 1:~;"the expectation value over t:he trial spa,(e,
In other words, the sta~,io~ary signals admit a fl'equential decomposition into uneorre/ated random variables. The stochastic independence between spectral increments in the sta~.ionary case means that di@:)ilg frequermy bm~ds share no energy~ This can be regarded a~ a consequence of the ideal frequency tocalizatiol~ of the Fourier decomposi~io~a itl conjunction. with the permanence of the frequency (on~ents of a stationary signal. Generalizations. The cI~s of stationary signaIs is too restrictive for giving at(omit of most of the ordinarily ob,~erved reai situations. One possible generalizat.ion consists in departing from ~.he charact,erizatior~ of the sta.tiolm.Ev c ~ e in terms of the doubly orthogonal decomposition and relaxing at least oI~e of these orthogonalities. Let us firs~ choose to retabl the (:omplex exponentiaIs as the decomposing flmcti(ms. Then v~ obtain a lmw representation (due to Lobve), which is qui~e the same as in the stationary ease, except that the spectral increments are no IolNer uncorrelated. Rather, they Nlfill the reiadon
E { d X ( u } d X " ( ~ ) } = q~,-(u, ~) d~ dt/
(1,42)
(3~q;t~r 1 The Time-Frequenqv Problem
39
where the spectral distribution fimction ~[L,(~,~) can ha~'e a support of positive measure; hence, it may exist on a larger set than just the mai~l diagonal of the frequency-frequency plane. For the existence of such a decomposition we must meet th.e requirement (Loire's condition)
The corresponding signals axe called harmonizable, Their (nonstationary) autocovariance flmction is dual to the spectral distribution function by means of the Fourier-type relation
[12
This equation generalizes tile Wiener-Khinehin relation of Eq, (1.37) between the autocorrelatkm function and the power spectrum by Mlowing nonstationary signals. It: naturally reduces to the earlier equation in the borderline case of a stationary sigmfl, for which 'l~r(z~,~) = ~(~, - ,~)F:~.(~). Another eonceivable possibility of generalizing the stationary case is by retaining t.he double orthogonality, but replacing the complex exponent.iNs with other functions fbr the decomposition. This leads to the construction of representations of the form :r(t) :=
/~ ~2;~Z,(t
,,) dx(,,)
with a comparable orthogonality relation
x Such decompositions (m~ned after Karhunen) are possible, indeed. They reduce essentially to taking the eigenflmctions of the mttocovariance kernet as functions tbr tile decomposition° From this point of view, the re~ quirements d the stationary case endow tile autocovarimme kernel with a convolutive structure, making it act like a linear filter. Because tile eigenfunctions of tile linear filter are the complex exponentiNs, the stz~ionary case can again be revealed as a borderline ca:~e in this hu'ger class, While one can certainly find generNizations of the stationary case, it also becomes clear that they bring about certain difficulties regarding their interpretation. In the first case (harmonizable signMs) we formally Mhered
Chapter 4 2~me@~'equeney as a t~radigm 4.3.2. M a x i m u m
;/45
L i k e l i h o o d E s t i m a t o r s for G a u s s i a n P r o c e s s e s
Turning t,o the original formulation (Eq. (4.62)) of the problem~ we no~ supw~se that x(t) is a Gaussian random process, so that E {x(t)} = ,(~),
~.~,(t,,~) = E {~.,.:(t) ~-. - ,U:)]
[~.(~) - #(s)] * } .
(4.66)
It ix known that the detection problem under
whidl is
More precisely, we recall t.ha~ x(t) admits a decom-
doubi~7orthogond; ttm.t is, tile relations # , , j ~ } = ,~,~ &,,,,
E {Ix,, ..- , - d [z,~
/ are verified, where A, and p~.~(t) are the eigenvalu(;-s and eigenfunctions, respectively, of the autocovariance of z(t). ttence, they at*e defin~M by the int.egral equation
f ,.:,,(t,.~)~,,,(s)ds = A,~.~(t),
~:c (r),
(7} The coe~cients x,,, y,,, and #, of the decompositions d x(t), y(t), and #(Q, respectively, axe the projections o~tto this basis of eigenflmetions ( ~ xn = (x, ~ ) , e~c.). 2Ne optimal det(x~.tor (in the s~use d a maz~:imum likelihood estimator) is obtNned t~. a comparison of the decision statistics A(y) = A~,(y) + Ad(y) to a thr~hoht; in this definition we put --
~ ¢1
.
.
.
.
@(Y) = 7o ,=0 A,~ q % (14,67) J
X
T~==0
•
Chapter 1 The Time-fYequen(w Problem
41
which is true for all signals y(t). In particular, if we let .z/(t) be a complex exponential with frequency ,J and insert the actual form of the autocovariance functiol~, then we obtain
j
'~~ .,,,.(t) de. G ( . )
> 0.
9*.
Here Fa(u) denoCes the Fourier tratmform of % ( r ) . The integral is nonnegative owing to m , ( t ) > 0. VvZ,can therei~re conclude that Fa:(u) > 0. lfence, the positive definiteness of % (r) has been proved. Remark 2. In order that; a locally stationary signal be hazmonizable, its local autocorreIation iunetion %@) must be ~sociated with a stationary signal of second order. Moreover, the Fourier transtbrm Ma.(u) of its modulation function m~(t) must be absolutely inte~able. Indeed, as an immediate consequence of the definition of locally stationary signals, we find that the spectra] distribution flanction has the form "
<
2
/,"
We can themibre write Lo6ve's condition as
/;r
Ie,;(~,.~)l<S'/a,, =%.(0)
,,7
IM,...(,,)I <S~,< +<~,
ex:
which is the announced resutt. The notion of totally stationaw signals, as already introduced here, cart be expressed by a modulation in time of a stationary autocorrelation function. Another conceivable w w is to impose a modulation directly on a stationary signal. Then the resulting signal, called unh%rmly modulated, b~s the fbrm ~,,,(t) = c(t) :~(t) , ( ~.45) Here the deterministic function c(t) expresses the effects of the modulation~ and :g(t) is assumed to be weakly stationary. We can immediately conclude that its "loeM" autocovariance/hnction equals T
T
T
T
If the modulation is "slow," dmt is, if the varia~tion of c(t) is small in compa,rison with the statistical memory of re(t) (which can be quaaltified by its radius of correlation %, or the half-width of %(r)), then we can write
Time-Frequen%'/Time.. SCate A~mt3~sis
42
Consequently, we obtain as a first appI~:,ximai;iou
(
c ~+
c t ~
~'~(-'~U)
whenever the values of the local (orrelation are (essentially) nonzero. We can thus see, given this level of exactness, thai; the uniformly modulated signals are ctmstructed along the same lines as the locally s~ationary sig-. nals. The two definitions actually coincide with each other in the extl~me c a ~ of a white nd>~ signal (¢hus havii g a microscopic ton'elation) whose amplitude is modulated. ThN situation of a. slow evolution of a stationary antoeorrelation timetion (stc,w compared with its radius) is usualiy called quasi-stati(Tma(~,. §1.3. T o w a r d s
Time-Frequency:
Several Approaches
The f?~)urier duality mak(~ both descrip[ions of a signal, in time and frequency0 n(~essary and insu~cient at the same ~ime. Evml though they c a r w the same hfformatiom they are both necessary, because t.hey represent, it in two complementary ways. Even though they carw all information, t h w are both insu~cient because they present it in a fbrm that is often too far from the physkal reality, so they cannot be exploited couvenim~:ly. However, we have seen before that there exist tools for breaking out of the rigid Kamework of a Fourier analysis in the strict sense: yet most often t,hey gs) only halfway We axe thus encouraged to perform a more important st.ep, namely the search for genuinely mixed descriptions together in time and frequent> InsofiJ.r ,m these, descriptions should be developed from the signal, they can certainly not supply aa!)" gain in infbrmation by going beyond the time axis into the time-frequency plane. The truly antmipated g~dn consists of a better inteBigibility. This means ~hat the change in the r q > resentationaI space correspm~ds to a better strueturflg of the informatiom at the eventual expense of at~ incre~sed redundancy. As there are mmiy nonstaVionary si~;uations and some invk)lable the~ oretieal bars, the issue of describing a signM simultaneously in time and frequency does no~ permit a unique and unanimously satisfactory answer. It suNces to glance througl~ ~,he literature on this subje~'t, in order to be (onvinced that it offers a bestiary of methods, which is respect, able by its number and its variety, combining domestic and wildlife animals, bea;sts of burden and racing horses, mficorns and raccoons. Before entering a de~ tNled discussion of the possible solutions, it might be useful to dose this chapter with an inventory of some guiding principles ti~at preside over both the choice a.nd the elaN~ration of a time-frequency representation. This Nso Mlows us to ~ some further notations.
(,?hat)ter 1 The I~me-l?rcquex~q}, Problem
43
1.3.1. Tile Time-Frequency Plane and Its Three Readings The wealth of the time-frequency plane, which serves a~ the spa(e tbr the transfbrmed representations, roots in the I)ossibility of different complementary red,dings. Has'ing two variables for t.he description at, our disposM, we carl consider them together with their cross-relations. More generMly, we can even envisage some gtobM quantities that are inaccessible by approa~:hes with respect to only one of the variables.
Frequency (time).
The first interpretation of the plane is obtained tV regarding the frequency ;~s a flmci;ion of time. This is connected with the idea of an evolutionary .~pectral analysis. As we emphasized before, the tool of the instantaneous fl'equency h~s some natura! limitations concerning its i~tterpretation when the analyzed signals axe of multicomponent type: By letting the signal %urst" into the time-frequency plane, one can a priori overcome this difficulty, because at any moment there is a whole range of frequency values available (and not just an as;erage), T i m e (frequency). A second interpretation, which is dual to the first, considers the time as a flmction of the frequency. This corresponds to the idea of a sequential monitoring of the output of different frequency chanimts, Hence a complete lfistory is offered for each fr~queney, and this gives us access to events located in time in a frequency-l)y-frequency manner. These first two re~ulings have one point in common, a,~ they prepare the ground for some naturM approaches to applications such as matched filtering or the separation of overlapping sigl~als. As an example let us consider two signals thai; overlap in time and in frequency, such as parallel chirps, but which have disjoint descriptions in the time-frequency plane. In principle, tim two dimensions of the time-frequency plane Mlow us to draw a separating curve between the two components, while this cannot be achieved in a purely temporal or frequentiM setting. Time-frequency. The timeqYequency plane admits a third interpretatMn that is more general and global h:~ the sense that it deals with truly joint objects in time and frequency. The time-.frequency duality, which certainly underlies every retevant description of a nonstationary signal, is thus brought go the surface .......not as a concept of "twice in one dimension," but rather "once in two dimensio~ls." This is clearly the most ir~structive point of view, but it also requires an intrinsically joint, imagination.
1.3.2. Decompositions, Distributions, Models Whichever rez~ti~lg of the plane is chosen, we can conceive several vv~ays of t~sociating a time-frequency flmction with a given signal. Diffbrent approa~:hes can be distinguished concerning tile nature of this association.
44
l'ime Jqequenc37" l'ime-.Scate Ar~at3sis
On th{ one hand, they relate to the physical interpretation of lhe representative time-frequency features in a signM, and, on the other hand, ~o the degI"ee of one's a priori knowledge about the signM. Decompositions. A s a signal {annot be arbitrarily cone~ntrated in time and frequency, it is tempting to regard the most concentrated signals (to be m e & more precise later on) as |.he elementary parts of every signal, the %u[lding blockg' of an arbitrary w>eform. The simplest, aaM most natural rule for the construction process is the linear superposition. Here the elemem.ary signals, also called %line-frequency a|oms," play the role of a b~sis of the decompositkm. Within the picture of a linear decomposition, the time-frequency representation is given by a {discrete or continuous} set of weights, each being associated with one atom. Moreover, th<~e weights should be accessible by a projection of the signal onvo the elements of lifts %asis/' Distributions. The preceding wW of decomposing a signal gains some information about its energy allocation il, time a,nd frequency as a byproduct. We can also pay particular attention to such e~ergetic chara,'terfzations from an angle of energy (or power) distributions directl> Dora this viewpoint i~ is natural to consMer a quadratic rule, which associates the signM with its (Nlinear or sesquilinear) representat, iom and thus generalizes t,he notions of correlation or p(gver spectrum known from ~he stat.iona.ry case. Let us remark that, it is not tmcessary to apply a quadratic rule for the a.~sociation in order to obtain an energy disgfibution. Other approaches of ~;higher" order are also e(mceivable, though leading in generM to more complications. Models. FinaIly, if the sVrueture of the analyzed s[Nmls is available a priori, it is interest, lug to incorporate this M~owledge into the modeling by means of, for exampte, a parameterization. In a nonst, ationary context we call the model timo-dservation. Understood ~s a savings, in this sense, the Nm of the modeling is somehow opposite to the aim of a decomposition or a distribution: While the latter are mainly directed at a bettor represengation of the data, accepting enlargement in order ~o gain better a~JalysN, the modeling pays attention to a be~t p~>ssible reduction of the redundancy ( e.g,, for the p u r p o ~ of coding)- This shows once more tha~: the modeling and the (nm~parametric) representation without an a priori knowledge are complementary issues: the assumed neutrMity of the t~presenta~ion can az~sist the process of finding a m o d e l which in return wilI furnish an even more precise representation, becaus, e it will be well Mapt.ed.
Che~pter I The Thm>l~}'equencv Probl~,m
45
1,3.3. M o v i n g and Joint, A d a p t i v e and E v o l u t i o n a r y M e t h o d s In order to ta.ke the no~lstationary nature of the analyzed signals into account, one proceeds by reintroducing the time as a necessary parameter tbr the description. This can be done with or without reference to some stationary methods, thus giving rise to amot,her cIassification of the possible approaches.
Moving and joint. The first approach, and no doubt the most natural and practical one, consises in using a local time window, which t~bllows the monmnt of the analysis mM has a limited horizon of observations ("short~ time") centered a~'ound this moment: methods of this type are cMled movlug methods. In the que~sPstationary case (or one that is supposed to be) the moving methods are most often some locMly applied stationary meth~ ods. This evidently leads to the imposit.ion of restraints, which may concer:a, for instance, the kind of "short-time" horizon or t.he validation of the ;~sumptions of the quasi-statkmaD ~ behavior, a priori a.nd a~ po.steriorL VVe will see, however, that the moving methods can also include some "nonsta. tionary" approaches~ thus relaxing the concept of quasi-stationarity, which is most naturally associated with a given window of short duration. In c o n t r ~ t to this situation, and renclering a more general setting fea~ sine, the "nonstationary" methods for a time~-lYequency anMysis consider a nonstationary situation as it is, thus avoiding any a priori reference to the stationary or quasi-stationary ceuse. They can be called joint methods, as they treat the time and the frequency symmetrically, although they are explicitly based on a time-depeIMence.
Adaptive and evolutionary, 1~ In the setting of a p~rametric modeling we can adopt two viewpoints ibr the introduction of a time-dependence, The first, wMch can be called adaptive, consists of using a stationary model (with constant coe~cients) and adj,~ust.ing its estimation at. each instaat. Such an adaptive a~pproaeh contains the time in the algorithm for the iden~ tification. The second point of view, which is called evolutionary, uses an explicitly nonstationmw model, which means that its coefficients depend on the time. Hence, an evolutionary method comNns the time in the modeling (no matter if the algorithm for the identification itself is Maptive or not). Certainly the approaches thai have b¢en introduced herein are not mutuMly exclusive. One can imagine moving decompositions or joint dis~ tributions as well as joint decompositions or moving models. In particular, the large variety of approaches reflects the protean nature of the timefrequency problem. One cm~ expect that this large variety is a.ccompanied by a not lesser multitude of solutions: this is true, indeed, and is the subject of the next chapter.
46
J i m e of)'( ~q~, e n c3,/~1 ' i m o- S(:a le I n a ly:sis
Chapter 1 Notes
t.1.1. The lAmrier a.nalysis gaw~ rise to considerable literature. One might took at BracewelI (1978) fbr a ;signal" point, of view. and G,~squet and Witomski (1990), Dym and McKean (t972), and K6r~mr (1988) for a lnoi,~ mathematical poin~ of view. 1.1,2.
The Heiserlberg-Gabor uncertainty principle probably appeared first in work by ~Veyt (1928) (in its mathematical form, which is cotton,,ted with tim Fourier duality, rather than in it.s qualitative physical interpretation)~ It was proved in work by boi,h Gabor (1946} and Brillouin (i959) where further aspects were discua~ed, especially those concerning its links with quantum mechanics. Numerous generMizations of the inequality have been proposed, such a~ the extension to discrew sequelmes (Pearl, 1973) or to measures of concentration associated with L'-norms (CowlhG and Price, 1984), Surveys on this subject are offered by Ber~edetto (t990) and l%lland and Sitaram (t997). :~The ea:~ of real signals is especially treated in work by Kay and Sitverman (t957) and Bor(hi and Pelosi (1980). 4 As far ~s quantum mechanics is concerned, a profitaMe source is L@yLeblond (1973) where a critical discussion of the notion of "uncertainty;' is given. 5 A classical reference for a detailed discussion of quan~mm mc,::hanies and its operational formalism is Cohen-'I3nnoudji, Diu, and LMo6 (t973). ~' A presentation of the operational formalism of signal theory can be found in work by Bonne~ (1968). The use of this formalism in the time-freq~mncy context is contabled in Flandrin (1982) "~2~ will re~urn to this issue in Subsection 3. 1.2. 1.1.3.
7 The samplh~g theorem is proved ir~ almost every book o~ signal proc~s~ i~g~ One can find it, for example, in Duvaut (1991), Section 2.3;I. The flmdamental articles dealing wi~h t,he simultarmo~zs energy concentration in time and t}equency are by Slepian and Pollak (I96I) and Landau and PoHak (1961). Another very interesting presentation (which in fact b> spired us) was done by Dym and McKean (t972) and Papoulis (1977).
Chap~.er I The Fime-bi'~,que~(?" t:'roblem
47
Among ~he precursors of ~he formalized approach I)3' Stepian, Poilak, and Landau, one shouM mention Vitle arid Bouzitat (eft their works fl'om 1955 and 1957) and Fuchs (195.:1). We also recommend the reference by l)onoho and Stark (t989), which generalizes ~he notion of simultaneous concentration to other domains tltat~ b~tervals (and exploits it for pim~blems of signal reconstruet.iot?). ~' See, for example, Riesz and Sz.-Nagy (1955). 1.2.1.
10 The firs~ atte:mpts {,o give a defiifition of an inst:antaneous frequer~ey go back to Carson and Fry (1937} as well as Va,~ der PoI (1946). Nowadays the classical defiifition is the one by Vitle (1948). It relies on the concept of anNytic signals wh(~e s~eds can be Ibund il~ Gabor (1946). One tail also eonsuk Bo~sha~sh (1992a,b) fbr a more complete tret~tment of this issue. ~ A detNled s~:udy of the conditions of the ~ma~yt,icity of a (:omptex signal aI~(t the problems in connection with the Th~:~)rem of Bedrosian (1963) is given [~y Pici~fl)ono arid Martin (1983). It also contains a bibliography on this issue. A more recent paper is by Pieinbono (t997). ~ The example of the linear chirp with a Gaussia:n envelope is broadly discussed in Kodera, Gendrin, and de ViIledary (1978) and Gendrin and Robert; (I982). ~:~ An introduction to the method of stationary phase can be found in Papoulis (1977), For a more mathematical pre~,~ntatkm one should h)ok at the book by Copson (1967). 1.2.2.
14 The general notion.s re,iated to random signals, whether stationary or not, m'e discussed in a large number d books. For example~ one can find them in Blanc-..Lapierre and PicinboI~o (1981) or Priestley (I98t; 1988). More pi~cisely, ~he concept of harmo~fizable signMs is due to Lo~ve (1962), the Ioca~ilystati(mary signMs were introduced by Silverman (t957}, al~d the uniformly modulated signals by Priestley (1965). 5 This remark is due to Jai~sse~ (private communication). 1,3.3.
l~i Tile di,stinction "adaptive / evolutionary" is borrowed from Grenier (1984), to whom we refer for a deeper s{udy of the parametric timefl'equency aspect, ~s well ~s to his work in 1987. This issue is not disc-~zssed in this book.
Chapter 2 " S Classes of Solutmn
We h~rte emphasized in the first chapter a number of problems that arise in coiounction with the search for a tim(,>frequency description of signals. This second dmpter is devot~ut to tile discussion of some clas;~es of possible solutions and an inventory of their properties. In a certain sense the following t)resemation can be regarded ~ a "nmitiresohltion" a~tvance, At first, we &aw the maiT~ features of a general pa,norama from an historical perspective (Section 2.1). TheI~ we introduce more a~d more details of tile principM approaches. These ca~l be grouped into three large sets according to (linear) 'atomic" decompositions (Section 2.2), (bilinear) energD" distributions (Section 2.3) and p~ver dis~ribu,~ion t;anctions (Section 2.4). tn each case we make every effort ~o begin with general principles for co~> structiag clmsse~ of such solutions. Only afterwards do we specify mm:e details concen~fig the form or the properties resulting fl'om some ~u.lditional constraints. More precisely, Secl,ion 2~2 deals with the question of decomposing a sigt~at with respect to a, family of elemental" signals, which are well localiz~d both in time and frequency, l'his point: of view (ca,lied 'atomic") is di~-ussed in a, general fbrm in Subsection 2.2. i, Then we present two particular app~xmches, n a m e d the slmrt-time Fburier transf?~rm or Gabor decomo position (in Subsection 2.2.2) a,nd the (time-seMe) theory of wavelets (in Subsection 2,2.3). An interpretation as a "ma.tched filtering" is discussed in Subsection 2.2A. By giving up the lir~earity of the representation~ we concentrate on the construction of ciax~ses of })ilinear solutitms in Sect.iox~ 2,3. Here our approaz'h relies on elementary principles of covariance. In particular, we show that simple const,raints relative ~;o translations and/or dilations yield an infinite number of (parameterized) repre~ntations in the time-t~equency and tile thne-sca.te plane. These repre~ntations are divided into two large (:lasses, which are cMled Cohen's cl~.s and the a n n e cle~s.s, respectively. 49
50
"l lirn~"-G~('qt,e*~q~;/Time-Sc,~Ie
A ~i~@',sis
St;aztiiG from this genera] fl'amework, we systematically explore ill Subsection 2.3.2 tile properties of the resul~ing distributions (a,mong which the spectre,gram, the sealogram, the Wigner-Vitle; Bertra~J.d and Cho/-Williams distributions come fir.st). Fhese propervies can be expr(~ssed explicitly in terms of conditions on the parameter hmction, whic~ defines the represema~iom This p~-ves tile way for tile design of particular representat, ions sagisfying a catNogue of specifications, Bu~ it Mso makes ~;he exclusive character of certain desirable properties evident: th<~e issues, which al~ discussed in Subs~u:tion 2.3.3, justify the central role of the Wig~er-Viltc distribution in the thne..,frequency theory of signals. Finally, we change our viewpoint and turn to a stochastic set, ring in S~ction 2,4, Here we iCemize the principal paths that cml be followed in order ~o gNn a time-frequency descriptior~ of nonstationary signals a n d / o r their power distribution. Anticipating some profit from our ir~vestigN~ions of the deterministic case, we retail il~ Subsection 2.4.1 the two main strategies (orthogonaliV or fi'equency in the Cram6r decmnposition), which are induced by t,he presence of nonstationary fea,tures in ~/~e signal. This leads to two different N:miJies of approa(:hes. One of them gives priority to the double orthogmmliey (evoluUo~mry spectrmn "ii la P r i ~ t l e y " in Sui×se(> tion 2.42), and the other o~e supports t.ile physical interpretation of the spectrum (Subsection 2A.3). This latter approach em.phg~sizes once more, but in a different light, the role of the Wigner-Vi!le transff~rm, whose special properties in the stocha~stic case wilt be explained. Finall3q Sul)section 2.4.4 describes some links and some intersections of the two different approaches. {}2.1. A n I n t r o d u c t i o n w i t h H i s t o r i c a l L a n d m a r k s Before we begin a more systema.tie construction of the large classes of solutions for the timeqi'equexmy descrip¢io~l of sigIxMs availN)le today, it might be u,~efSHfirst to explore lhe pioneer work in this field and to consider it in its hkstorical context. Since hist~ory (and especiMly the development of ideas) is only linear in a rather superficial wai~, one should not be astonished if sonm d the Mwmces cited in ti~is short, overview look chaotic from time to time, 2.1.1. S h o r t - T i m e Fourier a n d I n s t a n t a n e o u s S p e c t r u m ~ It was recognized a long time ago that stationary frequency analysis is of~,en insufficient,; one cad even sa:y ti~at ~,his is a recurren~ theme in the scientific literature, both in theoretical and ~::Lppliedwork. According to Pimonow, it was Sommerfeld in his doctoral thesis in 1890, wire used the "il~stantaneous spe(:mC for the first time. There one can also find tile premises of a great deal of work that today seem ~ natural that no speciM patronymic is
ChaI)ter 2 Ch~sse:~o["So]utions
51
gi~x:,n to the (orresponding methods, Their conm~on denominator is the replacement of the global Fourier ana.lysis, which loses all chronological information, with a sequence of local ar~alyses with r~specl~ to a moving observation window, However, the actuM boom of t,his type of approach had to wait until the middle of ~.;he I940s and the invention of the sonag;rat)h. This marked a considerable breakthrough, as it created tile concept of an "instantaneous spectrum," which ,,~s still questionable theoretica.lly, in an oper~tional and evideni 12)rm. It also left a deep impression on the following generations and their wa,y of apprehending the time-h'equeney problems. (This mark is still very strong today, when tLsers fighting tbr a "good" time-frequency representation want most of N1 a 'super-sonagram.") S o n a g r a m and s p e c t r o g r a m . By construction tile sonagraph operat;ed in the ~equency domain. By bla&ening a sensitive paper in the course of time, it, recorded the output pc~ver of a bank of bandpass filters working in parallel. The filter bank was virtual, insofar aa the signN under consideration w~s recorded on a rotating magnetic drmm only one filter was needed for scanning the useflfl frequency range due to the synchronous heterodyning of the rotation. The corresponding output couM be writt, en, frequency by t?'equency, on the sensitive paper st:uck to t~he drum. Such an operation can be modeled ~s dTt, 2
+x
(2.1)
where X(t/) denotes the Fourier trans%rm of the analyzed signal z(t) and H(~,) is the frequency response of the analyzing filter with no heterodyning. Certainly such a~ analysis lends kself to severn v~riants. For example, the way of seamfing the fl'equei~cy ra,nge can be varied The forementioned scheme is classically understood as a sweep t\v shifting a lowpass filter. l~)rmally, it corresponds to a uniform fitt,er bm/k. However, one can also imagine more generM situations, where tile impulse response of each filter may depend on tlm anMyzed frequency. Ch'~ssical examples are the constant-Q filters (the quality factor Q of a fiker being defined as the in,,x~rse relative bandwidth, i.e., the ratio of its central frequency a~d its p~ssband). They can be described by
* .....
~
S
(~(I/Q/. / ) ) c
d~,
~
where H(t.~) is a bandp~ss filter with (entrat frequency z.,~. (Although this is an old principle, it is rea~dily connected with the concept of u:a.veIets, as we will see later.)
'l imo-l~requeucy/Time-Scalt:: , Analysis
52
Let tLs mentkm that some intermediate situagions were also proposed in addition to ~bese two concepts. One of them is ~he Frequency Time ANalyzer method (FTAN) tV l~evshh~ et al, whkh is defined by , i-
,5":,.(t,~.,) .~
(~,~/q,)2 /
i2
X(rQh,~,((r~_....~/,~., • , ) t ~ , O / t , .+.,,, ) J ~: i2,,+,dr ~
bk~r each of dlese cas~:s there exists a dual irtterpre~ation in the time domain, which il~troduces the impulse response of the fil~ers. It is this inlerpretation that imbues the not.ion of local analysis (in time) with a proper meaning, The k~ealiv is linked to a temporal horizon, fixed by the duration of the hnputse r~ponse of the filter. In this regard we can write ~he sonagram equation (2.1} in an equivalem form as
s~..(t. ~4 = , '
:~,(s) h"(, - ~) ~.-~ ......
(2.2)
This exhibits its compm, ational structure as a "short-time observation of a signal through a window + local frequency a,atysis," and it is ehe reas(m tbr eMling it a spec.tr(gr~mL Note that improvements to glm t.ecimicat environment, have led to a preli~renee of t be ~;emporal point of view in most modern introductions of this method (after the devek)pme**t of calculators and Mgorithms for the f;~st ~burier transfbrm took place in the middle of ~he I960s). Restrictions. The time-frequency amdyMs by the spectrogram/sonagram is indispensable arid should not be underestimated. It. has served, and is of current, use today. ~ a hams for a considerable amount of investigalAons of r~atu'rN signMs, esp(~:'ia,Ily sonic and ultra~.mnd (q)eech. music, animal commulfication~ echolocation, etc. 2). However. it is worthwhile to mention that its structure carries essen~.ial restrictions that are t,ypical for all l:burier-Vpe methods, if they are used in a nonstationary context. As it operates locally by employing filters (in tinm or in frequency, depending on the [nterpreta.tion), it necessarily faces a trade-off between the temporal and the frequentia/ localization. We cal~ interpret those two as antag~> nistic r~:~:solutions: the time-resolul;ion of a wi~dow mmlysis, such a~s the spectrogram, gets better when its window becomes shorter; however, the [requency-resotutiou degrades at the same rate. because the t:ourier a~mlySiS is collfitwd to ehe same short tAme-window. Com'ersely, analysis by a filter bank with more selecti~:~ filters has a better fl'equency-resolution; but ghis implies a lesser time-resolution because the impulse responses of the filters haste a longer durat.iom
Chapter 2 Class(~s of Solutio~s
53
This type o1! restrictiol~ was already k~low~, whe~t the sonagrapb was invented. Originally it ofli~red ~;wo possible selections of filters: one for a "widebm~d" (300 Hz) a~d a~?other o~e lbr a narrowbarld (50 Hz) amflysis. These specia.t values were chose~ in accordance wilh their main application of speech signals, hi the wideba~d positio~, the fine temporal structure of the speech was retained (as manif~sted, for example, by the temporal periodicity of the pitch); this couId onIy be gNned [)5' sacrificing the localization of its tbrmantic structures. The narrowbas~d position improved the latter, but deleted the noi;ion of a temporal appearance of the pitch, replaxting it with the corresponding harmonic structure. The sonagram/spectrogram soo~:~ acquired a stat~dard form and became a classical tool for lhe interpretation of signals. (Certain experts were even able to read messages ti'om the sonagrmn~ :~) "~t there appeared diflbrent attempts to improve it, while mN~:ltNning its mail~ features. ~,\~ already mentioned the eonstant-Q varia.nts or more general ones with a varying fl'equency-resoh.ltion. Their advantage (and also their inconve~ nie~lce regarding application,s) was to get a.,x~, from the use of a unique filter and replace it with a bank of filters of dilibrent: resolutions. Then o~e could hope to match e~u:h charl-mI~eristic of the signal with a.t least one of the titters. This point of view is better ada.pted to the perceptual interpretat, ioi~ of speech signals: We know that the ti'equency response of the inner ear operates lil.:e a uniform filter bank at low freque~cies and a. batik of constant-Q filters at high frequencies, ~s a first approximation. 4 However, i~ other situations where a good resolut.ion at all frequencies is desired, the eonstant-Q structure has no advantage (e.g., tsr the ea~e of rotating machines with a rich harmonic structure). Some other wasps of improving the spe(trogram h~~ze been explored as welt (and will be discussed in Subsection 3,2.1). On.e tries, ibr instance, to improve the joint resoluti(m of a spectrogram by including the phase i n f o f ma.tiol~, which is u s u a l b igt~ored. Nevertheless, all such methods belong to the same paradigm of an "instantaneous spectrum," which is obta.i~ed via some window analysis. Two major theoretical contributiolis appeared almost simultaneously with the im~ntion of the sonagraph, The first dates from 1946 and is due to Gabor, He introduced the notion of the decomposition of signals into minimal grains of infi)rmation (or time-f~equency "atoms"). The second dates fi'om 1948 and is due t;o Vitle. II. is concerned with the time-fi'equency decomposition (or distribution) of the e~ergy of the signal, not the signaI itself. As a matter of Net, these works laid out two important directions tbr approaching the time-fi'equency problem. They have remained the two reference directions for most a~rk on this subject until today.
Time@)'~'qu~mcy/Time-Scale A~m]5:sis
54
2.1.2. A t o m i c D e c o m p o s i t i o n s It. is a rather natural idea to decompose a signal into a family of elenmntary sign~ds and regard it. as their linear superp~sition. Fk~r example, this is the underlying principle of every discretization of a signal, The choice of ~:he elementary signals for the decomposition may depend on the a priori knowledge about, the constitutkm of the s i ~ a l (examples are tim cardinM sine function for sampling of bandlimited signMs, complex exponentiMs for the Fourier expansion of periodic or periodized sigl~als, or eigenflmctions of she autoeovariance kernel fbr Karhunen-Lo,?ve repre:~m~ations, etc.). In a time-freque~ey context, we can look upon a joint decomposition as a sampling (combined in time and frequency), which is a~sociated with a tel'gain paving of the plane G a b o r . The t,mte followed by Gabor wag the following. As we have seen in Subsection 1.1.2, no signal can be arbitrarily localized both in t,ime and frequenc?~ Therefore, we should consider the most co~mentrat~ut signM (which is the Gaussian) ~ an e~ementary signal, which carries a minimal amount of illformation. Such a time-frequency ~'atom" shouM be ~he ukimate paa'ticte, the "building block," of each signM, which can thus be written in the fbrm + -x
} :~:
In this expression the 9n.~(t) represent, the different time-frequency atmnso They as~ obtained Dmn a single Gaussian (with widths at and &., and ira> mMized, so that b;t ~Sz/= 1), whose cen*er is shifted to the point (nt0, mpo). The G:,.[n, rr[i aa*e tim sampling vMues in the time-frequency plane. Hence, each coefficient G~[n, m i carries informar.ion, which is localized to a neighborhood of a time-frequency ceil of unit area. The basic idea consists in paving the plane by means of such information (:ells, and this should be doue in a complete and nonredmldant way. ttereby we ~ussociat.e with each elementary signal a natural surface element of minimM area in the timefrequency plane, which represents an information "quantum," N~) called a fogon by Gabor. intuitively, some informatkm will be lost if lhe mesh is too loose (t0z~o ~> I). Conversely, a mesh that is too tight (tovo << I) induces redundancy and an unnecessa W large rmnNer of coeNciems. A choice toy0 ~ 1 should represent a good compromise, as it yieI& a number of useful eoeNeients close to the "dimension" of the signal. Driving our intuitive arguments even fl~rther, the actuM form of the mesh d o ~ not matter a prhn'i, if only the correct are.a, of the hgons is imposed. This ailows us qualitatively to regard the sampling % la Gabor" as an interniedi~te re~ alization between the extreme cases of "Shannon sampling" and "Fourier
Cb;.,.p~er 2 Cl~s,ses of So/uric ns
55 Fourier
Shannot~
-i ii i . . . . . . . . . . . . i!~i l tI
.[
l
time Wavelet
Gabor ...................................
i
.....................
Figure 2.1.
.............................................
Timedi'equency discretizations.
Symbolic representation of different, discretiza.tiol~s of a signa| by pa.ving the time-fl'equency plane: sampling in time (Shmmon) or frequency (Fourier); rectangular (Cabot) or dyMic mesh (wavelets). N)r all types tt~e area of the different cells (togons) is the same, sampling," which either advantage the temporM or the ffequential ~spects (cf. Fig. 2.1). Variations. a As appealing as Gabor's initial idea was, it initiated onty a few applicatioi~. The main reason was that, in its originN form (Gaussian Nnctions on a rectangular mesh of the plane with a minimN densi V of t0~/0 = 1), the N-~ticipated decomposition was not coupled with a nice matbematica.l structure. It Niled to provide a basis; in other words, the coe~cients could no~; be obta.ined by a simple projection, that is, e + :.~
G:, [n, m] #
. . . r~
v(t) 9 , , , ( t ) dr.
Quite rapidly the need was felt to a.ugt*mnt the redundancy of the
Thne-Frequenqv'/Time-.ScMe A na/ysis
56
vepresentation by taking a denser mesh, and the extreme consequence was to pass to a continuum. This was performed tD- H e l s t ~ m in 1966, and it vve~s further generalized in the following year by Montgomery and Reed. They ex~n ga~¢e up tlhe idea of a Gaussian for the decomposition. In this way, the initially discrete approach by Gabor lost its specificity and returned to the bosom of the short-time I:ourier analyses Ahhough ~)rne isolated attempts foUowed using variations of Gabor's idea (for example, the Gaussians were replaced with prolate spheroidal wave functions), atomic decompositions only reappeared on the stage of signal theou,; at the beginning of the 1980s. r{especting the nonorthogonality of the decomposition, it was Bastiaans in 1980 who first suec~eded in showing that the Gabor coefficients can (f()rmaliy) be computed by mea~ls of a duM basN. He also emphasized the bad numerical condition of the solution. Balian (tis~(~'ered a Gabor-type basis shortly afterwards (rectangular mesh with minimal densiV), but the construction led to bad localization properties in time a n d / o r frequency. This, of course, w ~ just the opposite of the initial plan. The ;'coup de gr~ix:e" was delivered by a theorem (Batian-Low), which states that there is no way out: There exists no basis associated with a rectangular mesh of minimM densi V, which consists of functions that are well localized in time and in frequency. Memttime another issue came into existence. It is e}o~ly associated with the concept of wavelets (or ondelettes in French) arsi was introduced by Moriet and Grossmann around 1983. The originN idea we~s specified in terms of a. continuous representation VCavelets. e
%(t, a)
= ,
f
~ ;>,5
.(s) h:,,(s)&. (X5
Hence, it is fbunded on a principle similar to that of Helstr6m's approach. Differing from that, however, it uses a projection of the signal onto a family of functioi~ with vanishing mean value (the wawelets}. These are derived from one elementary function by translations and dilatior~s, that is,
By construction, the name of a rJmc-sca/e representation is better suited for such a wavele~ transform. Nevertheless, one can find a tim(> frequency interpretation of the transform for wavelets, w h o ~ frequency response is locMfz¢~.t to a sinai} neighborhood of a n(mzero frequency z~. Then the rel~t, ion ~, = u0/a yields a suitable identification of the scale parameter a and the fl'equency vm'iable ~/.
Chapter 2 Classes of Solutions
57
The family of wavelets acts like a continuous basis. This means that there exists an exact inw~rsion formula (also called CNderon's formula and known to mathematicians in another context since the 1960s). But it also ha~s certain advantages as far ~ its discretizatkm is concerned. If we stick to the idea of information ceils of the logon type, we will find out that employing a dyadic paving of ttm plane insteM of the rectangular form renders the existence of orthonormal bases possible. Then the correspond-. ing decomposition has the {brm x(t)= ~:::::
- - c ' , ~ , r~ t :'= - -
cxc
where, the wavelet .~5(t) may be a flmction that is well localized in time and fl'equeney, and where the collection {%,.,~,(t.) = 2"~/24~(2'~t- r0; n, m <~:7Z} constitutes an orthonormal basis of L2(IR); that is, v:,,.,,.. (*) de = <,.,, e.,.,, The discovery of such bases (by Meyer, Daubechies, and MaHat around 1985), which eI0~" many more desirable properties and c~nne with efficient computational schemes, marked the beginning of intense activity in (.his field, 2.1.3. P s e u d o - D e n s i t i e s
We already ment;ioned that almost in parallel with Gabor's work there was an article by Ville in 1948 that prepared the ground for another approach to the time-fl'equency problem: l~>llowing this route one deals with energy distxibutions of a quwdrat.ic nature (just like a spectral energy density or power spectrum). Thus there is an intrinsic difference to the linear decompositions of a signal into elementary components. The energy I):z of a determi~istie signal x(t) C L2(g4) can be written in two equivalent ways, namely <
:=
l . ( t ) ? de =
Ix(,.)l-
('*2a
This confers the rank of energy densities upon the quantities Ix(t)l 2 and {X(u)l 2 (regarded as temporal or spectral densities, respectively). Therefore, it is quite natural to search for a mixed quantity p. (t, ~,), which allows of an intermediate point of view by providing a joint density, such that
E~, = ~ ' p , ( t , v,) dr&,
~lime-~7"equency/"Tin-,e-ScaIe Ana/)'sis
58
The inscantalmous spectra, whidi were briefly discussed in Subsection 2.I.t, are possible candidates for such distribu~Aons, provided {ha{ their wi~ldows sa{,isf~v some miId normalization conditions. However, thW represent very particular c~ses, a~d there is enough room fbr defining more generat bilinear repr~entatior~s, which are not ob{,ained a,s by.-products of linear transforms. Moreover, it can be easily verified that they do not satisfy a natural constraint tha.t seems }ust.ifiable, We require from a timefrequency representation that it contain tt~e ener~: densities (in tinm al~d fre.quency) ~ its margi~als~ thagt is, e
,
e,
p~,(t,
)
t:,,,-tt,-) ~tt = t x (~')t" ~
I
'
,~) & = ix(~)l
2
,
Assumil~g that the ins{a.ntaru~.ous spectrum f)~:(f, 7/) "deploys" the energy of ihe signal into the tw~di~mt~sioim] time-fr~tuency pla~m, the first constraint express<~s the Ii.~llowing: An imegration para|ie[ to t,he time axis leMs to an accumulation of the complete history of the signM, so it should yield the global spectrM intbrmation. The second constrNnt is d u n and can be interweted acvordingly. V~qgner-Ville. r
Ville proposed {,o define an ~'instauta.tmous spectrum" by
~.lh(t,v)=
t
~
m" t - ~
dr.
(2.a)
This quantity satisfies the constraints of the marginal distribut.ions. Here the signal x(l) is supp~.~ed to be analyt, ic. The motivation ik.~r this deftnition is its anMogT with a probability density flmction. Fhe preceding expression is nothing but ~.he Fourier transform d an ~acceptable form" of the characteristic flmction for the redistribution of energ3,. We mention as a curious fact. that the method employed by Ville s~_~?ms to be inspired by the operational fi)rmaIism of quantum mechanics. However, there is no reference or remark co~merning tim equivalmtee t.o the (position-momentum) function that w~s earlier proposed by Wigner in 1932. (Le~. us note in passing that, W i t h e r sNd almost nothing about his motivation t2yr giving this definition, either. It seems that it was proposed in an ad ~loc nimmer, a~s onty a footnote contains a hint. that. it w~s "found t:g L. Szilard and the present author some years ago for another purpose" Ville's defimtion ,exhibits some theoretical advantages. Let us once more consider the academic example of a linear chirp. One can show that the Wigner-Ville distribution idea[|y localiz~ to t:he instai~taneous frequenc}: More precisely~ it verifies the relation :r(f) = exp{i2~(*e0t + (/7/2)f2)}
~
/4~.(t, t~) = f ( p - (P0 + ~f)) -
Chapter 2 Classes of Solutions
59
This property is achieved regardless of the slope/~ of the modulation. It establishes a defiinite advantage ove.r the spectrogram. Meantime, Ville's article has another surpAse in store. The issue of energy distributions in the time-frequency plane fills orfly a. ~x~rysmall portion of the text. As its title tells, the paper is devoted ma.inly to tile theory and applications of the new notion of analytic signals. Surprisingly, the connection between these two concepts is not even mentioned, and nothing is said about tim localization of the joint representation tbr signals, whose frequency modulation stays close to the rule of their instantaneous frequency. This point had already been ctarified ~" Bass in 1945, but it. was given for Wigner's %nction and in a context of statistical mechanics: Pursuing the analogy with a probability density flmction, he proposed to interpret a. "local inolnen£unl" by means of a conditional expectation ~a,lue. In the terminology of signal theory, this reduces to identifying the instantaneous frequency with the center of gravity of the Ville distribution
t.~:,.(t) =
~ l'K~,(t,
/(
....ix l/IG(t L')
.
However, there exists a natural limit insofar ~s the interpretation of the Wigner-Ville distribution as a probability deiisity flmction is concerned, In fact, it turns out that it can attain negat.i~h~ values. This evidently bans it from being used as a probability density, and it contradicts Ville's expectations. There is no doubt that this w~s the main conceptual obs{acle against Vitle's definition for more than twenty-five years. Another red,son tbr this "forgetting" can certainly be found in tee difficulty of putting the method to work before the generM advent of calculators. When this bar failed and the Wigner~.Ville distribution reappeared at the end of the 1970s, some of the theoretical reservations also faded away, thanks to the offered possibilities. In 1980 there appeared a series of three articles by Claasen and Mecklenbr~uker, which marked a turning point and the beginning of a true renaissance of the Wigner-Ville distribution, s In the meantime, one h M realized by the facili W of simulations that it obeys a principle of quadratie (and nonlinear) superposition ac(ording to the relation =
+ % ( t , . ) + 2 Re
where
wi,:,(t,,,)
=
:.:.
t
y"
.........
Hence, a combinatorial proliferation of cross-terms, or interference,% is inlplied, which often scrambles the readability of a time-frequency diagram.
Jime>I~}eqne~cy/Time-Seate Anatvsis
60
All through the ~980s a large amotmt of ~ork was devoted to modifications of the initial definitiort in order to improve the readabilky of a timcfreque~my represent, a.tion, without giving up much of its good ~h~x~reticM woperties. We will presel~t a more detailed discussion of this issue in Section 3.2. Page.!1 Ville's endeavor of definb*g a time-freque,my energy disWibution w ~ not the only o~m. Contemplating ~,he fact that the notion of a. global spectral dei~sity e r ~ e s all time dependenc< Page proposed in 1952 to look at a causal specgral density; this ,~ma~:~s d~at at each instant, it can alone be computed based o17 the past of the signM. He derived a definition of an iT~stantaneous spectrum, which relies on the temporal variation of this quam.iW, t.ha¢ is,
t!r(t,~/) = a~
a:(s)e- ~' ~"~ds
,
(2.4)
It sat&ties the condition of a ca~zsed marginal distribution L
.f
~
9
/),:(.s,~.~)ds= ,[f x:r(,s)c i2r"S ds "*
lt~ t955, Blanc-I,apierre a.nd Picinbono suggested some variations on this theme. T h e y made use of "trm~cation operators" in time (so that; causaliey is ensured) and certain ;'bandpass filters," which were defined anMogously by interchanging the time a~d [reque~ey varia;bles. (In ail
Rihaczek, lo This last expressiol~ actua,lIy turns out to be the real part of what is nowadays called Hiha,ezek dist.ribntiom It: was defined iI~ 1968 by Riha~czek and is b~sed on a completely different argument, whidl might be of inverest to ~he reader and will be reproduced I~ex~, Let us consider two
sig~mls, g~{*) and :r2(t). We define their (complex) energy of interaction by the inner product
Chap~er 2 C!as,'es o f Solutions
61
V
Figure 2.2.
Riha~:zek disLributiom
The "complex energy density" of a signal x at a point (t,v) of the tim~>frequency plane is defined as the energy of i~teract, im:Lbetween the r(~triction of a: to an infinitesimal interval centered at t and its filtered version Lhrough an infinitesimal band centered at v, normalized l~y the (infinitesimM) area of the correspondh~g t,ime-fl'equency cell. More specifiea,lly, if we let :r~ be tim restriction of a given signal z to a , infinitesimal interval a T a b o u t an instam, l, and z2 be the same signal z f l t e r e d t h r o u g h an infinitesimal b a n d at~ a b o u t a frequency ~/, then we obtMn (by a p p r o x i m a t i n g the int, egrMs fly l{iemaml stuns)
z ( s ) ~,B X 0 4 e -~>'~"'~ ds
This Iea,ts to an int:erpret~ticm of the expression
i%(t, ~J) = z ( t ) x*(zJ) e -~'~'~
(2.5)
~s a complex energT "density" at tile point (t, v) (see Fig, 2,2), T h e preceding quanti V is called Rihaezek distributJon.
62
't'imeo.I~)'eque~cy,/Th~e-S(.ale
Atmlysis
Extensions. Although the tbrementioned pseud(~densitie~ were initially defined for deterministic fi~ite-energiy' signals, t~heywere used (a,t least for= malty) for larger cla:sses of signals, such as distributions or random sitnals wi~h eventually finite mean power, However, we should say at this place thai some special approaches ibr random signals exisL which define a (pseud~)power spectrum. ~ (Among others there is the 'evoIutionary spectrum" of Priestley in the middle of the 1960s, which is ba~sed on the Cram(?,r decomposition of a nonstationary pro('~ss.) We will come back to ~,his issue toward the end of this chapter. Unification. I~ Because of the diflk,rent origins of the preceding definitions of time-frequelmy representations, they look very distinct a priorL Moreover, none of them is outstanding in a particular way, It is therefore desirable to find a consistent fl'amework to enable us to compare them with each other, The first attempt, aiming in thN direct, ion w~s made i~ the previously mention~t work by Btan(>Lapierre and Picinbono. But the veritable synthesis was managed t~y Cohen in i966. He proposed a generM form; which covers a!l the fbrementioned definitions (spectrogTam/sonagram, V~lgn~rVille, Page, Riha(zek, ere.) and which s~ill gives free rein to otlmr proposMs. The definition is specified by means of an arbitrary ~rarameter function .f(C r), t,hus taking tile form
(2.6) The original objective of the s~called "CoheIfs class" w~s to define joint repre2~entations of position-momentum space in quantum mechanics, But Eseudi~, and Grda in 1976, and later Cta~.sen and Mecklenbr~uker in t980, were responsiMe for its reappearance in the time-frequency context of signal t.heory. By the facility of i~.s parame~;erization and the large m~mber of possible interpretations, it provides the theoretical reference framework witlfin which m(xst of the wnrk in the 1980s wa;s performed. A major exception is given by the clas~s of represent, atioi~% which have been suggested by both Bertrands since t983. They can be thought of ~s extensions of the Wigner-Ville distribution to wideband sigr~als~ They received a growing imerest in paralIet with the development of the wavelet methods because they are bilinear generalizations of |hose. One possiMe justification of their definition ea,n be given by a con~ struction in tomography, whi& N~.r~puts a i~ew light on tim Wigner-Ville distributio~l. Let us ~.~ssume that we wish to cor,struct, a time-fl'equency dNtributi~m p.~.(t, ~,), which has the "correct" marginal distributions, that
63
C h a p t e r 2 (~lasses o f S o l u t i o n s
is, such ~ha~;
/
'+ ~ p~.(t, ~) dt ........ [X(~)[ 2 ,
px(t, z.,)dT,, = ix(t)t ~ .
These entities can be viewed as the squared absolute values of the inner products
in which the analyzed signal is multiplied b y the ideal analyzing signals of one pure frequency and one impulse, respectively, As a generalization, it is conceivable to use a,ny linear chirp as the analyzing signal. This corresponds to a straight, line ~/= ~'0 +/~t pointing in an arbitrary direction of the time-frequency plane. Au appropria.te extension of the constraint of a correct marginal distribmion reads as
x:
where the member on the right is a, mea~sure tbr the interaction of the signal and the amdyzing chirp z,,,~,;~(t), expressed by the complex inner product, The posed problem thus reduces to the inversion of a Radon transtbrm. It turns out that its precise solution is the Wigner-Ville distribution. Next we (:an retain the elegant idea behind this tomographie construction, and cha.nge the underlying geometry. We do this by replacing the straight: lines of int.egration with other curves in the time-frequency plane. In particular, whe~ we choose hyp~.~rbda.s, whM~ pla;¢ a central role in connection with scale changes, an analogous construction (to the preceding inversion of a RMon transfbrm) le~ds to the definition of B e r t r a n d ' s uMta,ry d i s t r i b u t i o n + ....
X
\,
sinh(~;/2) ) e
~\
i2'r~;
d?
.
sinh(?/2) ,1
This is only one of" the possible definitions in a huge class of a ~ n e distributions, which is comparable to Cohen's class regarding its richn~s and diversity.
64 2.1.4.
Time-f'req~mncy/'Time-Seate AnMysis The Parallel to Quantum Mechanics
We repea,tedly mentioned the formalism that brings together sig~tM theory and qmmtum mechanics. It is thereibre not, ~tonishing that the same mathematical objr~::ts appear in both_ fietds, even though trot. only their interpreta~iot~, but also the reasons for their definitions may be completely different. Different concerns, l:~ A s an exa,mple v~x~consider a special distribution of Cohen's class, by choosing the parameter function = S!!LS 2
.
~T
~,~ will see in Subsection 3,243, that this choice, in a tirne-{]'equei~cy context, ~ motivated by a concern for readability of the representation in the pla,ne, It only became known in the mid-1980s. Nevertheless, the related distribution carries the name Born-Jordan. Thks ira.dillon g o ~ back to Cohen's article in 1966, where reibrence was ma~:Ie to an article by the two physicists Born and Jordan, which appeared in 1925. In this inaugural,lug article of quantum mechanics the issue of joint rewesentations was not raised directly. But, among other problems, and in a much more general way, the authol.-s deMt with tile %llowing question: How can (me associate with a tSmction, which represents a physical quaat, iW~ an operator w h o ~ expectation value &~cribes a}t possible measurements? From this viewpoint, the introduction of joiilt represem.ations eorresport(ks to the goal of attaining a different description of a m e ~ u r e m e n t - not as an operator e x p i r a t i o n value, takee over the possible states, but a:s an "ensemble average" of the cla-~sicM flmction with respect to a "proba~ Mlity densiV" of these states, When we trans%r this into a time-freqummy terrninology, it reduces to an impIicit definition of a joint rewesenta,tion by the idemi V
( ~(~, ~) >~, = "'//g(t, ,,) p~: (t, ~,) dt d~, Here the function g(t, ~) is a~s~.)ciated wit, h an opera£or ,O(t~~) composed of the elementaEy time-frequency operators t and ~ (defined in Subsection 1.1.2). It turns out that sudi an association, or %orrespondence rule," cammt be unique in general, Tile nonuniqueness stems from the fi~ct that there cannot exgst an unambiguotLs and well-defined joint distribution that is built m:~ two canonically conjugate variables ( ~ s i t i o n - m o m e n t u m or time-frequency), because their associated operaaors do not commute, Indeed, as a simple example we can look a,t the three functions tu, ut, and (tt~ + ut)/2, which are identical, of course. However, rite operators {0,
Chal)ger 2 C'lausscs of Solutions
65
P~, a.nd (~i, + bi)/2, which are obtained by simple subst~itutions, are not the same. This is true owing to the commutation relation 27r ' On the other hand, it is d e a r that the arbitrariness in writing the o~> erator ~)(t, P) is associated directly with the chosen definition for p:~(t, ~): The parameter function f(~, ~-) of a joint distribution is, eherefore, indica-~ tire of the choice of a correspondence rule. It is under this form that the time-frequency distribut, ion a~ssociated with the fbrementioned choice of f is rooted in the proposed rule by Born and Jordan in their 1925 article. This point will be furtlher discussed in Subsection 3.1.2. Some intersections, t4 A number of definitions, properties or results that can be useflfl in signal theory were developed in the literature on theoretical physics a~ld can be looked up there. The converse is true, but to a smaller extent. One is urged to draw the conclusion that there were very few links between these two arenas until recently. As we mentioned before, the Ville distribution wa~s given without refert;nce to the one by Wigner. The same thing happened regarding the Rihaczek distribution, which was preceded by a suggestion by Margenau and Hill in 1961. Conversely, definitions from Page or Levin did not resonate, it seems, in qua~ntum mechanics. FinMly, as another amusing observation, the pursued objectives in each of these ~reas and their own culture qualifies the ~mtions of a "naturN" or an "intuitive" point of view. Looking first at the community of pre~:titioners of signal processing, one ca.n assert that the t.ime~frequemg paradigm h~hs its l~ots in the "intuitive" i~otion of an evolutiot~ary spectrum (viewed a;s a short-time spectral analysis). It was a slow process to move it closer to the more fimdamental concepts such. a~s energy distributions, the Wigner-Ville distribution being the pro(otype. In quantum mechanics, one can observe a completely opposite situation: The primN and "natural" object w~s the Wither distribution (this lasted until the 19708) and t;hus it ws~s quite some time beyond the trivialization of the spectrogram (and once more without a~xy reference t.o it), before some position.omomentum distriNltions % la spectrogram" were proposed. Itl addition, some new difficulties arose in this context. The necessary ~uloption of an externM quantity for the state (the equivalent of the short-time window') needs to be endowed with physical rank and interpretation.
l~imc-b}'cqaeilc> /Tim~+Scate Ana(vsis
66
~}2.2. Atonfic D e c o m p o s i t i o n s The main principle miderlyi~ig every atomic decomposition is the consideralion of an arbitrary signal as a linem- superposition of elemel~tary signals ("atoms"). In the time-t?equency comext, oae requires that these atoms be "well" localized in time and freque~ey, so ,:hat each of them is al~ bMivisible entity ili tim sense of Gabor's no~ion of a togon, Moreover, one generMly deman
2.2.1. P r o j e c t i o n s and Bases = - G e n e r a l Principles Discrete bases. Let~h ........(t) E L~(N,) be a time-fi'equeiicy aWm, which is localized to a neighborhood of a point of the phme indexed by n and m, T h e collection {h,.,~(t); n, m ~F 2g} tbrms a~i orthonormal basis of L2(IR) in the cl~ssieal sense, if ,.#-
b~,'.,' It) dt = br,./ 6 . . . . ,
(2~7)
and if every sigmd x(t) with finite energy E+ ca.n be writ, ten as
:r(t)=- ~
y'~ 1~&~,-,lh~,,,,(~->.
(2.S)
hi this ca.~ the (oefficients of ~tie decomposition constitute the joint rep= resentatioll of the signal. They are simply given by the projections
t~:,,[,.,,,,,,] ......
:,{t) h;,, (t) de.
(z9)
They aztually define a measure for the energy of interaction of the analyzed signal with each of the analyzing signals~ the various atoms. More generally, the energy of interaction of two arbitrary signals a'(t) and y(t) is given by
L~+, : ./~.7
Z
Chapter 2 (:lasses of ,otut~ons
67
Hence,. th:::oh:Ient.i{;y of' energ~~. conservation
E,. =
Ix( )l dt =
follows. A time-freq,m~nw basis tht~s allows decomposition of a signa.I and represent, ation of it g~s a linear superposi~ion of atoms, As a by-product, it also yieMs a~t energy distribution of this signal. C o n t i n u o u s b a s e s . ~5 In the disere*,e case, the representation is supplied by a coumabte family of coeffieien[s They are associated with a discrete lattice of points in the time-frequency plane. Ol-m c~m vividly kink on a continuous representation as the limit of discrete decompositions (with or without respect to a ba~sis), when the dmusity of the lattice tends to infinity: Then the a~,oms of the decomposition formMly behave as if they cot~titute a continuous ba~sis. A j d n t continuous representation thus keeps to t,he principle of' being the projection of the signal onto an (uncountabIe) Nmity of atoms indexed by ]1,e. Hence, t,he ibrm
ht:~ b" ) ds
(z l)
emerges, where ~ ~ lI{ is an auxiliary variable that is (direx:tty or indirectly) commcted with the frequenc> Its &oice is relat.ed to the group G of transformatiol~s
which generate all atoms from one single element. Le{, us denote the Ha~r measure of this group by #o, Then the interpretafioIl of the collection {ti~>,( t ) s, A ~ IR.} as a comimmus basis amounts eo an mvemi.m t}~rrnula"
= . / f Ly(:% A)
d#c;(,, a)
(2.:12)
provided that this fatuity is admissible, The inversion formuta can be read as a contirmous decomposition of the sigrtal. In order fc.,r such a relation ~:o hoid, we must require that
68
T i m c - b ~,.xiuen
l'his, in turin yields the clo,sure relation, or Ihe re~~flution o f the identit3,
i//,,,t~:,(t) t,f~V')dt,,,,(s,,\)
= ~.(t .... t').
(z.la)
A multiplication of both sides of Eq, (2,12) by h~,~,(t), followed by a~l in~;egration Ooives '
,:r(t)I~5 x,(t)dt ....
L,.i~.,,\~h~(~)IC,>,(i.)d~ d#,.-;(~:;\)
|:I.~
Hence we find
It;:
where we have i:i~roduced :,he so-cal:ed r e p r o d u c i n g I:~'rnel of a.nalysis
]/,.--x
,=
(
K(s,a;~'., 5 ' )=. . . . . h~x(t) h ~ , x , , t ) d t .
(2.14)
This kernel is nothbig b m the represeutat[ou of one a t o m by means of all others. It dlus provides a measure for the energy shared by different atoms. At the same time, i¢ plays the role of aa~ evaluation flanction~fl associated with an a r b i t r a w point in the plane As ally COII1iIluOtlSrepresentation is infinitely redundant, it cannot be a.ssociated with orthogonal atoms (i.e., atoms with vanishing inner product or zero energy of interaction). Nevertheless, the reproduchlg kernel alR)ws us {.o perR)rm simila¢ operat.ions a~s in the case of an ordinary basis. Especially, one can derive the i~)metrie retaliou
.~:~,.~=//' ,
A (s, A) Q(,s, x) d#~:(,s, X)
(2.:I,5)
|;12
mid the energy conservation E,: =
/;l
tLJ, bs, A) - d#<;(s, A') .
(2.16)
II 2
Further:more, t.he squared modulus of a lilmar representation is an energy distributiorl by anaiogy wigh the case of discrete bases.
Chapter 2 CTasses of Solutions Frames. ~
69
The discrete decompositions, for which tlm family of atoms is
ove1~comt)te~,e and non()rthogzmaI, lie in between the continuous representaX ions and those obtained by the projection onto an orthonormal b~asis. One typically runs into this situation either by discretizing a continuous representation by means of sampling the time-frequency plane, or when tile atoms are given a priori rather than being constructed under an orthogpnality constraint. In both cases, there is no obvious red,son why these procedures should produce a basis. Nonetheless, it would be nice to be aJ:~leto work in such a context in a controIled way. This can be achieved if one weakens the notion of orthonormaI bases and uses that. of f?ames (or %tructures obliques" in the French terminology). By definition, a family of atoms {h,~,~(t); ~, m ~ N} constitutes a frame if there exist two constants A and B such that 0 < A < B < oc,, and so that the set of projections L~.[rt, m] = {z, h,~,) satisfies tile inequalities
AE~ < The interesting feature of such a structure is its similar behavior as a behsis (see Eq. (2. I0)). This can be observed particularly well when A is close to B. For this purpose let us consider the quantity
+"
Z Z
which, apart from a constant, woukt yield the discrete representation of the signal itself, if the h,,~,(t) were a basis (cf, Eq. (2.8)). In the present sit:uatioI~ one can show t h a t
This implies, in particular, that th.~ 1deal case, when A = B (%ight fi-ame"), gives rise to the exact inversion formula
fbr the frame decomposition. Since r = ( B / A - 1)/(B/A+ 1) < 1 still holds in any case, it is sufficient to iterate the preceding construction in order to reduce the reconstruction error ~s much ~s desired. This reduction of the
Time-tq'equ( ~qy/ l'ime-Scat( Arm~v:ds
70 error is all the faster, as t h , ratio
I3/,I is ck>ser to l, h.qeed, it su~ices t:o
sta,rt from
whmx!~ e: (t) is the initial re,.ons(ru(tion error; so i~,s energy E~ is bounded by rE~,,. A further decomposition of thi:s error leads to
e~(~) ..... < ( ~ } + c 2 ( t ) with E2 <~r'2E~:, mid so fort~h. After k iterat, ions we obt, ain E~, 5 r~:E~:, m~d hence E:, -~ 0 a~s k tends :,o i::finiU¢. It follows that t,tm exact reconstruction is obtained as the limit
2.2.2. T i m e - F r e q u e n c y E x a m p l e s S h o r t - t i m e Fourier. tr The most obvious example of ~ continuous t.im~ frequency decomposition (A = ~) is given by tim short,-time Fourier transtbrm, Reeatl t h a t the ela.ssical Fourier anatysis eraases M1 chronological notion and projects the anMyzed signM onto a Nmity of everlastir G monochr(> matic waves. By w~ay of eou~r~st, the short-time Fourier ana,lysis, ~ its nmne already indicates, introduces a t:emporal dependence by replacing the pure was.es with locNized :'wave packets" of the form h.,,,(s)
=, h ( . ~ -
~) e :" . . . .
.
(2.17)
A simple ilhlstration is given in Fig. 2,3. One can properly say thai: at each instant the "window" h(t) selects only a segm~ent of the signal by its restric~,ed horizon, prior to the Dmrier analysis. Hence we obtain a mixed representation, ,joint in time and fie,queney, which can be written as
I%(t,u)
= {z,,,,.)=
a:(.s) h*(.s-
t)e-~2~"~&.
(2,,~s)
This representation bears the D'pieN restrictions of the F'ourim' tranMbrm described in Chapter 1: An improvement of the time Iocalization can only be obtained by shortening the window h(t), whieh in return diminishes tim frequentiaI k}calizat.ion The eonvense is also true, and can be rendered
C'h~t>~er 2 C~'a~es of Sotut.h)x~s
71
)l "YYV
-w,
T!)~,
....
,i~! I I' I
time F i g u r e 2.3.
Short-time ~))urier transh)rm: time-frequency atoms.
The azmlysis by the short-time l?burier transform (Eq, (2A8)) can be regarded as a projection of the a~lalyzed sig~al onto "time-freqllency atoms" of td~pe Eq. (2,17). Each of these atoms is obtained from a unique window h(t) by a translation i~t time mM a frequency modular;ion. The figure depicts two examples of such atoms (the solid lithe shows the real part and the d.~he~t line shows the modulus).
even more appa~l~t by writing out the duM representt~Zion of the short-time l?~mrier transform ~(t,~)
;x
= ~: ' ~ ' ~
x(~) H*(~
.) ~ ' ~ .
(2~19)
ttere the l~%urier trmisform H(u) of h(t) plays the role of a spectral "window" gliding over the spectrum X(u) of the signal :r(t). This second reading (time as a fimction of fl'equency) makes the short-time Fourier transform appear ~ an analysis by a continuous bank of uniform filters with constant bandwidth (of. Fig. 2,4). One verifies without difficulty that it reproduces the spectrum of the signal in the extreme case of an infinitely selective filter (i.e., if H(x.) = 50")). Besides these two interpretations, the short-time Fburier trmlsfbrm is just the projectiot~ of the analyzed signM onto a Nmily d atoms, which axe MI derived from one unique element (the window Nnction h(t)) by" time- a n d / o r frequency-shifts. It thus relies in a fimdamental way on the a~:tion of the corresponding transformation group, which is called the gSeyl Heisenberg group. This group is unimodular, and its naturM measure is d#wl..~(t, u) = dtdu. Tim closure condition of Eq. (2.13) shows that the short-time Fourier transform is an admissible representation if the analysN
Time-f?'cqueucy / Time-Scale Analysis
72
0,5
"0
0,05
O,I
0A5
02
F i g u r e 2.4.
0.25 (}.3 frequency
0,35
(1,4
0.~5
0.5
Uniform filter ba~k.
In its representatkm (Eq. (2.19)i) the shor(-time Fourier transform can be regarded as an analysis by a uniform filter bank. In this bank, each filter is derived from a tmique template by a frequency shift. The figure symboUcally depicts some filters of such a bank, window h(t). hms unit ener~,y_. ~,.. In other words, we have
/7....
th(t)t2 dl = l.
~
:_r(t)=
[~(s ~) h~¢(t)dsd~ .
.;i
More generMty~ let us consider two short-time l~urier transforms Fa:(t, 7~; h) and , ~ ( t , ~,; 9) b0~sed on windows h(t) a,nd g(t), respectively. T h e n the condition ~s
"+" h(t) ~ ( t ) dt =~ 1
(2.20)
ensures t h a t ghe identi~:ies of "mixed" reco~struct.ion
z(t) =
//
£.(s,~;h)g,~dt)dsd~ =
N
t :~ : As, , s~" , g ) h<(t)d,sd~
hold as well as the generalized "energy conservation"
. / y F~(t, ~,; t,) ~:);:(t, ~; g) dtd,, .-.---E~..
(2.2I)
Chapter 2 Ch~'s~s of Soluth)ns
73
Obstruction established by Balian-Low T h e o r e m . As a continuous representations, the short-time Fourier trazlsform carries the intormation of the signal in a~n extremely redundant form. This is very nicely expressed by the identity
~2
where the reproducing kernel
J<(t, ~,; t', ~/) = e ~ ( ' - " ' : ' ~ Fh(t' - t, ~ / - ~,; h) is defined according to Eqs. (2.14) and (2.17). Therefore, the evaluation of the representation in one point of the time-frequency plane sets forth all values of this representation in a neighborhood of this point.. The size of this neighborhood cannot be reduced arbitrarily, ~s it is linked to the temporal extension of h(t) and the frequentiat extension of H(~,). This suggests that it shouM be possible to reduce the redundancy, without arty loss in information, by a time-frequency discretization in accordance with the extension of the reproducing kernel. Discretizing a short-time t~burier transform results in tile calculation of the values
(z22) with 'n, rn ~ ~, where to and t'l~ represent, tile mesh sizes in time and frequency, respectively. These latter values define a rectangular lattice of points, to which the coeNcients of the representation are attached. Intuitivelyi a mesh that is too tight introduces some useless redundancy. Conversely, a mesh that is too loose is likely to disregard some information. It is possible to give a precise formulation of this intuitive argument. In fact, one ca~ show J9 that a n~:essary condition on the lattice must be satisfied in order that the family of atoms
constitutes a frame. This condition has the simple form
and it can thus be written as a constraint of a minimal density of the lattice in tile tim~-frequency plane. Taking up tile preceding heuristic argument
74
"l'ime,. /'b:~quem~37"f'ime-Scale Analysis
again, it is tempting to use a dia:retizat, ion of the short-time lAmrier transl))rm at the cTitic~d density tol/o = 1. This is particularly interesting, as one wishes to obtain ar~ orthonormal basis radler titan just a fl:ame. Unfortunately, thm'e appeaxs an obstruction right at this point, whidl is estaMished 1~" the s~>cMted Balian-Low Theorem. ~ It alarms that, at the critical denMt~g, there exists no l:}'ame (and he~(~, a t),rtiori, no orthonon~M basis) which det~nes a discrete shor~-time Nmrier transform, and whose atoms are well localized both in ~ime and &equenc:y. More precisely, if the family of atoms h(t - at0)e ~ ' ~ / ~ a frame of L2(N), t.hen we either have ........ ~
t.~ ih(t)I ~ dt = + ~
or
constitutes
~'~ tH(~')I" d - . . . . i-,x~.
Note that the Batian-Low Theorem does ~aot exclude the theoretk:M possibility of all e(xmomical discretization of a short-time Fourier transtbrm. But it does rule out the possibility of gaining an atxmfic d(~.:omposition by sampling the time-fl'equeney plane at the minimal rate; that N, we cannot retNn the spirit of a, local time-frequency analysis if such a discretization is us~z'd. A simple exmnpte for this sit,ua~:ion is furnishe~t by the pair %haraeteristic ikmction of an interval/cardinal sine function." An orthonormal baosis can either be obtained by the choice h(t) =- t-" o I/~ li0.t,, I(t) (where 11 (t) is the characteristic function of the inten~fl I), or by emphwi~N its Fourier track,form 1/2 sil~ ~tto h ( 0 = t° ~t
insteM, ~5,%eobtMn that /, ? x ,,~
'~ > {II(i,~[ ~ ~.l. = + , x ,
..... X
in the first case, which is the price to pay for the very good localization in time (compactly supported window). Likewise: by d~e Fourier duMity, we have
jf-':~: t 2 lh(t)[ 2 dt in the second case. It is surely possible to find other solut.ions that in a certain sense represent a more balanced compromise between time and fl'equency localization. Noim d them, however, can truly reconcile the orthogonMity with the k, calization at the critical density. 2i
Chapter 2 C/a.s~'es cff Solutio,~s
75
G a b o r and wtriants. The fact that there exisl,s no frame of localized atoms at the crRical densi W is the major drawback thal rules out any numerically stable reconstructkm in this situation. This happens, in pat:ticular, if one uses a Gaussian =
exp
-
(t/to)-
as the window function, thus tblk,wing Gabor's original idea. This choice looks quite natural because we know that it attains the minimal product of the time and frequency widths among d t signals, if these values are measured by the second-order nmments (cf. Eq. (1.3)). IIowever, precisely because these two moments are both finite, the Gaussian cannot produce an orthonormN ba.sis at the critical density (which Gabor certainly knew), or even a fl'ame. It: follows thae the evaluation of the coe~cients G,~[n, m] in a Gabor decomposition cannot be reNized by a simple projection onto the Nmily of atoms g,.,,(t), which are defined by
Likewise the reconstruction of :~'(t) by a combination of the 9 . , . ( t ) cannot be dram using these projections. However, we can circumvent this problem f?om a purely theoreticN point of view 22: The solution relies on the idea of a d u d basis. Givml the family of functions g,,.,~,(~), the dual basis is defined as another family %.,(t) such that:
.//'+'~ a........(t)
,v,;,,,, , (t) d~ = a ....... ,t,,~,~, •
By employing this dual basis (which has an explicit form in t.erms of the Zak transtbrm 2a), both relations
an d
q-~
x(t)= Z
--}.. x~
Z <x,.g,,.,,,>?,,,.(O
are verified. From a practical point of view., however, the explicit computation of %,., (t) reveals the limitations of this procedure. In fact, Bastiaans showed that the flmction 7(t), frc,m which tile %.~(t) are constructed, has infinite energy and possesses a "spiky" structure, which makes the numericM computations very unstable.
[l'imc,-l:r~,quenc/,?:Timc,-Scale AnaIv~sis
76
There is n.o opportm~ity t.o use ~his exact, but unstable approa~:h i~ practice. However, one ca.n develop approximate aad more s~able solutions, if one is willing to give up the idea of a minimM density and accept a certain amount of redundan.cy of the representation. All this is well understood nowadays and quantified in t:erms of the associated [tame bounds. Daubeehies ha~s a~'tuaIly computed the constants A and B (hence the ratio B/A~ which determines *he stability of the reconstruction) as well ~s the dual window filImti(ms ?(t) for several sizes of the lattice be3a:md the critical densiv, She found that ~(t) tends to the Gaus~ian or to Bastiaans' function, when t0*~0 tends to 0 or 1, respectively. Furthermore, one can de-. scribe the wecise rate of growth of the ratio ]3/A, when t01/0 appmachc~ 1. This provides a measure tbr the b M rmmerical condition of the solution. :e*~ 2,2.3. T i m e - S c a l e E x a m p l e s Continuous wavelets. /%r the time-scale case (A = a), the natural tra rLsformation group is no longer the \~\~@-Heisenberg group of time-frequency shift.s~ Here we use the group of translations and dilations, Ms~ called the affine or ;ax+b' group. This group acts on a flmetion h(s) by generating the ~,ransformed enti V a
' h
-
,
(2.23)
which corr(~ponds to a shift. (and a normalizat, ioa) after stretching ()t,i > 1) or compressing (ial < 1.) the flmctiom It, is crucial that such a transtbrma, tiou leaves the shape of the %netion im,ari;~nt (of. Fig. 2.5). Remark. The scMing parameter a is usuatly supposed to be strictly positive. As a generNization, one can Mso introduce negative scales, which play" a role simiia:r to that of tim ~mgative freqtteneies i~* Fourier analysis. By the projection of an arbitra:ry signal x(s) onto the Nmily d Nnctions {ht~(s); t.,a ~ IR}, o n e obt, NILs a tim~scale representation of t.he signa,l, which ha.s the form
//
. . . . . .
.
(--j-)
ds.
(2,24)
It is called the continuous wm,~.qet t,rans~)rm. ~5 In order to define an effective repr~entation, this quantity must be invertible so ~s to yietd x(t) =
[/f 71(s, a) h~. (t) :I#: (s, a ) . IJ [(2
Chapt,~ r 2 .
.
.
.
.
CIa.~se:>'oI" Solutions
.
....................
77
.................. . . . . . . . . . . . . .
r ......................
i
...............
f
"'"*---
I" ~x
-I i time F i g u r e 2,5,
Wavelet anMysis: time-~ca/e atoms.
The wavelet anatysis (Eq. (2.24)) can be regarded e~ t.he projection of the signal onto "time-scale atonm" of the form of Eq, (2,23). Ead~ atom is obtained fl'om one wavelet h(t) by a translation in time and a di}ation. The figure depicts two examples of such atoms (so|id line for the wavelet and dashed line for its em~elope}. Here l** is the canonical measure of the attine group, which is ,lot unimodular. Rather, its left-invarim~t measure is defined by 2~;
d# ~{t, a)• :
(it da ({2
Hence, we can rewrite the c|osure relation d E% (2.13), which is direedy associated with the in',:~rskm fbrmula., a~s
6(t~t')= //)'
th(~)
h* (t' \--~[-7
a2
jl 2
=
h(s)
](h,]
da
By gaking Fourier transR~mm of both sides of this equation, we flirther derive t h a t
if+:×: :.
,o dl.~
:
(2.25)
This col~stitutes the admissibility condgion of the wavelet transform. Igs physical interpretatkm is ra.ther simple, In fax% in order for the foregoing integrM to be finit, e (which is the important part:, while the vMue
R
Tim~>-Frequenqv/Time-St ate Amdysis
~..
t is only rela,ted to the normalization) one needs to check ks convergence both at: infinity and at the origin. The first condition is very mild, as il reduces to the requirement that the spectrum of h(t) decrease at leas~ as fast as )~i '!/2. The second condition is more severe, aas it,imposes a sui~-able annihilation on the spectrum of h(t) at the origin, in order go obviate the possible divergence stemming from the measure This second condition implies, hi particular, that the mean value of h(t) vanishes: that is,
/
' + ~ l,(,) dt :-: H ( 0 ) .... 0 .
(Z26)
Hence, the function shows at least some oscillations, and this is the reason it is called a wave/et. As in the c~se of the short-time Dmrier transfbrm, the admissibility condition of Eq. (9~.25). can be generalized to a mixed condition H(~')6'°(~) ~
2.,.7)
= ~,
which furnishes the ld~nt~tm~ of the "mixed recons{;ruetion"
2:(t) = ,
/i
J~
71~:(s.,a:g) h~a(t) dsda .... "-
g2
-
7~:(:s,a;h)9,~,~(t ) dsda (g2
on the one hand, and the generalized "energy con~rvation;' G. = .
.
// .
ds da
.
T~.(s,~.:.q)T2(s < h ) ..........=~.
.
0.2
(z2s)
on the other hand.
By its n~:essary extinction at tile origin of the tYequency domain and its decay at infinity, a w admissible w~>er_4ethas the character of a bandpa,ss filter (in a broader sense), We ca.n dins regard the wavelet transform as a continuous bank of const.an>Q filters'. This can be better explabmd, perhaps, by, rewriting the definition Eq. (2.24) of the wavelet transt)~rm in its equixva.lent, form 7~.(t,a) = a L'~
o'f oo~%::yc
X(t,)H~(axe)e*2~:"~& ,
(zeg)
J
which operates in the fl'equency domain. Obviously, tile spectrum X(•) is mukiplied by the scaled Fourier t.ransforn~ [a{ ..... H ( a , ) of the wavelet.
(~hapter 2 (.'/asses
i. . . . . . . . . . 1
{
I. . . . . . . . . . . . . . . . . . . . . . . . . . . .
of S~dutions
.....................
.........................
79
~.................................
~
. . . . . . . .
................
~
~
.........................................
i~ i
!
0.5i-- ~
0
I!
O05
0, I
F i g u r e 2.6.
0. I5
0.2
0.25 frequency
0.3
0.35
04
0AS
0,5
Filler bank with constant quality factor,
In its tbrm Eq. (2.29), the wavelet transform can be regarded as al~ analysis by a filter bank with co~Jstant, quality factor, or constant-Q fil~,er bm~k, In this bank each (bandp~s) filter is derived from one model by a frequency ditatkm or compression. The figure symbolicMly depicts several fillers of such a bank~ Suppose H ( ~ ) possesses a central frequency le0 and an equivMent band [l~o- B / 2 , ~'0 + B/2i. Then any vMue a # 0 of the scding parameter defines a filter with an equivMent band [ ( ~ - B / 2 ) / a , (~'0 B / 2 ) / @ Hence, t.t(z~) is a, template filter, whose c e n t r d frequency and bandwidth are modified by the act, ion of the atone group, while its quality N.etor Q (t.he inverse of its relative bandwidth) b'0
O = remains
LJ0/(t,
= 5-i,;?;
eonst;ant.
Although the wavelet transtbrm is a time-scale representation in the first place, it also admits a ~;ime-frequency interpretation by considering the variation of the sealing p a r a m e t e r a as an exploration of the frequency axis. This interpretation part, ieulm'ly matelms the situa.tion, where the analyzing wavelet is unimodM and localized to a neighborhood of a frequency v'o, w h i & can be used as a reference for the "natural" scale a = 1 (el. Fig. 2.6). The previously evolved argument a.llows us t.o look on the associated wavelet transfimn a,s a flmction of time and frequency by means of the fcormM identification 7e = I~o/a. Viewed from this perspective, the resolution of the wavelet, transform depends on the point of the evaluation and varies ms a flmetion of the frequency (ef. Fig. 2.1). This is opposite to the short-time Fourier transtorm,
'Time-b}'equencv/Time-ScMe Analysis
80
which ofli:~¢s an identical resolution at each point of the plane, Indeed, an anNysis with constant-O fil~,ers has a firm frequency-resolution at low fre~ quencies (i.e., for big a), at, the expense of its temporal localization due to the big dilation of the analyzing wavelet,. Convensely, at high frequencies (for small a) the compro~sion of the wavelet is in favor of the temporal resolution, but it reduces the fr~Nuency-reso}ution. Whichever point in the time-frequency plane is considered, a trade-off of Heisenberg type (cf. Eq. (1..4)) between time- and frequency-resolution always pe~>ists. Here it takes a local form, which func~,ionally depends on the evaluated frequency according to 1
~'~., (') ~? 4 - 7
At~.~, 04
"
More specifically, the const, a,nt-Q property of the wavelet a,nalysis implies that =
- -
z< ,
=. - - .
t.~
A.f,
.
(2.3t))
Vo
Let, us finally note that a wave}e~ transform and a short-time Fourier transform contain the same information, as both ~*~pres*ent the signal in a on.to-one correspondence. ~A,~can thus pa~s from one transfbrm to the other without ~osing information. A straightforward catculat, ion shows that they axe connected via the pNr of relations
F,:(t,
'
,:
~)
=
/]
r~(s,
a;
o) <~',~.,~ot.)
ds da a2
where ~(t) (and O(t), respectively) denotes the window (the w-avelet) of the short-time Fourier transform (the wavelet transtbrm, respectively). The atoms ~<(t) and ~.,(t) (or ¢~,,(I) and g,~.(t), respectively) emerge from Eq. (2.17) (Eq. (2,23), respectively). Discrete wavelets. The forementioned equivalence betw~m the c(mtinuous short-time F~mrier a,nd wavelet transforms becon:ms dissymmetrical when we attempt to discret;ize ~;he repres4mtat, ions. A naturaI w ~ ~of paving the plane relative to a wavelet, transform (el. Fig. 2.1) is to use a discretization that is denser in time (and therefore wider in frequency) for Nl points at a higher frequenw. Tiffs type of nonuniform mesh
{ (t, a) = (,~to<; "~. a o: ' ~ ) ;
n., m E ~ }
,
to>O, ao>O,
Ch~:~pter 2 C/asses of Solutions
S]
l
I
0t ................................
.
~. . . . . •.....................7 ............................
/
\
/ ]
-1
-£5
F i g u r e 2.7.
0 time
0.5
Mexican hat function.
A simple example of a wavelet is the "Mexican hat," the second derivative of a Gaussian. leads to the definition of the :l:~c[n, m ] = a~...."~
discrete wavelet transfi~rm
x(s) h* (ai')~s - nto) ds
,
n, m ~2 7Z .
(2.31)
The fundamental point, to be made here is that such a construction does not sufh~r from an obstruction of Balian-Low type. This meal,s that there exist "good" fl'ames ( i.e., the condition number B/A of the frame is close to 1) assoeiat, ed with the wavelet transh)rm, that are well localized in time and fi'equency. 2r One possible example is supplied by the Nnction h(t)= (2~tz) -1/2([-(t/6t)
2) exp (. ~(t/St)2),
called the "Mexican hat." (second derivative of a G~ussian, cf. Fig. 2.7). It verifies the relation At • Au = 5/47r, Moreover, the choices of a0 = 2 and to = ~t in Eq. (2.31) amount to a, ratio B / A = 1.116. The choice of a0 = 2 corresponds to a decomposition into dyadic scales (only one sequence of coemcients per octave). Combined with the chosen sampling of the time domain, this defines a r~ther thin Iattice in the plane (though it is difficult to make sense of the notion of density in the time-scale
Timc-t:requ( ~]q~t/Time-Scale Am]8<sis
82
1,5 |~ .........................................
~ ................................................. i:}
-t I
- 1
-0.5
0
Figure 2.8.
(} S
Morlet waveleL
Tim Modet wavelet is obtained by a modulation of a Gaussia~. The real p ~ t of such a wavelet is drawn ~s a solid line and its modulus as a dashed line. The number of oscillations of the Mor/et wavc!et ~s usually chosen, so that a good compromise between the quality factor and the respect of the admissibility condition is realized (see the text). case). One can show that a small degreo, of c,versampling in scale and in time improves notably the qualky of the decomposition. So by introducing two voices per oc{ave (i,e,: two sequences of coeNcients instead of one, which is realized by intert.wining two different dyadic decompositions) and taking t,} = 6t/2 (whidi means doubling the temporal sampling rat, e on each scale), the condition number B/A drops down to 1.0002. This results in a. practically negligiNe error, if one uses the fYame just Ilk{: an orI.tmi<,rnlM lx~sis. ~s Remark. This behavior is not a speciat=ty of the "Mexican hat." Ig is caused tV the w<,; d paving the p~ane, which is associa~:ed with a wavelet decomposition. It was tbreseen, and even experimentNly verified, by Morlet at the beginning of the 1980s. 2~ HN own defiIlition of a wavelet (today cNIed MoHet wavotet, ef. Fig. 2.8) w~s inspired by the baltic G a b o r signal. It h M die form of a moduIated Gaussiaa \
h(t) =
2"
I,,4
(-
t, 2
+ i2
0t}.
Strictly speaking, such a wavelet is not everi admiasible (in the sense of
( hapter 2 ('ht:s~cs
83
Eq, (2.26)), because H(o)
= ,+,2 : : ~ : )
'
e+'
[+7(2+to~<,)+)# o.
bT)r fixed to, however, i::. be(;olnes asymptotically admissible when z/o tellds to infinity, But in this case the flmction h(t) tends to look more like a wave than a wavelet,. The intermediate solution adopted by Morlet consisted d imposing a smMI (absolute) value em H(0), while only allowing a smaI1 re:tuber of os(illations of h(t). Hereby, the constraint H ( 0 ) i / m a x [t/(t/)i = 10 - ° led to a relation of the form 2rrt0t,0 = 5.4285. A cla;ssicd choice is to use a value between 5 and 6. M u t t i r e s o l u t i o n a n a l y s e s a n d o r t h o n o r m a l bases. The di~retization of a wavetet transform ca:: do even be~ter than providing good frames: In certain caxses it can be associated with truly ortho~mrmM base% composed of time-scale atoms, which are welI localized in time and fl'equeney. This point is of grea.t importan(::e and definitdy deserves a more comprehensive expbmation. As it: is somewhat b%'o:td the scope of this work, we sketch only its principles here and reik:r to other books for a compiete treatment of this subject. ~0 Let us restrict our attention m the c~se of decompositions of real finite energT signals into dy~die sc~des (a0 = 2). Then we set the problem of finding a wavelet '~)(t) ~ ~ , such that the family
{ .~/-,..(t) = 2"i2-~,(2'"~t
.- ,,)
;
~,,m c ~ }
is an orthonormal b~csis of L~(R). We t M s require t h a t
and
This problem admits at le~ust two solutions. The first is provided [V the Haar s:ystem qI ,ifO~t< 1/2,
~/?(t)
=
--1
, if t / 2 ~ t < l,
0
~ otherwise,
and the second tzv the Littlewood-Palq~" decomposition
q~(~')
=
i
, if 1 / 2 ~ I~JI < 1,
0
, otherwise.
(2.32)
8<:l
Time-l:7"equeuO/T ime-S{ al(~ 4nat>;sis
However, in both cases t I ~ basic waxe~l~et is badty localized i,~ one of the domabks and has a vet3 l~a regularity its ¢" ill fact discon~ilmous) in l h e dual domain. (These Iwo protmrties are of com-se linked ~o each other, as a function is Slllooth of order r, i.e., it has ~........ i COIltilKlOttS derivatixes, if i{;s spectruni decreases like ~eI~'" ai. intil~i~y ) tt was one of' the m a j o r a x c o m p l i s h m e n t s in wavelet: t:heor) to show thal a construction of bases wiCh be{,ler localizat.iou and reguIarity properties is possible, and t h a t one can cont¢ol either of these features to a certaii~ extent.
Muttiresotution analysis.
T h e COllSl;rllci.iot~ of orth<:m<>rmaI wax~elet bases relies on tim notion of multiresolutiou auatysis iu a central way. :;~ This notion gives a formal description of t:he intuiti,,e idea t h a t e,mry signal ear~. be constructed bv~ a s.m(es' ,,-s'~:(-, ~fim:m:nt,':~" -,~ e which m e a n s by. adding d.: tails to an a.i~Proxima~ion, m~d bv fterating this process, More pr(~:iselv. a. muttiresolution analysis of' g2(ll/) is defined to be a ~,quence of' nes{,ed subspaces
so t h a t
(i) (ii) (iii)
N [J
:r(G ~ E .
........ { , , } ;
V . is dense in L-e(]R) ;
"~=~*
r(2t) < E . + i
(iv) there exists a f~mctio)-~ g: so ttia~ {~:(t - ~) ; r~ ~ 7Z} is a basis of I"i~. W i t h each ~,;~ we can ~Lssociat.e a t.ime-resoh:iv.ion of 2"~, and the approximation of a signal re(t-) at lhis resolution level is o b t a i n e d by the pro~ jection ont, o ~,he corresponding subspace. Due to tim propert;ies (iii) and (iv) in the ibregoing list, a basis of g;, can be derived fl'om lhe ba~sis of 1/{~ in the following way: S~arting from the single function ~,~(t), which is called the stmliug fimction, the basis of k;.,, is obtained by taking the t~mily of dEa~es and trmtslates
It is e~.iden~ i,ha{: the coeflicients of the a p p r o x i m a t i o n s
(,Tmpt~r 2 C)%~es of Solutions
$5
which are a.ssoda.ted with ~he co:tection of all {hese bas~:s, s h a w a la,rg,, amount of infi)rma~ion. This makes it an extremely redundani represents-. don. Going back to the idea of describi~g a sigmd h~ terms of succes.siv~ approxhnatio:~s, a much more economical repn~.sent.ation consists of finding the ~nformatLon d~fferencc begween l w o consecutive a.pproximalions. This m e a n s t:ha~ we are interested in the de,,nit tha.~ nmst he added ~o the coarser approximation i~: order to pa.ss to the finer one. N)r ea~:h approximation space l/~.: this amounts to saying that the detNls belo::g co a spat:e I:V~.: which is t.he or~hogonal complement of E~ in E,,.~-i. Hence, we infer dw. reladon E...~ : ='
E~
~i:, W,.
whkh leads (o the d~;omposition
Therefore, i{ is enough to find a function ~;.,(t) (which is the proper u,a~leQ, so tha.~ {'~.(t ......~t); 'r~ ¢ 7Z} is a basis of t ~ . Then the set
{ ~:,..~(t)
...... 2~"~7%(2"q
- ..) :
~ , . ~ e ;g }
constitutes an ort.hoI~ormal bases of L 2(~).
ScMing fimction. As the scaling function ~(t) is a,n element of U~, it is also in l,'~. So there exists a set of coeNciem.s h[n], which give
~,(t? =. d~ ~ t :~
ld,~l p(2t- ,,), - "
(2.:-~3)
iN
a:::d
/7
h[.,d ~: v..,s
~(t) ~(2t
......,.)
,
~,'i,d = :: .
t)ec~mse t,he integer tra.slates of the scaling flmction ~(t) tbrm a ba;sis of ~}), the discrete filter with coe~cients hire] ha~ vet')." particular propert:ies. The t')Jurier trat~form of tl:~e t:~,o-scale equation (2,33} t:)r ~.(t) gives
(2.34)
e(~,) = II(,.,M) +0.,12) whel~ @ <7~
h'~ i c
.
Thu+',,f)'equencS,;/Thne-S( ale Amdy:*is
86
This trm~sfer function (which is periodic witI~ period 1) cannot be arbitrary, because the orthogonalhy of the shif~:s ?, !fit) in I,% implies that;
T~ : : : : :
'
m~(lthis
-
7X2
yields the result
I+(-
+
.)i ~ = ~.
(2.3s)
FSJrthermore, by bringing out the transf)r function ((f. Eq. (2.34)) in rids equation, using its periodicit5 and putting u = 24~, we obtain ,,, 2 <
=
2
IH(<
+ ~ ) 2 I+(C + :~)l ~
d- <',~
k :=" --
~,>
+,:~
=IH(<)I2 Z
+ "x,
I'I}(C +k)l~ + [H(<+:])i2 ~
l+(<+;}+k)I2,
Hence, Qle reladon
~m>)i- + In(,, + ~;i- = ~
(z3G)
follows for all u by an applicatkm of Eq. (2,35),
~A.~welet. Let us now return to the wavelet +~,()). As it is an elemen~ of l/~, it is also chm~azterized by a, discrete tilter with coefficients 9[n], so tb~t
C h a p t e r 2 Cta~sses o f S o h u i o n s
87
~rld g[n] = v/2
+:~
~×:E 4- g2[n i =
~',(t),p(2t ..... n) d t ,
I .
With the same nota£ioaal conventions as befbre, we find that
,~(.) =
a(~,/2) + ( . / 2 ) ,
where, G(~) is t,he (1-periodic) t,ransfk'r function of the discrete filter g[n i associated with the wavelet '~!~(t). Atso note ~ha.t .~(t) is an element of t4)~. Thus it is orthogonal to the space Te~),and this gives
+,'x:
=
,1,( ) ~ (~)e,
--
&7
~(~ + n) ~ (u +
d, .
Because this identity is true for all k ~ ~ , we infer that 4- ;x5
If we proceed as before, which is bringing out the Nnctions H and G in this equation and using their periodicity, we finally obt~Nn
a(, ) H" * (.)" + C;(. + 5:~)H*(u + 1) = 0 .
(2.37)
The discrete fitters h[n] and g[,n I form a pair of quadrature mirror N~ers ~2, employing the nomenclature of the theory of filter bamks. Fbr given H , the solution G to the preceding equation has the form
a(~,) = a(~,) H * ( . + where
A(~) is
½),
a 1-periodic flmction, so that
a(~,) + .~(. + ½) = o . One possible choice is A(-) ......... exp(i2rr~}, and this gives
c(,) = ~,~~<'~+~)H*(, ~-. ~)
=~
g["i = (-- 1)'V,[, -,~].
(2.3S)
Time-F)'eq ~eu~:y/Tim(~Sc~ te ;4 n a/3csis
SS
It still remains to show that the integer shifts of' the wavelet .t,'~(t), which is constructed in this wa,;, form a basis of llqj. We show in a fi~st step that @ 2~
@ ;x2
E
-
H~,~
+'~)]~ I~(~ +~ ) t ~
+iH(~)l ~ :
1,
Hence, we can conclude that e ( t ) ¢ ( t - h') dt =
I ' ~ ( . ) l 2 e ~2~''~k d~.
t a t us now surnm~rize our findings: Stz~rting from a scaling function ~(t) (such that ~he family { ~ ( t ~ n ) ; n c N} is a ha,sis of ~;) and coefficients h[n] of the ~s~ciated discrete fitter, tile w~Nelet
defines a se~ { "~,,.~(t) = 2m/'2g'(2~'t ..... ~) ; n, m C 7L }, which cor~stitutes a b ~ i s of L2(~).
Remark 1, The seating function p(t) and its associated filter h[n] have the characteristics of a Iowpt~s filter, while the wavelet '~'(t) and its related fitter g[n] resemble a ban@a~s filter. I n d ~ d , we inter from Eqs. (2.35) and (2.36) that: the relations t<5(p)I < 1 and IH(y)I < 1 hold. More~)ver, we can write • (~) = H(./2) '~(~,/2) = H(z,/2) H0//4 ) ~(~/4)
.
Cha.pter 2 Classes of SoIuth)ns
89
By iterating this fa.ctoriza~tion process and passing to the limit we obt~dn
~,(,..) = e ( o ) H
H (2 ........... ) ,
This indicates that '#(0) must be finite and no,lzero. Hence, it fbllo,vs that ~,(0) =
/,,i ,
~(t) de •
@o~
and this gives H(0) = 1. In fact, it also implies H(1/2) = 0. Now we can easily, deduce that G(0) = 0 (which yields 'g~(0)= 0) and [G(I/2)I = ~.
Remark 2. Incidentally, the relations of the quMragure mirror filters ira-. ply that (ef. Fig. 2.9) IH(,,)} ~ + la(v){ ~ = 1 . By using the nullity of ,;F(t/) at infinity (due to its lowpass character), this leads to I+(~,)t ~ = I~(2,~)l 2 + I~'(2-)] ~ = I ~ ( 2 . ) t ~ +
I~(4.)[ 2 + i~,(4.)[';.
Hence, a.gNn by iteration, we find that
I+(,,)I ~ = ~
I~' (2'"-)t ~
for every nollzero fl'equency. P y r a m i d a l algorithms. The introduction of the discrete filters h[r~.] and 9[n] is a major tool for the practical computation of the coe~eients of the
approximation
axing m] = [
x(t) ~.,,,.,~(t) dt
(2.40)
'1 im~-Fr~q~emK/Time-ScMe A t~a/ysis
90
/,/ •
/ ,
/
/ z
-0.5
?,
/
,,,
-0,25
0
0.25
(t, 5
frequency F i g u r e :2.9.
Quadrature mirror filters,
For orthonormM wavelet bas~, the fih~r coegicients h[T~ and gin} of the scaling fimction and the wavelet, respectively, ~'e linked to ea~.h other by the "quMratm'e mirror filter" relation. The figure depicts this situation symbolically. The (Iowp~xss) pattern of the scaling filter is represem~.t by a d~sh~d line and the (ba~tdp;~ss} pattern of its related wavelet filter by a soIid line, The "quadrature mirror" relation IH(u)t 2 + IG(v)" = 1 deri~~s:i from Eqs. (2,36) and (2.37) connecU both.
mid th<, de, rail
of an o r t h o g o n N wavelet decomposition. Ana/ysis. gives
A straightt})rward computatiol? bas~ct on the prec(~ling relations
a:,,[7~, m] =
:c{t'}. ~:
' "~(~
t
.....r~} dt
2X;
l E
h[k]/_
:c(t) 2':"~+~ '2 ~(2'~'~+I! ~,,,,,,(k + 2u))dt
Chapter 2 Ctazs6,. of botut~ons
91
We have thus proved {,he f'mmula @ :x.
]¢'::::: ---~i~>
One can show in the same way that
-
.
(..4,~)
This shows that the coeNcients of the approximation and the detail at a fixed resolution le-el ca,n be derived by means of a filtering, followed by a decimation, from the known coefficients of t,he approximation at the next higher Ievel. Operating sgep by step, one t&us achieves afks{, and recursive algorithm, which only brings two discrete filters into play for the iterative procedure. Such an algorithm is called pyramMat, 3a Regarding its practical aspects, the initialization of the Ngorithm is pertbrmed either hy a projection of the analyzed signal onto !/i~, if continuous data a,re given, or by mapping the available sequence of sampling values into |his sp~:e, if only discrete values are known.
,~ynthesis. The analysis" scheme can be inw~rted, thus lemming to a dual algorithm for the synthesis, ttere an approximation at a fixed resolution level is derived from the approximation and the detail at the next lower level. In order to establish the st~ructure of this algorithm, we need only observe that the approximation .r,~.(t) of a sigr, al x(t) at a fh'ed scaling level m is obtain(~l by the projection of x(~) onto V.-,. As the family {~n~,(t) = 2m/2~(2mt -- n); n C ~ } is a basis of this subspace of L~(~), we can write @ ¢.3~:,
= Z
gt~--cX,
Recall that G,,+l = t'~,,(:i)14%-,zand that tile ort;hogonat subspa(e l,l(. admits the basis {g,,,~. (t) = 2'w'2~/a(2"t -. r~); rz ~ Zg}. So we can conc]ude t,hat
xr,+,(*) = *,,,(0 + The projection of this equality ont;o ~2,,,,m+-1(t) gives
92
AnaIysis
Moreover, it is easy to see that =
/
<'~ 2' '" "~ " ¢ # "-" (2t-
k ) . C, "~* b ' s ~- #. ( W
+ t -
n) dt
x;
-==
v'
=
hit,
/{~
-
~4t) ~,_o,,~/'~t -
(,
2#))
a:t
2/~:] ,
and by an a.nalogous reasouing
Hence, we tiredly co~clude that the wam;ed reconstruction has the coefficients
<,@,,,, + 11= ~ h b - ~#1. ' 2~I,~..,i ~ Z .@' -- 2A!d,-[~-;,q, ( z a ) In contrast to the aIgorRhm for the anMysis, which pertbrms a filte> i~~g followed by a decimation, the algorithm %r the synthesis operates by interpolation first, folhYwed by a filtering. Bo(h Ngorithms for ~he anatysis attd the sy~thesis a ~ schematieMly represented in Fig. 2.10. One can observe tha4 glmy bring two identical ca.scades into plas. Th~se rela~ed to the a;mlysis (A) cot~sist of the two fi~ters H and (7, fothm~t by an operation of decimation, which onty k~.,pns every other output sample. Conversely, the cells associated with the synthesis (S) first pertbrm an interpolagion of the input (by inserting a zero beg-weeD two consecutive sa,mptes), a~ld then a4@ly the fibmrs H ~ and G ~, which are the transposes of H a,nd G (iD the sen,.~, that h q n = h[--,z] and j[.] = £ - @ . Some wavelet bases. Relying on the general framework for the c~mstruction of wa.velet b~ses that we ,just. developed; we can now turn to some special examples.
Ham'. The simplest example of an orthonormal wavelet b~ks e m p l w s the scaling f~mction ¢(t) =
i
,if0}i~t<
0
, otherwise,
1,
Ch~tpter 2 C/asses of Soh~tions
93
~, -2] •
a[n, O]
din, -It
•
•
dfn, -ml
d[n, -21
a[n,
-21
A F i g n r e 2.10. Pyramidal ~lgoritlm~ for analysis and synthesis by orthogonal wavelets, The analysis and synthesis by orthogonaI wavelets le~/(t themselves to an efficient computation composed of idet~tical units, which are ~tsed iteratively, For the analysis part, each unit A maps an approximation to a coarser a,pproximation and a detail by the action of a sca]b~g filter H and a wavelet filter G (with impulse respons~s h[n] and g['@ respectively), both followed by a decimation by a factor 2. For the synthesis part, each trait, S operates o~ an approximation and a detail by producing a finer approximation. It first pertbrms an interpolation (inserting a zero between two successive samples), then a filtering by h'[n] = h i - . 1 and ~/[~] ~': .q[-n], respectively, and finally adds the obtained output signMs.
-m]
94
Timc'-t~?eque~cy/Time-Scafe Amdvsis
which essentially reduces to the investigation of piecewise c.xmtinuous sig:uNs, The relatext fiher is given by h[rA = ~
¢(t) ~(2t .....n) dt .::~ 1/v~5 0
, if n .... 0 or 1, otherwise.
This leads fi~rther to
.q{.~,]: (-:I
.... ,~1:
{ +I/v5 -I/.,/5
, i f n = 0. , f i n : 1,
0
, otherwi~<
Consequently, the ~ssociated wavelet has the florin ,~J>(t) : ~ ( 2 t ) .... ~ ( 2 t - 1 ) ,
or in explicit terms
{
+t
{;)(t):
if 0 < t < 1/2,
-I
,if I/2~t<
0
, otherwise.
I,
We can thus recognize the Haar wax~let of Eq. {2.32). Battle-Lemari~. Even if the Haar system forms a b~~x~isfor L2(N), it empko's a wavelet that is not smooth (in thct discontinuous) and badly locNized in frequ(mcy ( i.e, its spectrum only decre~k~es like [z/i-1). More regular bazes can be constructed by imposing some eonstrNnts on the scaling funetiom For ii~stance, we can demand that this function generate a space of •Eptine flmctions: Then the f a m t y of Battk~Lemari6 wavelets is obtained, which appeared as the firs~ generalizations of the Haar wavelet. 34 Let US define the scaling funcdon
p(t) =
{1 0
-!tI
,if0~Iti
, otherwise.
It is continuous (with discontinuous deri~tive); ho,~,ver, the collection of its integer translates does not form an orthonormal family. Nevertheless, we can derive a function p+(t) from p(t), which has this orthogonality feature. It ks defined in the frequency domain by putting i .,]ii
Chaptor 2 (7tasses of Solutions
!)5
The foreg~}ing ~ries has an explicit form (axs a trigonomet.ric polynomial) :m and riffs e~mbles us to firm the fl'equency response of its ~lat
i sin ~r~/yv+~ i <
t¢'(v)i . . . . "
7f/J
i
--
C(1 +
, l,,i)-~,',+,)
if N is the degree of the sptines. Because the considered sea,ling funct.ions ~(t) dec~o~ exponentially, one can show that the same feature remains true for ~+(t) and t,he wavelet, g,'(t), which are derived from it. By taking higtmr and higher degrees N of the sphne we can t t ~ construct wavelet ba:ses, which are generated k~:"a function that is we11IocaIiz~ hi t/me (due to its exponential deca)0 a.nd has a high degree of smoothness. The regularity of a wavebt is c l ~ e t y associated with i~s properties of eamxqlation ( i.e, the number of vanMfing moments). More predsely, ~(t) being a member of the class C ~ impli(~ that
f
)" + ~ t a; t~;, (t) dt = 0 ,
k = O, 1 , . . . . , r ,
or equivalendy da"~ (0) = 0
k = 0, 1 , , . . , r .
dvk
Daubc,'hies. We have just scer~ that starring from spline ftmctions leads to w~velet bases that are well localized, yet have an unbounded support. This means that they are defined via di;screte fibers with an infinite mm~ber of coefficients h[n]. The newty set problem to be sotved now is, how one can find wavelet,s that satisf~¢ the stronger property of having compact support. Such solutions invoive only a finite number of coeNeient,s in the related filters. :~s For this reason, it, is naturN to begin with a scaling flmction d m t itself is compa,My supported, even in its orthonormal form. An immediate consequence d this restriction is the fiu:t that H(~)
1
+~
Time-f:requ~nq}?STime-Senle Analy.~.s
96
must. be a t~qgonometricpo]ynonfia,L It is a 1-periodic funet, ion and must fulfill (according to b;% (2.36)) the "quadrature mirror" relation
iHO,) + IH( + })I : I. The problem b(~:omes even more constrained, if we further demand that the solution have a certain regularity, becanse the necessa.ry smoothne~s of the s p ~ t r u m of the wavelet at the origin carries over to the frequency respot~e of its (bandpass) filter. In accordance with the ~'(luadra~ ture mirror" relations of Eq. (2.37), one can show that the (lowp~s) filter H must have a zero of multiplicity r at the frequency zJ = 1/2, if it corresponds to a wavelet v:ith r vanishing moments. Therefore: ig admigs a factorization
H(z,) where L is a 1-periodic function and belongs to C ~, if" H does. In case of a finite impulse response, L must be a trigonometric polynonfial. As the quadrature mirror relation only brings t,he squared modulus of H into play, we can put
where tile polyncmfial P must satis(v tfrc relation (:~ - z ) ' P ( z )
z " P ( I - x) = t .
Daubechies showed that such a polynomiM is given by
After a spectral factorization of the resulting function IH(u)t 2, this provides a solution to the posed proMem: it defines a compactly supported wavelet with 2r nonzero tilter coetficients and a reg~flarity, which grows linearly with r. A commonly used practice for the spectral factorization insists o~ the property that all zeros of the mmsfer function lie inside the unit circle. This corresponds to the minimum-phase solution. Mem~while, other choices are possible. One can c o ~ t r u c t a filter that is closest to a symmetrical (or linear phase) ~lution. The perfect linearity of the pha~e, however, is incompatible with the compact support, hi case of a (reM) orl,honormal basis. 3v
C:lmpu~r 2 Cla,s~se~so£,~olugions
97
Some examples of (x)mpactly supporv.ed (Da.ubechies) waveiet~sa.re dis-pb~yed in F'ig~ 2,11 (right column). Their related scaling flmctions are also included (left column). The simplest example, (he Haar waveLe~ of Eq. (2.32), is drawn on rap. It is chax~u:terize(t by only two coefficients and is discontinuous. The next: two exampIes ,~se filters of 4 and 16 nonzero coefficients, respectively: They clearly demonstrate ~he influence d a growing size of the fiiters on the regulariV d the b,~sis flmctions. The shown solm tions of these two examNes correspond to the &oice of a minimum-phase faetorization. The I~st example displays the result for a filter with I6 eoeg ficients and a spectral factorization, which leads to an almost symmetrical solution.
2.2.4, A "Detection-Estimation" Viewpoint There exists a |arg~ w~riety of problems in signal processing (sonar, radar, ultrasound ~an, nondestructive testing, telemetry, etc.), which can schematically be fornmlated in th~ fbtlowing way: (i) a know~l signal x(t) is emit;t~l for ~questioning" a system; and (ii) a ';r~ponse" is received in the ibrm of a signal y(t), whose modifications relative to x(t) car~;y some information about the system. Although such a scheme is initially directed at problems of the "detection~es~imation" Wpe, it turns out to be useful as a formal framework for investigating atomic deeompc,sitions of a signal as well This allows us to look upon the previously discussed ~)lutions from a different angle. Ambiguity functions. In order to be more precise about the "det~:tionestimation" point of view, let us first co~ider the classical problem of a statistical test with Nna.D" hypotheses
{
H,
:
y(t) = b(t) y(t) = x0 ,(t) + b(t)
Tile nuI1 hypothesis in this model signifies that the obser'~-ation y(t) contains only noise b(t) (supposed to be zerc~mean Gattssian white noise). The alternative claims that there is a userS1 signal xo,~(t} superimposed with the same Vpe of noise. The first problem arising here is the detection, that is, the determination to which hypothesis t,he known observation should be attributed. A second problem arises when we assume that the signal xo~,(t) to be detected is equal t.o a reference signal x(t), which is known apart from an ur&nown parameter vector 00. Then the problem of estimating this parameter vector is set forth, provided tb~t the alternative Hi has been selected.
9S
Time-~)'equenc)'/25me-Scale An:dy~'is 2 coefficients (Haar) ,.%
0 -1 0
4
8
-4
0
4
8
4
8
4 c(xefficients, minimal phase 1
~~ 0 l........................................................... ~-1 4
8
d
f
-I -4
0
16 coefficients, minimal phase
O
0
-lt 0
5
10
15
-5
0
5
10
15
16 coefficients, quasi-linear phase I r
C
1 l
-1
' 0
5
t0
Figure 2.11.
15
-5
0
5
10
t5
Daubechies wa.~vetets.
The Daubechies wavelets form a t~mfity of compactly supported o f (houormM wavelets. They are parametcrized by an index that controls the size of the filter, the support of the wavelet, and its regularity.
Chapter 2 Classes o~"SoIutior~s
99
In the forementioned setting, the theory of optimM decision :~s (e.g., in the form of a maximum tikeiihood) dictates that we should retain the alternative Hi if the quantity
,,,ax{ a(O) = (?,, 0
exceeds a certain threshold. The strategy of an optimM detection thus comprises the following steps: one correlates the observation y(t) with a battery of copies of the reference z(t), which is modified by conceivable vtth~es of the parameter vector 0; then one decides that the searched signal ix present, if the largest of the obtained wdues ties above a. threshold. Intuitively, this maximal value should be obtained when the tested vector and its true value cob~cide, because then the (noise.. free) correlation is maximal~ One can actually show l;hat this ix t;rue in general and that a proper estimate for 0 is
0
Remark. As the procedure of detection-estimation compares the observed signM with e~ known reference signM (in the sense of their correlation and tbr each vMue of the paranmter vector), it is called a matdxed lilt~ering. Two simple examples of this approach are fitrnished by ra.dat" and sonar. :~ In both cases, a known signal is emitted and its echo is recorded. Apart from the additive noise, and ~s a first approxima.tion, we can consider the differences in the structure of the emitted a~nd the received signal as caused by the existence of a propagation d d a y (reIated to the distance between the emitter and the target) and a Doppier effbct (induced by the relative motion emitter~target). Therefore, the vector 00, which pa.ram~ eterizes the modifications of the emitted signal z(t), is a vector of two components: one for coding the delay and a second one for the Doppler effect. Let tts momentarily suppose that. the emitted signM he~s a narrow band a n d / o r the relative velocity is small (whieh is most often the case tbr r0,lar). Then the Doppler effect is manifested by a global frequency shift of the signal, called Doppler shift., tIence, we can use the transformation
as a model for the underlying modifications of the emitted signal. Here r determines the delay, and ~ denotes the Doppler shift. Certainly, this transformation can be rec(~gnized ~ the action (Eq. (2.17)) of the group of time-fl'equency shifts, which a~ppear in the short-time Fourier transform
100
'17m~,-t:7"cques~etv,/77mc~-Sc;Ue A~sab~e~is
and Cabot analysis, h~deed, if we come hack to ~he statbstieal decisio~ prohlem, we. immediat.ety obtain {using, e.g.~ the uotation as ht E% (2.21)}
Thks is tormally equivMent to a short-time Fourier transform of d~e obser.ration with the window flmction being the emitted signal. Bearing in mind the nature of the observation as "sigr~N noise," we can finally write A(0) = A~(~ .....~0, r - r0) e-i~:t~-~'~' + fluctuations with +- ::y;.
..i.,(L-r) =.
('r,.,.F, rl.
:,.:It):J (t .
=
;,-)<.-~.,u.dl,.
(2.45)
This timction is called (narrowb~nd) ambig~.~ity bract:ion in the literature o~ radar (or ambig~fity fnnc~ion ir~ transladolx, or agNn is~ Woodwardls sense) '~% It, essentially measures a t,ime-[7~,q~le~qg corretation~ t~h~:~tis, the degree of resemblance of a signal to its different tra.nslat~.~s in the plane. Playing t.tm role of a corre{a~ion function, its maximM vMue appears at the origin, tlence, one can see that, apart from the additive fltlct:uatio;ns, l;he decision statistics are maximal near the true value of the delay-Doppler pNr in tt.u? ambiguity plane. The bet~er the emitted signM is jointiy localized in time a.nd frequency, the bel.ter is ~he esi;imation of this pa,ir a priori. If we extend Ibis definition I,o iwo interact, ing signals, the short-time t%urier translk~rm can be regm'ded, in a deteciion-.estimation contexg, ~s a cross'-ambiguitiv timcdoss of the signal a~}d the window. This teas
.(0 The
]DaritllteU?,r 1] is a
~
.:o(t)
c()ntpl~;s:sJoli
=
,/4 ~.(.,~(t- ..)) .
/~l(:~of o r
DOppJes:r a t e .
Herein
we call
recognize the action (Eq. (2.23)) of the aN~e group. Hence, an anal(Nous line of arguments az~ before leads to
x(0) = 77,0-,
:i/~7;:~').
Hence, the wavelet transform of the observ,,d signal na~.urally appears in the decision statistics, if a sh-nple Doppler shift, is not satisfactory (e.g.,
(7tmpter 2 C!lasses of SohltiolJs
t 01
wideba.nd signals). File analyzing wavelet is fi~rmally Mentified with tile emitted signal. The corresponding ambiguity fl:mcl.ion has the form A . , ( . , ~) ~: 'i~, (~, 1 / . ; . )
.~,(t) ,,* (.,,(~. - ~ ) ) d ~ ,
: ,/;i
(2,4G)
....:XJ
and we obtain that
~(0) = &- 0;~/~0, (r - r~)/'~lo) + fluctuations. This new flmction is (:ailed wideba~M ambiguity &netion (or a.mbigu; ity function in compression). ~i It, also has the properties of a. correla:tion flmction in a.n Mline sense: it measures the degree of resemblance of a signal to its dilates and translates. By an extension of this definit:ion, ~he wavelet transform can be regarded as a (wideba.nd) cross~.ambiguity &nction of the mlalyzed signal and the analyzing wavelet. So both eases, narrow- and widebmld, 1W open an equivalence of the mathematica! structures between tile decompositiou tools, which are the shortq:ime })~mrier and the wavele{, transfl)rm, and the decision t:ools, such as the narrow- and wideband cross-ambiguity functions. (We note in passing, and for the sake of recalling a, previously used notation, that both (auto-)ambiguity functions a.re nothing but t;he rq)rodudng kernels of the associated linear decompositkm.) As a consequence, many properties of the considered representations can be fbund in the (a priori remote) literature 0 1 1 radar and S O l l a r , A t o m s and m a t c h e d filtering. Once we have recognize(t the equivalence between the structures of decompositions and decision statistics, we can easily find a natural interpretation. For this purpose, let us continue regarding the analyzed signal as being "received" and the anNyzing signal as beiug "emitted." Then tim decomposit;ion of the signa.1 into atoms amount.s to identifying, at which place and by which weight, the different atoms of the signaI ma~fifest themseh't~s. This means that we test for all hypothetical components ~tmt can be derived from the elementary anNyzing signal t:~v the action of a natural group of transformatkms. It thus corresponds to a problem of detection (are there atoms?) and estimation (where and with which weight:2), The use of "atoms" (in the sense of wetMocNized signals in time and frequency) as elements tbr the decomposition corresponds to the goal of emitting signals that permit a. fine estimation of the transforma.tions ilk dueed by the questioned system. We observe, by pushing the analogy further, that a. "good" emitted signal should ha:ve an autoambiguity timelion that is maximally concentrated about the origin. In a context of signal
Time-t;"eque~(37:Time-Scvde Ana/ysis
102
tmatysis, this comes t:o requiring theft the dif[iereni atoms approxinmie the, situatiolt of a O~sis: their cr~>~s-a:mbiguiW functions shouM be zero or net-. Iigibte.
Rema, rk. "I;he coi~stra.int of a concentratioi~ of i~s ambiguity flmction does not necessarily imply that the signa~ ii:self is weil localized in time a:t~d frequency. The chirps with large duration-bandwidth product Nit in this ca~tegory }~y an effect cMled "pulse compressio~l?' This is weI1 known in detectio~t tbeoLv- Moreover, one cm~ recognize that, without havii~g a large dura.tio~bbandwidth product, the l)aubechies wax~elets with miI~imal t ) h ~ e (and sufficie~tly high reg~flaxiry) have s~!t~ a s~:ructure of a chirp (see Fig. 2.11). Although the mathemati(at structure of linear decompositions and (cross-)ambiguity flmctions is the same, they still feature major di:tf~rences concerning the useful range of their variables. This cain be welt explNr~ed, in part.ictflar, lk~r the Doppler eft>or. Indeed, the Doppler rate relative to a rMiN velocity w is expressed by t + wit *~ -
I - w/e
when c is the spe~d of the propagador~ of ~.he waves in {.tie medium. Hence, in the ease of eIeetromagnetic wave~s, fbr which c = 3- 10s m/s, a radiM velocity of 3600km/h leads to the smalt value of ~? ...... i ~ 6.66 . 10 {;. This justifies perfectly the approximation t] ~ 1 + 2w/c commonly used ila rMar. On the other hand, if we consider the case of acoustic waves in the air, for which the speed of the propagation is c ~ 340 m/x, a velocity of about t l 0 k m / h leads to .r] ~ 1.2. AIthough we cannot u ~ the preceding aNm~ximation for this value of ,], it still stays close to one. However, a wa~'elet ¢raasfbrm usualty anNyzes the signal over severn octaves (or powe.rs of ten); which uses a scaling parameter that. notably digresses from olm. Expressed in a shortened t.orm, one can thus took u n t o a wideband cross-ambiguity function a:s the fine exploration of a wavelet, transform in the neighborhood of the natural some (chosen to be 1) of the analyzing wavelet. A finaI consequence of the detection-c~stimation point of view fi)r linear decompositions results from their interpretation as ma~:d~ed filteririg. In fact, it iv intuitively clear that an atomic represer~t~tion is all the more pertinent (and economic in the sen.*a~ that it ha~s fewer significant nonzero coefiffcients), as it brings imo play atoms that preferentially exist in the anMyzed signal. Precisely at thLs point., a philosophy of matched filtering can be disec~eered. This paves the ",~7ty for less general and less rigid deeompositiop~q than t~hose, which are bound to an anatysis by a constant
Cha:pt;:er 2 C1<~;s-,:,s~e~,o£ Sotutio~,s
10',g
or constant-Q frequenc.3, band. The a.nticipated ga.in in efI~(ieimy of the representati(m nmst clearly be compared with ali expense, either cotmted ~s an a priori knowledge about the nature of the analyzed signM, or as the implementation cost for an adaptixs~ algorithm for the selection of the most pertinent atoms. ,re
§2.3. The Energy Distributions In contra~st to the linear (atomic) representations, which decompose a signat int.o elementary constituents, t,he objective of a joint e~i('~rg2v'distribution is to redistribute the energy of the signal between the two '~'iables of the description. General setting. As before, we ~re going to consider the two laxge (:lasses of time-frequency (k := ~e) and tim<~scale (A = a) distributions. The present goal consists in finding a distribution p:,.(t, A), so that
~:~ = i/" p~(t' a)d,<..(t, a).
(2.4.7)
R~
As tim energy is a quadratic entiW by nature, it suggesgs a sea~'eh fbr quadratic distributions (bilinear or sesquilinear), even though this is not a necessary constra, int. In other words, we are after generNizations of the welgknown temporal and speetrN energy densities £:~. =
i~.(t)12dt =
IX(~)12dL, .
V~e already noticed in Eqs. (2.21) aitd (2.28) {,hat some quadratic by~ products of tile linear de(:.ompositions, s~lch a~s the spectrogram or the seM(> gra,m, can play the role of such energy distributions. This results f¥om the identities
•
/7
iF~(t,,,)i2 dtd, ~ =
Si,
T<,(t,a)l:~
dtda a 2
== E , , <<
which axe valid subject to adeqtlal:e normalizations. However, the squa.red modulus of ~ linear tra.nsformation is only a very particular case of a bililmar transformation. It is, therefore, important to find more general solutions to the posed problem. A eoneeiv~ble approax:h is to start l¥om the generic form
a~=(t, a) = i) r ~'(,~,, ~,'; t, a) :~,(,,.) x ~' 0/) d,,, d'a', / J
(2.4S)
104
]l~me-Frequei~q,~q/Timc-,~:cale AnaI>'sis
which is parameterized b}, a kernel K. tn order to satisfy the, constraint of an energ3~distribution E% (2.47), t.he kernel K ha;s (;o be cilosen such that
. f / K('. E; t, ;~) d#c(t, a) = ~(~, - u') .
(2.49)
Cov~riance principles, After the general setting has been fixed, the a(> tual choice of the kernel for the parameterization requires ~.,he imposition of additional constraints on the searched distributiom One way to pro~ ceed is to specit~~ "natural" coILstraints lot ~i~e distribution p:,:(t, A}, and express them as aAmissibiiity conditions for its kernel K(u, u/; t, A). We g i ~ precedence to those col~straints, which are related ~o principles of c ~ variance; this means that the effect of certain transtbrmatioi~s of t,he signal can be observext on the signM itself a~ well as on its bilinear representation, In other words, if T denotes any transformation, imposing a covariance principle relative to T is equix~]ent to demanding that the diagram :r
T:r
~
~-~
Px
PT~ :: Tp~
be commutative. (The operator acting on the distribution must be considered as a natural exLension of the operator T acting on the signal. This wiII be made clear in each considered case separately.) Tim first covariaime principles considered here are naturally linked to the transformation groups of ~he time-frequency and time-scale closes themseb:e~. \ ' ~ wili soon see that such a point of view, as simple ~ i~; may look to ~_anow, allows ~he space of solut~ions ~o be reduced considerably. On the other hand it will still comprise a wealth of possibilities. In particular, we will show how t h e e covarianee principles generate cergain classes of distributions "'~ la Cohen." This surpasses the empirical remarks made before in connection with the issue of unification~ Instead, it rather replaces them with some constructive approaches, 4:~
2,3.1. C o n s t r u c t i o n of the Bilinear Classes T i m ~ f r e q u e n c y . Our go;~l is to define a general class of bilinear timefrequency representations. Let us therefore consider the operator of timefrequency shifts, whose action is defined by x(t)
-~
:~,~,~(~)
= x(~ -
~') e ~''~
.
Cfml)ter 2 (T/~;~'es ,~f Solm:ious
I05
hnt osmg the covariance with respect to this o~erator ll~e~$iis tha,t a time-frequen.cy representation must "tbHow" the signM, whe~ it is shifted in the pMne. The constrair, t can thus be written as
p<,,(t,
~,)
=
! i/ - .'). p:~(t - t,~
(2.so)
When we inser~ the pa~'ameterization equation (2.48) of the distribution by its biiine~r kernel i:~to this relation, it is ee,.sy t,o show that
......
,
..~
,=
-
-
x
( ,. ) d~z d~,
The equality of t h e ~ two quantities for eve D" sigmd implies that the kernel must verify the relation *'"/ -4~",~,'a i
~s:,'t~ I + t ; t , 1•
If we now x t ! = t and
W ) ( ~• 2 ~ / p.!
K(u,u';t,u)=
*
t~'"
z =h~(t,
/ tt ' ; t - - t % u -
//'}, .
= t / a n d reorganize the variables, we find t h a t
K('~t~-t,*t'--t;O,O)e ~-~2~''('~-'/~ .
(2.5I)
T h e kernel, being a priori a function of 1bin" vaxiables, theretbre turns into a t~anc~ioJa of only t~wo independent variables. This is an immediate consequence of the preceding constrNnt of covarimme with respect to timefrequency shifts. Rem~trk. There N an analogous situa.timi in the space of ~inear operators: imposing the co~ariance relative to the temporM shit, s cuts down the cla.ss of M1 linear operators to the subspat:e of linear fi/tens. 44 tf we use this fbrm d ~,lm kernel in the generic expression of Eq. (2.48) and perform a change of variables, we obtain that
p:,. (t, K,) =
K
s ......t ~
~,
s-t-5;0,0 '7"
7"
H)6
t'ime-Frequet?c> /Ti)~]e-Sca]e Analysis
Now we can rewrite the integral with res•~)ect_ to r in it:s fi"eque.ucy, ' " form~ By Ietting
H(t,~e)=
K t
6't
~;0,0 e . . . . dr
and W,,(t, ,~) ....
(
:l," t - r,) e...~:~,,~d.r
t+
/
we end up with the final result
p,(t.,.,) = f f 1fts .... t,¢ /d
~,) W . ~ i s {) as
........
(Z52)
Recall born Eq, (2.3) that U;~,{t., ~} is the Wigner-Vilie dis~ribut, ion, It is easy to ~,? t.hat the tbregoing germral form can also be written a~
C:,(t,u;
f)
d;s dT ~a
(2,53) where
f ( G r) =
II(t, ty) e -'2~i~'~ <~dt&J
(2.54)
R e
Once again, we recover tlle definition of Eq. (2.6) of Cohen's cias;s, which wa~spreviously proposed t~s a means %r including most time-Kequency e~mrgy distributions in one common framework, t.{,~?have just showr~ theft this claos's can ac*:u~dly be coustrucu.ed, and ~hat it eomp~rise.s atI bilinea# rcpresen~atior, s that are cmv)riaut relati,w to time-fi'equenqy shi#.s. Remark, The chatact.eriza~ion of" Cohen's class given here supposes titat the parameter function equation (2.54) does not depend on the signal. Then t.he transformatim, of Eq. (2.53) is bilinear. We should note, however, tha._t, this characterization is more res~.rietive than t.he definition, whk'h was initially p r o p o ~ d by Cohen. The originat tbrm oJlows f to depend on the analyzed signal. Nevertlieless, we will keep to the fbregoing terminologT t:hroughout this book, sanctioned IV usage, and ea,/i the Nmily of equat;ion (2.53) Cotton's class. Let tus teeN1 the cor~straint equation (2A7), which was fixed ab inirio and shows that the bilinear representation is an energy distributiom According to Eq. (2.49), it can be expressed as
Ki'~,-t,'d-t;00)e
"
(ttdu=a(u .......d),
Ch~pt:er 2 C:tass~s of Solutions
107
In the setUng of the developed parameterization, this reduces to
~
' II(t,
zJ) d t d v = 1 ;
or equivalently to f((L 0) = 1 .
(2.55)
~A~ always take ~his simple normalization condition [or granted. Then Cohen's cl~s is exa.ctly the set of all bilinear energy distributions, which are covariant with respect to tran.statkms. Let us finatly note that the definition of Cohen's etass ~s the correlation of t:he Wigne>Ville distribulion and an arbitrary parameter function can be expressed as a produc~ in the conjugale Fourier domain. More specificMly, we obtain that
~
C~(t, v; f } e2~'(gt+"T)dt du .~: f(~, 7") A:~:(~, r) ,
[{2
where
&(~,r) =
/11.17()() ,r s+ ~r
a'*
r
s-
eCe~e, ds
(..o7)9:
denot(~s the Fourier transfbrm of the WignerJVille distribution. Recall that the 1,~st quantit, y represents the (symmetrized) ambiguDy function of the signal Except fbr a pure phase, it is the same as the expression in Eq. (2.45). Time-scale. The definition of a ge,mrat cla~s of bilinear time-scale representatimls parallels the Ume-frequency case. It is enough to l~plaee the co,a~rianee relative to the shifts with the covariance relative to the afline group. Its acUou on an arbitrary signal x(t) is given by :~(t)
-,
z~,,,,(t)
=
I~'1-~"2
(:!_::: ~,;
~" \
t'~
j .
Here the constraint of covarianee reads a8
p~.,~.(t ,,) =p,. ( t _ t '
~,)
(2.5s)
and t,his reve~Ls the identity t .... t t
a)
'77me-.FT"
108
for the kernel Because the last equation must~ be verified for arbi{.ra,ry values of the variables, ~he ;special choice ff = t and ¢s~ ~= c~ le~is to tee simplitled form
a,
Again, we can see tha~; only two degrees of freedom are left for the parameterization. We finMly obgNn, by insert, ing this reduced form into the general Nlinear expression, that
p~(t,o.) = Df lil (~.-:i~ ,a~) ~,~.~.(s,~)dsd~ ,
(2559)
where we use the quantity II(t, u) ....
[4" t + 7 ' t - 7 : 0, 1
e <~""~
dr
.
This parameteriza.tion constitutes the "aNne cl~ss" of bilinear representations, which are covariang relatix~ to the action of tralislationsdilations. 45 In order go gain veritM)le energy distributions, we must demand that Ill
' ~
r
•
"
J.ld rt ~ Here '~9(~, u) denoLes the partial Fourier trausform
~
+ ¢x;,
:lYe
We cast fiirther employ tlie marginal w o p e r t y of the Wigjler-Vilte distribution, which asserts that lI~(s,
~) d,s
,
¢
2
Then the adn~ssibitit, y condition takes it:s final, form
f
+ .... ~
.... ',#(0,
du ~,)..~}
=
1 -
(2XJO)
Provided that this eondilAon is satisfied, the a£flne class is the set of all bilinear energy distributions, which are covariant, with respect to
Cha,pt, er 2 Class~s of Solutioi, s
t 09
trmMations (in time) a,nd dilations. Just like Cohen's cla~ss, it admits several equivalent forms depending on the represeiltational space chosen for the free parameter function. In p~xticular, we Obtain (by introducing a notation for E q (2.59) similar to Cohen's class equations (2.53)(2.54)) that
(~2*(t'a;f)=/(/[l(S-t' I;f'~(s'~)dsd~, a~) ,
a
""
"8--t
(2.61) The last expression favors tile ambiguity plane. By an analogous calculation, one can also conclude that
(2.62)
This represents the time-sce~le distribution in terms of the Fourier transform of the signal.
2.3.2. The Troika of Parameterizations-Definitions-Properties The unifi(ul tbrmulatio~s, which are offered by the bilinear classes (Cohen's and a n n e class), reflect three main interests: (i) It is possible, by an appropriate specification of the parameter function, to cover most definitions of known energT distributions;
(ii) It is relatively easy to trmlslate a given constraint of a joint representa:tion into a corresponding admissibility condition for the parameter flmction; arid (iii) It is possible, by the use of such arguments of admissibility; either to specit~' the properties of a particular representation a priori, or to construct a class of solutions relative to a whole set of given specifications.
1 t0
'l~ime.,f~eque~u:59/'f~ime-,~calc Amdy,~is
In D~ble Z t we present sevelal examples of defini~.ions of time-frequer~cy distributions, which can be recognized as special members of Cohen's class in temps of a correctly specified parameter function. Tile given examples do not exhaust all conceix~,ble definitions, not even tlm existing crees; the table should rather be regarded as a list of representatives of most clas~s of solutions. This means that other definitions can be derived from t h e ~ by more or less minor varia~A(ms, 4~ Definitions.
l~k~rthe complex-valued representm:ions (such ~usthe Rihaczek distribu~ lion), o~xe can evidently consider the associated quantities of tile compiex co~ougate, the real part or the imaginary part. l_~?~rthe example d the Rihaezek represmltation a.nd its by-products, one can show that
This defini~io~x is usuMiy attributed to Ackroyd, We intentionally include the class of s-Wigiter (or generalized WignerVitle) distributions in the tabM as it, permits a smooth transition from the Wigner-Ville distributio~ to the one(s) by Riha~::zek by means of a singie reM parameter s. Its value is usually re:~tricted to !s I < 1/2. One can reMily check that the (.*a~es .s ..... 0, + 1 / 2 o r - 1 / 2 c o r ~ s p o n d to tile Wigne>Ville distribution, the one by Rihaczek or its complex corijugate, respect, ively. The Born--Jordan, aa,d Chd#Witliams distributions appear in Table 2. I. It is importm,t ~o note, however, that they are special examples of an import, am. {:lass of representatio~s: here we refbr go those repr~entations, whose parameter flmctions behax~e like an apodization kemet (in the ~ n s e of spectral analysis) operatAng on the N*oduc~ of the variaMes. More precisely, we have
f({,r)
= ~(~r)
and
t~2(c~.)t < ~,(0) = l .
This gives free rein to variati(ms of the form of the apodization, which can be achieved by replacing t,he cardinM sine or the Gaussian wi~h any other form deemed approwia~e. As an example let 1
we can regard ig a~s a Bugterworth parameterization by analogy wigh filter theory, Here and in the case d tim Born, aorda~ representation, we observe ghat tile Wigtm>Ville distribution appears as the limit, when the %'quivMent width" c~ tends to infinity. Otherwise, the representation related to
Ctm,pter 2 (.la,~ses of Sotuti(ms
1I 1
Table 2.1 (~Oh , n 8 cl&s8: 8o1?;1(~ examples of pa,rameter flmct;ions and t,he corresponding gime-frequency representat;ions
f(~, r)
(.,;,.(., v, f)
1
Name
.
m t+~
'"':"
/7"('
t--~
.
Wigner-Ville
dr
(' ~0"*(' .......('~)'),--""~"~ s-Wigner Rihaczek
z ( t ) x * (,.) e - ~ > ' ' '
:./~<:, *('~)u(~(t -
at
slit rr~r .
i[
-.<.-~_
"
iT] ~ ,St ....i',-II2
Page-Levin
~)) ~-">~""
l
..... 2
Born~Jorda,n
j]
CY
(
T
)
-i2~r~r
(,ho>Wflh~ms i.:f .
z(.~) h*(s - t) e ''i>'" d s a
spectrogram
l
C(~)h(r) separable
1.] 2
Tim<>Fr,~{tue~lo'/Thne-Sc~ie Amdysis
an apodization kernel of product type is a smoodwd version (in time and frequency) of the Wigner-Ville distrib~ltion. The smoot.bb~g etfec~, ix more significant, a priori, when die regulating parameter a is small. This i n t e r pretation by a smoothing ca~ giv<~ rise to other variatiot~s of the definitions of B o ~ - J o r d a n type. bi)r ii~sta.nce, we can make its time-f~equency behavior dissymmetrical. While ig was governed by a single panm~eter c~ in its original form, we (:all gtms employ transformations of the V p e
Let. us fina]ly remark that the use of separable parame*cer functions, that is, such that: f(~, r) = G(~)h(r), offers another natural way to deal with the issue of smoothing (or at least of convolving) the Wigner-Ville distribution. It is clear that this modification can Nso be performed widi any other distribution. ~Ihis g i v ~ rise ~o so many modified versions of a give~i representation, ~ there are reasonable changes of the parameter function ~'eording to f(~,r)
--~
G(~)h(r)f(~, r ) .
The representations, which are b~}sed on so-called "cone-shaped ke> nels," were introdueed by Zhao, Atlas, and Marks. They cqin be considered in this co~ltext, e~ their parameter functions have the form sin rr{r f(¢~, r) = go(r) l'ri --------~r~r Hence, they are %rmaily equivaient ~o smoothed xu~rsions (in frequency) of the Born~Jordaii distribudo~i (and this explains why they do not appear in Ta.ble 2.1 ). Table 2.2 offers a similar panorama of dme-seMe distributior~ of the a n n e class. As before, we only include some large classes. One of them is the ~scah,grmI~, of c o u r ~ ( i.e., the squared modulus of the wa:vetet transform). It plays a role in ~.he dine-scale context, similar to that of the spectrogi'az,l ila the time-freque~ey sett.i~g. One Nso ree~yeers the Wigner-Ville distribution again, which ix the core of the generM class in itself. But its reappearance in the time-scale context necessitates the introduction of an arbit.rary nonzero reference frequency t/(~. It is needed for the formal identification "scale = inver~<,~of frequency" in terms of the relation
This type of identification can also be extend(d m other distributions. In order to see how it works, let us start over from t.he definition, of the af{ine
Chapter 2 Classes of Solutior~s
113 Table 2.2
Aftine class: some examples of parameter functions and the corresponding time-scale representations
f~a:(t, a; f)
f(~, r)
(? - i 2 ~ u
Name
/'+'~ (t + 2) x*(t 2) e-'i~,'(,'~./~>drWigner-Ville
r
Lx X
- - "
A;(~,~)
scalogram
G([)h(r)
a,
separable
~ ( { r ) e "'i2~'""
C(~') e -~2'~H(e)"
a
Cohen product kernel
/%<) anne Wigner
class in the ambiguity plane. We suppose that the parameter function ha~s the form f(~, r) = ~2((r) e -i2~'')" . Then it immediately iollows that
R 2
Time-l:hequency/Timt~ScaIe AnM)'s'is
114
This last extm~;sion [s no~bing but ~he timc-&equeI~©, dist:ributio~ in CoIron's class associated with the parameterization ~((T) and die ideutifica~ tion v = v0/a. The correspondb~g repre,~'ntatiol~s tie in the intersection of both classes (Cohen's and affine ctass), ttenc~ they are amertaMe to hoth the tim~s fl~equency and the time-scale interpreta:tMn. A particularly interesth~g cta.ss of time-scale representations% which a
priori mNntain the dNtinction bet:ween scale and frequency, are the called a n n e l/t~)2gnor distributions. Their pan~neter flmctions axtmit the generic, form f ( G r) ::: G(~) e '2~t(~)',
(2.63)
where G(~) m-id H ( ( ) are two acMtrary t e n flmctions. By menus of the identification a = v'o/v, vve ca,n associate with every time-scale distribution of this V p e a gime-frequeImy distribution
(_-?~.(t,~/: f) := f?;,~(t, v~,/v; f ) ,
(2,04)
The details of the most current defini{hms that belong t,o this class are given in 'l;~bte 2.3 tog~M~er with their parameterizations, (Note that the t,aine a n n e or widebs~nd time-frequency representations is M~) used in tile literature.) az Let ~_kst i m b e r assume (and this wail be implicitly done in the sequel) that the signal, on wtfich these different distributions act, is given in its analytic tbrm ( i.e., X(>,) is zero for all negative frequenck,,s}. Then we can veril}: that alt the~se parameterizations satisfy the nec~ssary conditMn
Constraints, T h e ~ are many conceivabIe constraints t,hat, can be imposed on a, joint representation. The most important ones are associated with: (i) The nature of a representation and its physicM meaning, which meets the con(ern for the interpretation of the mathematical object;
(ii) the properties of covarialme or the compat, ibility with respect to usual or natural trm~sformations in signM processing; this meets a eo~cern fbr c(d~erence with the descriptions that exist ouly in time or in frequency; and (iii) the possibility of exposing the specificMty noustationary properties, which concerns the ne~d of theoretieM and practieM e~xploitatio~, We are going to investigate stone of these constraints aud establish the rela*ed Mmissibility condition in eemh ease, We begin with a consideration of Cohen's ck~ss and then turn to the atfine class. 4s
C h a p t e r 2 Cta,s,:es o f Solutions
115 T a b l e 2.a
Affine Wiglter distributions (cf. Table 2.2) for analytic signals
Name
c dt, z , / )
a~(472.o) coth(4:/2z,,~)
~,,
Bertrand
sinh(~/2~0)
.... sinh(!,/2) X u s i n h (, 7 / 2 )
X"
,.,, (1 + (~/2,.o) ~) ~'~
,.o (t + (~/2.o) ~) ~/~
u.
u ' sinh(7/2)
.
e
""2 ~ " " ~ d ?
Unterberger (active form)
(:1 + (~/2,.,,) ' ~) '~ ~'/~
7
.
,
Unterberger (passive form)
k "~j
D-dist, ribution
-(~/~,o) 2
.+ ?
e-i2rruT~
77m+>N'<~queucv/Time-Sc,le Asm:ysis
1t 6
(:ohez~ 's clr~ss Energy. In order for a time-frequency representatiol~ to be an energy distri}mtion, we m~tst have t~:, :,
/J
.,:
(:, u; f ) d t
dv
(too5)
.
II 2
As ~ e n before (cfi Eq. (2.55)), this condition is equivalent to a simple constraint on t;he parameter fimction reading a~ f(O,O)
=
I
.
Anot, her v,,a;y go show this equivalence relies on a well-known property of the Fburier transform: Integrating a fimction over the whole space yields the value of its Fourier transfbrm at the origin~ "\~k~recalt fTom Eq, (2,56) ~hat i;he Fourier transli>rm of (],,(L u; f ) is gi~x'n by f(~, v) A+(~, v), and A~:(O, 0) = E,~, boris b:,,. definition (cf. Eq. (2.57)). Hence, the relatiort
/t
'd
•
C'c(t, tJ; f ) dt dz: = Jr0,0) E,:
follows immediately, and this yields the desired result.
Marginal dist~tibutiotts, Let us now [lnpo~ the additional constraint that the "ut~ivariate" energy distributions (spectral energw density and instantaneous power) ca1~ be obtained as the marginal distributions of the joint; representation; that ~s, d,~(~,t,;f)dt
tX(~:)i °
and
C~(t u;f)dx.'
!x(t)l ~
(2.66) Then we can e~si}y prove ~ha~, t;he respective i,:lenli~ies f(0, r) ::,,,~1
and
f({, 0) .... 1
(2.67)
must hold. Le~ us co~lsider the se(ond constraint, fi~r example. The definition ecjuation (2°53) of Cohen's (lezss leads directly to
/;
C,.( , , / ; f ) d u
i + it
and tMs gives the announced result.
f(~, 0)
. . . .
7
,
Chapa,r 2 (i'I~acs~so[' Solutio~s
117
'lhe constraints of corre( t marginal distributions have several interpre... rations al~(t consequences. Certainly, the firs~ inlerpretation results from the analogy of energy distributions and probability de~lsity functions. If both constraints are meI, we can recover the a priori de~sities from tt~e joint density by means of its marginal distributions. As a second interpretation we can regard the fact that a joint represe~> tation "deploys" the energy of a signal between the variables of time and frequency. As a consequence, the integration with respect to either variable must restore the global infbrma:tion relat;ive to the conjugate variable° Therefore, an imegration over time corresponds to collecting the complete history of the signN into one vahIe; all chronology is thus erased and only the global aspect of the spectral density subsists. In a dual manner, integrating over the frequencyaxis suppresses tim specialty of different spectral contributions at the considered instant, and thus (from a n energetic print of view) comes back to the instantaneous power. Let:. us finally riot( in this context that attaining at; lea.st one of the correct marginal distxibutions is stronger than the constraint of the (global) energy conservatio/~; this can be expressed, tbr instance, by the relations
c : , ( t , . ; f ) d t =. I x ( , ) f i ..}.. x ~x~
//
///
:N,
t/? or equivalently f(0, r) = 1
~
f(O,O) = I .
Reality. The (local or global) integral constrail}ts of elmrgy distributions guarmliee by no means that a representation of Cohen's class be realvalued. [n order for this property to be met, we need the representation to have an Hermitian Fk)urier tralsfornl, l~.ecall that the latter has the value f ( t ~, T) Aa: (t~, T), and note that by Eq, (2,57)the ambiguity function A~ (~, T) itself is Hermitian. Hence, the condition of reality of" the representation is equivalent to f(g, T) = f*(--~, "..r).
(2.68)
Posit.ivitiy. If one finally wants to carry on the same interpretation~ it is tempting to require an energy distribution to be nom~egative everywhere; then it would acquire the rank of an energy density function, .~v In order {o h~vestigate this property, let us introduce the cha~ge of the parmneteriza,tion _ - >
=
e
,~
(2.69)
F i m e- l))'eq u en c~7'Ti m e- Sea te 4 n ah=sis
] 18
and the notation y,.(s)
= :r(s + t) e -'2':'~
Then the condition of positi',ity, imposed on ~,he genera.l form of Cohen's cle*~s, reads a.~
(,, r) >.,,
.~ !Jt~
5
ds d r >- 0 .
Ilence, F(.s, r) must be {,tie kernel of a positive definite operator. If we r ~ t r i c t ourselves to the class of square integrable parameteriza,tions, then this kernel admits a decomposition T h=l
with
+ :'~
Here the ht:(t) ff)rm an orthonormal family of functions in L2(N.). By taking Fourier t.rmlsibrms with respect to the first variable, va~ ca,n infer from the previous relation that
..v (~, ~ ) .
(z70)
This signifies thal~ the a&soci~.t.ed positive representation has the form 2
x ( s ) h ; ( s - t) e ~ > ' ~ &
(zn)
k=i
subject, t,o the normalization condition
k:= 1
Therefore, tile posit.ivity of the representation rules om all representations of Cohen's class except for linear eombina,tions of spectrogran~s with positive c(~fficients (at least: if we assume that ~,he parameterization is .~uare integra.ble).
(,hapt~r 2 Cta,sscs ot Soluticms
l !9
Causality: The evaluation of a joint representation of Cohe~fs class at a fixed instant, employing its most generM definition, involves d~e future a.nd the past of the sigrmL Meanwhile, it is bgitimate to raise the issue of whether there exim causM solutiol~s in this class; this metals tha:t at. each instant t they may refer onIy to the past of the signal given by the set of values {x(u.); u < t}. An answer to this question can be given by rewriting the generM form of Eq. (2.53) as <,,(t, . ; f )
:
'if .." ( u. . .+ 2. .,s. . t,'~:- s
:r(.,). * ( .'~). e. . .
i2"z ~ ( u - s )
.
&ds . (2.72)
Then the causality of the representation is equivalent to t;he condition
F \('tL~s 2
t, ~ ....... s )
=
,~'
(u4 s t,u ... s) U(t_u)U(t_s ) ~
....
,
which must be met by the parameter function; here U(t) denotes the Heaviside unit step i\mction. This admissibility condition can further be simplified to
F(t,r)=F(t,r)U(.-4 .........[2!~) •
(2.73)
'P~king partial Fourier transforms of both sides leads to
f(~, T )
=
-
pv
e "~;l:'q d 4
•
(2.74)
The l~st constrain< which ensures the caus,nlity of a bilinear timefrequency representation, can be compared with the classical relation (due to Kramers-KSnig) guaranteeing the causality of a linem~ filter, 5o As a corollary, the parameter fimetion f(~, r) cannot jUSt have reaI or ilaaginary values. Indeed, by separating the real and imagina~'y part, the MmissibiIity condition is equivalent to the two identities m~ {f(~,T)} = - . ~1 pv f! d
lm {f(~:,,)} = + -
7"f
pv
"b~x.,
Jim {f(~,~')}eos0r~T)
*x_
"x;
d~"
[i~,, {f((,~)}(:o~(~¢~)
Timc:-I;?'eqt~e,~c}/'Time~Sca]eAnatvsis
] 20
Let us suppose, liar instance, that hn {f(<;', r)} = 0. We immediateiy conelude that the quantity Qr(() = Re {f(~,r)}cos(rrGr ) ha.s a vanishing Hitber~ tr~msform. This implies, in return, that Q,-(~') must vanish everywhere, hence the t e n part d f(¢, r) vanish~ as well. Unless f(~, r) is identically zero, it must therefore have stric~,Iy complex values. The simplest, example that is compatible with the constrNnt of causali v is provided by the function f(~,
r) = e ~e'~i
.
It corresponds to the Page distribution (cf. Table 2.I), and we recapitulate that it was explicitly constructed (in Subsection 2A.3) t~sing an argument, of causality.
Invertibitity. The bitinear transformation, which associates a signal with an ener&v distribution, can define only a veritable representation, if no information is lost t)y this operation. Therefore it has to be ~nvertible; in other words, the,~ must be a one-to-one corr~pondence between the Fourier transform of the distribution C,(t, u; f ) (whictl is f({, r) A~.({, r)) and the signal x(t) itself. One e ~ i l y checks that. the ambiguity function A:,(~, r) is invertible (apa~ from a pure pi~ase), since z( )z (0)
A,({,t)e'*'~* d{
(2.75)
Hence, the invertibili~;y of (?~ (t, u; f ) follows, if f({, r) h~q no zeros. Then we caal fonnMly write
x(t):r'(O) = j];[ F-I ( . s - 2 , t ) C ~ ( . ~ u f ) e ~ ' ~ " ~ d s d u
(Z76)
Rz
by' putting F
~ (t, r ) =
tf +'~
1
~:~2~,d~.
J._:,~ f(g, r)
5l>ar,slations. Tile mos~ important covariance to be called ior refers to t,he translations in tile time-frequency plane. Obviously, it does not give ri~ to any restrictions within Cohen's class whatsoever, ~s this cia~ss w~esprecisely built on this covariance. Dilathons. \%,qfile we stay inside the ciass of time-frequency representatioi~, a ~ c o n d natural covariance can be imp(~sed relati~ to dilations. Let us consider the transformation
Cb.apter 2 C/asses of Solutions
t21
We know t,hae a dilation (k < 1) or compression (k > 1) in the time domain produces an opposite effect in frequency, because
.x~.(,,) = ; 7
x
V
.
Consequently, the naturM constrMnt of covariance relative to dilations is met if both the time and frequency behavior of the joint representation are conjugate in the sense that C~.(t,t/;f)=Cx
kt,~;f
.
(2.77)
We first observe that: the ambiguity function equation (2.57) ha~s this properr;y, that i~s,
Hence, the wanted condition simply reads a~s
f(~,T)= f ( ~ , k T ) ,
v k.
(2.r9)
This eliminates one degree of freedom from the corresponding parameter function: It. is no longer a function of two variables { and T, but only a function of the product ~r. Therefbre, it must have one of the tbrms
It is remarkable that as soon as such parameter functions define energy distributions, that. is, ~.(0) = 1, the constraints of marginal distributions are automatically satisfied due to f(~, 0) = f(0, T) = %O(0) = 1.
Fii~erhN.
~Aqien we look at a linear filtering with impulse response h(t) a:nd frequency response H(t,), it is a well-known f~mt that the relation bet w e e . the input sign.aI x(t) and the output y(t) is governed by a convolution in time and a multiplication in frequency.
y(O =
h(t - ~) z(s)<~.~
~
z(.)
= H(~)X(~),
(zs0)
Requiring a joint time-frequency representation to reflect these two points of view can formally be expressed eus C~(t, lJ; f ) =
Ci~(t - s,~ ,f)Cx(s,t,: f)ds
(2.sl)
Time-b?~equenqy //Time-S¢ ale A~m]}~is
1.22
Starting over from ~he most general bilinear form (Eq. (2A8)), which was parameterized bv ~he kernel {m u : t , u), this constraint ~,. equY~alent to K('I~ + w , ~{
wq ~t,
~,)
h "(,u, '~':t - .~, u)h'(w, w': t. u) d.~ .
=
It, follows that K~:,,('u. 'd) IQ~.(-.~', w') = K ~ . ( ~
'~.~,,.'+ w')
hy putting
If we u.se the explicit form (Iskl. (2,51)) of I'((~.~,d ; t, u) inside _ol en s class and bring out the parameter function f ( ~ , r ) ia the latter equation, we obtain
.
,
,
BL~,(u,u)=f((,u
- u ) exp
{
-i2n-
(,,,,'
~--~2 " +'0~
........u ' ) ) }
.
This yields the admissibility condition
f ( ( v) f(& r')
f(G r + 7") .
(9 '%~'
The parameter flmction thus submits to the form ,f.&,
) ....
(2.s3)
,
where G(() may be an arbitrary function. By inserting v = 0 into the last expression, we confirm that f ( ~ O) = 1. This shows, by itself, that. the marginal property of the [nst~amaneous power is a by-product of the compatibility with the linear filtering, Mor~ver, the second property of a correct marginal distribution (spc,;wal energy distribution) is g~sured if and only if G(0) ......0.
Modulations.
These are dual to the lin~,~r filtering: the modulation of the signM by means of a multiplication by m([.) h ~ the form of a product in the gme-domain and a coovolution in the frequency~domain; more precisely, we
y(t) = m(t)z(t)
~
YO') =
Mr.
- 0 Xti) d(.
(2.S:~)
Chapter 2 Cla:sses of Solutions
123
The correspo~ding time-frequency constraint reads as
Cv(.,v: f) =
/
+. , x :
(2.8a)
G,~(t,u-(;f)C=,,(t,~.,.t')d~,
and one c~m show ~ alreasty mentioned that the associated admissibility condition is given by (2.86)
f(~, r) f(~', v) = f ( ( + ~', r ) . This requires the parameter flmction to be of the tbrm f(~, r) = eS~('~) ,
(2.87)
where 9(r) is an arbitrary flmction. If we insert ~ = 0 into Eq, (2.87), we easily confirm that f(0, r) = l. Hence, the property of the correct marginal spectral energy density appears as a by-product of the compatibility with modulations. The dual marginM proper V regarding the instantaneous power is e~scertained if a.nd only if g(O) = 0.
Remark. The simultaneous compatibility with filtering and modulation necessarily leads to the form f(g, 7) = e ~'~**
(2.88)
of the parameter function. It thus guarantees a fbrtiori the covariance with dilations and, e~s aftereffects, the correct marginal distributions.
Sut)ports,
It ks desirable to demand from a joint representation beforehand that, it should preserve the time and frequency support of a signal. The corresponding constraint can actually appear in two different forms aecording to the considered nature of the support. The first form (conservation of support in a wide set~se) only insists upon
{
,.(t) = o ,
/tl > T
= O , 1.l > B
C , ( t , u ; f ) = O ,Itt > T ,
(2.89)
===> C , ( t , u ; f ) = 0 , tul > B ,
If we consider the first constraint, for ir~tanee, Eq. (2.72) implies that the parameterization must satis~,
F(s,r)x
s+t+
s+t
ds=O,
It1 > T .
t 24
"1'ime- Frequene3(/~l'ime-Scale A rmt3csis
An the integration over s is restricted to the domain --T -. t 4 IT[~2 54 $ 5::. T - t .....{rt/2, the va.tues of the integral for t > T (or ~ < -.-7, resp~'tively) ,aill refer *~o the interval .s < ....F I / 2 (and ,s' > r l / 2 , respect;ively). Hence the nutli V of these values is ensured if we h a ~
'"Is, r ) = o
Iri f?.~r {,q > .-5-.
(2.90)
In the same way we can establMt the admissibility condition relative to the dual constraint, which reti~.rs to the conservation of the frequency support (in the wide sense). Then we obtain the condition IF! 2
'
where ~'((, V)
nt.t. ~., ,-
de =
f((, r) (~i~*r,.,~" dr .
R e m a r k . The const;raint; of the conservation of the supports is responsible ibr fixing ti~e usual domain of the free p a r a m e t e r s in the s-Wigner distri[mtion. Indeed, one can easily compute
V(t, r) :: ~(t + st)
and
~:'(5 z.,1 = ~(.
in this cause. In order t.ha~, the equatiorl t it.> V / 2 , ,,,e m,~st h~,,ve ,~I ~; U z
<)
87- =: 0 be mtsotva,ble for
The conservat.i(m of supports in the wide sm~se can ~ raged as insufficient, when one colLsiders signals witdt disconnected supports. T h e Wigner-Ville distribution m a y on{x: more serve aas an exampIe, It satisfies the fbregoing conditions of Eqs. (2.90} and (2.91). Therefbre, the corresponding representation of a eompactb~" supported signal
x(t) = ,r(t) 1 _~,.+-c(t ) identically vanishes out, side the same interval (here 1,(t) is the characteristic flmction of the intervat I). However, in the opposite case when the signal has the tbrm
the rdat, ion
I'E,~(0,v) = Jj/'/2>r x
Chapter 2 C/assc~sof Sohztions
125
exemplifies ~hat the distrRmtion can have nonzero values at an inst:an~ where lhe signal vanishes. II is, henceforth, possible to envisage a stricter constraint (conservation of supports in the strict, sense), which demands that :,:(t,,) = 0
=~
C:~,(t~, ~,; f ) : 0 ,
X(v'o)=O
~
C,,(t,~,o;f) ........0 ,
(2..~2)
In this case, one forces the time-frequency represent.at, ion to vanish at all (time or frequency) points where the signal itself is zero. The Riha.czek diseributiml of Eq. (2.5) provides a sohtion eo this problem. However, it is not the only possible one, In order to find the general admissibility condition, let us first express the temporal constraint a:s
z(t~)=0
==*
F(s,r) z s+t0+.~
z* s + t 0 ......
ds=0.
If we now define the auxiIiary signal
y(t) = :~-(t) + ~(t - t,)
with
~ ¢ t~,
we c~m deduce the equivalent relation
~,~ F ( s , r ) S,
s+to+~ ~j* s+to.-.~ ds=O.
By e.~:panding the terms we arrive at /
7"
\
F i*:t ..... tt', i" ~, r) ~r(tl + r) +
F
( t~,---to-~,r
;r *,, (~
~) { V ( *• ~ - - t 0 ,
This (:an only be true tor every signal z(t) and ew~.ry delay r ¢ O, if the time-time form of the para.meter flmetion satisfies
ttence, we can write the admissibility condition aas
F(s,r)=g+(r)6
s+ff
+.q.(r)6
(") s ......[
,
(2.93)
where g+ (r) mtd 9 - ( r ) m w be chosen arbitraril> al One can readily check that. the Rihaczek distribtttion corresponds to a solution of this type, en> plwing the functions g + ( r ) = 0 arid 9o, (r) = 1, More generally, the set: of
1 mle-f req~ienqKi 7 sine-Sea/( Anatvsis
126
al! such solutions is composed of ~,!1 linear combinatioris, which are formed of smoothed versions of ~.he tlihaczek distribution and its complex conjuga.te, the smoothing taking place in the h:equency domain. In a du~d f~JMfion, the constraint of conserva.tion of the frequency support in the strict sense leads t.o a ~Aution of the t}7;~e
~)(~,u)=G+(~)6(~,+~)+G..~(,)6(u~-~)." ~"
(2.94).
[;nitazqty. Tile Fourier transform is an isometry of L2(N). Th~s the inner product of two signrds remains the same, no matter if we represent it in the time or the frequency domain,
iT it is often desirable to ensure a rela~;ion of the same type, mutant,is mut.andis, when pa.ssing from the time or fre(plency axis to the time-frequency pla.ne. In view d the bilinear character of t;he distmibutioIKs in Cohen's cla;ss, it is natural t.o identit~ the inner product d two time-frequency distributions with the square of the ~LsuN inner product in the time domain (referring to the signaN) or the frequency domain (their spectrum, respectively). Hence, the condition of mfitarity takx~s the fbrm •
, + :x,
#
]2
=j
"
The MengiV of Eq. (2.95) is usualiy called
Moyal's formula. 52
In order to investiga.te this constraint, let us rewrite (by Parseval's relation and Eq. (2.56)) the ieft~hand side of the previous eqm~tion as f)C,(t,v;f)dtd,~
=
t{ 2
l]'(¢,r)[~A,:(~,T)A;(¢,-r)dCdr It ~
Note that the definition equation (2.57) yields i •
"
'
l,
+ ~';>
dt
2
--2>c+
~%%~can thins concIude tha~; ihe condition of )mitarit:~ ~ is equivalent to
f (If(~, r)l 2 - 1) A,:(~, r)
r)d£ dr = O,
,
Chapter 2 Cl~.ssc~ of Solutions
127
and this requires the parame~.er function to be unimodular, tha). is, f ( ~ , 7)1 = 1 .
(2.97)
It. should be noted that the fl~lfilhuent of this condition guarantees that the parameter fimction ea,nnot vanish. Thus the ~sociated )~presentation is inver¢ible (apart from a pure phase}.
InstaI~tmleous fi~uenqy.
We were already' in search of an analogy bet w i n time-trequency represent, ations ~ d (joint) probabitit.y distributions when we looked fbr an interpretation of the marginal distributions in terms of the a priori probability densities. Carrying this analogy farther, wc, ~re now interested i~ the local behavior of a time-frequency distribution. [;))r example, we can look upon the fi'equentiM cross sec.tion at a fixed instant t~s a c:onditionM prc)bability density with respect to this instant. Co~sequently, the center of graviV d such a section has the sa,me nature as a conditional exp¢~:tation value. It thus provides some information about the (average) instantaneous frequentiM contents of the signal A proper requirement sigtfifi~ that this information matdfes the instantmwous fre~ quency, as it was defined tbr analytic signMs in ISq. (t.21), (Intui{,ively, in. case of a single-component signal with frequency modulation, a '*good" time-fr(:queney representation should essentiNly live nero" the cmwe of the instantaneous frequency.) For a signai x(t) let z~.(t) be the a.ssociated analytic signal. The ira-. po~:d constraint can be written as
J
'~'- uC,, (t, u; f ) du _
2%
C
, ~
u;f)d~" ,
,
1 dargz~: (t) . 27r dt
)
Expanding the nnmerator leads t.o
j!i I-~ u C
(t, u; f ) d u =
N2
i " ~01 >' '- ~,, ]Jill~ ~ ( s - L 0 ) ] z , , ( s ) [ ' d s . This gives =
(t)
(2,98)
Fime-.bY~:,que~qK/Timc-Scale Ana,I~sis
19z8'
on the assumption that the parameter function satisfies the %llowing adrnissibility conditions (the first of which guarantees a correct" marginM d i a tribution):
.f(6o) = : ~,:,:: ~J(~ o)= o. "
(2.99)
07"'
Bringing out the modulus and the phase of tim anNytic signM yields
dzzt , ~ "
T()..;;:~)
=
(
"~
dargz~ ") ,<,.(,)t.
(0 + iiz,,(t)! ~ ( t )
~ t
Consequentl), we arrive at; 11m27r
J,-)t~-(t,)z$(t),,
= Iz;r(t)l~
2)
ttt
Hence, the ibrementioned admissibilk,y conditions (Eq, (2,99)) ensure the satisfa.ctioa of the given constraint: the instantaneous fl'equency is equal m the center of grax'k,y (or locM first-order moment) of the distribution.
Group delay. We can Mso consider the local first-order moment (or the conditional expectation) relative to the frequency and proce~:t in a similar manner. Then we find that the group delay is obtained as +:× t C~:, (t, ~J;f )
(It
~2
t dargZ~. (.), 27: d~'
(2,1oo)
provided tha,t the t~fllowing t~x.) conditions are satisfied: f(0, r ) = I
and
@[u,r~=0. u~
(2.:ol)
Stationarity. In Chapter 1 we appealed to the notion of stationary deterministic signals. '~I "h e~y were defined as the sum of components with constant, amplitudes a~d instantan~ms frequencies, namely
~g
This definition is connected with the idea of a steady-state or time-invariant system. It is reasonable to demand tYom a time-frequency repre~ntation
C h a p t e r 2 Ctas:se,s of Solutio~s
129
of a st;adot~ary deterministic signal to be indepe~ldent of the time a~s well. This is expressed by the necessary condition
OC~.(t, .;
f) = o
8t
The definition of Cohen's claoss (Eq. (2.53)) and straightt0rward computations show that C,.~ (t,/,; f ) =
a,,,a,,, 71
H
T?~
s -- t,
2
u
•
-
ds.
(K:
ThereNre it is (:lear ?:hat t,he searched property imposes independence of the time on the parameter Nnction II(t, 1,). ONy the fl:equency variable remains at our disposM. Consequently, the parameter Nnction f(~, r) must ha:re the fbrm This condition, in flint, is very restrictive, a~ it gives up the concept of energy distributions (f(O, O) is not defined) and is only formally a,ccepta:ble. It defines some smoothed versions (in frequency) of the spectral energy density, that is, C'~:(t, p; f) =
C(~ .... ,J) IX(~)l 2 d~.
Locdization. For a single-component frequency-modulated signal, our intuition suggests that the time-frequency representation should be concentrated near the curve of the instailt.aneous frequency ~'a,(t). w e restrict ourselves to the case of a unimodular analytic signM (i.e., its instantaneous amplitude is constant and equal to t), The "ideal" situation would be l~hat; Cz,,, (t, ~,; f ) = ~(L, - , x ( t ) ) • (2.102)
Let us first suppose that the analyzed signal is a linear chirp characterized by
,<~(t)
= ~0 + , ~ t ,
Some direct computations show that C ~ (t, u; f ) =
f(/~v, 7-) e i2~('/'u'(t))r d r .
2.1.03)
tience, for a nonzero slope fl of the modulation, the only solution is given by f ( c T) = 1, and this charaztefizes the Wigner-Vitle distribution. On
Time-t.i'oqu~,nc//Tin~.,-ScMe Ana!ysgs
130
the other hand~ ibr a pure frequem:y (fl .... 0) every definition according to a pa,rameter function with f(0, ~) = 1 is suitable. This evidently a g r ~ s wRh the fact t,ha~, this same condition guarantees a corr~:t marginal distribution in f?equency (cf. Eq. (2.67)). If one goes further and wishes to maintain a perfect loeaiiza;tion to a more general curve of an instantaneous frequency, which is heir.her con~ stant nor linear, there exist no solntions in ColLen's class. Neverttml~s, one can find a formal solution whose pararneter flmction may depend on the analyzed signal, (This is obviously outside the scope of bilinear distributions0 Indeed, for a (unimodular) anMytic signM with arbitrary phrase the condition of ideal localization reMs as '
2 , -
"
"
If we consider, for instance, a parabolic chirp (i.e., a signM with cubic phase), which is characterized by
some simple calculations leM to the (x)ndition
This defines a parameter fl.mction, which depends on the curvature of the instantaneous fYequeney. Remark. At first sight, the representation associated with Eq. (2.104) fimctionaHy depends on the anMyzed signal by an a pricn'i unkm>wn paranmter (the curvature). However, this parameter cma also be written in terms of the moments of the usual Wigner~Ville retmxsentation, tn t~ct, one can show that =
0 : It) .....
+
where
Thus the Wigner-Ville distribution provide~, at lea~t theoretically, a way to find estimates of the characteristics of a quadratic rule of the frequency modulation. Inserting those ~stimates into a parameterization of Cohen's ch~ss iv likely to yield a better tocMiza~iom 5:~ The a;ssociation of coI~straints and admissibility conditions fbr the rep.~ resentations in Cohen's clash is summarized in Table 2.4. The eol~strNnts can e~sily be checked for the special representations in Table 2.1 by inspectiIN their parmneter f\mctions. For a subset of these distributions the ~,rified conditions are marked in Ii~ble 2.5.
h~pter 2 CJa~ses of Solutions "
}
~
t31
;g
Table 2.4 Cohen's (:lass: typical constraints on C:~,(t,~; f) and t;he corresponding admissibility comiitions for the parameter funct.ion f(~, r)
Constraint
Admissibility C;ondition
f(o, o) = i
l Energy Marginal in time
f(.(, O) = i
Marginal in ~equency
f(O, r) = 1
I{eMitv
f(~, ~) = f*(-~, --r) •"{~ "7N3
Positivity
f(~, r) = E c~ A¢~:(~,, * "")~ c~: > 0
Causality
(if f(~, r) ~ 19(~q2))
e ~¢1< ct¢
...... pv
Invertibility
f(¢, r) ¢ 0
Dilations
f({,r)=f(~,kr),
Convoluti¢m
f ( L r) f ( L r') = f ( L r + r')
Modulation
f({, r) f({', r) = f(~ +- {', r)
Va,
Time support (ir~ the wide sense) taYequeney support (in the wide sense) Unitarky
t¢I :,. j / : ~
{v{ > T
f ( - ~ , r)e"~e~~"dr = 0
..3~ ~
If(L r)[ = 1
Instantaneous frequency
f(~,0) = 1,
Group delay
f(0,~)=l,
of
-~(~,0) = o Of,o,, =r~o ?gv
132
°ITme-t) ~q~zcnqK/Tin~e-Scah .~Analy.s/s Table 2.5
Cohe~fs class: a list of constr~ints lh~.t are verified by several represemat, ions( S :,,,:spectrogram; WV = Wigmer-Vill< R = Rih~'z,~k; P : Page; BJ .... Born-Jordan; CW = Ch{ff-Williams) Distributions Constraint
S
WV
R
P
BJ
CW
Energy
Vi
v/
v'
",/
,/
,/
Marginal in time
/
v
V"
v....
v
/
,/
MarginM in frequency
,/
H(,ality
;'
\/
V
/
V
.
%/
Posit ivit,y CausaliV
VJ'
Invergibi|i ty
,v/
Dilations
~/
V/
v/
,/
\/
v/
V
V
Modulation
,v/
\/"
V/
Time support (in the wide sense)
,/
,d
~/
,/
i
~<,1'
¥/
/ b `+
/
/
Convolution
Frequency support (in the wide se, Lse) Unita,riky
V V
V
V/ /
V"
V"
/
Instantaneous h'equency
V
Grout) delay
v
/
V'
/
V
~i
V/
Alfine class
F h e mmlvsis of the time-soMe case runs h~ paIallel ~,o the tint~:>frequency ca.~< We therefore list only some of its specific properties, Enor£?:
A time-sca,le repr~:~-~entation is an energy d b t r i b u d o n if ....
a; f )
d~ d(t
(2,105)
As seen before (eL Eq, (Z60)), this corresponds to tim admissibili V condition dt/
/
+" ~,,(0,,,) I%] : 1 .
(2,10s)
(~hap{er 2 ( ~ s s u s ot Soh~tions
I33
l%r the spe(ial (a~e o[ {tie scalogram, whhh has a parameLer functJton
t'(~, r) = A/~(C r), we compute th~:~t
Hence, the Mmpiified condition
/.~. x
db'
?
'~
(=.m,)
IHU')i~ i,;{ = ~
follows, a i d this is the usual admissibility eoltdition (cf. Eq. (2.25)) of a wavelet analysis. The constraini, equation (2.106) must be slightly modilied, when we turn back to the time-frequency in{;erpretation in Eq. (2,64), wlml~:
c,,(t,,
; f) m. d,, :-~. U,, •
(2.ms)
In order for this condition to be met, by the parameter funcgioli we Imlst haJ~'e -,~
e(o, ,,) i,,--i = ~,% "
(2.:t09)
For the affine Wigner distributions this leads to the constraint I H ( 0 ) i = >, C ( 0 ) .
(2.110)
One easily (Jlel{k8 tha, t this identit,y is verified by all distributions in ~;[~> ble 2.3.
MaxT;'h~al distributions. Let, us first look at the marginal distribulion rel~ ai;ive to the sc~de parameter by integrating over time. By simple manipulatio,m we obtain that
j/']~" ~L(e, a; f) dl, = '+~
i
-j- ,:~
=
f
,,(o,,..i
+
T
,-)..], ,,.),. f
,
i2rr~!T~
134
7"ime-f~reque~cv3/Time'-Scate
Analysis
Provided that th.e parameter fun~ tion satisfies ~9(0,~e) = 5(u .......u0)
<=:::~ f(0, r) = e - 2:~" ~ ,
(2.1!I}
this implies that the marginal distributkm has the form (2.112) '
-(7/
J
*
And thus it. really corresp~mds to the spectra~ energy density of tile signM subject to the identification "frequency = hlverse of s~.-ale," At this point it is appropriate to mgCke some remarks concerning different types of distribut, ions: (i) When we consider the intersection of Cohen's class and the a n n e cIas~s, we end up wifE parameterizat, iot~s having the form
Hence, the foremm~tioned identification is fully jus~.ified, and t,he correct margina2 distribution is attained if
This coincides wit] the previous condition (of. Eq. (2.67}), which was obtai~md in eommction with the (time-frequency) aa:latysis of Cohen's class. (ii) For the class of affine Wigner distributions, a~ given by Eq. (2.63), the wan~ed property implies tha~ (3(0)=1
aad
H(O) = r,o .
(iii) Finally, the required condition can nex~r be met by a scMogram becalve .#,(0,,J) = IH(,,)t 2
~
f(0.~)
= ?h(r).
Here the related marginal distribution is biassed according to the retation
7
/7
(2.11a)
Clmptr,r 2 Cfas,ses of Solutio~ls
135
Let us next turn to the ma,rgimfl distribution relative to the time. By integrating over Nl scales we obtNn that
,
c~o
"'<"°':>
.
.
.
'_2
.
I{? x
e~2:~d*<'t)d{du.
Here the rnagginal distribution will provide the correct vahle {l.(t, a; f) D77 = tx(t)12
(2.114)
.. q y . ~
if we have
f <'~7
717 ~' " +
~:
dr = lx('~)?
...-
In other words, the condition
nmst be satisfied for all frequencies ~. In the particular case of the aft]he Wigner distributions, considered in their time-frequency form, imposing this condition is equivalent to
Jf<x~ 2_+L(_ sL~%)A/exp
~,o
This yields in (:OIlSeqllellce that l:oG(a~)
= a2
d
(TC;
H
~)
which by a proper simplification gives ~(:, C ( ~ )
=
H({) -
dH ..
~ -,7-(~)
ag
•
(2.116)
136
7]m(>Freq u(,tJqK/<77me-Sc'al¢, Ana{)'sis
F~eali(~: Just like the representations in Cohen;s (l~ass, a time-scale distrb bution in the a n n e class is real-valued if and oniy if its paramet,er function is Hermitian. For the a~l~e Wigner distribu(~ions (Eq. (2.6a)) this is simply exwessed by the parity relal, ions G(~) = C;(-~)
and
H(~)= H(-~).
(2.117)
Localization. The localization of a time-scale representation can have severa.l meanings, t h e first is ass)ciated with the time only, and centers round the condition (introduced l)y P. and J. Bertrand)
Let x(t) be a signal mat,citing the forement,iol~?d model, The fl°equentiM form (Eq. (2.62)) of the generM definition of the a n n e chess ieads to the expression =
-
"
~
(Tis C i2'-r~( t - ~'°}Ac \2)
)
"
<"
which renders the desh~d resull, 2) -!/2
, !~;,2
,~,,~(~,z , ) 9
-
~
2
rtJ., .
. . .
1,0
.
(m:l:i~9)
I~r the special cla.ss of affine Wigner distributions this const.rMI~t rc~tuces
to I/2
hence
,G~
)
(~)~
I~1
(2,120)
A s ~ o n d and more general constraint requires a time-s(Me representation of a signal to be perfk~ctly localized to a certMn curve ill the plane, which describes tile group delay of the signal With respect to the identification u = no~a, this is equi:va,tent to demanding that a signal of the fornt
XO/) = u I/~ e ~e~('> U(u)
Chapter 2 C:tasses of Solutions
137
have the time-fi-equeucy representation ( t i 2= I ddv ~ (v) ) U(v) .
O:,.(*:,v;f)=v6{t-t,(~,y)u(v)=,e,5
,2.121) (" "
In thN form, the problem boils down to the ideM localization of a timefrequency representation to an arbitrary "chirp." 5~ Without fllrther re-. strictions, we can perform a direct computation arid o b t d n
"
•
,
!ea,.,~
\ 2 /
/
Hmme, the d~ired condition is verified, if the inner integral with respect to v has the value
l~k)r the ciass of af~ne Wither distributions this leavts to the pair of constraints .......
;)
a<,
2 ,/2 "o a ( < )
=
(!;:e5 (2. t22)
"(~)
We can therefore deduce severa,1 consequences according to the type of tile considered rule of the nmdulation: (i) Constant, group delay: 4~(v) . . . . 2r~vto. The first, equation of Eq. (2. ] 22) automa.tical.ly holds in this c ~ , and the only constraint refers to the second equation. ~¥~ thus recover the previously established result fbr the purely temporal localization (el. Eq. (Zl20)).
(ii) Linear group delay: ¢,:~.(,,) = --.-2~ Ode + ~ 2 ) . ~*~ can easily prove the necessity of H({) = vm and conseq~mntly, • ~
2
1/2
Time- t'?Tequenqy/Time-Sevfie Analysis
J.38
T'he corresponding representation is nothing but a, modified version of the Wigner-.Vilte distributiom as the (unique) specification of the parameterization leMs to
&,,(~, ,.,;f) ~ ~,W~.. (~-,, 4
-
In the previous relation v~e employ the signal (of a pure phase) Xl,,,2(.) =
' "~~ X 0') ~..-,
We t,h u s r e c o v e r a characteristic feature of the Wigner-Ville distribution, namely its perfect localization on linear chirD< In the time-scale context, where it was just re-established, this property has a nat, ural geome*¢ic in~ terpretation, h fact, if we rewrite the const, rain~o equation (2, t 22) regarding h'(~)
as
•
:'~ ~ } . 7
, a
'
') 1 ' ) 3
it can be viewed as an identity of a finite difference and a deri~a,tive. In ease the pha~x~ is a quadr;stie polynomial, both values are the same if the difference is centered around the point where we take the derivative. This mee{s exactly the condition from before with H(~) = *J0.
()] ,
(iii) Hyperbolic group delay: ~< (~/) ..........27r !t/to + c~log ;~? In this ease, the first co,mtraint is verified if
In conjunction with the s ,cond ~,ondition, this implies thai
~,~,2~ thus recognize (see Table 2.3) t,he dlar;~:terization of the Bertrand distribution, which turns out to be t,he only solution to the given problem. Hem'e, its importance relative to h y p e r b d i c chirps is comparable tz~ the rank of the Wigne>Vitle distribution relative to linear chirps. (iv) Grcup'-~delav. "by.. l/~.,~ " • 4L~(I/) = --2w (~40 ~.. a).~, An analogous eNculation as before leads to the specificalions
s(~)=
(
,.,~+
and G(~,)=~.
Chapter 2 C];~ses of Solutions
139
Here the definition of [ nterberger's distribution is revealed in the ~ c a l l e d %wtive" form (of. Table 2.3). Not<-: that the related "passiv(:" t;:>rm of the dNtribution remains essentially localized to group delays of the type "l/zJ~"; however, their localization cannot be perfect owing to G(~.) ¢ t. (v) (~roup delay "by 1/V ' ~~"
:
,I,,.(,,)
= -2~
(~,to
+ 2a~/i)).
Here we obtain the pair
H(e)
[
= ~0 I I-. ;i~;~;
and
G({)= 1 -
as a s<)lution,, which, by definition, characterizes the D-distribution (cfl Table 2.3). Unitarit, y. In respect of the natural me~sure of the affine group, the constraight of unitari W (Moyal's fornmla) for tim{>scale representations is e x - ptx:ssed by t.he equation
'f+<~
., . ,,,..,~ ~,,. a2<,<, ...... ./-~.~ //, ~,..f,. <,, s> <,>
x(t).~F(t)
,2
~,,
•
(2.124)
By reasoning similar to ghat :reed for the marginM distribution in time, and by some further calculations, we arrive at the condition
/,+ ....
f
(,) ( a~,~. f*
c,.{,--
-i,~ = 6 ( r - r ' ) ,
(2.12<)
which must be flflfiIled f;.~i< " exeiv" ~- L" frequency ~. For the afl}ine Wigner distributions, written in their ~ime-frequency form, the same line of arguments leea~ing to the marginal time distribution can be used here, It shows that Eq. (2.125) is equivalent to z
dH
(zI2~J)
Group dehv.<. Once more, we proceed by a:mlogy with the tin:e-frequency ea~:: after ~soeiating the scale with the frequency in t:he usuM w<>~. The nunmrator in the expression of the local first-order moment can be
7'ime~}equenQ.'/Time-Scale Analyuis
140 computed ras
Let us restrict ourselves to the case where the condition of a correct marginal frequem:y distribution is satisfied (i.e., f.~(O,u) = ~ ( r , - ~0); of. Eq. (2.111)) and where {~?.!'(0, u) = O.
(2.127)
0~
Then the desired resuk, which identifies the group delay and the IocM first-order moment, follows in the form:
/
' ~ t f~:~(t,a; f) dt ~X4
=
t,, ,~,
.
(2.1,s)
/[ "i~"~~%(t, a: f) dt For the subset of a ~ n e Wigner distributions, the additiona| constraint equation (2.127) simply reads as
dG
dH <0'
5~rrowhand limit. Finally, let us reconsider the foregoing constraints ~-.. sociating the group delay with the loeM center of gravity. Here we wkst~ to investigate the behavior of t.ime-scale distributions in tile limiting ease of narrowband signals. Provided that, the preceding conditions are satisfied, the time-scaie distributions reduce to the usual Wiglmr-Vitle distribution of such s~gnals.
Chapter 2 Classes of Solutions
141
In order to prove this assertion, let us make a simple substitution and rewrite the affine cbLss (E% (2.62)) a~s O.,(t, ~; f )
"
\~
2.)/
Assuming a narrowband signal signifies that the useful range of the integration with respect, to the ~-variable is confined to
This implies that |he pnrameter function intervenes only by the behavior of .~:,(a~, ~) in the neighborhood of a~ = 0. When we limit ourselves to an e x p a ~ i o n of first order, we can thus write
B~sed on the hypotheses of Eqs. (2.111) and (2.127), namely
f . , ( 0 , ~ ) = a ( ~ , .......~'0)
and
~a'~ . , (0
~,)= 0 ,
we gain the approximation
f b ( ~ , ,,;
~
,~ x \ a -
x"
+ j
dg = ~4,-. t, a ,
Recall that the group delay is a purely local characteristic in frequency. Hence, it is not surprising that it leMs to the same constraints, which yield an interpretation of the Wigner-Ville distribution a.s the limit of representa.tions of narrowband signals. In Table 2.6 we give a summary of the discussed constraints and ad.. missibility conditions. The marks in Table 2.7 point to the conditions that are verified by the Mfine Wigner distributions in Table 2.3.
142
"
,
"
"
"
Table 2.6 Affine Wigner disgribut:ions: typical coI~straints and the corre,sponding admissibility conditions for the pa,rameter fl.mctions H ( ( ) and G(~) C(mstraiut
Admis:sibifity C o n d i t i o n
Energy
. . c ( o ) = IH(0)i
MarginM in time
.,~c(~) = H(~) --- ~
MarginM in frequency
ReMity
a(o)
c(~) =
=
l
(,t-,)
H(O)
dH
=....::.', : ,,<,
H(() : H(-~.)
Time IocMizadon
~?:,2.,
(~)2
Unitaritv
o :, dH. , . . C " < ) = H(~) ......( ";i~~"( 0
Group delay
G(J) = 1
H(o) =
,,,. ;
Narrowband limit
c(0) =
H(O) = -o ;
~d G( ( )) =
dtt .
dG
dtt
(0)
=
ff
2.3.3. R>esults of Exclusion and Conditional Uniqueness Each constraint of a time-frequency representation can be expressed by an admissibility condition tbr the free p~m~meter fllnetion. It is therefbre ea~sy to construct representations that. are Compatible with a preferred iist of speeific~'~tions. It. is not. alwa0,s possiNe, however, to achieve the envisaged goal, as certain constraints can lead to mutuMty exclusive conditions. This c~m be o b ~ r v e d by inspecting Tables 2.5 and 2.7, becat~se no distribution verifies tile Nll act of properties considered there. "~'k~wilt next explAn some of dlese impossibilities, and we wilI Nrthcr im:estigate the issue of uniqueness of a representation relative go a certNn eatalog"ue of constraints.
Chapter 2 Classes of Solutions
t43 Table 2,7
Affine Wigner distributions: a. list of constraints that are verified by several represmatations (B = Bertrand; Ua = Unterberger, "active" lbrm; Up = Unterberger, "p~ssive" form; D = D-distribution)
f
Constrah~t
B
[Energy
v/
Distrib~xtions Ua Up \/
Marginal in time
D
v/
~/
~/
,,/
Marginal in frequency
¢
,,/
v/
v/
Reality
,~/
,v/
v/
v/
Time localization
v/
,v/
Unitarity
v/
Group delay
v/
v/
v/
v/
Narrowband limit
,/
~,/
v/
\/
.j'
W i g n e r ' s T h e o r e m . '% The prototype of impossibilities, that the timefl'equeney ret)resentation~s are always confront.ed, is a theorem from Wigner (whi(h applies beyond Cohen's class), It is remarkable in that it puts an impossibility right to the core of what might be expected as minimal constraim.s of an energy distribution. Its formulation is very sin@e: there ex-
ist:s no timc-freque~,cy representation that is bilinear, has correct marginaI distributions& and is no,metafire evezywhere, In order to prove this result., let us suppose that there exists a representation p:~,(~, t,) that satisfies all three conditions, We must show that this leads to a contr~tiction. Let x(t) be a signM, which vanishes identically outside a certain time interval, l%r ex~'ry instant t outside this interval, our ~ussumptions imply
0 ....... 1:4012
p:c(t,-)&.
= •" : ~ 7 )
As the representation is supposed to be nonnegative, I,he expression oil the right-hand side of this e(tuaiioIl can only ValtkStI, if the representation is zero on the whole (:ross sect.ion. We c a l thus deduce from our tapotheses that the represent:ation vanishes wherever the signN vanishes.
Time'-Freque~cv/Tim(,-Scale Ana/ysi,~
144
Let us next ~onsider two signals :r(t) and y(t), which are zero outside
two disjoint intervMs T~ and 7); hence
If
we
form the line~r combination :dr) = ,,:~:, (t) +
bz,~(t),
the first hypo{;hesis (biline~~xity) ensures that the resulting representat.ion has the tbrm
(with obvious notations used here). Beta.use p, (L L,) is z~ro at every instant outside ~t~, the nonnegativity of p~:(L ~.') can only be g u a r a n t ~ d for M1 a and b, if the mixed distributions p~2(t, tJ) and Pzt (t, ~.,) bodl vanish outside T~. The same argument, when aNdied to 7), leMs to the necessary rmllity d p~2(t, ~) and P2~ (t, t~) outside 2r~. It thus follows that Pt2(t, ~) and p2~ (t, ~) must vanish everywhere, and we obtain that p~.(t,.) = l,~i2p~ ( ~ , . ) +
tv?p2(t.,,.).
If we next u ~ the second hypotheses (correct marginal distributions), we end up with fx0J)i 2 = I~,12 }x~ (v.) 2 ÷ t)i2 tN2(L.)I a ,
and this demands that by the cor~struction of x(t), As this evidentIy corltradiet.s the assumption of a finite duration of Xl (*) and z2(t), the proof is completed. There are several consequences of Wigner's theorem regarding the three exclusive properties: (i) I:f a bilinear time-frequeucy representation has ¢~)rrex t marginM distributiol~s, i~ musg attain negat:ive (or comptex) values. This h a p p e ~ for the dist.ributions of Wigner-Ville, Rihaczek, Born-Jordan, ChG-Williams, etc.
(ii) If a hi/Junior time-frequency representat:io~ is p;~itive, its marginM distributions cannot be correct. This is the case for tile s p e c m g r a m s and a|l their positive linear combinat~ior~s. (iii) If a time-frequency rcwesentation has correct marginal distributions, while being nonnegatiw~ eyeD'where, it era-mot be a bilinear transforma.tion of the signal.
C~:mpt,er 2 Cla,sses of Soh~tions
]45
Let us note, however, that there is no obs{ruction to the existence of solutions of the Iast type, if one a.ccepts leaving Cohen's cla=ss (in the strict sense of our present definition), or if one allows the parameter function of Cohen's ct~ss to depend on the analyzed signM. *v~ewill turn to this issue in Subsection 3.3.3. Some results of exclusion. Inspecting tim different properties of the various time-frequency representations of Cohen's class, the positivity seems to plaT" a completely singular role, e~s it, excludes most of the others, r,~sThis point can. be fbrmalized by considering several examples.
Positivity and marginM distributions. As already seen, the positivity of representations in Cohen's class is incompatible with the simultaneous attainment of correct marginal distributions in tirue and frequency If we restrict ourselves to square integrabte parameterizations, we can prove this result without appealing to Wigner's theorem. For tMs purpose let f(e,r)=Ec~.A~.~:(e,r)
wit},
cx, > 0
and
k=l
Ec~':=l k=l
be the parameter function of a positive representation. Taking the constraints f({, {}) = 1 and f(0, r) = I relm.ive to the marginal distributions into account, we obtain AL(e,0) = k= 1
A L (0,
= t
k:= I
Hence, tim simultaneous identities lh~(t)l ~ = 6(t)
and
IH,~(.)I ~ = ~(,.)
follow. This system of equations obviously ha~s no solulion.
Positivitv and dilation, filtering, or modulation. We find as a corollaw of the previous result tha£ the positivity is ineompatibte with the covariance relative to dilations (or change of scales), the filtering, or the modulations. This comes t~om the fact that each of these constraints has the attainment of the marginal distributions as a by-product. Positivity and supports. A positive distribution cannot preserve the support of a signal, not even in the wide sense. Indeed, if we consider a signal that is restricted to the time interval [---7', +T}, then the conservation of its support by a positive distribution lends itself t.o
/
~,x ~ , ( t , v ; f ) dze = O,
itt > T .
Time-Freq ue,ic~y/'l'imc-Seale A ~mI~,'sis
146
This ca~ be combi~ed wi~h the general rela,tioTl of representations in Ct~ hen's cb~ss~ which holds that
c ,:(t,~ ,j') d~
F(s
. t, O) l:4s)f2 ds
Hence, the parameter function must be such that. f(~, 0) is constam. This implies that a eorrc~ct~ marginal time distribution is attained. Hence it is iI~eompatible with the positivity. A~t analogous conc}usion can be drawn fl'om tim cons*ervation of the frequency support. Remark. This impossibi}ity is rather intuitive, in N~:t. In all cases where a posi~;ive distribution is a positive Iinear combinatkm of spectrograms, it is based on a set of "window" analyses that have nonzero supports (in time and frequency) by construction. Consequel~tly, the supports of the distribution are enlarged by the size of the largest "window." In the limiting case, when the ai~alyzed signal is a Dirac impulse, the positive distribution has the tk)rm C~(t, l/; f ) = E
c~lh~(t)!;"
}c=i
Hence, its supI)oT~ cannot be a mflI set in tAte time domain.
Positivi~v and unit~rity. It is img~ssible tbr a nonzero time-frequency energy distribution to be positive and unitary (i,e., verify M%:N's formula). Indeed, if the a:~cond condition is flflfilled~ then j/.f C~ (t, u; f) (;'~(t, ~ , f) dt d~-"=
./,_~~ :!'(t ) y~ (t ) dt '2
holds for every pair of signals x(t} and g(t). This must remain ~.rue when
x(t) arid y(t) become orthogol,.N on the real line, wtfieh brings about the nullity of tile right-hand side of the previous equation, It thus implies that
C~,(t,~,f) Cu(t,~,; j) dtdu = 0 , and this can happen only for a nonnega~ive distribution, if it vanishes ident;icMly. However, this was excluded by our assumption.
Po.sitivity arm inst~mame(ms frequen(:y or group delay,. The i:~stantaneous fl'equency and tile group detav cannot be obtNnc4 as the local iirst-order mot:rants of a positive distribution. This is an immediate eoesequence of the fact that t h e ~ constraints imply the aetaimnent of the correct marginal distributions befbrehand, and this is hnpossiMe in the case of a posii.ive distribution.
" S of Sotutious Chag~ter 2 (Jasse,i'
147
S o m e r e s u l t s on conditional uniqueness.
If w e accept, giving up the positivity, some of the remaining constraint.s can be satisfied sinm/taueously. A larger number of such constraini, s will naturally cut back on the set of soluti(ms. One can even identify certain combina±ions of constraims that. imply {he uniqueness of the representation. ~\/e will explain several such situations as an illustra.tion. s-Wiguer. The s-Wigner distributions are unique soh.ltions of diti;~rent collections of constraints: (i) 'l'he s-rWign~" distributions are the only tim(sfrcquency distributJons ld~at are u n i t a w and eompat, ible with filterings aud modulations. Indeed, it. has already been mentioned (cf. t~;q. (2.88)) that the h~st two constrail~ts require the special [brni f(~, 7") = e a~r of {,he parameter flmction. Combined with the condition of being unitary, which requires lf(G r)l = I, this shows that the fi'ee parameter must be purely imaginary, so o = i2rrs. ( ii) 2'he s-Wigner distributions are the only time-scale distributkms that are unit~ary and have; a corl~et mar, filmt distribution in time. Indeed, a(> coming to Eqs. (2.126) arid (2.116), imposing these two properti~s simultmxeously amounts to dH .o
=
......... 7 ( ( )
arid =
.o
).
-...
Therefore, we must have 47(~) = I, thus lea,ling to the differential equation dH
Its solution has the form H(~,)=-s~+.~,
sEIFi,
and the related parameter flmction can be written tm f(g, r) = e ~>'(< "~'> . This defines an s-Wigner distribution in its time-scale form, where the scaie a is a~ssociated with the frequency ~, = *Jo/a.
Tim<'-Fmq~e**cy/Th~,,~-Scab., A ~~a,t<~is
I48
Wigner-Vi]te. Severat combinations of conditions ensure the uniqueness of th.e Wigne>Vilte dis*ributiom Here are some of them: (i) 7he Widower-Villa distribution is ghe only s-Wigner dfistribudon tha~: is rea~vdued. As x~- ha~:e see~ in Eq. (2.(18), the reality of a distribution is a result of the Hexmitian syr0metry of hS parameter function f ( ~ 7). Ill case of an sJAqgner distribution it ~bi[ows that
and this (:an only be true if s = 0. This is precisdy the Wigner-Vflle case,
(ii) The $}~gner-Wlle dis'tribution is the on!y s-Wig, mr disa'ibution whose local first-order moments furni,sh fhe instantaneous frequen W ;rod the group &day, h~deed, ~s an s-Wigner distrlbution ha;~ correct marginal distributions, it su~ees to verify the conditions com:erning the derivatives of the parameter fimedon in b:qs, (2,99) and {12.I01). We simply obtain
0f(
3r ~,0) .... i2~.<
and
of,_ ,
~}~:fo,r) = i2~r.~r .
Each of dl.ese expressions can o~fly be zero if s = 0, and this correspm~ds to the Wigner-Ville distribution.
(iii) The Vf'Tgne>Ville di.stributkm is the only distribution in Cohen % ehkss that is compatible whh filterings and yiel& the ins~antm~eous ti,~iue~( T as its local .first-order mome.ut. Indeed, according to Eq. (2.83), the flint condition imposes j(c
r"
....
e (;{~>
Hence, the seco:ud coudition in Eq. (2.99} takes the form G(() : 0. This gives the announced result.
(iv) The Wigner-Ville distribution is the on)v distributkm in Cohen's cl~ss that is compa4,ible wi~ih modular.ions and ~qelds the group delay as its k~cat first-order moment. This result is dual to the previous one and established analogously. (v) The W~gner-Wlle dLstributkm is the only distribution in Cohen ~ class that is ideally cor~centrated on N~ear chirps. This was shown in the preo ceding subsection (of. Eq. (2,103)). P,~ge. The F~ige distribution is the onIy distribution in Cohen is cl~a~s tha.~: preserves the temporal support (in the wide sense) a.nd is causal, mlita~> and ccm~patiMe with modular.ions. Indeed, according ~o Eqs. (2.97) and {2,87)~ the h:~st two cons~raint.s demand from the parameter fimetion to be of the fbrm . ( ( G r ) = ~ ~'~(~' = : * - F ( s , r ) ...... ~ ( . ~ + ' ~ : ) ' ] . k z /
Chapter 2 Cta.ss'es of 5olutious
t49
In these expressions 9(r) is a real flmctio,n As a consequence, ~.heconstraint equation {2,9(}) of conservation of ehe temporal s~pport gives Ig(r)t ~ lr!, and the constraint equatioa (2.73) of c~,usality implies that tg(r)l ~ It. Hence the result fifllows.
Bertra~M. The Bertrand distribution N ~he oMy a/~ne I-V/g~er dis~ribmhm that N ~mitao, and locMized in time. In fact, accordii~g m Eq. (2A26), the first condition requires that ,
,
~dH.
and with Eq. (2.120) the second condil:ion by itself implies that
.,{ c2(~)
-({t
• "
-
k2)
If we introdnce the auxiliary t\mctions
the~.
t, w o
constraiuts tead t,o the differential ~tmttion
c(~)
dV c
V(~). dU
~(~5) o<
-r:: (,) .....
ag
1 .....
l./i)
u(~-) v'(~).
By putting l.I;(~) = U(~)/V(~) this gives dW
,
1
Hence, we derive that; w({) ,
.
=
H(~)
- , &{
'2 )
U(~) + (~/2)
-
',-e',,
Ce
,
which implies 14'(0) = 1. This also gives C = 1. It ~bllows that
. . ==. (~/2) . . eoth(U2uo) . It(~)
a,nd
c (~) " -
(U2-o) sinh(U2,,~ ) '
This defines the Bertrand distribution (cf. Table 2.3}~
150
Time-1" ?'equency /'Time-Scate A ua]ysis
D-distribution. The D<listribution is the only real distribution in ~he arlene cla:ss that has a corr~=,ct ma,:gina/distribution in time and is localized in ~:ime, Indeed, according to Eqs. (2.116} a.nd (2A20), mee~.ing both constraints (and the svssump{;ion that H({) > ~ / ~ leads to tile idmLtity r
,..0ct~)
= H(~) -. ~
,)
: H~(~)
........
Hence, we infer the differential equation
The constra,int of reality requires that H ( - s v) = H(~), and it su~ces t.o find a solution of the previous equation for ~ 2 0. ~qe tluLs obtain that
H(e)
577
:
:=> c:(e)
=
s -
m~d this gives the result.
§2.4. T h e Power Distributions The time-frequency (or time-scale) represent~tions considered ~) far were int,ended mainly to deM with &~terministic finite energy sig,~als. Formally, they can sgitI be used in a wider se~ting, for example, including generalized flmctions. When we now wish to consider random signals, the first thing to be done is to give a nmre general no~;ion of tim power spectrum by making it time-dependent. As in the de~.erministic case, this general~za~.ion is not. unique a priori, and a multitude of possibie solutions exists, e ~ h of whk:h comes with specific advantages and disadva~ages.
2.4.1. From D e t e r m i n i s t i c to R a n d o m Signals Decompositions and fluctuations. A first approach, taking the st.ochastic character of the signal into account, is bas~,d on our knowledge of the deterministic c~se. When we consider linear decompositions, the stoch~.as~ tic character introduces fluctuatioI~s of the (x)effieients of the decomposition, RecalI that the decomposition acts like a tinear filtering of the sig~al. Hence. it is ea,sy to relate the fluctuations of the coefficients to those of the
15l
C?ha.pa~v 2 Classes of Solutior, s
analyzed signal. Let us look more closely at the contim/ous decomposition equat, ion (2,11), for irlstarwe. It is given by L~.(t, A) =
j/i. ~<;~>
x(s) h;>,(s) d s ,
where ), E IR, is an auxiliary variable denoting frequency or scale. Provided that the fourth order moments of x(t) exist (and x(t) has zero nman), we can writ.e ["
E {tl,:,,(~:, a)l ~ } =
~g t * " I I .,.:~,(.~,.s ) h. t:, (s) h~.~ (s) ds ds .
(2.130)
Here we mal~m use of the autoeovariance function r:,:(s; s') = E { x ( s ) x * (s')}. The foregoing quantity defi~,es a verit, able power distribution, because the property equa,tion (2.1:1) of tlie continuous bg~ses implies that
jJE
{i1,~(t, x)l ~} @,~(t, a)
=
.
j[
:.....
.+ :x.
.....¢x.-
6(s -- s ) da ds
E { :45-)~,~ d~. "
(~.131)
~ I
Distributions and e x p e c t a t i o n values. Let us recall that the scpmred modulus of a linear decomposition is just a special case of a bilineam representation, Hellce, another possible way of defining a time-dependent (power) spectrum is by taking all ensemble average of an energy distribution with respect, to M1 possible realizations ,,,,,~(t, ~)
,
E {#,,(t, ,,,)} .
(2.132)
Accordingly, we can generalize Cohen's class (Eq. (2.53)) to the stochastic setting, Assuming, as before, tha.t the fourth o r d e r moments of the analyzed signal exist, we obtain a general class via the definition Cj:(t,v;f) . . . . . .
t,r)r~
s + }r,s -
e -~2'~'' d s d r
(2.133)
Tin~e-t:r~qu~v~<~y/Time-Scale Aua]ysis
152
The choice of the parameter function can be guided, mutatis m)Jtandis, by the Mmissibility conditions that were estaNished for determinist.it signals. Special eonst, raints for the stochaastic situations may be added a.s well. So it seems natural to demand that in case of a stat, ionary signN (whose stoch~stic properties are time-invariant) the spectrum reduc*> to the power spectrum in each instant. Because tim aut, ocorrdat&m has the form r . ( t , t ~) = 7:~.(t - t ') in this e~ese, tile preceding condition is equivalem: to the identity
-t,r)%(r)e
~
d.sdr = P~(~,)
which yields f(0,r) = 1 ,
(2.134)
V~~.thus recox~r the condition of Eq. (2.67), which ensures the correct marginal frequency distribution of a deterministic signM, C r m n 6 r and beyond. FinMly, a t,hird a,nd less ad hoc possibiliV is t,o go back to the fundamental characterization of stationary sig)mls and to find was"s of deviating fl'om it. We already mentioned in Subsection t.2.2, that a random signal x(t) is stationary in the wide sense, if and only if it admits a spectral representation (called Cramdr decomposition)
/ q-:N~ x(t) =
c ¢2~''~ dX(~.) ,
(2,135)
It underlies a double orthogonalit,y in the sense that --<~=
,
(~-)
(2.136)
E {,IX(,.) ~tX'(¢)}
.... ~(,<
- ~) G0.)<.~ d . ,
When we next wish to consider nonstationaI 3" signals, the offered alternative is simple. Because we cannot maintain both the spectral repre~ sentation and ~,he double orthogonality sinmltaneously, we must give up at least one of them. This defines ttm two large cla~ses of approaches leMing ~o an explicit construeticm of time-dependent repre~ntations: tho~v whid~ prioritize the corthogonality, and those giving preference to ~he frequential interwegation, 57
Chapter 2 Classas of Sohttious
i53
2.4.2. The Orthogonal (or Almost Orthogonal) Solutions Karhunen decompositions, We first; describe the soIu~ion that maint, ains the. double orthogonality of t.he decomposition of a nonstagiona.D signal. Ii. is obtained by repla~'ing the complex exponentials wi~h other b~sis funet.ions
~,(t, ,,), so that x(t) =
e(t, ;/)dX(u) ,
(2A37)
and the orthogonality relation (2.138) holds. Such solutions exist and are ca,lled Karhmmn decompositio~s: They resu|t from employing tim eigeufuuctio~s of the autoeovariance kernel ms bask functions. In fa.ct., we can immediately derive from Eq. (2.137) that the autocovaz'iallce has the form
.,.~ (t, s)
= ff,~,(t, ~) ~:,*(s, ~) E {dX(,,)dX"(~)} ~2
(2~139)
Tim mukiplication of both sides of this equation by "~,(s, u), integration over the time variable s, and an application of the orthogonality assumptions of Eq. (2A38) furnish
rz(t,s)tg(s:u)d.s = -- jX,
,,~,(t,~) ~] e:~
~-
(,(s,u)¢, (,,{)ds rz((:)d[ %3
= r:,O~),e(~, ~f). The advantage of such Karhunen representat, ions lies in their double orthogonality, Their major drawback, however, stems from reptaging the complex exponentia.ls with the dgenfunetions '~,)(t; u), as the variable u of the deeomposition can no longer be in~rpret.ed as a frequency. Keeping
Timo-F>equenScale Ana(>sis
t 54
this reservation in mind, the decomposition of the autocovariance leads to the identity
E {Ixtt)i- } =
/7 Id~,tt,,/)l'ra.(u)d~.
Each "spectral" contribution I',-(z/) is ~hus weighted by a time-dependent function, a.nd this leads to lhe forma;l definition of a time-dependent "power spectrum" by K , , ( t , ~,) = le(t, t')l 2 F~(zJ) . (2.14{1) \¥e thus dispose of a truly time-dependem, quantity, which is nora negative everywhm~. More importantlyl it reduces to the ordinary power spectrum in the stationary case, Hc*wever, its spectral interpretation is questionab[e, Nevertheless, the "Karbunen spectra" provide a prototype of repre~mtations for nonstationary signals, on which variants of more tamgible interpretations can be built,. Priesttey spectrum. 5s Su(h a viewpoint was introduced by Priestiey. He started from a formal (but not uecessarity orthogonal) setting of a Karhunen decomposition, supposing that the basis functions submit to the g,meric torm {2.141)
'~'~(t, t~} = d ( e , . ) e .2~':'~ .
The stochastic processes, which give rise to such (not necessarily unique) representatim~s, are cailed oscitlatoW. Physicall> they are directed at taking the temporal evolution of the different spectral contributions of a signal into ac('ount~ by providing a model in a frequency*by-Dequency manuer. Here the flmction A(~, ~,) operates like an amplitude modula.tion of each complex exponential. If A(~, ~e) is slowqy varying in time for ea*:h frequency, the introduction of the notion of osciUatory processes stands for a compromi~'~ between orthogonality and frequential interp,~tation because the basis functions '~(t, z~) are almost, orthogonal under this ~ssumption. Indeed, we infer that
•
}
"X
and this tends to 6({ - ~,) when A(t, •)A*(t, ~) tends to 1. This tast condition ca.n be rephrased by reflerring to the notion of quasbstationary signals, which signifies the fact. that A(t, ~) s}owty varies in t.ime as compared to tile oscillations of the frequency z,.
Chapter 2 <~,ses o[ Solu~ions
t55
The corresponding t.ime-dependent spectrum
(2,i,-12) is derived tYom the special form of ~.,'~(t,z/). It is called the evotutic, na<;~; spectrum (in the seI~se of Priestley), Just like the Karhunen spectrum, it has several interest:lug properties: It is nomiegative everywhere, reduces to the ordinary power spectrum in the stationary cause (A(t, r,) = 1), and has a correct marginal distributioI~ in time because =
,,)}~:~ :,.(,),i,., = s {i:~,(t)I } .
I~hrthermore, we can observe a satisfiu:tory behavior for simple nonstatiormry signMs. Fbr this purpose let us consider a uniformly modulared signal of Eq. (1A5) type, hence,
~(0 = 4 0 : 4 t ) where c(t) d e n o t ~ the modulation and x(t) is supposed to be we~ldy stationary, Then we obtNn the representation
This defines an oscillatory signal characterized kty A(t, ~J) = c(t). Consequently, d~e t ~ o e i a t e d Priestley spectrum has the vaIue
~v(*,,,) = ~:2(t) L(-),
(m143)
and this readily describes the anticipated behavior of the spectrum of a stationa.ry signa,1, whose amplitude is uniformly modulated ia time. These a~dvantages of the Priesttey spectrum, however, should not let us fbrget about its drawbacks, which hax'e restricted, and will always restrict, its use, One of them is the fact that the class of oscillatory signMs on which the definition of the Priestley spectrum relies is not well definext. In fact: there are no simple criteria by which we may verify if a given signN is oscillatory or not. Even wor~? b the la~:t tha.t the class is not stabte under Mdition ( i.e., the sum of two oscillatory signals need no~ be an oscillamry signal). The introduction d the evolutionary spectrum was proposed i~~ l.he Fourier domain. We can Nso pursue a complementary approach in the
Tim(:~b ?('quertC(/Tin~e-Scate Aria@sis
t56
time domain. 'Ihis ofI)rs a new imerpre~ation in a different light- In iSct, let us consider the F\~urier decomposition of tf~e modulatir~g function
A(t, l/) =
h(t, s) e ~ ' ~ ds .
The sta.ti(mamv signal x~(t), which can be constructed from the spectral increments dX(t,}, is
' LT; Hence: we ob~Mn the identities
A(t, ~,) e '~"~ dXO,)
z(t) = •
--
: :xF~
--
:x.
=
h(t, s)
f +:x: =
h(t, s) x~(t
-
e -~'~''
s) ds
ds
e ~2'''~ d X O , )
.
This signifies that a,n osciliat.ory nonstationary signal can be regard(M as the out, put of a linear timc-vm)qng filter who~" input is a stationary signal. The impulse respoI~se of the generating f h e r it nothing but the Fourier trailsform of tbe modtflati~g funct.ion defining the ~ c i l l a t o r y signal. This interpretat.ion is interesting flora a physicM point of view as well. It suggests folkrMng a sIightly different approach than the one by Pri~ttey, ~s it directly refers to a temporal dccomposi~,ion of the signM. Wjostheim, M61ard, G r e n i e r approach. The generM idea behind this modified approach is to start from a Cram6r Vpe decomposition of a ~onstationary ra~ldom signal, which is an exgension of the Wold decomp(~sition for the sgationary case, For ghe sa~kc of simplicity, we will be contented with a heuristic description using discrete signals only, in order to avoid the problems of multiplicity related to t,he contimlous time. a.~ It is known that every discrete stationary random signM {x[n], n C 7Z} M m i t s a decomposition
Chapter '2 (,Tasscs of ~5ohltsons
;i57
(called *61)ld decomposition), where e[n] is a discrete-time normalized and white noise, such. that E {e[n] e[ml} = ~5,,... This signifies that every discrete stationary signal can be ~ .pr(,scnted as the output of a eausa,1 moving average (MA) filter, with an eventually infinite memory, whose input is white noise. An extension to the nonstationary ca~e yields the correct tbrm :':{"'7 ....
~
h[,,.,,,,.} 4 ' " ]
(Zt4~l)
ff)r every discrete nonstationary signal. The generating filter varies in time, as could be foreseen. Because the input nois~ is stationary and white, it can be written as
f t,1/2
.,'
~["1 = , , < / ~ <':'~"~""dS~(, ) where
E {dE(, ) d/£*(()} = 6(r, - ~)&,d~ . We conclude, that the signal mini itself h ~ the form
2:[?Z] =
.]~-.tt2
"
e
'
by defining
<.,,.)
=
< . , . q ~>,',.~.
x~ ~71 "~
•
%,=?
Such a decomposition is the formal anMogue of the one by K;~rhunen (Eq, (2,137)) for the discrete case. Proceeding in the same way as tbr the Priestley spectrum, we can further define an evohstiomu;y spectrum (now s , of Tjestheim and Melard" ~;0) by in the sen,.e~
o , [.,,.1 =
~
<.,,4,
e i27r~/m 2
•
(2.145)
~';q, ?K Once more, this defines a real quantity, which is nonnegative everywhere and reduc,es to the ordinary power spectrum when the signal is sta~ tionary. Moreover, we can prove thai
<,[,~,,J)d,~ = ~ J - 112
,m . . . . ~,c
Ih,['~,'*l ? = E { ! : 4 ' 4 7 } ,
~lime-F)o~tue++qy/7iime-Seale Analv'sis
158
and t,his guarantees a correct marginal distribution in time. Let us first look at uniformly modulated signals g[~] = c[,n] :r[n] (with :r['~t] being stationmy), The (tim~va
where h:,.[n] is the (time-invariaalt) impulse respon~ a.ssociated with the stationary signal z['n]. Hence, the definition of the evolutionary spex:trum of Tjostheim and M~lard leads directly to tile anticipated resuh (-):,.
=
r:,:(,,).
In contrast, to tMs example, cases exist lbr whicl~ the obtained results ean~tot be considered satisfactory. For instance, if we look at a piecewtse stat;ionary signal r ~ {z~[7~] f o r r ~ < 0 zJ~1 = - z,_,[n] for 'n :> 0, where z~[~] and z2[n] are two different stationary signals with respective power spectra Ft(~/) and F~0~), we have that
but only lira
O:,:{n, ,J) = F2(~/)
F~i~ure to reproduce the individual power spectra clearly sten~ Dora the b;usically causal character of the MA representation, on which the definition of the evolut.iom~ry spectrmn mlies. A second drawback of the MA structure underlying the Tjostheim and M61axd spectrum is the di~eulty of its estimation due to ghe order of the model, which mighe be infinite a
pNorL In order m gain access to a more flexible tool, Grenier f;:l introduced a modified version of the T.j0stheim and M6Iard specgrum. So ~ to erMicate the possibiy infinite oMer of the MA model, it is repiaced wigh (or wa.lizeM by) an autoregmssive moving average (ARMA) model of finite order P
q
2=0
j=O
Tiffs new form is amenable to different assumptioEs, More precisely, a (causal) MA system w~th impulse response bin, m] admits an AF[MA
Chapter 2 Classes o f Solut:mns
159
representation of the foregmino'~ .... type,.. if and only,. if there cxl,.t , "~ two integers p alld q, and p fun(:tions aj [hi, such that
m > q , j=O
and ao[n] = 1. The obtained A R M A equation is called "in synchronous form," It gives rise to a representation of the signal in the tbrm of an observable state by means of the relations
-a,I,,-l! 1 0 ... o,}
[
- a ~ [ ~ - 11 0
y[n]=
.
/ b0f<
~ ". ..".
b,[,,] (}
e[n]
y[n-l]+
1
',, -a,-,, [, °- 11 0
-.-
0
\<,,-~[~1
and
x[n]=(l 0 where Y['I = (> b] Y~[',} '
. . ,
O) y[n],
> d " 0 r.
T h e way the time indices (which were arbitrary a priori) are written, this represemation depicts a situation which is frozen at the moment "n, Hence, the nonstationary model seems to define a tangvntial stationary model at each moment ,n, which is characterized by the coefficients a~ I n - I] and bj In]. The spectrum of this tangentiN model can be used as a naturai definition of the rational spectrum, or relict; by Retting
•
i d~(z),4~(z,,-..x)
.
(z14r)
with P
A,,(~) = ~ a j [ ' , , , - ~1~-j , j :: 0
q
~,(~)= Zbj[~]~J. j =0
This spectrum preserves the majority of properties, which are known to be satisfied by the "[~;~stheim and M61ard spectrum. Moreover, it also meets the c(mdition of sectional locality. Another important point is related to tile possibilities that it offers regarding its estimation. In pa~ticu!ar, let us consider the c ~ e of a timedependent AR model, that is, such that b0[n] = 1 and bj In} = 0, j ¢ 0.
7\in~e-t+(?que~,.::y/lime-Scake A ~m/.},:sis
160
Let us f'~lrther assume that; l,he coefficiem,s themsetv(> can be decomposed relative to a ba~sis of functions {f,:['rz]; k = 0 . . . . . K}, K
k;:0
Then the defining equations of the e~s)hltionary model can be rewritten in the equivalent form
:r[n] +
- " ~ ~,-v)" 0 = e[n t
(2.148)
where 0 = (a>,...
,c~;<, a 2 ~ , . . . ,
a~,,,') 7
•
This shows that the projection of the time-varying coefficients onto a basis of flmctions causes the scalar ~ml~statkma.ry model equation (2146) to be mat)p~t onto a vectoriM stationaw model equat, ion (2.148), The timeqm,ariance d the t o e , c l e a t s of this new model (the exja.'s) Mlows us to employ estimation techniques (ba~sed on o~le observed samNe) a~s in tim classieN stationary ea~e. The only arbitrariness lies in the choice of ~,he basis functions, and this depends a priori on the supposed type of the nonstatio~mry behavior. Because the parametric methods are beyo,~d tim scope of this book, we wili not dwell on such evolutioimry models, Instead, we refer the reMer to other books on the subject matter for a more complete study of this problem. G2
2.4.3. The l~'equeney Solutions Harmonizable signals. ~i:-~ Tile second feasible stra.teg2: regarding a relax~ ation of t,he stationary assumptions is described here. It, involves re~.aining the idea of a frequentiat decomposition, while iI~troducing a correlation between the speck,rat increments. On the assump~im~ tha~ the spectral distribution flmct~iot~ ,I)~. is (absolutely) integrabte, ~:his corresponds to an investigation of the harmonizaMe signNs (defined in Subsection 1.2.2), Their autocovariance f'unctim~ admits the representation
(2.:1 m) This function and tile spectral distribution function (D~:(u~,~'2) constitute a pair of Dmrier transforms. This eaI~ be regarded as an extension to
Choppier 2 C'h~sses of Solutions
161
the nonstationary cruse of the pair "autocorrelation ftmction .........power spe(> trum," which is described in the Wie~er-Khinchin Theorem. It certainly reduces to the latter in the stationary case. Indeed, for this case we recall from Eqs. (1.33) and (1.41) t;hal
rx(t~, t~) = %.(t~ - t2)
and
~:~("~, ~2) = ~('~
.....
~'~) G ( ' : , ) .
Hence, the identity equation (2. I49) tlu-ns into
The harmoniza;ble signals admit two equivalent descriptions of second order. One is given in the time domain, the other in the frequency domain. Therefore, it is natural to look upon the time-dependent spectrum px(t, x/) that we tEy to construct~ here as a third description, lying in between the two and being a mixed function of time and frequency, l~:~ra better explanation, let us consider tim Fourier relation that connects r~. (t l, t2) and ~ : ( l ~ , ~ ) . Then we require the searched spectrum p~:(t, r,) to support a linear relation with the autocovariance (or the spectral distribution), so that the new description remNns of second o r d e r . The problem thus reduces to finding a distribution, which ensures the commutat.iviIy of the diagram
/ where the transformations are defined by total or partial Fburier transforms. Hence, we tk)rmulate as an appropriate requirenmnt thin, the relations
r,(t~, t2) =
p,dt, t,) e ',2~'" d~
and
pz( t, •) =
~l~:~,(az~ + b~, c~, + d~) e ~2~r~ d~
be verified for all linea:r transformations of the coordinates
tl
(2.150)
162
Time- F;'equer~c3?ITime-Scale Analysis
~tIld
Furthermore, the invariance of ~he area elemenls defined 1U the ~mrmalized time-time or fi'equencv-frequency ba~sis (whk:h is physically needed for the energy conservation of the transformatioI~s) imposes the additional constx~dm, of isometry
(2.152)
Idet Mt I = idet M,,I = 1 .
The two pa.rtiat Fourier tra.1~sfbrms thus brought into pIay are como patible with the two~dimensionat tra,nsform connecting r~: and
(01 0)<(;
°1) = :lL,,
leading to
The frequency transtormations can thus be derived fi'om the time transformations and vice xersa, Using Eqs. (2.150) and (2.151) and the forementioned paramet,eriza{,ion, the autoeovariance equa{.ion (2.149) can be rewritten as
(2.153)
×
exp
i2~ ( a + 3 ) r
+.-.-N ............~r ........('~ +6)r,t +-
~t
dud~.
By assumption, the searched distribution supports a linear associaticm with the spectral distribution function (and with the autoem~riance). By analogy with the determi~istSc case, imposing the compatibfiity with the time-frequency shifts leads to a ~.lass of admissible definitkms p~, It-)
=
e,~T<, ........j,,~" "<" ..... r)¢),.
(-
~,~, +
::
I2.1>~)
Chap~er 2 ( ' tass{:~ '~ of Solutions
163
where f(~-,r) is an arbitrary parame*;er flmction. [;nder the given h3pothesis mM for the c(msidered transtbrmations, we can deduce that. tile autocovariaiice functio,1 has the form
'<,:(l:~.,t,e) ...... j j
j,~,T)e~
, .....
,,-.~-
d~d~,.
When we compare this expression t.o that of Eq. (2.153), we find ti!e necessary condit kms ctd , ~ : 1 ,
-
........................i , '2
or equivalently o
3
-
..................
2
1 ~
-
-
2 ~
:y +
6 =
2(~ , -
1).
Thus the admissible distributions fall in the general class of represe,it.ations, which are covariant with respect to translations and haste a parameter function of the form
f(~,'r):exp{i2~[((t-o-~)~T
..... 2(-?'.~ 1)fit]} .
As this function must be independent of t and z,, we must further insist on 0 : 1. This is compatible with the condition of isometry, because it implies det al~ . . . . 1. We end up with the set of parameter v~flues {(~, 17~== 1--(~, ~ = 1 ,
}.
6 = - I
Subject to the given constraints, the class of ax]missible representations is eharax'.terized by the paramet;er fimcdon
i
(2.155)
Here we recognize tile pre(ise form of a distribution of s-Wigner type. Its explicit representation can be given in two equivalent ways, namely by
w ~ I t , ~) = or
0~. (, - (~ + ~) ~, ~ .... (~ .........~) ~) ~ , - ~
~
(2,156)
:l'ime~.bi'e'quc.~qv/7~mc-Scate Ana!ysis
164
These two equa~ior~s define a (::lass of represei}ta*ions that depend on one scalar parameter s. As in the determinist}< case, the specification of this p a r a m e t e r leads to dif[erent definitions and ca~l be guided by some additionaI constraints. In part,k ular, the cases s = 0 a,.~d s = 1/2, respectively, define the Wigner-Ville spectrum
(ztas) :
r,~: t +
~-, t - -
and, by anMogy with Eq. (2.5), the R A t , v) =
'
Rihaczek :Spectrum
~I, ~.(. - ~, . ) e
~e~e*
(< (2.I.~,9)
In geImral, dmre is a one-to-one correspondence between the s-Wigner sp(~:tra on the one side and the autocovariance grad speetraI distribution fmmtions on the other side because
w<<' ((;!,
,>(t,,t~) =
<(,.~,.~) =
/7
w?'
.~) t,
<.
~
(t, (~ - ;,),,~ ~ (~ +
~ ~,,
dr.
In particular, the relations of marginal distributions
(2.160)
W ! "~> t ~)dt
t~Ad for all values o[ s. Remark. These two properties could have been extra.polaled directly from the deterministic c~se by meal,s of tim ibrm of the parame~.er timction: It suffices to notice that f ( ~ , 0 ) = f ( 0 , r) = L SimilarD, we can assert that the > W i g , m r spectra are compatible with the changes of scale, linear filterings, and multiplicative modulations. T h e y finally reduce to the ~suN
C?he~pa"r 2 Classes of S,~lutior~s
165
power spe~ trum in the cease of stationmT signaIs, and h~we the property of ...... ) i f ls I < 1/2. conservation of supports (in the wide s~nse We (:an ~hus see that the s-Wigner specira feature a h~rge ,mmber of attractive properties. The mNn missing properties (in the general case) are the reality and the positivity. However, we wilt see that the imposition of certain of these constraints (or of new ones) is enough to restrict the free parameter s to be O, XWigner-Ville s p e c t r u m . Within the chess of s-Wigner spectra, ~he task of finding conditions ibr the uniqueness of the Wigner-Ville spectrmn is trivial, ~ it. reduces to the search for constraints that require s to be zero. Several such constraints can be imagined, aad they lead to the fi~llowing assertions.
(i) The Wigner-Ville :~)ect,rm,~ is the o . l y s~.3qgrler spectrum that is reaI~ valued. Indeed, if we. look at the definition equation (2.157) based on the autocovariance, the requirement of reality of the spectrum reduces to the identity
whid~ nmst be attained for every signal. This leaves as tile only sMution s=0.
( ii) The Wigx~er-ViIle spectrurrJ is the on.(y s-Wither spectrm~ whose local hrst~order momel:~t (center of gra,vity) in t.he frequenqV domain is the m<,a,~ instantaneous ['requenqy: In order to obtain this result, it is suitable to adapt the definitions of instantaneous frequency and gToup dehky to random signaN. Let us re{onsider the instantaneous frequency of a, real signM. As we have seen in Subsection 1.2,1, the usual definition %r a deterministic signal proceeds by computing the derivative of the phase of the analytic signal, hence 1 d a r t z~, ,,,,(t) ~. 2~
dt
(t)
wi{h z r(t) = :c(t) + i H {z(t)}, and tt denoting the Hitbert tramsform. Equivalently, we find that
1 d - - - arct~-~n u , ( t ) = -27r dt
hence,
~7(L)
J'
t66
l imr. b)~quc'nqVl/(/fme.-Sca,le
A~mlysfs
An appropriate definition of the instantaneous frequency of a random signal can tlms be given by the stochastic quanthy 1
"t
[ dz~. •
.,,(t) ..... 2~L { :=,(t)~,}
if the random signal is differentiable in the sense of a quadratic me~!m.'~ Its expectat, itm value equals {[zr(t)' 2}1
{0r,
.
.i
}
(2.161).
Consequently, a direct computation shows that
,i~+~
-~) I [ = 25 k (~ -
This yields
the
(:~
Or.
2,))/£)(t,,~)-
(1
&% ,] 2s):...~.:?~-(t,s)t 08
J .~=:t
.
announced result
] [ . + :x: !~,-(t)I ~ L. ~- vW!;; (t, ~,'1, > = E {.~, (t)} ,
(2.].62)
provided that the condit;ion ,s .... 0 is satisfied, Remark. Again one could have draw~ one's h~pirat&m directl.y fl'om ~he deterministic case by noticing that
f ( ~ , r ) =,>~ ..... "
~'
i')r:({'0) = i2ms{ .
Ptfis quantity is zew if sad onty if s -= 0. As in the deterministic case, i¢ is worthwhile [o remember that the uniqueness of ghe Wigner-Ville spectrmn makes sense only reta¢ive to the imposed constraints. Some differe~lt viewpoints coutd lead to ottmr solutions with their own advantages and disMvant.ages, As an exarnNe, the additiond constraint of causality in the [;;aurier relation between the spectrum and tile augoeovariance fimction would lead to the definition ~'0 (~'
")
~':'7 ( t '
t
r'"" r )
....
~ " ~.lr
,
This is the counterfmrt of the determiifistic Page distribution for the e~se of random signals. Apart: from the causality, however, the Page spectrum
Chapter 2 (3~><,,s of Solutious
167
does 1to~ possess as ma~y good tbatnres for applications as the Wigner-Vitle spectrum. In order to emphasize these good properties o,w.e more, it might be im:eresting to illustrate the behax4or of the Wigner-Ville spectrum and its possible fbatures in signal analysis by looking at some simple and typical examples.
Loea.lly stational:y" sigmds.
The corresponding tbrm of Eq. (1.44) %r the
autoeovariance is T
T
It leads immedia,tely to the resutt
and this agrees nicely with the physical interpretation of a stationary power spectrum, which is modulated in time.
U1dform~ modulated sig~lals. In this ea~e we have
W,:(t,v) =
t +
c
t .......
? ~ ( r ) c -':2~'T d r
.
2~2
Consequently, if the evolut;ion of c(t) is slow in compariso11 with the radius of correlation of x(t), we obtain the approximate form w~,(~, ~,) ~ e2(t) L 0 " )
•
This can almost be identified with a Priestley spectrum (cb Eq. (2.t43)).
Filtered white noise. Let us consider the quantity ~:(t) ,=
?
h(t,.~)dB(s),
E {gB(t)gB(s)}
= G2~*(t -~- s ) d t . d s .
CXT)
Physically it corresponds to the filtering of a white noise ("deriva.tive of Brownian motion B(t)") by a filter with an (eventually tim~va,rying) impulse response h(t, s). In general, it also furnishes a Cram6r representation of nonstatfionaI3, random signNs in cont.inuous time with nmltiplieity one. Fbr such signMs the Wigner-Ville spectrum can be written as
~Ar (t,u)=~2
h t + 2 , . ~ h? t - T
e_~,,~,dr
Time-bb.~Tucuc\v/"Time-5"caIe Ana(vsi5
168
Its exp[icit form depends on the nature of ~he considered fih:er. Le~ us first look at the speciM ease where
hit, .~)
h~(t
....
,~) (/(t) (.,:(s),
-
whidl corresponds to tile g,ctivation of a time-inva.ria,nt linea,r filter with impulse response h~(t) at the initial moment t = 0, Then v,~ obtain the simple form of the sp~x:trum
W:,: (t, ~.,)
. . . .
t,I), (s, ~.,) ds [7(t) ,
~.2
(,.163)'~
The preceding relation shows, in par~,icu|ar, that the evolutiona,ry character of the response t,o a white noise is restricted to a duration T, provided that the fih,er has a finite impulse respo~se with the same duration T, Beyond this transient period, the steamy state of the filter ks effectively attained: It is manifi~.sted in the Wigner-Ville spectrum by the time-invariance of a certain frequential pn)perty, which is closely relate~:t to the power density of d~e frequency response of the filter, hMeed, a straightforward argument shows that t > T
~
W~(t,~)=<~i~(~)
2 .
On the other hand, if we consider the perfect integrator h~(t)
=
u(t)
,
the transient period lasts tbrever, and it aatua,lly defines the Brownian motion or Wienm~-L6Kypwcess
B(t)
h(t, s) J~(st u(t).
= tl
In th.is case~ the Wigner-Vilte spectrum admi~,s the simple (positive} form 9
= 2<, t
,
2
t sin 27rtuit \ " L" t
''( )
(2. o4)
Here the time-dependence is evident and expresses the nonstationary character of t;he Brownian motion, Let us also observe tha~t the W[gner-Ville spectmlm puts us i~go a position to define the notion of a mean spectrum of a nonstationary signM. ~,:~,~simply let
Chapfer 2 Ch~ses of &dutJons
169
In the case of the Bmwnian motion, a direct, computation shows that ~r 2
IbO,)
:
. . ,,
4"~ -
(2.1{;{~)
L'"
and this establishes a nice corresfxmdence with. the beha.vior ~'qike 1 / f :>' of the "empirk:at" power spectrum of a Brownia.n motion.
Impulsive noise. A last and simple (bul, ptGsically important) example of a nonstationary signN shall be eo~sidered next: It is called impuMve noise m~d can be regarded as a fikering of a Poiss(m point process. Its represmltation in time haas the ge~eral form +-x.
~:(t)=
~
<>,
h(t
Iv ...........~ i
where h(t) is the impulse respomse of a linear filter, at, d the instants t;: are distributed according t,o ~ Poisson rule of variable density A(t), It fbllows that, the associated Wig~er-Ville spectmn~ acquires the form WAr
L,) =
~'G (s, ~,) a(*: - s) & .
Hence, it is the convo}ution of the Wigne>Vilie distribut.ion of the impulse response of the filter and the density of t,he Poisson impuI~.~s. In et'~qe of a shm~ variation of the density in comparison with the temporM support of the fil~er, the spectrum has the approximate vatue v¢,:(<,,)
;,: x ( , ) ! H ( , ) I
~.
It thus exposes a locally stationary si~.uation, which of course reduces to the stationa W c~se when the density is constant. When we consider the intermediate case ),(t) = .X U ( t )
,
where a consta:ng density is used from the activatior~ of the filter (initially at rest) at time t = 0, we obtNn that.
Hence, one ca~t recognize a situation which is fi)rmal}y equi~Jent to the tiltered white noise (cf. Eq, (2.163)), ex(~.'pt tha~, a replacement of the power s p e c m u n of the white ~oise with the density of the Poisson impulses has been performed.
170
1 m~,-1~requen(T/ I u n e - Y c a k A n a l y s i s
2.4.4. S o m e Links B e t w e e n the Different S p e c t r a Atdiough the differem spectra discussed befbre were obtained by arguments of a different nature (ord~ogonali V vs harmonizabitity), we {an still try to establish ce~Mn links between them, at: lea~,t in special cases. C o n t i n u o u s time. ~r, Let ~Ls first dwell on the doubly orthogonal decomposition of equation (2.137). We ah>ady noticed (of. Eq. (2.139)) l,hat the a..~sociated autocowariance function can be written as
,,~:(t, ,~) ~:
,,.:,it,,0 e* (.~, ~) r,.(~) <
By inserting this expression into the definition of the Wigner-Ville speetnlm, we derive the result j.7 ~
r
r
Written in a different %rm, this is the same a;s
>z
where the notation for the Karhunen spe<trum K:~.(t,{) = l O ( t , { ) t ~ I',:({) ~sed as before, and where we let V ( t , u ; ~ ) = I0(L~)}2 .......,~ ~, t ~
,g
'¢,* t ........~ , (
e-'2*~;'" d r .
We ha:ve thus established a fbrmal relation that enables us to pass from the Karhunen spectrum, whM~ is b a ~ d on *.he double orthogonatity, to the Wigne>Ville spectrum, which wioritizes the frequential interpr~,tation. Let us next extrapolate this rehttion to os(illatory signMs. T h e y are defined, as we recall from Eq. (2A41), t\v lett,ing
V,(t, ,J) = X(t, ,,)e ~2;'''~' . Then we (an find the analogous identity
w~(t,~) =
v(t,~,;() z~(t,O<
:
(2.16,9)
(;'I,ap~-r" 2 C1a~sses of Sob.itiol~.v
171
here E,:(t,{) is the Priestiey spectrum and
V(~':.z/:g)=, A(t ~"
A
"
+
L
(
c
r
A ~ t-
g
)
e -i2"~y-z)r
dT"
(e,,ro) Let us restrict our a t t e m i o n to oscillatory signals wittl sufficiently sIow variation, so that we can use the approximation T
C
.....
Then the identity equation (2.169) can be simplified t.o
w.,:(~,,,) ~ z~,.(t,,,).
(2.171)
This simplification is in tandem with previous statements regarding the equivalence of both spee{xa, t.he Priestley and the Wigne>Ville, in quasis t a t i o n a w si~,uations. Discrete time. If we sta.rt fl'om the Cra.m6r decompositiml (which underIies the ~[]estheim and M61ard spectrum) of a real and discrete signal
xI~,! = ~
h[~..,,,;
+*l,
we obtain the r{,pr( . . . . .8(Jlt,at, ' " ~lt n i,,i~(j,k
,.~,.[:i,kt= ~
)
h[:/,~,1t+<,,q
for the autoeovariance. Having such a model at, our disposal, we can incorporate it into the defirfitions of spectrm which are ba~¢d on the a m o covariance. In particular, if we consider the discrm,e-time version of the Wigner-Ville spectrum, which rea~:Ls as
k; =, #
we derive that
(2.173)
Time-fq'equenqv/~l)'m,:~Scate Analysis
:t 72
This expression should be compared with the Tj~st.heim arm M~lard spec~ trum (Eq. (2.145)), which can be rew-dtten as
e~,[...,~) = 2 ~
~
h[,,,.~
- . .
~:1<~..~.+~,1~ ''~'*' .
.
.
.
(2.:~ ~7 4)
}C~--5X~; Wsra::, :We
T h e ~ two representations reflect the same st.ructm'e. They are only dNtinguishable by the blend of their indices for the snmmation: this exhibits the %ndamentally causal (noncausN, respectively) character of the Tj0stheim and M6tard spectrum (the Wigner-Vitle spectrum, respeetfw~ly). ~h% can extend ~his comparison by referring to anot.her causal representation, which belongs to the same general chess as the Wigner-Vi[Ie spectrum: the Page spectrum. By anMogy with the definition in continuous tinm, let us use the relation
as its definition in. the discrete case. This leMs to an expansion of the form
Again, there appears a structural shnilarity and a distinction in the blend of indices for the summagion~ Let us regard the Cram4r &composition. of a nonstationary sigl~al as being char~terized by an (infinite) lower triangular maGix
h[n H=
1, ,~ - 1]
h[,~,,.
......... ~]
h[,,~ + L
~, -,,
0 hi,.,.;,,]
(2.1r6)
1] h[.,~+ 1,,,.1 t,,[,~ + 1,,, + 1.1
The three spectra (Tjostbeim-bl6lard, Page, and Wigner-Ville) are distinguishable by the information content, of H that they use. More precisely,
Chapter" 2 Classes of S(fiutions
7M
173
P
Figure 2.12,
WV
Difl>re~t spectra.
Symbolic comparison of the information used by the definitions of Tjostheim-M61ard (TM), Page (P) and Wigner-Vi/le (WV) tbr the construction of a time-dependent spectrum, which is based on the Cram&" decomposition of a nonstatiom~ry discrete signal, In each case, the square represents the matrix H, and tim portioll dra~vn iD bl~:k d(s piers the part of that matrix used by the analysis. for calculating the spectrmn at the moment n, the approach by Tj0stheim and M6Iard uses only the elements with a row index n, the one by Page also brings the elements of the upper triangle into play, and the WignerVille spectrum is the only one that, by its noncausal character, uses the irflbrmation of tile lower triangle as well (cf. Fig. 2A2). This highlights the differences that can result from caTJosator noncausal approaches. The causality can intervene oil two different levels: (i) The causality of the system generating the signal: Only the past innovation can affect the present state; and
(ii) The causMity of the transformation, which determines the spectrum of the signal at each moment: Only the signM's past can affect the present value of the spectrum.
174
Time~-t,?vqt~cr~c37'Tin~,>Scah, k ~alysis Chapter 2 Notes
2,1,1,
1 It is diNcu/t to locate precisely the first appeara.nce of the methods of 'qnstantaneous specIra" in tim literature. The Mlusion to Sommerfeld is borrowed from the book by Pimonow (1962), wilere one can lind an extremely important bibliography on v.he subject, which includes in particutar measuring apparatus, lit is surprising, however, tha.t there is no refbrencc to the hmdamenta} article (Vitle, 1948).] The first ar~.ictes on ~.he %ound spe~trograph" or sonagraph were Koenig, Dram, and Laccv (1946) and Pot~er, Kopp, and Green (1947). We mentioned the FTAN me~hod proposed in Levshin, Pisarenko, a.nd Pogrebinsky (1972). But we should menl:ion, that this variant is x~,ry similar~ in principle, ~;o other approaches such as GambardeI}a (1968). There are many examples of applications of spectrograms/sonagrams. The most. significant and best M~own or~es, for which these methods were invented, concern humm? speech, or in a wider sense, any audiooacoustic signN~ There are books such as the one by Potter, Kopp, and Green (I.947) devoted to the "visibte speech," or Pimol,ow (1962}. A more general sy~lopsis of ~tm applicatkm of time-freque~lcy methods to speech signNs can be tbund in d'Ale~andro a.nd Demars (t992). :~ The reading from a spectrogram was particularly well dew, loped by Zue and the group at. M1T {e#ee, for example, Zue and Col< 1979; or La.mel, 1988). ,I As far as the auditory models in terms of filter banks are concerned, we refer to Fl~magan (1972) and d'Alessandro (1989) for a first study. 2.1.2.
The firs~ continuous generalizations, which followed from the initial work by Gabor (1946), appeared in Helstr6m (1966) and Montgome~T and R e ~ (1967). The decomposition into the prolate spheroidal wave timer&ms is due to Lacoume and Kofman (t975). The origine~l articles are t;he ones by Bastiaans (198(.I) and Balial~ (1981). ~:~Although "wa~elet theory" emerged only reccnt~ty as an established sob entitle field, it, has Nready undergone a rich history. An important: impact comes from the f~-t. that. it offers a forum vo workers with rather different backgrounds (physics, mathema.tics, signal, etc.) One of the inaugurating a~rtieles was prese,~ted by GrossmaIm and Morlet (1984). A~a excellent introduction to the theotT, coupled with a Nscina~ing historical perspective, can be found in the small book by Me?,'er (1993a).
Chapt:er 2 Claess'e,sof S(thltior~s
175
2.1.3.
r The Ville distribution was defined for tile first l.ime in tile article by Ville (1948) quoted here, and the ~Vigner distribution in Wigner (1932). Bass wa~s one of the first who devoted much of his work to t~he Wigner dkstribution (during the same period as Ville, but obviously without any connection i.o him). tte introduced the notion of the locM momentum in Bass (I945). s The original articles are by Claasen and Mecklenbrguker (t980a,b,c). They constitute the first complete synopsis of the useflllness of the WignerVille distribution to signal theory. They were fir:st published in the Philips dourn,~l of Research, and were later more widely disseminated in international conferences ( C l a ~ e n al~d Mecklenbrfiuker, 1981). Indirectly, they invigorated the more confidential work carried out at the same time by some French groups (of. Escudi~} and Gr~a, 1976; Bouachache, Escudid, and Komatitseh, 1979; and Flandrin and Escudi6, 1980). One can ,.tssess the development of the studies in this area that tbllowe.d until 1985 by con~ suiting the cumulative bibliography in tile survey article by Mecklenbrguker (1987). It can also be noticed that this date marked a slowdown of the exponent, ially growing activity experienced until then. As a m ~ t e r of fact, this slowdown was intensified by the growing interest in the new wavelet theory. On the other hand, but only later on, the excursion to the land of the wavelets gave a fresh impetus to the methods ~'~)~la Wigner," which is to be regarded eus a natural bMancing Mt~ect. One can thus experience a renewed ii~i;erest in these methods since the beginning of the 1990s, a,s is now testified by the compilation of ref~rences in tile survey article by Hlawatsch and BoudreauxoBartels (1992) and in the volume by Mecklenbrguker and Hlawatsch (1998). v The original article is by Page (1952), but the same ideas were reintroduced several times, see, tor example, Grace (i981). The related "anti~ causal" version w~e~ initially introduced in Blan(>Lapierre and Picinbono (1955), then independently in Levfit (1967). 10 The Rihaz:zek distribution w~s proposed in B,iha~:zek (1968) ba,sed on purely theoretical arguments. As far as we know, the original publication w~, left totally isolated and has not led to any follow-up studies, not even by the author. An attempt was made at the beginning of the 1980s to use it fbr an application called a "phonochrome" (Johannesma e~ M., 1981). But it only t%und mediocre success due to the b M visual quNity of the representation. Extensions or modifications wez,~ proposed more recently in Hippenstiel and de Oliveira (I990) and Aug~r and Doncarli (1992), ~ There is a very interesting dkscussion of the sheer notion of a "spectrum" of a nonstationary process in Loynes (1968). [One should also read the
176
Time-Frequen<:~y"Time-Sc~le A m~*~.>'4s
discussion that follows this article, l Among other genera~ and ~vailable reibrences we also might mention those by Priestl(,y (1988), Crenier (1987), or F h n d r i n (t989a). :~2 The original article is by Cohen (1966). It was presented in the setting of quantum mechanics. The same author gave. a "signal" tbrmulation later on (which was introduced in Escudi6 and Gr~a. (t976) in parMlel). Cohen (1984) he~s fl:trther proceeded with the development of its properties. Since then he hms made numerous contributions. A good ~mree is Cohen's review article (1989) and his book (1995). There also exist other general articles and/or books dealing with this subject, for example, Longo and Picinbono (1989), Itlawatsch and Boudreaux-Barteis (1992), Boashash (1992c): or Mecklenbr~iukcr and Hlawa~tsch (1998). The approach by J. Bertrand and P, Bertrand was started ii1 P. Bertra~id (t9~3), and the tomographic construction w~s especiMIy laid out by J. Bertrm~d and P. Bertrand (1987); the same authors gave a recen~ survey of ):heir work (1992a,b). 2.1.4.
1:~ Besides the already quoted and more recent work by L. Cohen (1992), there is a comparative approach of ~'quantum mechanics/signal theory" in Mourgues (1987). The referenced work by Born and Jordan is f¥om their original article (1925), l~ The quantum theoret, ical distribution, whMx anticipates the Rihaezek distribution, w~s propo.sed in Ma.rgenau and [till (1961). On ~he other hand, the reason that there was never any analogue of t,he P;tge distribution comes from t,he difficulty of giving a suitable im;erpretation of the notion of ea.usMity in .space. (The same difficulty occurs in image processing.) A similar difficulty arises, though diffbrent by nature, when one tries to transfer the notion of the spectrogram, as it was done in Kuryshkin (1972; 1973). We refer to Ivanovic (1983) for a. possible interpreta(ion in terms of
prol)ensit:y, 2.2.1.
1.-> I'Br a more rigorous treatment of the approach presented here. and to know more a.bout the t,heory of represent at,ions in connec~Aon with the transformation groups, the reader is referred to Grossma~m, Mortet, and Paul (1985; 1986), for example. These reiin'ences also provide an introduction ~,o the notions of invarbmt meaxsure and reproducing kernel. Is A more detailed descriptk>n of ~;l:retheory of fram(:s (%truc~:ures obliques" in l'~'ench) and their applications are given in Daubechies, Grossmann, and Meyer (1985) and Daubechies (1992).
Chapter 2 Classes of Solutions
t77
2.2.2.
lr Among the abundant litera.ture on the short°time Fourier transform we quote the surveys in Allen and Rabiner (1977), Portnoff (t981), and Nawab emd Quatieri (1988). Is It is diNeult to identify the roots of t.he condition for the mixed reconstruction equation (2.20) precisely. It "seems to ha~*e been part of the common knowledge for a very long time, just like the tales of the brothers Grimm" (A. Grossmann, private eonununieation). >? The conditions on the density of t.he sampling lattice for the short-time Fourier transform are discussed, for instance, in Ba~stiaans (1981). 20 The obstruction by Balian and Low w~s originally described by Balian (1981). A complete proof is given in l)aubeehies (t992). 2~ We can refer to Jensen, Hoholdt, and austesen (1988) as one of the exampies for the construction of orthonormal bases % la Oahor." Nevertheless, it must be noted tha~: the Ba]ianoLow obstruction can be circumvented by using IaTilson bases. They emplc)y suitably adjusted sines and cosines instead of the comple:x exponentials entering Eq. (2.22) (Daubechies, Jaffard, and Journd, :I99t). ~ The solution can be looked up in B~stiaans (1980). 2a A summary of the Zak transtbrm, viewed from a signal perspective, can be found in Janssen (1988b). 24 Gabor's original article (1946) Woposes an iterative computation of the eoetticients of the decomposition. The issue of how the dual btusis depends on the density of the time-frequency lat.t~ice is explained in Daubechies (1992), See also the collection in Feichtinger and Strohmer (1998). 2.2.3.
2~; The continuous wavelet transform wa~s originaIIy proposed by Morlet who called it "cycle-octave analysis" (1982). The first theoretical paper, which explores its capabilities and the admissibility conditions, is by Grossmann and Morlet (1984). A general presentation of the suhject matter is Torre'sani (1995). 2~3 One can consult Grossmann, Morlet, and Paul (1986), tbr example, for a description of the anne group and its im~ariant measures. ~r The conditions for the discretization of a wavelet transform are extra> sively investigated in HeiI and WMnut (1989) and Daubechies (1992), 2s The quoted numerical examples are taken from Daubechies (1992).
t78
Thne~Freque, nq~?l'l'ime.~.Sca]e Anats,sis
2,~ The principle of a,n a,hnost perfbct rec,onstruction by ()yet'sampling and the use of several "voices per octave" is used, tbr instance, in KronlandMartinet, Morlet, and Grossmann (1987). :~ Our presentation *~ithNlly tollows Daubechies' book (1992). One can equally gmn profit from reMing the books by Chui (19!)2), A. Cohen (1992), Abry (1997), or Maltat (1.998), not t;o forget those by Meyer (I990; 1993a,b), of course, which are just ms e×haustive as introductory to the subject. These different, hooks also deal with important extensions and/or variants of the theory (such as wavelet p~(kets, local trig~mometfic toases or MMvar wavelets). We will not touch on m~y of those in this book. :~l The concept of multiresolution anNysis was introduced and perfected by Mallat and Mayer (see Mallat, 1989a,b). :~2 The "quadrature mirror fiIters" were originally introduced by Esteban and Galand (1977). They we.re the subject of many development,s, which can be associated with t.he names of Smith a~d Barnwelt (t986), Vetterli (1986), and Vaidyanathan (1993). The conlteetion with wax'elets was first observed by Mallat a.nd explored more deeply by A. Cohen (t992). These aspects receive close attention in the books by Vetterli and Kovacevic (1995) and S~rang and Nguyen (1996). :~:~ The notion of a wramidal algorithm, such as the Laplacian pyramid in Burt and Adelson (I983), existed in image processing before the advent of wavelets. Daubechies (1988a) was the :first who brought these works together with those by Mallat. :~ It should be remarked that a first basis of wavelets was discovered by Meyer (1990) shortly before tile int;roduction of the Battle-Lemari6 wavelets. At first sight it looked unique, as f%~.ras its const,ruction relied on some "miraculous" properties. This is very well explained in Lema¢i6 (1989). 3:, See, for example, Daubechies (1992) or Chui (1992). :~; The first construction of orthonormal wavelet bases with compact support is due to Daubeehies (1988a). 37 See Daubechies (1992), where other consequences of the compact s u > port are also investigated in more detail (such as t;he necessary growth of support as a fimction of regularity). As Nr as the Ioeation of the poles for a. best possible approximation of a linear phase is concerned, one emi also look at Dorize and Viflemoes (1991). Let us finally remark that we only considered the c ~ e of orthonormal ba.~es here. More flexible solutions can actually be obtained in a biortho~)nal set~;ing. This is described in Daubeehies (1992) as well.
( hapter 2 Cb~sse.~ o f Solutions
I79
2.2.4. :~ "~V~quote Wbale~ (1971) and Van Trees (I968) a~s examples of general books dmt. deal with the optimal decision theory (deteet;ion/estimatitm) and related concepts (maximum likelihood, matched filteri~lg, eteo).
;~v There axe two references in t;¥e~leh that specifically deal with the p r ~ e~ssing of radar mid/or sonar signals: Le Chevalier (1989) and Bouve~ (t992). A more mathematical N~proach is given in Blahut~ Miller, and WiIcox (1991). Here one also fit~ds a thorough treatment of the modehng of echoes in radar or sonar from the equations of physics. .~0 Tile definition was introduced by ~i.~odward (1953). The notion of the ambiguigy %notion initiated an abmidant literature, particularly durir~g the 1960s. As far as we know, however, there exists no book that is specially devoted to this subject. Pot learning more about it, one can still profit from the books by Vakman (1968) or Rihaczek (i96(,t), whkh contNn ma:ny illustrations, 4~ The original concept of wideband ambiguity Nnctions goes back to Kelly and Wishner (I965) a~(.t Speiser (t967). A more protound investigation of t.he properties was performed by Altes (1971; 1973)4:~ An iterative ~qection algorithm for choosing the most pertinent atoms in a large dictionary w ~ proposed by Matlat and Zhang (1993). The paper by Att(~ (1985) deals with the same issue a~(t includes a.n e~:~silya~:ct~sibte description. 2.3. 4:~ No doubt tile principle of the,se constr~lctive approaches appears for the first time in the literature on quantmn mectlanies, ~s seen in Kriiger and Poffvn (1976) a~d Ruggeri (t97t). Later it. is presented iI~ different contexts of signal th~)ry, such as in P. Bertrand (1983), Fla.ndrin (1982; 1987), Hlawatsch (1988), or RiouI and Flandrin (1992} 2.3,1.
~ See for example Duvaut (1991). it, Our presentation here %llows Riout and Flandrin (1992) fbr establishing tile general %rm of a class of bilineax time-scde distributions. We have chosen it tbr at least two reasons. First, i{; accentuates the parNlel with Cohen's class. Secondly, it simplifies its inte*t~retation while explic-. itly depending on the scale parame{er (,, which also plays a role in the wavelet transform. However, when we endow the ~ime-scale class with a tim~>fl'equency interpre{.ation (an issue to which we will crone back later),
180
Ti*~i(-f)eq~e~lc3'//Jime-ScMe A~a]ysis
we must observe d~at a suitable (hange of the parameterizat, ion cm~, i~ certain ca~es, lead to an equivalent fl~rm~lation as Bertra~M's class, which is considered in J. Bertrand and P. Bertrand (1988; t992a) (see also Foot~:~ote 47).
2.3,2. More compte~,e lists of definitions cal~ be fourM in the surveys by HIawatsch and Boudreaux-Bartels (1992), Auger (t991) and Boudrem~xBarrels (1996). IIistoricMly, the Aekroyd distribution waa proposed iIl Aekroyd (1970), die ,s-Wiggler dfst;ribution in Ja~!sser~ (1982), tim dist;ribution with a separable kernel it~ Martin and Flandrin ([983) and Flaadrin (1984), the one }~,"ChN mM Williams in t989, by Zhao, Atlas, and Marks in 1990, and the Bu~terwor~h represml~ai.ion in Papandr~,m and Boudreaux-Bartels (1992). in a sense the Born-Jordm~ distribueim~ was never really defined in its explicit: fi~rm, as we mentioned in Subsection 2.t.4, prior to its actual use in Flandrin (1984), ,~r The colmept of "atline'~ or "widebamt time-frequency" distributions is due to J. Bertrand and P. Bertrand, They were the first: to conslmet a family of i;ime-frequency solutions based on a c(wariance prim:iple reIatNe to the a n n e group (see for examp}e P. Bertrand, 1983; J. Bertrand a~M P, Bertrand, 1988; or Boasha,sh, 1992a,b). The <:lass of "aN~e Wig~mr distributions" defin<~t here, wi{h a differem parameterization equatior~ (2.63) than the one used by 3. BertraiM and P Bertrand, fits imo the same falmly of representations, ' ~ , should iiote tha~: ~he generM cla:~s of these d}stribt> tions inchMes ~,(~vera,1definitions, such as tim one by Unterbergm: (1984), which were propo~d in a totally different context (symbol caJcuhis). is The systemat:ic eNflorati<m of the properties of a distributio~ bazsed ott its parameter flmct;ion was first started h~ Cohen (1966), then purs~ied and complm.ed in C l ~ s e n and Mecktenbr/(~iker (1980c) and subsequent work. Nowada~,s the (more or less complete) lis~. of dtese results is contained in almost all surveys (see tbr example FlarMrin and Martin, 1983a; Flandrin, 1987, 1991; Aug,,r, 199I; Auger and Doncarli, I992; Rioul and Ftandrin, i992; Hlav~'atsch and Boudrem~x-Bar~,els, 1992; Cohen, !995: BoudreauxBarrels, 1996), Except fbr some special cases we do r~o~ tD" to retrace their geneMogy systemat h:ally, *'~ The properties of positivity of the time-li'e(#tency distribmkms were especially studied by aansser~ (/984b; 1985; 1988a; 1998). ~,i) ~ e : for examp[e, Pieinbono (1977). 5, This result and the a,rgmnen~ k)r its proof are borrowed from Auger (1991).
(;'tmi)ter 2 CL,~ss(!,s o f Soh~ti(ms
18t
:;~ Moyal's {brmula, first appeared in Moyal (t949), ~';~ The chosen approach w~:s proposed in Ftandrin and Escudi4 (1982). But one can ~s well consider other ways in which a good localizat,ion in the plane may be guaranteed. Some of them will be explained in the co~t:ext of atone Wigi~er distributions (E% (2.122)), We wish to mention that there exist further solutions that are based on a. rmdt.ilinear extet~sion of the Wigner-Vilte distribution, for example, from Boasht~,sh and Ristic (11992) a,nd Amblard at~d L~:ounm (1992). ::"~ There is a more systematic and formal treauneut of this problem in J. Bertrand and P. Bertrand (1992b). Tim p~sent objective is t,o simplify the approa~:h in order t.o ~ender its geometric interpret, ation more easily accessible. This will be further developed in Subsection 3.2,2, 2.3.3.
~'~ The original proof of Wigner s Theorem w ~ given in Wiguer (1971), Its presentat.ion and a complete discussion al~: contained in Mugur~Sch/i~chter (1977), :~G A rat, her systematic di~u~sion of the results of exclusion owing to the positivity of t~he distribution is given in Ja.nssen (1988~), 2.4.1.
57 Among the g(merat references that provide a synt.het.ica.1 description of these two large eh~ses of mediods we reti~r to Grenier (1987), Flmidrin (I989a), or Flandrin and Martin (1998). The paper by Matz, Hla.waJsch, and Kozek (1997) offers a complementary point of view, 2.4,2.
:;s Tile notkm of the ew:~lutionary spectrum (in the sense of Priestley) was introduced in Priestley (1965)~ It is also contained in Priesth~y (1981; 1988). The discussion of the problems connected with the definit,ion of osciilatory sigJmls catl be ik)und in Battaglia (t979), 5.(~bbr a deatription of the g~,neral notions of &;('omposit:ion and m u}tiplicity w~ ref(,r to Cram4r (t971). (;t~ See Tjestheim (1976), Mdlard (t978). ~l For an overview over ~.hese works one can eoilsult Grenier (1983; 1986; ~;2 Among the other approa,'hes, which are not itmntioned here, the adaptive me~hods come first. When they are used in a nonstationary sett, ing,
182
Tim~-b?+,que~cy/'Thr~e-,~cale Atmfysis
they aIlow one t,o follow tile spectraI paramet:ers in time. There is an abundant literature d~a~ is concerned with adaptive sig~al processing. We content ourselves with the quotatior~ of Michaut (1992) from a general poir~t of view, NIrthermore, a synthesized description of the adaptive algorithms, used in an explicitly tim<~frequel~,eycom.ext, is given in Bgusseville, Fla~tdrin, and Martin (1992). 2.4.3.
~:~ The approach based on tile harmonizaJ~ility was introduced by Martin (1982) at. t.he beginning of the 1980s. aI~d then further developed during subsequent, years (see, for example, M~rtin and Flandrin, 1985b). \:~;i-, should nmntion, however, tha~. the resulting definitions lead m the earlier solutions of Bastiaans (t978) a.nd Mark (1970), at. least; in some cases s~mh ~us tim stoeha4tic exte~sion of the Wigner distribution. ~;4 This definition is due to Broman (1981}. ~;5 See Hammond and Harrison (1985).
Chapter 3 Issues of Interpretation
The objective of this third chapter is a more profound study of the porchtiM, the characteristics, and/or the limitatio~s of several time-frequency toms t~hat have already been construct.ed. At the same time we attempt to give some guida.nce concerIfi~g their application (which representation should be chosen?) and facilir,ate their interpretation (which ilfformation can be extracted?). As not nil approaches can be covered in detail, we tbeus on the Nliue~ir reN'esentations (Cohen's cta,ss a.nd affilm class) in t;his d~apter. More precisely, Section a.1 describes ditferent viewpoints concerning the bilinear el~ses of time-frequency and l~ime-scNe representations. Sut> s(~:tion 3.1. I is devoted to a physieM i~tterpretation of their various t)m:am~ eterizations, depending on wire,her ~.he parameter Nnction N expressed in the time-~equency, time-time, h'equency-frequency, or frequency-time plane. This emphaMzes the complementarity of the interpretations in terms of energy dist;ribut,ions, on the one hand~ and correlations, (m the other ha.rid, ft. aiso leads to generMizatiol~s of the welt-m:~derstood cause of stationary signals. Then we take a diffierent look at the parameter %nc~,ions of tim bilinear ct~ses in Subsection 3.1~2 (which can be omitted during a first study). Her(> we explain how the arbitrariness of a time-frequency representation, as eonveyed by t~he multiplicity of its parameterizatioiLs, is closely ~J.ssoeiated with the noneommutativity of the elementa,13" operators related to ghe time ~md freque~ey variables, This arbitrariness in the dloiee of the parameterization is compared with the ~mNiguity in ~-~sociating opera,t ors with more g,~neral ~ime-frequency functions. Subsection 3. t.3 partially ~tses this appro~eh in order to reconsider the link betweet~ the two concepts of frequency and scNe This problem will be (briefly) discussed by other approaches tike the Mellin transform a~s well. 183
184
Tim(~-b)++,que~e3:/Time-,~cak , A ~m(vsis
In regard to its numerous theoretical propereies, the Wigner-Vitle traastbrn, plays a central role in the time-frequency theory~ A better identification of its properties of "visualiza~io~l" is therefore indispensable, if one wants to use it for t.he practical purpose of anMyzing signais wit,h an unknown structure, This issue is addl~s~d in Section 3.2. He~. we atta*:k the mmn problems ~ssociated with the special structure of the Wi~er-Ville distribution from a geon?etric pNn*, of view. First (in Subsection 3.2.t) we compare the Wigner-Vilfe distribution with the spectrog~:am. We show tg elaborating on simple examples, that it has certain advantages m,er tim Iatt,er (supports, localization), ttowever, we aNo und.eriine eertNn improvements of the spectrogram (ree~ssignment). Then (in SuL~section 3.2,2) we show that. the same reason (nonIocM bilinearity) that guarantees the good theoretical properties of the WignerVille distribution also affocts it with interfi~renee structures (or cross-terms). This can reduce its readabili V and useflflness, l'his point is st.udied in detail by anMyzing two simple modds. One is related to ~:~parated timefrequency "atoms," while another deals with freqaency-modutated signals, We succeed in ~tablishing the construction principles of the Wigner-Ville transform explicitly (either exactiy or by approximations by the method of stationary pha~?). \~¢~ Mso discuss possible generMizations to other representations and fllrther implieat.i(ms of this geometric approach cormerning the IocMization in t,he plane, The analysis carried out in Subsection 3.2.2 provides a rationMe for the reductkm of the h,terti;rences. Further considerations of this issue from several angles are the con.tents of Subsectkm 3.2.3 (use of the anMytie signal; atomic decompositions, f£xed or Mapted smoothing in the bilinear (lasses), It is often desirable to reduce the imporeance of the crms-terms in a time-frequency representation tbr improving its reMability; On the other hand, Subsection 3.2.4 addresxses the complementary issue, tha~. these terms al~) carry ,some useful information that should be p~x~served for certain reasons. T h w allow us to find some pha:se information, and their pure existence leads to a better understanding of what a "component" of a signal might be. We can think of the reduction of interferences as a problem of geometric estimation of the time-frequency structure of a signal. Another problem of statistical est,imation is attack~-t i~t Subsection 3.2.5: Here we seek for estima~.es of the WigDer-Ville spec.trum (which was defined in Chapter 2) from one observed realiz~tion. Under re~sonable hypotheses such as the stow evolution of the nonstationary properv;ies, a remarka,bte observation cm:l be made. It tells that the general bilinear classes provide some natural families of estimators for the Wigner-Ville spectrum. This is come,ted
Chapter 3 Issues of Interpretation
185
with the geometric approach from bef.ore, bu~ relies on a different, perspe(> tire, We investigat,e tim stochastic properties of first a1~d second order of these estimators. Thus the arbitrari~ess of the parameterization is no longer expressed in terms of localization or reduction of int,erferences, bul in terms of bia~s and variance of the estimator. The eventual existence of negative values in a bihlmar time-frequency representation is a severe obstacle. It sometinms renders a l.horough interpretation of the cross-terms impossible, mKt it also prohibits a strict analogy with the notion of a joint probability density functiom Several problems related to this question of positivity of the representation are discussed in Section 3.3. Subsection 3.3.t begins with an inventory of certain difficulties that arise from the nonpositivity. They occur, in particular, wtmn we {ry to measure the (local or global) dispersion by means of the second-order mo111ents,
in Subsection 3.3.2 we investigat~e the issue of positivity based on the analyzed signal. We show l;hat in general, |he positivity is the exceptional case tor deterministic signals (while justi(ying that the sit;uation is less critical for the spectrum of random signNs). Finally, Subsection 3.3.3 addresses the probIem of positiviV by considering the distribution itself. Two different solutions are developed. The first consists in leaving the bilinear setting, which permits (without violating Wigner's Theorem) reconciling positivity and marginal properties. The second solution can be fbund inside Cohe~fs ctass. It is expressed t~" a smoothing of the Wither Ville distribution. This point is related to the forementioned stochastic approa.ch, showing that a su~cient degree of disorder can assure the posi~;ivity of the specl,fttlH. §3.1. A b o u t the Bilinear Classes The bilinear classes lend themselves to many interpretations, each rooted in some special interest. \:~%will describe the most important ones in t:t~e following Subsections. 1 3.1.1. T h e Different Parameterizations The bilinear ela:sses (Cohen's and the af~ne cl~ss) are, by definition, bitinear forrns of the signal that depend on an arbitrary function of two variables. This function can evidently take different (though equivalent) forms according to the type of t;he al;tribufed variables. Using the same notations as in Subsection 2.3.1, we obtain the following diagram in which each arrow represents a partial l~burier traztsform:
t 86
"l'im~,-I~}'equ~ncy;/Time-Scate A n a l y s i s
l,~(t, ~)
¢:(¢, ~) j/
",,
f(~, ~-) ttere t and T (~ and ~,~ respec~b:ely) are the time (~md frequency) variables. In order to g u a r a n t ~ ~he cot~istency of this alia.gram when pa,~sing fl'om oi~e function ~o another, we must fix the fbllowing cow~:engions for ~.he siglis of the involved pm't.iM or tot.al Fourier transforms: :
Tle . . . .
d~dr
(3.1)
d~ d~
(:~.3)
dtd~
(3.4)
i/I[{2 .,:,(~,.)
= jj
,. , , . ,
,..
.-=/)t'n(t, #}. = ~ J -.. :~c F(t, r) =
~.,(~. ,-,, j)~i2,~e~_ d~ =
f(¢, 7)e ~2~'~¢td( =
(3.5)
II(t, r/)c - ~ 2 ~ ' ' dr,
(3,6)
.... .'~
[ ~ '*2% ,
L
F ( t , r ) e ~2..... d'c
N
x~
f(~, r) =
jf+x
F(t, r)~
f
...... dt =
-i- 2'~
/" ......O(~;,,),;
~"
d~..
(3.S)
We.' thus have four para.meterizations a.vailab|e (time-frequem~y, t i m ~ gi~?, t?equency-fr(~quency, and fl'equen(;y-t;ime), which resuh in as m a n y different ways of looking upon i:he bilinear classic's.
Chapter 3 ,/~ssucsof l~terpretatio~
tS7
Time-frequency. We can write the bilinear class~ employing the time~. l:req uenqy parameterization I] (t, v), which yields (of. Sut:,~ction 2.3.1)
•
f)
=/flI(s
...... t,
......,) ti~:(s, () d:s d~
~,:(t, a; f ) ==j~f rI (.~--,a~"l s- t [,,(~(,,~t d~d ~ . {£ // ....
(3.9)
(:,.t {))
Likewise they can be defined as the correlation of tt, e paranmter timelion with the Wigner-Vilte distribution of the signal. This hlterpretation results from the fact tha/t, both bilinear cla~ses can be written in the form of an immr product in the tirne-frequency plane,
Pz where a reference object (the Wigner-ViHe distribution of the signal) is compared with a family of analyzing objects. These analyzing objects are construct~xt by the action of a natural transfbrmation group according to a
= -
-----~ U(s,~) --,+ U,.(s,~)
=
U ( s . t,~5-- v) ,
In this sen~, and by analogy with the disct~ssion of Subsection 2.2.4, one could call the repre~mtati~m p~. (narrow- or wideband) quadratic crossambiguity Mnetiou relative to the time-frequency object t.hat is the WignerVille distribution of the signal, not the signal itself. Remark 1. The chosen correlative %rm was quite arbitrary. It could ~s v:ell be replaced with a convolutive form, at lea~t for Cohen's clav~s. An interesting feature of the correlative form is its hmnogenNV: it: Mlows a description of both classes by p~,~cisely those transformations that constitute the underlying group of the represer~tations. Remark 2. No matter if we regard this operat.io~ ~ correlation or con~ volution, we are not Mlowed, in general, to interpret it as a smoothing of the Wigner-Ville distribution; that is, it might not con~spond m a bivariate filtering of towpass type in the time-frequency plane. (As an example consider the Riha:zek distribut, iou (cf. "]'able 2,1) for which ll(t , v) = 2exp(- i4rcvt).) I ~ must therefon refrain from drawi~}g any conclusion that says that the Wigner-Ville distribution has better localization properties among MI representat.ions in the bilinear elaxsses by virtue of the previous relations, because the other distributions look like smoothed versions of it.
188
T~m~ -F}~eqt~et~cy/Time-Scafe A nalysis
Remai'k 3, Let us fina.llynote that the central role which the Wigner-ViIle distribution seems to platy within ~,hc bilinear classes, is quite ~trbitrary as well. Both classes can be constructed in a similar way by start.tug from any other invertible ~presentatiom We can us(:, for instance, m V s-Wither distribution (ef. ]~-Lble 2.1), denoted ~l:,:.' (t, ~.,) here, and obtain ¢-p~,(t, A) = .//__ II ia (u, s ) lI.~}":(u, ~)du d~, lq2 .
I,s'~
We on}y need to modify the time-frequency parameter fimcgion a~:cording to
1](<~ (t. ~,).
=
F ( t - sr, T ) e i : ~ ' ~
.
x~
In partieu}ar, while the previous relations yield the expression
for the Rihaczek distribution l~z(t, ~/), we can aiso write, vice w~rsa. t%,(t,
r.,) = ~, o],] e
. . . .
.
This simple exmnple demonstrates that the ~totion of a gene,,~ating dist;ribution of the bitinear classes must be further quMified. Moreover, i~. illustrates the Nct t.hat correlating an oscillatory kernel with a function, which its
C,:(t,-~.D=.
/j
F(s-t, Qz
(
(
s + ~"
~-
A first interpretation of these relations, a:a in the previous c~se of a time.Dequency representa~tion, is in terms of a "correlation" or "immr product;" in the parameter pta.ne. ~br this purpose let us introduce the w o d u c t q:,,(.s, r) = :c :~ +
x
s -
.
(3,13)
Chapter 3 Issuos of Imrertnvt.atio~
].89
Then we can express every representatkm in the bilinea:r classes by t,he action of a linear operator on this function, nanmly p:,.(t, ~)
=
b~(c,r)%:(s,r)dsdr
where :~
F(s,r)
-.~ !<,.(s,r) = F ( s -
t,r)e ~''~ ,
k=a
The weceding form gives r i ~ to another interpretation a,s a "Fourier transform of ~ local aut,ocorrelation function." Indeed, we can rewrite C:,. (t, u; f ) as
C,(t,u;f)=
{ ds < -)x.t s...- -)]
F(s .....t , r ) x s + ~
e -i2r''~dr.
If F ( s - t, r) were the constant 1, the tern, in brackets wouhi simply reduce to the deterministic autocorretatio,~ flmction
Taking its Fourier transfi?rm would thus yield the power spectrum of the signal; that is, C , ( t , r*;f) == I":,:0e) = {X0J)t 2. The presence of tile additional term F ( s - t, r) # 1 leads to two modifications of this scheme. The first is related to the first variable ai~d indicates tim* the evahmtion of the autocorrelation function is "local"; this means that it mum. be a~sociated with the current da~e t. The other modifica.tion a~:counts i~r the existence of the second variable r, whose role is an eventual weighting prior to the FOurier transtk)rm. For rendering these ideas more transpaxent, let us consider the special case of a separable parameterizal.ion n(<
=
,
in which 9(t) (and H(-..u), respectively) ascertain a temporal (or @equential) smoothing° Then tile a~ssociated representation takes the form
C j . ( t , t / ; f ) = . , _ ~ h(T)
--t) x s + ~ 2 x*
s--,~r ds (~!
i2~ur
at.
190
Time-t~?equenc~/Tim(~-Scate A m d y s i ~
Hence we recm,er the t w o es,~mtial and well-known ingredients that arise in nonparametfic spectral estimation by a 'weighted correk~gram?' 2 The representations of Cohen's class th,,L~ appear em natural extens{ons of the power spectrum to nonsta*ionary signals; making it timmdependent by virtue of a local estimation. tn order that this point of view can be carried over to the affine (:lass, we have to ~:t~ ~he formal identification ~.J = z~o/a. This can be adfieved by putting ll(t, u) = n~fft, u - uo)
where I-f~(t, u) is a time-freq,mncy function of towpass type with respect to tlle variable u. Calling its part.ial (time-time) l;burier transtbrm IV~)(t,r), we obtain 9.:(t, < / )
The term in bra&_ets once again measures the local and weight~t autocorrelg~tio~ function, The only difference is the scale-dependence of the local evaluation of the autocorrelation function and the weighting, which precedes the Fourier traI~sform. Wtlen we consider the forementioned set> ~wab~e came, we can write
=
,.
r) e
~ ,
dr
wid~
~ (t,r}=ta1-1/2
..... g
~
x
s+
s-8
ds.
J -2"*4
We thus see that the time-horizon of the slnoothing brought into effect by the funct.ion 9 is not fixed, but depends on the scale (being shorter on finer scales). Likewise the u~fifl range of dela.ys r, on which the Fburier transform operates (amd which is fixed ~V the windmv h) is a fimetion of ~he ~'ale within the same proportions. From a spec~rM Imint of view, we capture the observation that, the representation performs a broa~ter frequentiM smoothing ag higher frequencies.
Chapt(.r 3 Issues of Interpretation
19[
F~eequency-frequency. The preceding de~::ription in the time-time sett.ing h ~ a naturM com~terpart of l)'equency-l}~quenqy type, Here ~,~ use the pa,rameterization ~:~(~,t~), leading to the expressions
C;(l:,~/; f) = //'~,:;(G ( -,,,,~u) X ( ( - ~ ) X~ ( ( + ~ ) e-"2~ d~ d( , (a.~4)
= Ia[
9~.(t,a;f)
O(a~ a()X
( - ~ X* (, + 2
d,~d¢.
(3.~) By the definition of
/),
we c,an write
in the general c~e. Here the t~ransformatioI~ rules are given by
Although a frequential autocorrelation flmction is tess frequently used than its temporal counterpart, an anak, gous interpretation like the one i~, the time-time case can be given here, mutatis mutandis. ' ~ content ouns&ees with the remark that. a frequentia.1 autocorrdation function emphasizes the spect, ral periodicil.y of a signal, while a usual aut~oeorrelation function underlines the temporal periodieit3: This point becomm i m p o f taut in the study of so-calDd Wclo-sgationa.O, signals, which are characterized by periodic stocha,;tic properties without being stationary. :~ l~¥equeney-time. The fburth ir,terpretation concerns the fix,quency-time parameterizalion as given by the flmetion f({, r). The symmetriea~l and)i-
guiO." fimction A~,(~, r ) : :
~: .~ + ,~
.~ - ~- e ~ "
d,s
(a.16)
piws a prominent role in t,he cor~,~spmtding relations. I~ coincides witt~ the two-dimensimrml ~ u r i e r t r a ~ J o r m of the Wigner-Ville distributiom t.ha~ is,
({, r) = / / I G . ( t , ~.,)e i2"('~*'et) dt d~e . |{g
(3A7)
192
t imoJ~?~cque~cy/q°imc-~9:;~l~ ~ Ana(:~qsis
Owing Co ehe fact th;xt che I:imrier tralts%rm maps a com:otution into a product, w(; obtain (observing the conventions of the signs) that
<~ (t, ,.,; f ) =
],]
/(~, ~),4,, (~, , ,
(a~s)
~s~<~>,
tt::
All representat;ions in Cohe1~'s class therefore admit a particularly simple interpretation: They are jt~st the tw~>dimeasional Fourier transform of a unique object (:,he a m b i g u i v %notion) weighted by a,n arbim~.ry parameter fm~ction. This poirJt of view completes the previously mentioned app:mt~:hes, insofa,r ~ the ambiguity flmcgion naturally exgends the notion of an autocorrelation fl,mction to a nonsgationary contexL in which temporM and spectral properties must be t at~n imo ac'count simultaneously. Let us dwell on riffs point by recor~,idering the usuaI (deterministic and symmetrical) autocorrelation %nctions. When operating in the time-domNn, we can obviousiy write x
s
7
;,:
s -
d.s =
~'(s)
~:;0('~)
d.~ .
This amounts to measuring the level of resembla~me (in tern~s of at: im:er product) of die signM with its different copies that are shiRed in time. In a dual fashion, if we opera:.e in fl~quency, we obtain
,,{+:"
ix" i -
;x_
t
df =
I+
J:is) x ; _ ~ i s ) d.s.
'~2,
Consequentlyi a combination of both shifts, in time and fl'equency, r~mders a "mixed m~tocorreta;tion time:ion" ax'cessible. It Ls identical to the syrnmetrica} ambiguity function, apart from a phase, a~s we ha.ve
J
--- 2 ~
The preceding discussion justifies anew the seructure of the representations in Cohen's cla~s. In a %ta~ionary" situation (where stationm3 ~means that the specgral woperties of the signal do l~ot change in time) the power spectrmn provides a sufficient energetic representation. This is a univariate time:ion wigh a f?equency variable, which is the t;\'mrier trai~fbrm of
Chapter 4 2~me@~'equeney as a t~radigm 4.3.2. M a x i m u m
;/45
L i k e l i h o o d E s t i m a t o r s for G a u s s i a n P r o c e s s e s
Turning t,o the original formulation (Eq. (4.62)) of the problem~ we no~ supw~se that x(t) is a Gaussian random process, so that E {x(t)} = ,(~),
~.~,(t,,~) = E {~.,.:(t) ~-. - ,U:)]
[~.(~) - #(s)] * } .
(4.66)
It ix known that the detection problem under
whidl is
More precisely, we recall t.ha~ x(t) admits a decom-
doubi~7orthogond; ttm.t is, tile relations # , , j ~ } = ,~,~ &,,,,
E {Ix,, ..- , - d [z,~
/ are verified, where A, and p~.~(t) are the eigenvalu(;-s and eigenfunctions, respectively, of the autocovariance of z(t). ttence, they at*e defin~M by the int.egral equation
f ,.:,,(t,.~)~,,,(s)ds = A,~.~(t),
~:c (r),
(7} The coe~cients x,,, y,,, and #, of the decompositions d x(t), y(t), and #(Q, respectively, axe the projections o~tto this basis of eigenflmetions ( ~ xn = (x, ~ ) , e~c.). 2Ne optimal det(x~.tor (in the s~use d a maz~:imum likelihood estimator) is obtNned t~. a comparison of the decision statistics A(y) = A~,(y) + Ad(y) to a thr~hoht; in this definition we put --
~ ¢1
.
.
.
.
@(Y) = 7o ,=0 A,~ q % (14,67) J
X
T~==0
•
~l'imoJ')~*,,que~nqv/~7~m,:,-Scate Ane@sis
I9't
does ~ot justin, ~such an idea. A major defect of the ambiguity fmmtion is its lack of positive definiteness, which is one of the essentiaI characteristics of any true autocorrelation function. Indeed, the assumption of this prop¢ rty would imply that its two-dimenskmat ~burier transform would be nonaegafire everywhere, As this directly concerI~s tile Wiguer~Ville distributim~, tim hypothesis cannot; be true, Remark 2. Giving preDrence to the af}me class is sometimes justified for sigalats with a wideband struct, ure. However, we slloutd obsvrve that the alline class is b~sed on the same narrowband ambiguity functio~ ~,~ Cohen's class. Hence, it does l~ot invoke a u4debemd generalization of this functicm, although this would be deemed more appropriate a priori. Let us thus keep in mind that t;he afffue character of this class is only introduced by the operation of the weight fimc~,ion, which multiplies the ambiguity function by a scale-dependent measure,
3.1.2. P a r a m e t e r i z a t i o n s ~ O p e r a t o r s a n d C o r r e s p o n d e n c e R u l e s We already mentioned the fbrmM [ink between the time-frc~quency represen~ tations of signals and tim positiou-,momentum representations in quantum mechanics in Subsex:tions 1.1,2 and 2,1.4 Drawiug our iiL~piration from the approaches in the latter field, we caI~ in t~mt introduce an operational calculus to signal ~heory, It is based on the two elementary operators of time [ and frequency iz As we readily recall, they act by
(/.)(t)
:~ ,:c(t),
(,>aO(t) ~
I (Iz(t) ~2~ ,~t'
.
(3.2(:})
These two (self-adjoint) operators do not commute. On the contrary, they fulfill the relation •
i ?
(a2
)
where t (t<.not~ e ',~ the i&mtitv... Without going b~to tim details of this operationa! approach, we will briefly explain some of the issues originating from this perspective on lhe {}~[{owing pages, We will aiso dwell on the i~timat~e connection bet,w~:~n the lloncommutativity of ~he elememary operators and ~he ax'bi{;rarine~s of the choice of a joint representation via its parameter function '~
Why operators? Before we start dealing with these questions, it rnigh.t be helpful to explain ~o which extent ail operational formalism can be betmfi
(Theq;ter 3 Issues or" h~tcrpretatio~
195
for instmme, the projection of a signal cmto a basis of the represen{ational space, or t.he construction of a family of baMc signa.ls from one reference element, and many other situations encountered in Chapter 2 that. are liable to be described in terms of a linear operator ac~ing on a signal. The second reason of equMty great importance concerns the transtbrmarion tha:t a. signal uxMergoes and which can be expressed by the action of an operator axs well: filtering, dispersion, deI~y, Doppler effect, etc. In a noustationary cow,text the description of such situations gains a very particuIar significance (and diNcull.y) if both the temporal and fl'equential aspects are considered. As an example we already m.entioned the ambiguiV functions. The stone happens with liaear time-w~.rying "filters?' Their response to a signal x(t) can be written as ~ 4 - LN~
x)
The different forms in time and/or frequency of the "biva~'iate impulse response" h(t, s) carry some complementary ilfformation (tim~dependent transfer function, diffusion Nnction, etc.). Finally, the third reason for an operationN description is supported by the Net that the expectat.ion value of an operator, defined by <~4}:~,= (;-~ir,x} =
(,4x)ts) x*(s) d s ,
(3.22)
N~
provides some information about the inherent properties of t~he signal. (Titus it is immediately clear that @a: and {P).~ coincide with the time and frequency centers of the signal, respectively. Likewise, {t/e):r and (f,2}, yield the time and Dequency spread in Gabor's sense, etc.) The operator of time-frequency shifts. As a first example let us consider the lilmar operator, which is the exponentia.l of ~2. Then we obtain (t) =
2"+
+
&'(V
r~ d~x
(3.23)
and this defines a temporN shift of length ;. We reMize ~ a by-product that the expectation value of this operat.or is just the autocorrelation of the signal., hence • -.} cTx~?
A~m]y~is
19(i Analogously, tile exponential of the opera{or ~ is given by
which corresponds to a ffequential shift,. By the noncomnmtativit} of [ a n d / , th~ situation becomes more complicated, when we bring both elementary operators h~(o play. As our lmxt example let us consider the operator
ci2UU+:O , Unlike {lie or,:tim~ry calculus of fltllCt[OllS (or COIF~l'El)4t:iYlgoperators), this opera~or (armor be fa.ctored in the most obxious way. However, a classical result (Lemma of Glauber and Jordan s) stat.es that the foih)wing is true: If the two nonconmmting operators .i~ and ;~ are such that
[,~, [.4, s}} = [B, {,~, nil .... o, that is: if they commute with their commuta.tor, then we may write eA+f~ := cA 6 t 3 g - - ( t e 2
4 l~}
By at)t>lying this lemma twit:e arid truing the relation of (non)conmmi;ativity of i and i): we obtain as one possible fa,c~:orization (3,25)
ei'2~tZi+ri:d = (/~r,: :; Ui2,-rS.{ ~/rrri ,
.onsequentiv, the explicit computation of the expectation ~flue of this operator by virtue of this fa~:torization ieads t;o ,
/:~,=
f"[
e:. . . . ~e"
e
....
.--:
:;C
>_
x
,
8 +
(
s+
x*(s) ds
:r(,s + r) :,:°(,s) ds
2;
S -...
8 ~2r~'< d8
Chapter 4 2~me@~'equeney as a t~radigm 4.3.2. M a x i m u m
;/45
L i k e l i h o o d E s t i m a t o r s for G a u s s i a n P r o c e s s e s
Turning t,o the original formulation (Eq. (4.62)) of the problem~ we no~ supw~se that x(t) is a Gaussian random process, so that E {x(t)} = ,(~),
~.~,(t,,~) = E {~.,.:(t) ~-. - ,U:)]
[~.(~) - #(s)] * } .
(4.66)
It ix known that the detection problem under
whidl is
More precisely, we recall t.ha~ x(t) admits a decom-
doubi~7orthogond; ttm.t is, tile relations # , , j ~ } = ,~,~ &,,,,
E {Ix,, ..- , - d [z,~
/ are verified, where A, and p~.~(t) are the eigenvalu(;-s and eigenfunctions, respectively, of the autocovariance of z(t). ttence, they at*e defin~M by the int.egral equation
f ,.:,,(t,.~)~,,,(s)ds = A,~.~(t),
~:c (r),
(7} The coe~cients x,,, y,,, and #, of the decompositions d x(t), y(t), and #(Q, respectively, axe the projections o~tto this basis of eigenflmetions ( ~ xn = (x, ~ ) , e~c.). 2Ne optimal det(x~.tor (in the s~use d a maz~:imum likelihood estimator) is obtNned t~. a comparison of the decision statistics A(y) = A~,(y) + Ad(y) to a thr~hoht; in this definition we put --
~ ¢1
.
.
.
.
@(Y) = 7o ,=0 A,~ q % (14,67) J
X
T~==0
•
~!'il~e-f,Tr~'quc~cv/Th~:~-,5>ab, Anaivsis
198 by employing the defimtion o((,
r)
=
<;(t,
~')
-. . . . .
.
When a~ identify proper terms and include tim dependence on the parameter funct.ion in our ,~otation, ~his furnishes the anticipated restdt G' t
f(~, r) 9(~, r)
.
3".28) ~'
]J
We have t,hus developed aI~ explicit rule ibr associating an operator with a ftme¢iom This rule depends strol~gly on she choice of a joint representation. This brings the parameter flmction f in.to play for the representation as well as for the correspondence rule. In Chapter 2 ~ , lbmM that certain desirable properties of the timefrequency representation are guaranteed by imposing suitable conditions on the parameter fimctiot~. AnMogously, we can a~sure a desired type of a correspondence rule belween functions and operators by adequate specifications of the same parameter function. Example 1. The association of self-adjoint operators (thus having reM expectation vMues) with reM~valued flmetions imposes the co~Mition f ( < , T) = f * ( . . ; . ,
.
RecM1 ttn~ the same condition implies that the represent.ation in Cohen's class is rea,l~,adued itselL
Example 2. Let us mex{. consider the special cm~e w.hm~, f({, r) = 1 and G(t, v) = t a:. Then we can derive
and this leads to the eorresponde,~ce rule
{One can show in the same way that an analogous rule applies to the frequmicy. ) Contemplating ghis simNe and sat.isNctory situat, ion, we mighg believe that every correspondence rule at least ensures
C.~m.pt:er3
1;ss~es of
h:lterpr~t:a.tSotl
199
However, one can only conclude, in generM, tha.t
,: ...+ ,f 0 . 0 )
+
•
o(:(0, 0) i2rr
0{
.....
?"
and a dual relation for P holds. [n tS~ct, the parame{er function needs to meet certain crit, eria, concerning its function value and its derivatives at, the origin in order lbr the desired relations to hold. Such conditions are verified, for example, by the correspondence rules stemming from the Wigne>VilIe, Ackroyd, or Born-Jordan parameterizagions. However, the precise rule derived for a spectrogram involves the mean vahm of its window h(t) according to Hence, it yields only the desired correspondence, if the window has zero II:eaIt.
Kernels. Let ns pause for a moment;, before we give more elaborate examples t~)r the construction of operaX>ors associated with particular timetYequeney functions. We cau use the preceding result in order to cha.racterize the action of the operator associated with a. given Nnction G(f, v) by a k(;rneL We again denote the dependence on the parameter Nnetion f t~; an index in our notations. A simple computation sh<ea,s t -= (Gsz)(t)
g - l ' - t N ' "2
/
Zr(~, s) z ( s ) d s
,
where the kernel %~'(Cs) of the operator G.r is defined by
?7(.s,t.)= ~''~:F(t<2s- - O,t,-s)?'(O,~
-
(3.29)
a:id =
=
C;(t, z
)~
~>""~"
One can observe that this kernel admits an interpretation as a mem~ vat,e of ? (in the sense of functions, not opera.lots) whose definition depends on tim parameter function f. <5 In Net, fbr specific choices of die paa'ameterization and by puttirig ?, (0) = ?'(0, s - t), we can easily establish the torm of the kernet as listed in ~iI~,.ble3. I. In particuta:, we conchMe that the first t.hree rules (Wigner-Ville, Ackr~'d, Born-Jordan) yield identical results tor M1 lh~ea.r functions % (0); in other words, it giv~.~sthe same operators ~if for M1 Nnctions G(t, v) of the form ,.,) = <:(,,) +
Amd3:sis
200 Table 3.1 Kernels of operators ibr cliff(rein (orrt~pondence rules
Wigner-Ville f(s;, r) = I Ackroyd
, f ( ¢ r) = cos ~ ( r
BormJoMan f ( { , r ) =
=,a(t, 8) =
1
t
~[~,',(:) + ?o(*)t
sin rr¢r -"~r{r-rai~ (t, ,s)
Page
f ( < , r ) = e e~ef:" + :3::
Spectrogram
=~s(t, 8) =
/
h :~(t - O ) h f s - 0)~,~ (O)dO
More generally, these three correspondence rules provide almost, the same results when t.M evolution of % (0) is quasi-linear ~ (:ompared to the temporal extem of ~(t, r) with respect to the variable r. Weyl calculus. Let us now investigate the ;Wigner-Ville' case more deeply. Here we have f(~, r) = 1; and the kernel ~yw(t, s) of the operator Gw, which ~s t~sx~ciated wil;h a flmction G(t, z~) kY mea~ns of the respect.ire corr~:~pondence rule, ca,n be written as ~w(t,s)=~\
2
's-t
.
(3.aO)
Conversely, if a~l operator Gw wit.h .kernel q~w(t, .~) is given, the ass~> d a t e d function G(t, ~) is obtained via the transfbrmation a f t , ~,) = /'+": ~ - - (- t~ + ' r2'wt - r )
e-~2..... dr.
(3.31,
.......3%
Adopting the language of pseudo-di~i~rentiM calculus, r tim function the }~>'¢1.~vrnbol of {he pseudc-differential operator G~w. Pas> ing from an operator to its symbol i8 the inverse operation of the c/ore> spondence rule (also cMled }~,i{~;1quantization). It coiLsists of assocb~ting a function with an operator. (We note in passing that the symbol of the covariance operator is exemt.ly the Wigner-Vitle spectrum a8 defined in Sab-
G(t, ze) is
se~:tion 2A.3.)
Chapter 3 Issu~ of b~te;73r~?tatio;;
201
Beeau~ the operator a~gebra is not commmative, the symbol of the product of two operators cmmot be obt~ain~t direct ly by ~;heordinary product of the symbols. Its eomputathm aetually requires the introduction of a twisted produc~ ~ which is noneommutative and defined by the quantization Awt3.,v ~ (A # B)(t, ~) ; here A and B are thesymbols of .4w and/3w, respectively, ff we denote the kernels of Aw and B3v by a\,,, aad i3w, the new kernel %~: of the operator produet Gw = A w B w is g~ven by '4 :~:
f
.Xd
= j [ f f A ( t + r .'~
r f. s
.
IU Cm~equentlyi maki1~g use of the fast (A # l~)(t, . ) =
"+':':3a~ ( t + ~, T t - 7") j(.:. ff ~_i2~,;.cdT ,
we can derive the I~iation, after rearrangi~lg terms,
(A # B)(t, u) = 4
fti~[A(t -,~3-, ;J
()B(t-O,u-£)ei4~(%"°~ldTdOd~d(.
(3.32) Moments. Let us now suppo~ that the flmetion G(t, u) under consideration has a power series expansion
C(t, ;,) = 7 ~ 2 - ,
~;
"
We wish to associate with it an operator G f in tile ~mse of the corr~ spondenee rule defined by the pa.rameter function f(~, 3-). Owing m the linearity of the general rule established before, we can immediately find
aj =
k=O
Z
If
,
~=0
where each ~f.; denotes an operator that is ~:~f~sociated with the ordinary product t~J. Therefore, the analysis of the quantization can be restricted
202
Time-fV>eq~u?n
As~al$;sis
to these latter terms, This results in the study of the moment;s of the joint represent~tiori~
as we
have 1./
(3.33.)
By a welbknown property of the FOurier transfbrm, the momem,s of a function are connected with the behavior of the derivat, ives of its Fourier transfbrm at the origin, By invoking the relation (3A8) between timefrequency representations and weighted ambiguity t\metions, it is ea~sy to
i.=~,, = 1 / O*:'+*' ) ~'Pf '* (i27r)J"+g t , } ~ f('{' T)A,~:({, 'r) (0, 0) • Hence, we ini~r the formal result
ji~~:=
J.
( o~
)
(3.3,1)
An application of this re!a,tion to the <*enc,ral case can be quite tc~:lious. The procedure can be slight}y simplified by observing that
(o,o) p := (t q :::0
= ~
~
,,/p tq) oe~'o~-,(o, o) (i2~),,+, #'~
"
p ";:t} q=:O
This enables us to deal with the (Weyl) correspondence governed by the parameter function f(~, r} = I aad the Wigner-VilIe distribution None. For this mat,ter we recall the identities
and =
e
D?
.
dt du
.
Chapt(,r 3 L~sucs of h~tezpre, tation
203
[tence, the Weyl correspondence imposes the associatkm ~...... e i2rr(U+'':b .
e ~2~(~~°=')
(3,35)
By expa,nding both exponentials (of functions and operators) into power seri~ and using the tinearity of the qua.ntization once again, we can find the searched result. In fact, we infer from
+ ~ (i2~r)~+~
~..
/~'=0 ¢:.:0
that the relation @ (2*;2 4~ (7~,
holds. Moreover, we can write ei2~r(C,[ r ri') ~ e ~
i ei2rrri) ~ f~,,~[
+r*# -X
f, Cx5
~ 9 ~ #+-l+rr~
A chang*~ of the summation indices and a termwise comparison with tile wevious expression Ieads to a:
j...,,,, = 2
(a.a6) m=O
Remark,
\fTn/
If we had used the factorization
instead, we woNd have obtNned the equivale.nt form g
B y means of t,hese expressions v,~; can now determine the operators
~ z ibr any correspondence rule whatsoever, Howe-~r, it may be simpler in
204
Timc'- t~'eq uen(-L./Time~Scah~ A na]~is
certain practical situations to fotlow a more direct approach. If we consider, ibr instance, the Aekr(.o,'d distribution~ we find
Z
"
~"
//z
tIe:nce~ the respective correspondence rule reads as
r h L furgher implies ~ fA
(3.38)
2
The Born-.lordan parameterization (for which f ( ( : T) = sin re,ST/re(v) on its part lea(is to the rule ~) /i.k~
k
I
p. ....
p,~
I
l (3.39)
which can also be written in the more compact fi)rm ~A:~
27r
B,~=i(k+l)(l
[t~* ,.I f/-~J.] 1)~ ' ~ "
(3,40)
(The equivMence is proven ~y an inductive argument based on the relations ~br the commut.ators ft~' P] = (i/2rr)k [~' I and it, d] = (i/2~r)l d - ~ . ) The differences between the variolLs correspondence rules originate fi'om tim properties of the underlying parameter flmctions a~s related to their derivatives at the origin. Hence, if we restrict ourselv~ to the moments of t.otal degree k + l < 4, we can recognize without tnaI~,~ complica~ tioI~s that the tlme.e rules, Wigner-Ville (Weyl), A & r ~ ' d (symmetry), and Born-Jordan (commutator), yield identical resulm This comes from the Net that in each of these c,~ses f ( o , O) = 1
and
~k i-If 0 ~ O r ~ (0, O) = 0
1 ~; ~: + ~ < 4 .
Chapter 3 Issues of Interpretation
205
V?e can thus use the simplest rule (which is the symmetrical one in this c ~ e ) in order to express the con~spondenees for all three cases t~y ~ ~ + i, ~t'~. 2 ,
tkl~t ~-~
0
(3.41)
The first difference between the three rules turns up when k = l = 2; then we obtain
04Z
0
(Wigner-Vilte)
-27r ~
(A(:kroyd)
-27c2/3
(Born-Jorda,n) ,
(0,0) =
(3.42)
0~ 2 0 f
and this results in 1 t 2 u""~ ~
. £2 1 ~2 ...9 ~(t '+
1 ~-
(Wigner-Ville)
~2 ~2)
(Ackroyd)
1 i)~(~ i,~ + i2 V) + 11 2 ~ [~
(3,43)
(Born-Jordan)
Hence, the moments in all three cases have the generic form
H~
where only the constmtt f~ ditI?rs according to
I
1
04f
/~ = 2 - 4 ~ r 2 0~ 2 O f
(0,0) =
t/2
(W lgner-Vflle)
1
(Ackrotcd)
2/3
(Born-Jordan) .
(3.45)
Dilations and ambiguities, 'File simplest example of a combined funct;ion of time and frequency is the product tu, It allows of an interesting interpretation in tern-is of dilations. 10 In fact, as ~_ready stated (cf, Eqs. (3.23) and (3.24)), the exponential of the operator associated with t (or u, respectively) corresponds to a shift, in frequency (or in time). Let Lus now consider the exponential of the opera,tor associated with tu and determine its action. If we work in one of t~he previous settings where t~
~....
(3.46) 2
'
Time@)'equet~qy,/Tiuie-ScaJe A md}:sis
206 we find the identky
i2=(l.~g ~?-) ~*i)..{i:'[1/'2 ~ VI'~} gi2n(h>g ~})i*:*
= v:;i
£
(tort1) ~*'
,.,~!
(~'I)Y"
~,t=0
for all q > 0: here we used the abbrevia,tion
~=d dt ' A str~dghtfi:~rward argument gives
Hence, we derive (~i~,~(:~
og,~,:,. •
t ~, =
(~lt),~
,
and this implies that for every signal1 x(t), which h~L~a power series expaasion, the considered operator acts as (3A7) Tile so-formed opera*,or, which is the exponential of the image of the tim~frequency function t1~, is jus~ the dilation by a factor rl. ~Ve came acros~ this operator earIier in the framework of time-scale re~'esentatkms and the wideband ambiguity functions° In order to determine the timefrequency function G(t, ~,), with which this opera,tot is associated (in the s e n ~ of t,he correspondence rule of Wigner-Ville-W~yl, for instance), it; is enough to consider the kernel g~ssociated with
A simple computation shows that, in t,his (ase
/
+':~
>,~ (t, s) x(s) ds
....................
. . . .Y C
This leads to the searched quantization of 'operator-symbol' type, which leads as
(73)w)ter 3 Issues of I n t e r p r e t a t i o n
207
where o: sohes the equatiou ~1 = (t .~ o./2)/(I ........( ~ / 2 ) , We can go further aiid express die wideband ambiguity timer:ion as the expectation value of an opera,tor eombi~fing the actions of a dilation and a translatkm, tt thus t,akes tim form = f e ~2~?~:' e;-':2,~(l''g'~)(i~>e>i)/~)
.~/'7i i f "- X: x ( t ) x ' ( ' r / ( t - r ) ) d t
?eqd
/:~" ~
{3.49) "
By analogy with die situation of the narrowband ambiguity flmction, we can propose a slightly modified definition of the widebamd a m b i g u h y function b~" employing the substitution * ~ r t ) e i2,'r{{ eirrri~'
--';,.
ej','rr~
e i2r~(
og ~])({'i}+9t)/2 (¢i=r~ ,
,so we simply replace the frequency shift with a dilation. This r~uh.s in a first s y m m e t r i c M definition
•
(l
">L
(
~
2.
(a.5o) M / (1 - ~,/2) a~ be%re. (From a physical point of let: rl = (1 + u/2)~
where w e view, if the p a r a m e t e r ~1 is interpreted as the Doppler r a t e o is identified as twice the ratio relatv~e veloeitv/velodtv of the propagation.") By using tim definition of the twisted product one can show er>~r" ~ e " i 2 ~ t u ~ e i . . . . = e i2~Qr+~'t}u .
(3.51)
This mear~s that {,he operator, whose expectation vahie is the wideband ambiguity function, has the symbol
G(t, z )
V
- a2/4
ei:~(~~°'t)"
-
Another simple relation can thus be established between the wideband ambiguity function in its symmetrical form and the Wigner-Ville distribution. It states that
(
>
x
Time- b lmque~qK/'Time-~5('ale Analysis
208
Finally, we can apply r.he I~Ymrier ~r~:msform, which maps the WignerVille distribution into ~:he narrowband ambiguity flmction (here deplored by .A~(~, r ) for discriminating it from the wideband ca~t:).s :. Then the transitior~ relation ~
a~;; ~((:,,-,-)
=
,,,/i ~- ,:,~/4/{ A!J (,;,, + ~ / /
,tt J~
~.} ,,-~"~
(3.52)
o , d
|t 2
connecting the two ambiguity flmctions follows. Remark. Another synmmtricai ibrm of the mixed operat, or of translationdilation is conceivable, t~}/2 e/2:~?P Ci2~(iog ,V~)(i'~+P~'}/2
leads to a ~ c o n d symmetrical definition of the wideband ambiguity function by ~2 (-'~
'~/('J'
~)=.,
+:,c X
-:.~
,/;i
t ~
T"
T
Note that this new operator has a s}ight.ly more complicated symbol thaI~ the previous one. It has the form 1 +~,t~2~4 / exp
i i ~.
v
where we put (~ = 2(v @ - I)/(-Q~ + 1), 3.1.3. Time-17~equency or Time-Scale?
The t,wo bilinear ciass,~s (Cohe.n's and ~.~ne cl~,~s) were introduced mainraining the distinction between frequency and scale~ although ~he latter h ~ a frequential interpretation as well (sec Subsection 2.3.1). It is therefore importaa~ to identify tim cont, ext, in which t.he idea of the scale leadx~ to appropriate tools. This Nso enables us to a]opl, several definitiol~s while dealing with the pursued objectives. Fourier scale. T h e formalism of the previous seet, i(m was b~sed on the time-frequency distributions of Cohen;s class. AnNogously, we eaI1 empl%v the time-scNe dNtributions of the a n n e class for the computation of expectation values relative to time-~::ale fu~ctions. This couN be pertbrmed according to the definition
{~ Gf/x' "~ / / " C7"t (,a)ftz(t,ct;f)
dtda a2
(3.54),
( Jmpt~ r 3 Is,sues of [m;erpretatSon
209
In this expression Of denotes tTl~eoperator, which is associated with the (time-scale) fimction G in the sense of a correspondence rule, which on its part is based on the parameter fimction f of the distribution ~ . . Although this operator is ~ssociatetl with a function of the time t and the scMe a, iI can be represented l
A:,:(~,r)c-i2~td~dv
R~
Hence, we obtain
(Gfh: =
f a~,a (71((, (1;0"()
el(d7
a"
~.
R. 3
by putting
A comparison with Eq. (3.54) yMds tile result
])i/ da (if == f (a~, ~ ) C:i (~, a) ~~'r(~T':~) d~dT ~ .
(3.55)
If we now look at the special case in which the function G(t, a) is just the scaIing parameter a, a simple cMculation brings out the correspondence rule f((), O) (3.56) This states that dm operator associated with the sealing parameter a of (he arlene class is tile same, apart fl:om the constant f(0, 0), as the inverse of the elementary frequency-operator.
'Fime-b?c,ques.lqr/Time-Scale Ana/ysis
210
Col~versely, if we staa:t; h'om tim i'unctiol, G(t, s/) =: z4i/z/, where s~0 is aal arbitrary 0~onzero) referente fl'equency, we can show that !4)
f(O, r) = 1.
. . . .
-t-,
V
~+
l,"t) -
-
(a.57)
This double viewpoint lea
..... (1'
I/ l/
(a,~s)
Mellin scale, A different perspective concerning the notion of tile scale consists of leaving the usual paradigm of the Fourier transform. Thm~ we can adopt arlot.her viewpoint that is better suited to the operations of di.., lat.ion/eompression, it is based on a kind of invax'iance property, which is ditf?ment fi'om the cow,wiance i~l the definition of the affirm chuss, We shall explain this procedm'e very briefly. Vge know that the shiR of a signM i,~duces a modification of its l~burier transform involvi~ig its phrase only: The modulus of the ~kmrier transtbrm stays invariant under this class of operations. If we now wish to obtNn a similar result relative to dilatio~ls/compJ~ssions, we must give up the Fom'ier tral~sfbrm as the ap.. propriate too/. One can show that the Mellin tl~msform 14 should be used instead. Its definition in terms of the frequential form of the signal, %r iliStaltc% reads as 7x2
~V(s) =
tI!'
X ( z , ) S '~~ L & j .
By means of this definiiio~ it, is easy to show that
(:'L59)
Chapter 3 /s:su~:,s of h~terpr~,tatiozt
211
In co~,trast to the Fourier tra~sform, which decomposes a signal in~:o pure complex expo~mntiats, the Me~Iin transform u;ses ~he elementary w~.'eforms //--~2a,,
~ - L ..-.i2~.* l o g t:, .
Their group delay is t~yperbolic and not constant; that is, they verify S
t~.(.) = -
//
(3.60)
It ff~llows that; the Me}lin parameter s (which cannot be associated with a dimension) can be regarded a~s a tO,perbolic modulatio~ rate, Thus it has the character of %ime multiplied |~y frequency?' Therefbre, and by Eqs. (3A6), (3.47), it acquirc,s the rank of a scale parameter denoted a.'s the
Mellin scale. We can use this definition lor the construction of new mixed representations, This was done, for exampte, by Marinovie. ~:' He actually drew his inspiration from the structure of the Wigne>V~lle distribu¢ion (construction of a bilbmar kernel by muMpheation of the dela,yed signM and the advanced sigI,al, then taking the Iakmrier transform with respect to the dispIacement), He proposed, mu~a~i~ mutandis, tim definition of a scdeinvariant ~'I'hguer distribution, which perlbrms a Melli~ tra~mform of a bilinear kemmI obtained by multiplicasion of the compressed and the dilated signal. This amounts to the definition
X(>dl/2u) X~(,~ ~:L"2~,~tA~2~* :~d,~.
~3_2:~:(s,, ) =
(3,6I)
Remark. A dill>rent way of introducing the same entity was proposed by Aires (called the (2-dis~ributio~), j(} It consists of a generalization of the duMity relaJ~ion equation {3.17), which exists between the ambigui V function and t.he Wigner-Ville distribution. One start, s fronl l,lm wideband definition of Eq. (3~53) and substif:utes a Mellin trans~))rm for one of the Fourier tra~st?~rms. It can ea:siiy be show~ that Eq. (3.53) can be rewritten
A~(2~ ( 1I, r)
::
, X(~t* V ~ ) ~.X"(~.f .'~" e~2"~<~
(3.62)
Hence, we iIffer the relation (3.63)
As a consequence, the same procedure can generate other distributions l~: adopt.ing any other suitable definition of a wideband ambiguity function.
212
Time-f?equ< n c y / T i m e - S c a l e Aualvs'is
The form of Eq. (3,50), fbr instance, admits the equivaleut frequm~tia[ expI-eSsion
: ' ) ,c.y t
2
(with Ic~t < 2), Then, using the relation. ~] = (1 + c~/2)/(1 - a / 2 ) fi:om the foregoing Subsection, we derive
t[7/,i
• ,
.
4(11
~
:,.
/
it]
-
-
]
[,2;[Ui-,5-
)
1],2~s_. I e
i2~e/r
[
=
/1
dTd,~
'
(3.~-~4)
, .~-,
2m.
X ( ( i - . c,./2),,) X*((1 + a:/2t,,) \!i7.7~5- 7
V 1 _ -t~/4
Among the infinity of conceivable definitions this provides one distribution, which is based on the i]'equeney and the Mellin sca.le. T h e distribution in Eq. (3,63) of Marinovic-.Altes is called a scMeinvariant Wigner distribution, ~s k has the property x.(.)
= x(~,..)
:~..
K , : (s,.) = w , . ( s , . . ) .
(a.65)
Furthermore, in c&se of a hyperbolic chirp it gives x(.)
. ~"
t:O.)
~
k%(s,.)
= ,X.~ - c) u ( . )
,
(3.66)
and this shows that it is perfectJy loca.Iized to the modulation rate by, means of it.s Mellin variable ~s~ If we go back to ~,he interpretation s = tt/ motivated }V gq. (3.60), we em~ also derive a time-t?'equemqv represemation }l~7~-(t,~) = W__:~.(tu,u) tbr it, This admits a per~iect localization to the corresponding rule of the group delay, which reads a,~a ,K~(t,,,) = ~(>
c) u(,J) .
(3.67)
It thus joins up with dm chara,c~eristic property of the Bertrand distribution (cf. Sub~ction 2.3.2). This point, of course, is not ibrtuimus, and recent studies have shown that bod~ dist;ributions (Marinovic-Altes a,nd Bertrand) em~ be subsumed under a common framework, the sc~cNled hyp e r b o l i c chess. Within this setting further generalizations and variations are conceivable, such as it web's doue for Coherfs and tim atfine c|eoss. ~r
('hap~er 3 lsmm:e of luteU~rc*:ati~m
213
Analysis and decision statistics, A fim'd a~d noteworthy poim to l)e made here is {,he importance of tahe c(mtext i~ which we refer to {he m~tiolls of fl'equency and state. SchematicNly, we can differentiate bet;ween lwo ~ypicN situs{ions: The tirst allows for the tasks of mm/ysis and the second for ihe {;asks of decisiom In the first case, the (frequelmy or scale) description results from a deliberate choice by the user, who is interested mainly in a better (omprehens{on of the tine structure of the signal (finding t.he modulation patterns, "looking" al the details on finer scales, detecting the similarities on dif[i.~rent resolution levels, etc,). In the second ease, the situa.tiot~ is completely different;, tlere the user is ~vquired to introduce a. description in t:erms of ii'equency or scale, if he wants to give a.ccount of a physical reality that he cannot control by himself (dist;inet.ion between narrowband/wideband i~ presence of a Doppler e[[~,ct,, scale efl~ct;s related to model shnulations, etc.). Thus the specific context of the second situation can lead to a particular choic~ for dm definition or imerpretadon.
!i3.2. The XNigner-Ville Distribution and Its G e o m e t r y The Wigne>Ville {:ransformatioI~ plays a primordial role in theory and practice of timeofiequeney analysis. Although it, is not dm only element that can be used axs a generator of Cohen's (or the a.ifine) chess theoretically, it yet occupies a cemrat position practically, caused IV the choice of its advantageous properties. The knowledge about this disl.ribulion, Lhe analysis of its fine structure, and the comt)rehensioi~ al)out the rea,sons R)r its "good" properties were the driving {i~)rcesfl)r passing from a nice mathenmtical object to a limfily of operational tools. We are thus urged to take a, closer look m this distribution,
3.2.1. ~Nigner-Ville versus S p e c t r o g r a m In order to emphasize some special properties of the Wigner-Ville distribution, it might be interesdng to compare it with the spectrogram, which constitutes the first intuitive prototype of a thne-,Kequemy ar~alysis. S t r u c t u r e of the distributions. a signal J:(t) is defined by I
&(t,,,) .....
/
The spectrogram (with window h(t)) of
, }. x
2
:,,(s)l,.*(s .... ~ ) , : , - ~ ' ~ d s
.
(a.(~s)
Its evaluation eomMnes a linear operation ([%mrier transfbrm of the weighted signal) with a quMratic operation (modulus squared). The opposite
Time@?'cque~OT"Tii~le-Scah,, A,la]>:sis
214
situa,tion is in force for t:[~e Wigner-Viile distrib~ttion, which is defii~ed t)y W,~(t,~')=
.r t
~
x" t
as we teeM1. It fiI~t uses a quadratic operation N-@!ied to ~he signa} and then a linear transtbrmation ([;burier tra~tsform}~ This constitutes an e a sentiai difference be~.ween the eorr<sponding structures. This differeilcc is even more Wonounced by the fact that ~,he Wigner-Vitle transform, in its origi~lal fbrm, does not require the introduction of a (more or less arbitrary) window flmetion, which is external to tim signal. In spire of d~ese differences, we can stilI bring both definitkms clo~r together in several respects. I?irst, of course, bo~h belong to Cohen's chess. ;Recall from Table 2.l that the spectrogTam a~d the Wiglmr-ViIle distribution have the paranleterizations f(~, r) = A;(~, r) and 1, respectively. As a. ( O l l ~ l t e I t c e We obtaiI~ (a.ro)
A second level of comparing the two is rooted in die shorbtime Fourier t,ransform, As a matter of fact, it~ is easy to rewrite both definitions as & ( t , . ) = i ~" ( t , - ; h ) F",
w,
(~, .)
(a.r~)
= 2e .2~'*~ v:,.(2t, 2-; x
),
(3.r2)
where we put x (t) = x(-t). This shows that. the spectrogram conies from a shorbtime Fourier transform with aI1 external window flmction, while the Wigner-ViiIe distribution can be regarded as the same type of analysis with a "window," which is persistendy matched with die signal. This second ~'window;' is im~hing but the mirror image of the signal itself'. In other words, the Wigne>Ville traIisfbrm amounts to the following two operations: (i) multiplieadoil of the signal at each inst;am t, by the complex conjugate of its mirror image about ~his instant, in order to generate the quantity T
T
(ii) laburier transform of q:~.(t,r) with respect; to the variable r of the lag.
Chapter 3 lssu(,s of Ir,terpre~,ation
215
Pseudo-~,¥igner-Ville. The computation of this Fburier transform may correspond to a possibly infinite time interval, and this clearly causes probIems for pra(:ticaI applications. We can therefore try to modify the original defilfition of the Wigner-Viile distribution by imposing a restriction o~ the extension of q~ (t, r) in t;he direction of r'. This can be a.chieved by multiplication by a window p(r), which in turn amounts to a. frequential smoothing owing to the identity
In c~se this function p(v) ca,n be factored as ....... h.,
it gives rise to the so..called pseudo-Wigner-Ville distribution. ~s While it stays in the spirit of the Wigner-Ville transform, it is a (movb~g) short-time analysis. Hence, it is also linked to the spectrogram in a certa.in sense. hi order to be more precise, let us introduce the shifted a~ld weighted signa|
.,:~(,)
= h~(~)~(s + t ) .
(a.7:~)
This enables us to define the pseudo-Wigm r-Viile distribution by
(3.74)
= wi,,~ (o, ,.,). Tile int;roduction of the auxiliary signal xt(s) (which can be a~asociated with a moving reiercnce mark) thus allows us to interpret the pseudo-Wigner~ Ville distribution using the preceding notion of a "local mirror image," which is nov., restricted to a short-time neighborhood of the point of the evaluation. Indeed, we can e~sily see that (3.75)
Time-th'equenqy /Tim4~Sc;de. Analysis
216
~k~r every moment the pseudo~Wign(r~VilIe distrihutioI~ is computed
fl'om ~x~ct!y d~e same intbrmation a,~ the corr~ sponding spectrogram; but the differe~ce remains that the latter t~.~ubthe tbrm 9
Both distributions, spectrogram and pseudo-Wigner-Ville, thus use the same ingredient (the s~gment of the signN selected by means of a shore;time window) and apply a~ t:burier transfbrm together with a qu~tdratie operation. The diflbrent order, however, in which these two operations a,l:~ performed~ leads to eomplet;ely different properties of the distributior~s. This fact will be fllrther explained by the two simple examples that fbllow. Supports.
The first example concerns the proper~,y of the temporal sup-
port. As we have seen in Chapter 2, the Wigner~Vilh~ distributi(m preserves tile supports of a signal in the wide sense; that is, a signM with finite (:tu~ ration (or bandwidth) ha~ a Wigner-Vitle transform restricted to the same duration (bandwMth, respecti\'ely}. We have also observed that the same cannot be true for the spectrogram. We shNt p r e ~ n t a simple justification of this difference using the compa,rative approach developed h~re. tn fhct, let a signal x(t) be given, which is restrff ted to a~ time intervat [~t':~./2, +7~i,./2~, and Iet us a,ppty a short-time analysis (spe~.rogram or pseudo-W~gner-Ville) based on a wind(~w function h(t) with support [-1)~/2. _...a'l;~//'~j. Resuming the definition of the auxiliary., signal :c~(r),. we can easily see that it has nonzero values Ibr all instants t in the interval [ "7r/2 ......)')~/2, +7:,,/2 + 21)~/2], which is the support of the sigmd enlarged by the supporg of the window. The same must therefbre be ~rue tbr the speetrogrmm by construegion, the spectrogram begins to be nonzero, when the useflfl part of the signal '~enters" the rnovh~g window. For a cen~e1~?d window this happens bef'~re the signal commences to exist; by an amount of half the width of the window. The same argumem applies to the end, where the spectrogram exceeds the termination of the signal by the same amount. This situation can be improved only by taking a shorter window, which in return causes a deterioration of the fr~quential resolution. Let; us compare this to the pseud(.~W[gner-Vllle distribution, AIthough the signal x t ( r ) is nonzero tbr all .......T:,:/2 ........"7h/2 ~' ~i~ t < ......~I~,~/2, the "crc~sproduct" zt(T)X2('T) vanishes identieMly, by construction, It. only begins to be nonzero at the moment where the center of the "/ocN mirror image" starts to ~enter" the signal, and it becomes zero again when this center "lea~:es" the s[gnM. Therefolx~, the pseudo-Wigner-Ville transform preserves the time support of a signal of finite duratkm. This r e m d n s true
Chapt~r 3 l,ssu~t~'of lnterpret;~tion
217
rq¢aMless of the size of the short-time window. We can thus use ~.he fl'eedora of enlarging the window in order to enhan(e the frequency resolution without aft~ct.ing the temporal localization. Localization t o chirps. J'<* A second example concerns the capability of a time-frequency distribution to be locaiiz(-~t to the curve of the tYequeucy evolution of ~ modulated single-(xmlponent signal (chirp). Let us therefore consider the analysis of such a signal, which h~as a.n instant~meous frequency denoted by ~4,.(t). The frequency spread in a fix(~l inst, ant t of a spectrogram results from two sources: the ma,ximN frequentiN deviation of ~-'~.(t) inside the window and the size of the window itself. The spectrogram is tiros confronted with the well-known compromise concerning the choice for the observation window: r h e only way to improve the sh~trpness of tile frequency anMysis is by widening the t,ime-window, which in return r ~ u l t s in a greater incorporation of the instanta.neous frequency. Conversely, if a better approximation of the ';quasi-stationary" ease is desired (small varia.t[ons of 14.(t)), one needs to reduce the duration, of the window, and this affects the sharpness of the f>equency analysis, The best situation fbr ~he analysis is thus obtNned, when the instantaneous frequency stays quaMconstant throughout t:he ~perture of tile ~;im~window. The situa,tion is to~alIy did?rent for the pseudc>-Wigner-Ville dist, ributiom In fact, b a ~ d on its definition
*'
T
T
'T
it (an be viewed as a short-time ti~urier transfbrm of the modified signal
the analysis being ba~sed o n the modified window
Let us suppose for I,he time being that the anMyzed signal is given in its analytic |brm. We flirt her ~e~mne d:,a~ its instantaneous amplitude is quasiconstant relative to the essentb~l support; of the short-time window, Then t.he modified signal g~@) ha~s the remarkable proper~.y that its instantaneous
Tim~-F~v,qucnqv/';Thne- S'<:MeA nM~,:'ds
218
} ,,
/
....
/
i i
" /,/
/
/
,
ii i i
i
[
ii i i
/
Figure 3 , 1 .
i
.....
Interpretation of the pseudo-Wigner-Vi/le distribution.
'File symbolic represemation in the |eft-hand figure illustrates the fact that the pseudc~Wigner-V[lle distribution operates like a short-time Fourier anMys[s at each instant, based on a virtual signM with oa~ instantaneous frequency (thick lb/e), which is the ~werage of the instm~taneous frequency of the anMyzed signal (thin line) and its mirror image relative to the considered moment (dashed line)~ This re~ults fit a perf>ct IocMization to a linear frequency modulation. The right-..hand figure depicts an actual}y computed distribut.ion~ frequency is given by
,,
{ (
-
(
z.,:~,(r) .... 2~ J-~T arg x t + 2 ) :~:~ t -
= ).-~
(
i~£ a r g x .t+
,),,
:5 -
a r g x t - , :5
(3.77) •
Hence, in the curse of a / / n e a r frequency modulation, ~he modified shorttime anMysis encounters a constant frequency ~, (t) during the whole period of the window ceDtered m'ound the moment t, More generally, the inherent trade-off between time- and frequency-resolution of the spectrogram1 is pushed by an order of m~gnitude, when we move to the pseudo-Wigner~ Vi~le distribution. Here the best situation for the anMysis is attained fbr a quasi-linear evolution d the instantaneous frequency during the aperture of the window (el. Fig. 3.1). (Another interpretation of this result refers to the equation (3.72) in terms of the short-time Fourier transform with a window function, which is adapted to the signM a,t every instant by taking its local mirror image.)
Chapt.er 3 Issues
of h~Cerl)regatio~
219
We can detcrndne the limitations of the spectrogram more a,:curately when -a~ compare its resolution for lb~ear (and quasi-linear) frequency modu|a.eiot~s with the pseudo-Wigner-Ville distribution that uses the same window function. Let us define a mea~sure tbr the efl?ective width ht(h) of a short-time window h(t) by tim qua,ntity
t2Ih(t)l 2 dr. Here h(t) is supposed to have zero mean and finite energy -~h~. Furthermore. we will mea.sure the frequency spread (at a mornent t) of a distribution p~.(t, ~,) (spectrogram or pseudo-Wigner-Ville) by
where we put
F~)r the sake of simplicity, let us consider the case of a Gaussian window; hence h(t) = e -~"~
=,.
~ t ( h ) = a -~7~
\~,~ further assume lhat the anMyzed signM :r(r) can be locMly approximatc~t by a linear chirp in the vicinity of a fixed moment t; that is, its instantaaeous frequency l~:~.(r) and instam.aneous amplitude ao;~.(r) behave like ,,.,:(r) = ~'o + / ~ 0 -
- ~),
<~(r) = 1
~,r
ir - t.l <
6t(h)/2.
A direct computation leads ~o
while at the same time &2(pt~,~)
....
t ~t~(t,)
"
(3,79)
These formulas are wdid for any slope of the modulation (of. Fig. 3.2). The slope fl of the modulation plws a naturM role in the spectrogram, being a control parameter of the "degree of the nonstationaw behavior of
220
t'in~*-f ?'equc,:tcyf Time-Sca]e AnnOy,sis
the sig~aI' relative Io the o b ~ r v ~ i o ~ window. Duri~tg the p(riod ~t(h) the chirp sweeps a. freq~mncy range of size iJ dr(h). As tile r~'solu~km of the I:/mrier a,talysis is of order 1/dr(h) for tiffs same period, the condition of a qtmsi.--st:ationa~T behaaqor tak(~-; the form
Hence, in the quasi-statkmary case the frequency resolmioa of the spectrogram is inversely proportiorm] to the vitae resolntiom This rdates {o ~he discussion in Sub,~ction 1. t.2 concerning the Heisenberg-Gabor m> certai~lty principle. However, for a ~onzero slope of the moctu~atior~ the choice fbr ~St(h) underlies a certai~ compromise (see Fig. 3.2). In fact, for gabbing better co~dbrm}~y to a qttasi-s~at:ionary behavior we must reduce the size of the window; but this comes with a det,erioratio~ of the fl'equency r~olu~ion. Conversely, iru:r~asing gt(h) corresponds to analyzing a signal with a Nrger frequency ba~d, which (asymt)totieal/y) grows lir~early with ~t(h}. In bmween these two extreme points, there always exists an opt.imat window size dt,,p~(h), R~r which th.e frequency spreM of the spec~;rogram attains a minimum &/,,~i~(&). co It is easy to verify that
However, in the (abe of a blind amdysis lhe existence of l,his optima~ vahm is of minor use, ~s it assuHms an a priori kttowledge of the local slope of the modulation, fI~ olher words, ~his uw:terlines tile ihct that a spectrogram is not detined intri~sica/13'; rather, it; requ res the i~-~troduction of an ex ernal fimceion (the arbi~.rary window function}° This flmetion is more closa~ly ~ssociated with the was the signal is o b ~ r v e d than wi~h the signal itself. This amounts t.o aa mmvoidable interaction of object-measurement type. Accordingly, the result essentially represents this interagtio~l, and it only becomes indicative of the signM when the iIlteractiou can be consid~ ered ntmdisrtlptive (which we denoted hy a qml~i-s~ationary behavior). In the opposite c~se; or when t.tm valid regiolt of a possible quasi-statiottary beha:vior is unknown beforehat~d, the wi~dow f?anction ~:¢s limits to the iIfl'ormation about the structure of the analyzed sigr~aI that ca~ be drawls from the represe~lt:ation. If two types of nonstationary qua~:~tities with very different rates coexist in the same sigmd, the a~m,lysis cannot account for both sinmll,aneously. It, proves blind to either of them dependb~g on t,he select, ed window, therefore, wk.h an a pri~x'i fixed window; all short-|iw,d events ( i.e.~ of a much shorter dnratio;~ ~ha,~ the wi~:tdow) lose their specific ~m£/~re o~m can even say, that in this case the sigrlal ana,lyzes the windc,w rather t~hm~ the opposite . . . .
Chap.or 3 lssl,es of tiaterprefati(m
9 ~1
4 ,r---[
t
3.5 {
~,I t
/
/ {
/
2.5
ft
'
1.5 '-.
.i~.
0.5
~ / ~
t %
0.5 (1 0 U
0.5
F i g u r e 3.2.
I
1.5
"
2.5
~
3.5
4
Resolutkms of various spectxoglams.
TI:o time- and fl'equency-r¢~, oluiio~ts of a specirogrmt~ of a/linear chirp with a Gau.~sian window are shown. The curves drawn as solid lilies correspond to ditt~re~t va~ues of Lile slope of the frequency modulatiol~ (,f) = O, 0.5~ 1, 2, 4); the dm~hed line shows the positions of ~he minimM fr~.~tue~tcv spread for e~'}t fixed 3 (stm the text).
S p e c t r o g r a m a n d rea~ssignment. T}m ut.ility of the s p e c t r o g r a m faces t.wo lira|tat,ions. Tile first, a~ ~ e n before, comes fl'om p o o r properties of the rep:t~sentatioIl (concerning its Ioealization i~ the plane, for example). T h e see(rod limital,ion is caused by dm fact ~:l~at; the s p e c t r o g r a m is a nolmnitm'y transformagioI~ of tim s i g n a l These bad properties result i~ a loss in infbrmation. One v,,a5" to improve ~.his situation is ~s) use the pha,se inform a t i o n of the underlyi~g short-time l~burier transform, which is completely ignored by a spectrogram. A possible way to do tiffs, while s~;ayi~g within the setting of an energy distributiom was upheld by Kodera, Cendrin, a~M de VitledaKv. Their proposed m e t h o d is cali~x| %lodified Moving W i n d o w Method," ~l Its argumer~t is as fi)llc~vs: ]:f we start over from the definie ,~(tion equatkm (3. ~J) of the spec~:rogram as a twi(x>smo(~llmd version of the Wigner-Viiie distribution, we can write
L,(s; ~, ~,) &
s,,(t, x,) = x.
TJme-F}'~quer~qD/~l~me-ScaJe Analysi.s
222 where we let q :~
f
&(s;t,
w , : ( s , ~ ) u,i,~(s ~- t,~ - z.,) g ~ ,
At, every ti'eque~my ~ the spectrogram can thus be regarded ~e, a superposition (in form of a time integral) of the terms L-(s; t, I/); in other words, the resulting value of the spectrogram at an inst~ant t is the mean vMue of these contributions~ Here the important point is that t,he time a~mage is not weigh~,ed by construction; that is, a constant weight is ~signed to a.tl values of I~: as a function of s, which itself ca,n vary largely. Rather thin, a.ssociating the fina,1 result with t,he given time t of the evalu;>:ion, it seems wiser to ~assign it to a~other instant, which gives a more representative picture of the time localization of' I., One possible choice for the instant of the reassigmnent is th.e center of gravity of I., defined by
G(t,~') ...... S,:(t,~i},
x. s l:~.(s;t, ly) ds.
A direct computation straws that tit, u) = Re
F~,(t,~)
--2~Im
,
s : r ( s ) h * ( s .... t) e -c~'~* a s
G(t,.)
&,
CozLsequentty, if we represent tile short-time Fburier trans~brm in polar coordinates,
G ( t , ,.) = {G (t, ,')1 e~p {i arg F,-(t,.)}
,
we obtain the rc~sult
G(t,~)-
1 O 27r &, a r g G ( t , @ .
(3.81)
Hence, the group delav of the observation of the signal ttlrough the short.time window turns out to be the r ~ u l t of the previous analysis~ In a dual fashion, we can write
/
4- x~
Clmpter 3 Issues of Interpretation
223
where
Then we obselwe that t.he center of gravity of this b.~st quantity, in the frequency-direction, is given by I
O arg F:,,(t, ~) ,
(3.32)
and this is the instantaneous fz~equenqv of the filtered signal. Therefore, the computation of a modified spectrogram (in the sense of Kodera et al.) consists in: (i) Computing a short,-time Fourier transform; (ii) Ea:aluating its local centers of gravity t~(t, ~) and t/,~(t, l/) in every point; and
(iii) Assigning the energet.ic contents to this new point of the time~freqtmncy plane according to the reassigImmnt rule Ii~:(t,,dI ~
-,
s,, ( t ~ ( t , . ) , , . ~ ( t
,d).
(3.33)
This operation has the advantage of "reatlocating" the energy of the spectrogram at points that are close to the rule of the instantaneous Kequeimy. Hence, if a linear frequency modulation
is analyzed by a. Gaussian window of width 1 / - , ~ as in the preceding example (of. Eq. (3.78)), we realize that arg F:, (t, z/) =
7r/~ [( ~ ' -
a ~ d /32
~°)2 +
2f
- ~40 ........(-~2t21
It follows that tile centers of gravity t~,(t, z/) and ~,~.(t,~'), at every point (t, zJ) of the plane, verify the relation (3,84) and this signifies tha.t they are perf?~ctly lined up with the inst.aneaneous frequency (efl Fig. ;3.3).
22~ before rc~ss[gnmcm
after rcassignmcm
/ ~i
i~
/
S
Figure 3.3.
Spectrogram and reassignment
Fhe figure presents t,wo examples of specl rograms before (t~ft (:ohmm) and after re~signment (right column), tn the c a ~ of a linear frequenc) modulatkm (h)p), the reassignment yiek~ a perfect alignment with the rule of *he inslantaneous frequency. Roughly the same happel~s for nonlinear modulatio1~s, e~ssumi~g that the chosen short-time window permits a local lh~ear approxinlation of the modular,ion (example of a sinusoidal instautaneous fr~:quency is at the bottom),
We can therefore construct a modified speet, rogram by making use of the phase information, whh:h is supplied by the short-time Fou;der t.ran~ form, This can be done, so that ~he localization to linear d~irps becomes as good as for {;he Wigner-ViHe distribution. Moreover, Auger showed how (me can eireumven~ a.n explicit computatior~ of the phase of the short-time t%urier tra~s%rm, A simple manipulation of the previo~s equations yields the result that the points fbr the reassignment are equivalently ch~racter-
Chapt:er 3 l,~;sue;sof Inteu~retntion
225
ized by
L
J'
This proeedm'e amounts to empIoying three window fun(tio,~s instead of' one: h(t) th(t) and (dh/dt)(t), In addieion, this formal representation ca,n be h~rther generalized to other distributions as well, i/,em~rk. For Eq. (3.82) one of the key principles of the rea~ssignment method is tile computa¢,ion of an instantam:xms frequency of the filtered signal, which is given by the short-time Fourier transform. Suppose we are mt r~.stcd in the fine analysis of the signal in a fi'equencv band (cnt .red around u. Then we can choose a. window with an (adapted) width T = l / u . so that: 1 0 1
measures the fluetuaUons of lhe frequency around the given cemer. This falls in tile ca.tegory of another method, called Differentia1 Spectral Anal-
ysis. 2"~ Discretization. % ¥~)~ already~ mentioned the a:hantages. , ,' • ,' " of localization due to the special quadratic form of the Wigner-Ville transform. AnotheI consequemeG~ which can be annoying if not taken care of correctly, eonc.xnse" the ~'"'tls(rctlzatlon,~ ' ' of the transform. In fact., let us rewrite the definition of the Wigner-Vilte distribution in the form [,i K-4-~ , -
W,,(t, u) = 2 /
x(t ~ w) a¢(t - v) e i ~ , ~ dT" ,
wt:fich leads to a more convenient way of discretizatiom We denote by
:,'[Td = the samples of the signal with a regular sampling period ~ (deliberately chosen to be I.). Then a possible discrete~time version of the previous expression is given by
I'im(-b)'uque~ucy/'[[Sme-YcedeAnai.y~is
226
This expression t.ur:cs ou~ to be periodic in the f'req::ency directk~n wi~h period i/2. This is onty half the period of ~he classical discrete Fourier transform of the same seque~ce of samples, which directly enters the dis~ cretization of the spectrogram i ÷x.
2
If the eomponeI~ts of the analyzed sigmd are spread over the whole frequency ba*~d [-1/2, +1/21], the discrete-dine form of the Wigner~Ville distribution wiil be aff~,cted by adi~sing in the bands [-1/2,-..t/'4] and [}-1/4, [/2]. This occurs, in particular, for real-valued signals tha.t are sampled with a rate (:lose to the Shannon rate. Another interpretation of this phe1~omenol~ of Mia.siIlg is ba-sed oit the fact that. the Wigner-Vitte distribution is the b~urier transform of a Ntincar kernel, which is the %ross~produci:" of the signal and its mirror ima.g~. Cons~l~mntly, if the analyzed signM contah~s a spectrum-line at fl'equency 1/2 e, for exmnpte, then ttw cross-product maps it into a. liiie at frequency 1 2~:. This expansion of the b'e'quenqv baud eattses an ef[ecti~:~ undersampling of the cross-woduct i~ connection with the subsequent Fourier transform. Thel*e are two principal ways to avoid this probtem of alia:~ing (ef. Fig. 3.4). The first, of course, is a~ oversamplLr~g of the (real) signal by at lea,st twice the minimal Shmmon rate. Starting with a useflfl band [ 1./4, }--1/di of the signal, the cross-product occupies the larger band [-1/2, +1/2}, a~m t,his allows us to use a di~'rete b~)urier transform with a unit sampling period. The second eottceivabte procedure N to use the ana-. lytic signal instea.h By omitti~lg the negative freque~mies (el. Eq. (1.22)), this amounts to a division by two of the useflfl band of the real signaI, which can then be sampled at~ the minimal ra~e. In both cases (ovemampled t e n or anMytic signal) the mm:~ber of samples must approximately be muhiNied l:ty two: The ov~rsampling resutts in interlacing two sampling sequences ag the Shannon rate, while the usual sampling rate for the analytic signal yields two samples per point;, defined by the t e n and the imaginary part. 3.2.2. T h e M e c h a n i s m of Interferences
Because tile Wigner-Ville distribution is a sesqui/inear form of the signal, it cmmc~t submit to ghf~,winciple of linear superposition. "Ihis signifies that the Wigner-Vil[e distributioi~ of the sum of two signals does I~ot yield the sum of t.tm individual dist.ributions.
227
ChaI:)ter ,3 lss~u:.~ o f Ii:~terpr~ taticm
E
2E
B
2~
.
.
.
.
.
.
.
.
.
.
+ ......... b .
z~:2
.
.
.
.
.
.
.
.
~
F i g u r e 3,4.
.
.
.
e
Wigner-V~fte and sampli~g.
The figure gives a symbolic representatior~ of the t?'equency beh~vior of the Wigner-Ville distribution of a continuo~s signM with a given spectrum (lefQ. The frequency behavior of its Wigner-Ville distributkm before sampling (center) and after sampling (right) is Nso shown. In the first line, the ~mNyz~d signal is real-vahmd and sampkxt at the minimal rate: This results in alia=sing, A first, solution to this proMem is illustr~zted in the second row, where the analyzed signal is reM wall, but the sampling rate is twice the Shannon rate. The bottom line present:s a ~ c o n d solution, which uses a sampling at the minimal rate, and operates on the analytic signal (thus omitting the nega.tive frequencies). ,hxst like (a + b) 2 =, ~x2 + b ~ + 2ab, one can easily verify tha,t
with the definition of
More g~nerally, we o b t a i n for any linear combination N
J:tt:) : ~ ~,~,x,,.(t)
(a.91)
rt~' I
the expression Y "
/
)
. , ( t , , ) .... Z I~.,~!-.;, (~,,.,) -~ 2 ~ n=t k=r~+ 1
(3.92}
228
'li;~ <-t r ~qu~; c3 /'l'ime-S< al,~ A,~alysi,~
Hence, the Wig~ >.Vi~le di>~{r}butio~l colttai ~s ..\(X i ) / 2 additional corn.. pouenCs ibr e~.sigmd with N Imsi~, eonsti~ ueuts. ' | h e y result from the i,~teractio~ of difl~weut components of tim signul. 11he5 are called h, tc,fbre.r~ce *~I'IL~S lot tWO d~[l'erell{ l>asolts. }2iFS{, ti[t~' l~}{?(}/aIl}8Ilt Of ~heil' creation is a,nalogous to the usual inlerf>l>n~e, whict~ cm~ I)( observed for physical waves. A seceded reason fi-~r tills terminology lies in the disturbi~g effect that these terms introduce concerning the readability of a timmfrequeney diagram. So t h w int:erf~ere wit;h the interpretation of the diagram, as they a,mounI to a <,mtbinalorial protif~era{aion of additiona~ components. 'Fhe presence of interDrence terms (or cro,~s-teru~s) ill a Wigne>Vitle distribmion ca,n be regain:led a,s a natural coHsequence, which is characteristic of any bilinear struc~me. ()n t;he other hand, this very structure als~ leads to most of the good proper~.ies of the distributio~ (such as toeMiza~ion). Alhough it first looks like a dra>back, tile presence of interference terms is the price to pay fi)r gaining other ad~antages. No matter what. it is, advantage or disadvamage, it is important ~.o understand fully t,he mechanism of their t reathm. It is indispensable R)r drawb~g the correct b> ~erpretatio~ fl'om tlle represent at,ion of an u.uknowu signal, or for reducing the importance of these terms if desired, or even for gaining some useflfl information from them if needed. "~ C o n s t r u c t i o n principle, tt might he useful to b~gb/with a simple exampie, in order to understand the fbrtnatio~ of the interf~'rence terms in the Wigne>Ville disl:ribt~ti(m. Iwt us consider a welMoealized sigi~a,t J:o(t) (a tirne-fi'equency %tom, '~ fbr example), a.~d let us derive the qual-~tith:,s
from it, They define t;wo new signals, which are timeq'requeney mmslaies of the original signal ~0(*) including a tenua~kms and pha~se factors. By construction, the signals r_. {t) and ,~.(t) are slfifled by an a n o u n t of (+At/2, -{.:5t//2) al.td ( - A t ~ 2 , ~,e/2), respectively The covariance of the Wigner-Vitle distributiou relative to t.im~,~freque~my shifts yields, for each signa.t separately, that W.
(t,~,)
W.
(t.~e)=o.
=
~2
~I'.,,
t .............
II,., t-{
z. . . .
r, ~-
,
~
.
Chapa~r 3 lss~es o f InterprotaCio,~
229
Lo{, tffs 1)ext tltl?n t o {;h(? 811ill
.~:(e) .... .,:~(0 .i
(t)
of these tim~>fi'eqummy aloms. Its Wiglm>Vitle distribution acquires the form
t.-u.:~. (t,~)~,,~.,.(t,,,).
~:,,(~,,.,) .... w , , ( t , , , )
Here the expression I~
/Jr:i ..... (t, v,) = 2 t/.e
(a.ga)
(t, u) denotes the cross-term, which is given by
{j,
z+
t
{
:r L
t ------~
e~.: i2=eer (t7"
}
. (3.94)
A simple computation shows that
L_, ,..... (~,,4 = 2 ~,.,.
w,,,(t, ,~,) eos[2~(tzX~,-....... ,.,zxt)+ (~,-
~.....)1. (a..95)
Vhis mea.ns that the interference term esserltially is a m o d u l a t e d versi(m of the Wigner-Ville distribution of the unshffted signal. It thus has an o,scitlath.lg st,ructure, which can be be~ter explained by lookiilg at the previous equa.tior~ r n o ~ closely, h~ facL the oscillations are detm'mined b;v the a,rgm~,ent of the eosin(>fimction: they depend mainly on the time- and fl;equencpdista~(c,s At and Au of the interactfitg components. Hence, we observe fnster oscillations in time (or i~:~frequency) for larger values of Au (At, respectively). Hegarded as a joint quantity in time and frequency, it is ac~uatly the timo-frequen( T distance between the components that intervenes in this
R~rmnta. The direction of tile osciliati(ms is perpeltdicular to ¢l,e straigh|. lixm conneetfitg the centers (t l-.. At~2, ~/+ Av/2) and (t ...~ A t ~ 2 , u - A v / 2 ) of the two distinct compotmnts (of. Fig. 3.5). The chosen example emerged from a symmetric shift in the plane; however, the covariance relative co time-hequeiwy shifts wiil also yield a similar result fbr arbitrary displacements. In l~,v:t, let. us consider the general case of two atoms
:r2(t) = ~t2 ;}:O(t -" t2) c ~2;'~>'t e ~+'~ ,
a2 ~ O,
Then the sigmd :r(t) = :r~ ( t ) + J:~(t) has the Wigner-ViIle distribution
280
+I+i,J~+++l ~x'(1u ++nc39/ I +im('-S c a Ie A t~a .!VsJs
} i {
/>t+2-::,
i
i ii
F i g u r e 3+8+ Wig~!er-Viile transfbrm of two atoms. If a sig~at is the superposition of two atoms, the Wigi~er-Ville distri~ bution consists of two smooth contributions {related to both atoms) and one interference term (related to their interac~iot@ The iatter is locate{! at the midpoiil(; of the line c(mnecthlg the time.drequency cen+ ters of the given atoms. Moreover, it has a:n oscillatory strud;ure in a dirt~ction perpendictflar to th~fl~ line, and a frequency proportionM to the distance between the atoms, The figure depicts this behavior f,ar pairs of identical atoms at {tit];{wei}tdi,~tances+
w i t h the cross-term given IB,
L: :,,+(t,
.)
= 2<
a+,
(a.gr) t+T~-+(B-
t~,~,
x.,~)
Here we employed the n o i a t i o n s
(3.98) At
=
t ~ .-- t 2 ,
&'? .... ~
.......:¢2
~,x =. tvt ..+ P 2 27 ti A~e ,
,
(s.~9) (a, m0)
We t.hus recover the principle t h a t the interference term is concern t r a t e d in a, n e i g h b o r h o o d of the point in the time+frequency plane, which is the (geometric) mklpoint; bet, ween the imeractiI~g (:omponents, V~:e can also r e t a i n the idea t h a t k s oscillatory s t r u c t u r e N governed krv the timetYequency d N t a n c e b e t w e e n she, componertts.
Chapter 3 lss~,u>sof Ir~terpretat&m
231
A different perspective from the ambiguity plane. 27, The same construction principle also admii;s a dual ilH,ei'preta~ioil i~ the mnbiguity plane. Indeed, for the signal z(t) .....x i(t) + z2(t) (from the preceding text) we o!> rain A~. (~, r) = ,~'~~~ (~, ~-) + A~.~ (~, ~') + &~.,.~ (~, r) where Dora the Dmrier relation of Eq. (3,17) between the Wigner-Ville distribution and the ambiguity function, we cal~ conclude that
f{ 2
Furthermore, let; us recall that any ambiguity function is Hermitian (which is dual to the real.ity of the Wigne>Ville distribution); t&at is, we have
Hence, the behavior of the interference term t~a:,~(t, zJ) can be understood by inspecting the cross-ambiguity function A:,u:,,~ (~, r). l~)~:rthe previously consMered example we find
A,,,,,.,~({, r) .... a~ a2 A,:,,({ + Au, r - At) eq2~;~*~+~<> ~ i .
(3.101)
This relation implies that the Fourier transform of the cross-term of the two signals x~ (t) and :r2(t) reproduces i.he ambigttity funct.io:ti of the basic: signal xo(t), ti'om which they were derived by time-fi'equency shits; the only change is its displncement from the origin of the ambiguity plm~e. The amount of the displacemeDt is identical, to the time-frequency dist.ance of the two signa;ls. Certainly, this result conIbrms to the interpretation of an ambiguit.y function in. terms of tinm-fi'eqummy corrdatio.as: If two comp,> nents are locMized, to neighborhoods of tile points (tl, u~ ) and (t~, u~) in the time-iYequeney plane, the essentiN contribution to their cross-correlation involves those shift, parameters, which put them in an overlay position; these correspol~d to tt~e time- aad fi'equency-separations At and Au between the two components. By combining the cross-ambiguity Nnction with its Hermitian copy and taking the |ocalized and eccentric character of A~:~-~:(~, r) i~to account, we recognize fi'om a different angle, why the cross-term I~.,~:a (t, u) has an oscillatory structure: It results fi'om the properties of the l~burier transform operating on a 'bandpass" %net, ion in the ambiguity pirate. From this
I imp'--b)'~'g~c,~c~o/Time-Sc?~.Iu An,'@-sL~
232
point of view, the intertbren(e term oscitlatos :::oi~ rapktly, if 1;he cross-
ambiguity function of the two components is located i~ a region [~mher from the origin, and this coincides with tile direct argument, of a greater distance between the components. Conversely, and opposite to the crosb-term I.~,.,. {t,u), tim "signal!" terms |V~-:(t,u) and l'l~,.:.(t,v) are the ~burier transforms of the auto~ ambiguity functions A;:::(~,r) aml A:,:.(~, r). (iJa.rryi:tg on the foremengioimd interpret~tion in t.erms of correla~.ions, these two functions essent,ially live close to the origin, which lends O:em a '1owpass" character in the amNguity pla.ne. This results in an a prio~ smooth sl.rue~ure of the signal terms, which a,re jt~st their Fourier transforms. I n n e r and o u t e r interferences. 2~ The const, ruetion principle, which was just established for disth:c~ ~'atoms," can also ~:~rve to discuss more complex situatkms~ which can be brought back into the form of a superposition of a~;oms. ~3Yewill denot;e :.hem as outer intcrfere~ces, wl?.e~l they l~sult from the interaction of components (atoms or sets of atoms) that are sig~?ificautly separated in ~he time-frequency plane. This point of view, however, might seem artificial in certNn easy's, where it does not correspond to an objective or physicMly re}evant decomposition of the signal, This cas~' occurs, in partieu}ar, for modulated chirps. While such signals cannot be nicely described by considering an objective separation int~o distinct components, their Wigner-Ville distributions can stiI1 possess ~xscitlating s~ruetm.,s. \:',ib shalt regard them ~ inner intortbrences (eft Fig. 3~6)~ Obviously, there is no clean borderline between these t,wo con(epts of inner and outer interi-brences. We adopted both notions only tbr the sake of (xmvenience. ():m can observe that they overlap and complement each other by inspecting the so-called i~tertbre~ce R:,rm~Ia of daussen. 2r For i:~s demonstrat, ion, let us only consider the envelope of the cross~te:rm of two signals z(t) and g(t), which we write e.~
where
It. immediately follows from this definition that the Wigner-Vilte traasform of ~;, (s; t, u), considered as a fimctlon of ,s with paralaeters t and u, is equal to
•
(
)
Chapter 3 Issues of Intecpretati(m
F i g u r e 3.6.
233
Wigner-Ville and inner interferences,
}br a noMinear frequency modulation (which is of simmoidal type in the example), the Wigner-Ville distribution gives rise to an interference structure, called immr i,lterference, whose construct.ion principle is based on a pointwise application of the principle depicted in Fig. 3.5. Likewise, the Wigner-Ville transh)rm of k!v(s; t, u) .... ,¢~(.~..-s;t, u) is given by s As a consequence, we infer fi'om an applicagion of MWM's formula {cE
eq. (2.9,~)) that
.]/71
~::~(s; t,,
/
)%(.s,t,
=
~) ~v~ (s,{)dsd{
"
This tea~ls to the searched result
tlQ~A,,,J)I ~ =
14':~ t + ~ , , +
w v t--. ~ , , -
dsdg, (3.1(}2)
Tim('-l'b?quemb ?/Time-Scate Ana@~is
234
which Ls known ~s {tie fbrm~da of outer ii~tert'erem':es. This tbrmula permits several interpretations. Firs{;, if die W}gner-Ville distributions of the components a:(t) and y(t) are concentrated in regicms around (~'-1,~/I) and (t2, ~.'2), ~heir cross-term essentiNly exists around the midpoint (ti, z~i). On the other hand, we can took: upon tim existence of an imerferenee term at a point (t, z~) as the result of the (infinite) superp¢~sition of pointwise interactiot~s, which obey the "midpob~t rule" of separ~ted atoms as atreMy established. As ~ong as the given components are reasonably separated, this qud-. itative deseript;ion explNns the existence of outer interferences. However, thet~ is no reason why the same formula sh~:mld not be applicable in the extreme case where x(t) :; y(t). Then one obtains the formula of irmer intc*r~terenccs
t{ a
This ~m~xmuts t)~r the interav:tiou of a single componeut ~dth itself, and is thus deno{;ed ~s inner interference.
Approximation by the m e t h o d of stationary phase. 28 The model of t.wo distinct atoms waz well-suited for the s~udy of ~he mechanism of outer interferences. As far as ffm(~r interferences are concerned, a useful model of nonatomic Wpe is given by a signal with a moduIated amplitude a~d fl'eq ueimy. Let us therefore consider a signal (supposed to be analytic) of the form
where a(~) > 0 denot;es its ins>.mtaneom~ amplitude and
is the ir~tantaneous frequency. 1.ntuitively, a time-frequency representation should reduce to a %ignaF "~ term, which is localized to a vicinity of the curve of the instantaneous frequency. The fact that a distribution cat~ be supported outside ~his curve (a.s ~ e n in Fig. 3.6, for example) results from the inner interferences. We shM1 now describe their geometry more precisely. The Wigner-Ville disi.ribution of the chosen model signM of Eq. (3.104) has the form
u,~:(t,.) =
L(T; t ) :
d~-,
(a.m,:~)
(',hapr~,>r 3 Issues of Intcrprc~tatJon with
Lb-;~)=~
(
235
t+.7.
"
~-
'
(m]07)
Suppose the va,riations of the amplitude L a:re slow in comparison wit,h th(~e of the phase ~. {This corresponds t,o endowing the mat, hematical definition of the instalitaneous freque~tey wi~h iSs im:uitis~ ptiysieM meaning,} Then the Wig'ne~Ville distribution at~,ains the tbrm of a.n oscillatol~, integral It, can thi.~s be computed approximately by the method of sta, tiormrr pSase. As we recall from Subsection 1.2,1, tim underlying principle of this method is rooted in the ~ssumpt ion that the significang contributions to the integral of a highly ~cilhting %nction emerge only fi'om neighborhoods of the %tationary" points, fbr which the deriva, ti;x!~ of the phase vanishes. By definition, the stagio~mry points of the integral (E% (3,106)) under eonsideratior~ are the solutior~s of ~he equation
.~7: (r; ~, 1,,) = 0
(3.los)
solved fbr the variable r. PrJ~ld~:.d that N solutions exist, and that they meet the condition ,: 2@
(k~4~(r,~; t., ~,) ¢ 0 ,
r~.=i
.... , X ,
the approximation by the method of stationary phase allows us to wr~te 7¢
I
]/2
H,'~, (t, ~,) ~--~(2~r)~'2 ~,~ ° ~ ' ~,~-')I ~., 7):7~;" (~,,; n= ,~ [
{
x L(~,,;t) exp ie(r,~;t,u) + i ,7~ s ~ ~;~t~,;t,~,)
}
.
In view of the definition (Eq, (3,[07)) of (I~(r; t, ~/), the explicit computation of t h e ~ stationary points boils down to finding the solutions (in ~r) of the equation "T " ,u =
~
under the side-condition
T
t,5:
---
,
236
'l h~wJq'~ quency/"Fhne-Scale AnMysis
It is a s~mpte fact that the s~ationary points appear i~ pairs: in other words, if v, is a solutiom then must be a solut, ion as wall. If we reta.in only the positive stationary points, we can once more rewrke die previous approximation K, 5'/2
w~.(t.,.,} ~
)
~
-
2
E:T{,,; t,
(:~:ll{})
Hence, the statiomiry phase approximat.ion yiel~is a result, which coincides with the collstructioIt principle of the outer h]terf~renc~.s: The Wigner-Viile distribution ha~ noil-negiigible rabies at Mt points (t, ~) of the time-frequency plane, whieii are midpoints of a liIm connecting any two points of the curve of ghe instantaneous frequency. Moreover, the approximation permits a fiirther quantization of the import.anee of the imier imerference~, hi t~<:t, let us consider a point. (t, l/), for which the im;ert}~renee resuIts from the interaction of exaeBly two points (&, t&(tI)) arid (t2, ~':c(t2)), so that t--
& + t2 2
P=
.:(t~)
,a~,{t2)
~
2
Hence, we look at a situation where only two stationary poiuts v ..... i ( t ~ t2) exist. Tim Wigner-Ville distribution has the approximate value
4 This gives
l,G(t..) ~4a(t~)c~(t~) '
-d~ (t2)-
dt .
.
.
.
cos A(t~,t(,l~ .
.
'
(3.1~l) where A(t~, t2) is die area enclosed by die curve of/.,tie instantaneous fi'equeney and the tine eomiee~hig the two imeracthig poil~ts, h fotlows that the oscilla%ions of the inner h~terference at a point of the plane a.re faster. if t.hb pohit has a g~a~er dk;tance f?om the instantan(~xis frequency.
Chapter 3 t,~s,~esof Ir~r<~rpretatio~~
237
This local oscillating behavior ca,~ be determined more precisely when we look into small perturbations i>t and &~ of ~he poinv (t, ~), at which tim Wigne>Ville traI~sform is evaluated, [f the distance t~ -,,,,,,~ remains constant, we filtd A(t~ ,te) ~ 2~ [2xv6t - .~t &,] + e o n s t . This reproduces our earlier resul¢, acz:ording to which the toca.] s{;ruct~ure of ~i~,(t + tSt, t, + &~) can be derived fronl a poin~a~'ise application of the pril~ciple of outer interferen('~s. it is clear that the approximation by the method of stationa.rv phase is only valid, if the slopes of tile instam;aneous frequency at the interacting points are different, The c l o u t these slopes are, tile bigger is ~.he ampli~ rude of the inter|'e~nce term. Divergence occurs in ~he extreme case of equal slop~:~. ~lhis equality of the slopes maS~ correspo~?d to two different scenarios: (i) Th.e interacting points coalesce (r = 0, lille of zero Iengt.h). Hence, the rule of the instamtan<xms frequency itself turns out to }x: characteristic of a divergent behavior of the approxirnat.ion: As we anticipatcd intuitively, the amplitude of tile Wigner-Ville distribution will be significant at all points of this curve, m~d t.he distribution has a nonoseiIlat;ing structure due to the relation A(t, t) = 0; and
(ii) The intera,:ting points are distiI~ct (r ~ O, line of nonzero length). This defines a "phantom" curve, which is different t¥om the curve of the instantaneous frequency, The Wigne>Ville distributioI~ has a big a.mplitude along ~,his curve. Its oscillating behavior is charac*eristie d the presence of il~terferences~ This behavior is depicted in Fig. 3.7, which dNpIays an example of a segment of a sinusoidaI frequency modulation, Singularities and catastrophes. 29 The curve of tt~e instantaneous fTequeney, along which the stationary pha~se a49proximation divei\ges, can locMly be imerpret¢
"1>ime..bTeque~( E '7'im<~-&v~h~A nal3z~is
238 ......................................................................
?
.....................................................................................
i i¸ F i g u r e 3.7.
Wigne>Ville and stationary pha~se.
(l~eft: theoretical too(hi; rigt~t: comput, ed Wig~mr-Ville distribu~km~) T h e shadowed region represents the set of points, which are the mid-.
points of ~ line com~ecting two arbitrmy points of the cmwe of the inst;ar~taneous fleque~cy (which is sim~soidal in this example, draw~:~ ms a solid litm) At these points~ the stationary phase approximation predicts a significant value of the \~,'~gner-~ file dis~ribn!,ion This at> proximation diverges i~ the subset of all points, which are midpoims of a line with endpoints on the c~rve of the instanta~eo~s frequency, where lbe s~opes of this curve are equal. This subset is composed of two tiaras, the instantaneous frequet-ley (solid libra) and a "phantom" curve (dashed line). r.
~
r,
that we ha:re to differentiate between several eases concernfi~g the number of vanishing consecutive derivatives of the pheube. T h e ~0proach essentially redu(:.es to employitlg a ~'germric" p h a , ~ which ta~ the same structure as the reat phase concel~fing its sii~gularities. This leads to a descript, ion of elm ~-~scillatory integrN, which remains vMid m,e~ when the method of stationary phase breaks dow~. At the same time it amounts to a classific~tion of the possible geomet, ries that can materialize in a distribution. These reduce to a small number of typical c~ses cNled the eb.'.mentaW catastrophe.s'. The n u m b e r of possible ca;ses depends on ~he dimension of the problem; tha~ is, it is connected with the number of paranmters that are needed to determiIle the phg~e. In c ~ e of the Wigner-ViIle distribution, co~lsidered ~_man oscillatory integrN, tim phase *l}(r; t, v') is a N n c t k m of the ~ r i a b l e r and two additional parameters, namely time t and freque~cy z.,. A remarkable result emerging fl:om this theory is that only two elemen~;ary catastrophes are dominant and structuraily stable (in the sense ~.hat smM1 perturbatiol~s of ~I~ do not change t,heir qualitative Ilatm~,), T h e fbttowing distinctio~ of the conceivable cases renders this result understandable, Mthough it lax'ks
Chapter 3 Issues of h~te~'pretation
239
any validity of a proof. There actually exist five families of points in the plane, as tar ~s the quMitative behavior of the Wigner-Ville dkstribution regarding the consecutive deriva, tives of the pha~se [s concer~ed. Only two of them correspond t;o stable singub~rities: 1. The points (t, u) such that
~i7 (r; t, ,,,) ¢ 0 . These are the points, at which the equation T'
7-
h&s no solution fbr T. Consequendy, they correspond to the. nrea~ where W,.(t, L,) is negligible, 2. The points (t:, v) such that
0~ t,,:,,) 0, ??7 (r; :
i}~fl). ¢0 9 0 - ;~; t, ,,) .
They define the a,re~a% where ll~ (t, t~) is neon-negligible and where the approximaeion by d,e method of stationary phase is appropriate. As expbdned before, these points correspond to t;he midpoints of lines connecting two arbitrary points of the il~sta,ntan~:ous f'requency, where the slopes of this curve do not coincide. Hence, they verif~-~,the reIatioms 7"
T
dt
T
-iiY
"
Tile Wigner-Ville distxibution l-m,san oscillatory struettl'e llear these points, whic.h is ehaxaeterized by a circular Nnction (eL Eq. (3.1 t l)). 3. The poings (t, u) such that
i ; - ( v t " ~,) : i}7~-0"; t, ~,) : o ,
i~
They correspond to the cum;~s where I,}~,;(t, z/) has laxge va,lues. The approximation by the method of stationary p h e ~ diverg~s here. These curves are composed of the midpoints of lines with endpoints on the instantaneous frequency, where the slopes agree and the absolute values of the curvatures
l'ime4:rt~quency//Time-Sc~de
240
4 na]vsis
do not coincide, the cmwatm:es themselves having opposite signs. More precisely; these point;s are chm'aclerized by
..
d~ ~
(h/,r (t "-
v"
t.i
~
t
-~-
T
¢0.
........
This situation corresponds m a so-ca.lied fold catastrophe. We already mentioned (without referring to this name), that t h e ~ points constitute two curves, one of which is the instam.aneous ti-equency i{,~ii; and another is a "phantom" curve emerging from it (cf. Fig. 3.7). In the neighborhood of a, fold singularity, the approximation is governed by the Airy function
3
'
}
} &"
By a third-order e:xpa~sion of ¢(¢; t, ~) around this point and by ignoring the influence of t.he instantaneous amplitude (or, which is equivalent, con.sidering it ~Lslocally constant and fixing this cons1~mt ~o I), we obt;a,in ~he tra~L~'itionaI a,pproxim~tion
with
• i )
"
)
I/";
Remark. If wc wish to take t,he insta.nt, aneous amplitude into &co:ran{, exp!icitly, it suffices t:o use the compatibility of the Wigner-Ville transform with produc~;s. Then t.he approximation
is .}us~ified. Fhe transiiiona} approxima~ffon thus appears as a: refinement of the method of stationary phase in a neighborhood of the points, where tim
(hapc~r 3 Issues oF iuterpr~:ta,tiol:t
24I
latter divelges, For a noMinear frequeacy modular, ion the form of the Airy fimction implies that the Wigner-Vi}le distribution has significant oscilla,tions inside the convex hull of the instantaneous frequency and that it decays exponentially outside (el, Fig. 3,7). ('vV)ecan filrther observe that the instantaneous frequency at an instant t is the center of gravity in the fi'equency-direc¢ion of the distribution, and not its maximum.) The velocity of the oscillations inside the convex hull is controlled by the local curva.tme of the instantaneous frequency: The oscillations are slo,a~'r, if the curvature is small, and vice versa. The ex(reme cruse of a. linear modu]alion corresponds to Ietting the curvature tetM to zero. Then the remarkable property of the Airy function I lira Ai ( f ' } =>=~(,,:) ~: ......+0
X(/
yields the anticipated result related t,o linear chirps, which sta.tes that
~,)~ (,, ,.,) ,~ w,, (t,,,
,,,:(t)).
4. The poi~ts (t, u) such that 0 2<>
~!'. (~; t, ,4 = -a~: (~; t,,,) ,.
0 a <1~.
0 4/~
0,
,,) # 0
These points are midpoints of lines with endpoints on the curve of tile insta~taneous frequency, where the slopes ave equal and the curvatures agree in absolute xalues and have opposite signs. Hence, the points are chaa'acterized l:g u =
t +
+
-
re
,)
dt:Z
t +
~T
+
t ...-
= 0,
3
dU
t+~
#~-
t-.-
.
By (onstruction, they must belong to a "phantom" curve of folds: [)l face, they ~vpresent its pucker points. The corresponding cata.strophe is called a cusp, Its dtaracteristics are subsumed by the P e a r c w functio** Pe(a:, y) :-
exp
i
#'u ..-...-:r~ +
du .
A ~aLr:sis
2:12
Figure 3.8.
Wig,m>Ville and cusp singulariti~>s
The Wigner-Ville distribution shows a cusp sirGutarity (which is h)caHy described by a Pearcey function) at each midpoint of a straight, line between two poll,is of the instantaaeous frequency, where the slopes agree and the curvaturcs have equal ab,~lute values and opposite sig~s. A cusp corresponds to a pucker poi~/t of a foki curve.
R~,mark. We should mention that the app{m.rance of a cusp, while saiying ih the setting of ffequen< y modult~tions, requires a superposition of distiller instantaneous frequencies; it thus refers ro a situati(m of outer interferen{ es
(el. Fig. 3.8). 5, The points (t,~), which are cen~er~ of a pertect symmetry or antisymmet.ry. These points are the midpoims of hlfi~.litely many straight lines. The local behavior has the form of a Dirac d~stribution (eL Fig. 3.9). H(Ywever, ~his last scenario camtot be considered on the same level a~; the pre',~ous oltes becau~c~ it describes m~ "ur~stable" sil~uat}otl, This Ixtea~lts t.hat glm s~ightest modification of t,he phrase causes {,he exceptional charactm: of the singularity m disappeaz and return to one of the previous cases, An armiogous situa,tion is known for the focus of a perDcr !ens: Every deformation of the lens, as smalI ,~s it m W be, transfbrms the ideal
C!hapter 3 Issues of' h~tclpretation
Figure 3.9.
243
Wigner-Ville and higher-order singularil;ies.
The Wigner~Ville dis{,ribut,ion shows a singularity of higher order (which looks like. a Dirac distribution) at each point, which is the midpoint of infinitely many straight, lines eommcting two points of the instantaneous frequency~ The figure depicts an example of such a situation (cause of a sirmsoidaI fi'equency modulation). fbcal point into a caustic (whose sections ar~e cusps). On the other hand, if we~directly start from a lens with a teresa.in aberration, a. small modification affects the qua.ntit~tivc nature of its caustic, but not~ its qualitat, ive feature. The obje~ ti-,ee of catastrophe theory is a classification of structurally stable morphologies. Accordingly, we are abte to conclude that. only two possible structures can occur in a Wigner-Ville distribution in the timefrequency plane (for signals with amplitude and frequency modulations); these are fold curves and cusps. The corresponding construction principles were explained in ca~ses 3 a~td 4 presented in the preceding text. Interferences, localization, and symmetries, We cm~ roughly summa~ rize the co~struction principle of ~he (oui,er and inner) interferences of the Wigner-Ville distribution as :tollows: Two points of the plane interfere, so that th~:~ycreate a contribution ~t a third point, which N the m i @ N n t of the stx~gflt line connectinN the two. Although this principle is so sirnpIe, it, still gives rise to several consequences and iuterpret;a:t.ions, which put a new light on the Wigner-Ville transform, The first conclusion in connection with this "midpoint" rule can be drawn from its reeursix~ application. Suppose we begin with two arbitrary points and pretend that t.he midpoint,, which is general~ed in each step of the iteration, belongs to the signal itselfl Then it can agmn interfere with Mready existing points. By an iteration of the same construction principle,
Time-b}~'qu~.~( q/ Fim,e-Scale Amd3si.v
244
we obtain a perf¢~ct atignmem of infinffely many po}ms on the s~raight }ine comm( t i n t t he two starting points. ]Ien¢ e. w~ can ~,rgue jus{ on the basis of the geome~;ry of the interferences, that the Wigner-Ville distributiol~ must be perfecdy iocNized to linear chirps. Turning this argument upside down, we conclude ~.hat. the Wiguer-Ville distribu{im~ can only be }ocalizcd to a. curve in the time-fl'equency plane (viewed as an instantaneous fr~luency), if the midpoint of any t.wo poh,ls of the cur~> lies on the same curve agahl: the only possible solution that n}e.ets this criterion is the straight lh~e. A second interpretation of the "midpoint" rule can be gained from an opposite perspective. If two poims create a new point 14y intertbrenc< then they must be symmetric~t abotn, this third point. More specificall), let. us rewriCe ttm definition of the Wigner-Ville distribmion as
t'~,tt,~ ) = 2
;r(2t.- r)e il~:'~''ri. - c'(r)dr. •
..<
so that it. ~,akes tile form of ;m operat;or expectation v;~]ue. \Ve next introduce the displacement operator (fbitowing Gia.uber's terminology), which is defined }g
A direct computation }eads to tl~ (t ~/)
2 (I),, 1I
"
(3,t14}
where tel is tile parity ope.rator (or Ihe reflection about the origin of the time-frequency ptane), ~:~ that
(fi:,-)(t) = :~:( ~),
(fix)(,,) :~ x(-,,).
(3.:1 ~5}
1| %}lows that the v~:due of the Wigner-Vi!te distribution at one point of the time--frequency p}ane (an be regarded as the expectation value of a parity operat,or, which is eemered around this poi~i~ by means of a con> position with the (tispla~;ement operator. :';~ Hence, we can concIude that the existence of lffm Wigner-Ville transform in a point is determined by the interac~Aon of components that are s3mmetrically distributed about this point, This confbrms to the interpretation of Jaussens's interfi~rence fi~rmula (Eq. (3.t03)). Iiestated once again, the value of the Wigner-Vilte distributhm in one point is a measure of how much the signal is %entered" around this pNnt.
ClmgJter 3 Ls,suc,s of" h-~teaipretatiort
245
Generalization to the ,s-'Wigner distributions. Several geueraiizat;ions of the previous result~s can be addressed. The first deals wilh an exlension of the geometry of the interferences to the class of s-Wig~mr distributions
]/]1• 5"< dT
C
Recall that the Wigne>Ville distribution appears as the speciM case s = 0 (of, Subsection 2.3.2). Without giving all the dermis, wc mention only some typical results pointing in this direction. When we look at the s-Wigner distribution of the previously discussed example (sum of two shifted at:ores, cf. Eqs. (3,96)(3.101)), we obtain .
=
"
,:~,(t-t~,u-u~)+
-
u . . . . .
~
.
This is the same structure as befbre, except that dm inter%rence comprises two t e I ' l I l s
{e
t,
= +
v-
(it ........ (t~ - sin),,..-
(,,~ + s=.))]
,
with
F ~ (t.,,): u.<;i])(<,,) ~;~p{,;[2~(~A,
- ,.m) + A ~
- 2,~m,(ism
- t~)]}.
Therefore, if the %ignal" terms are localized to neighborhoods of the given points (t t, ut ) and (~, >2), the t:,,~x~cross-terms have signifi¢ ant values near the points (tl + s A t , l..,~.......sAu) and (ti - s A t , u~ sAu). The location of ~hese points depends on the parameter s. In case s = 0 both terms are localized to a region about the same point (t~, 1/i), One can easily verifly that they are complex conjugates of each adler. Hence, ellis restates the previous result, see Eq. (3.97), tbr the Wig~er-Ville distribution. In all other cm~es, when s # 0, the distribution is complex...valued, and the crossterms ha:xx~ a "dista.nce" from their midpoint (ti, ui) which is proportional to s. In the emreme cases of the Rihaczek distribution or its complex conjugate (,s = ± 1 / 2 ) , the signM terms and cross-terms are located at four corners of a time-freque,ney rectalgle with sides parallel to the axes m~d with opposite corners (t~, vt) and (t2, u~) (el. Fig. 3.10). This behavior, which varies as a function of the parameter s, yields another geometric int,erpretation of the optimality of the Wigner-Vitle distribution regarding its localization among atl s-Wigner distributions, In
@
i
I
~';2~-a'~
d
Figure 3,10,
Rihaczek distribmion of two atoms.
If a sigual is the superposition of two atoms, the Rihaczek distribu~ tion contains two smooth contributions (correspo~di~lg to both atoms) a~(t two cros~terms (relative to their interferettce). These are located at opposite corners of ~he rectangle with sides parallel to the timefrequency axes, whose other two corners are the cenl;ers of the given atoms, They sh~Jw an oscillatory behavior with a fre~tuency propor-. tional to the distance of the atoms. The figm'e depicts this structure ibr pairs of identical atoms with different separation (only the real part of the distributions is show~). faz't, we explained the perfec~ localization of the Wigner-Ville distribution to linear chirps by co~meptualizing the cross-terms of a linear frequency modulat4on a,s staying on the same curve as defin(~ by the i~stantaneous frequetmy. For an s~Wigner distribution with ,~ ~ 0 this result is no Iong~er true: the reloeatio~ of the crc~s-terms amomlts to an expa:nsion of the support of the distribution. In fac~, in the case of the Riha,czek distribution the whole rectangle supported by the modulax:io~ between (t t, zq) and (t2~ t.,~) is occupied (eL Fig, 3.1t). Just Ks in the case of the Wigner-Ville distribution, we can look a.t t,he geometry of ghe s-Wigner distributions and the creation of their ir~terfi~rences from a c~m~plementary persp(~::tive. This is expressed ii~ terms of the formula for the outer iuterI~reuces, which has the fbrm
Likewise, we can deduce a similar t~)rmula %r the inner interfi~rences by letting g(t) = 3:(t).
Figure 3.11.
Rihaczek distribution of linear frequency modulation.
~i~r a linear frequency modulation the pofiltwise application of the c'~mstruetion princip}e shown in Fig 310 leMs to a Rih~°mzek distribution, which occupies the enth'e recga~g}e. Its sid~:s are parallel to the time[requency axes arid iI c~m~ains the line of the instantaneous frequency as its diagonal (onIy the ten part of the distribution is shown). C e n e r a l i z a t i o n to t h e aff:Ine distributions. Al~othsr gelterMization of our results deMs with the localization of a distribution to a tim~frequency curve by means of argmnents of inteHbrence. For the f i r e , f r e q u e n c y cause, considered in the strict seI~se of Cohen's (:lass, ~:~ have observed a~d j u ~ tiffed in difl>rent ways tha£ only the Wigner-Ville dis{,Hbugion admits a perfect 1ocaliza~.ion to a time-w~rying freque~cy modulation; moreover, this can happei~ only if the modulation is Iine~r. However, we saw in Subsection 2.3.2 that other types of perfec~ loeMization to more general curves can be achieved, For this purpose, however, we have to turn to distributions of another cta:ss, rmme}y the al~lle etass, and u:se ~heir time-frequency interpret.ati(m. As the a,fline distributions ha:c( a bilinear character as well, (;heir underlying pri~ciple of superposition a}sf~ genera,tes cross-tetras, Their geome{ric interpretation, however, differs va~stly from the corresponding principle of the Wigner-Ville dis{rib,s(ion. hi order to explain the mechaIfism t?~r the creation of these terms, we
make use of the ff)llowing geometric argument: If a distribution is known to be pert~;~::tty ~ocMized to a curve in the time-flequency plane, the~ bhe imert~rencc reiative to any two points of tiffs curve must |~e ~ocat,ed at a,n~ other pOillt, Of tile same curve, 'l'here{}~re, we can compute the cross-term of two (mMamped) pure frequencies in two cottsecutive steps, First, we determimic ils frequency location. In a second step we derive the correspondi~g location, in the time domain from our a priori kuow|edge about, the curve, o~ which the distribution iiw~s. :~ Let us consider, for instance, the Unterberger distribution in its active form (of. 'lgble 2.3). Its time-frequency representatMn (~e = ~eo/a) has the form
tt can easily be seeI~ that the cross-term ~>s~miated with two pure ffequencieL~ :rl(t)
=
.... t ,
e
is given by
= 2~,
t +
6(~,-,
u~.) X ~
- >,
e -e~'(~'
'/"!* ds.
This y~e|ds its perfect locMization to a frequency r'i, which must be defim.xi as the geometric mean
of the two i~teracting ffeque~lcies. We next use the fact (cf. Subsectioll 2.3,2) that the Unterberg/dr distribution is perfi~etly Mcalized to the curve of the group delay of the V p e ~/.,,,,.2; heI~ce, two interacting poit~ts (tl, ~Yl) and (t> t,~) define a corresponding curve, whose parameters G and a are determined by the system of equatiot~s -2
VV~ can t}ms derive the va,hles of ~o and ct and insert them inlo the equatior~
Chapter 3 Issa<s of lut:erpretation
249
This least relatio~ states that the i~terference point (ti, ~q) must: also lie o~ the same curve. The previous expression lbr ~{ can now be used in order to find the value ti ..... tl
+ t'2
I
~
-- >2
(3. ~ I s)
We have thus shown theft the underlying geometric construction principle of the Wigne>Vilte distribution does not immediatdy ex{end to other distributions in d~e atfine class, While the former "midpoi~t rule" is given by
{ t~v _ . ~,£',\
_
h
'tel
+ t~e 2
(3,11,9)
//2
2
'
the (active) Unterberger distribution is governed by I,he modified tllle
(3.~20)
tt is possible, of course, to follow the same procedure for other loc~dized distributkms of the Mfine class. In particular, we obtain t.hat the Bertr~mdand the D-distributions (which are localized to the group delags by z,:~ ~:md ze ' ~ , respectively) obey the coas~ru( tion rules { tp = t~ + t2 1 --~ + E (< - t~)
in + z~'2 ------+
5,. (-<- - --- 7.2)o
) (a.121)
libi
122
-
'
"
I/1
log(>,/<) and t~ .~ t.~
.
I ,.
v/i>T -
,/G ~
~[) ......... 5:....... + :~ t< ........~:~) ,/,>; + v ~ : (3.1~2)
As an interesting ~:',~ture of these construdions, we note that ea,:h of them can be associated with a different definition of a, "midpoint" between two given points. This is laid out more clearly wtmn the preceding equations are rewritten using the variaMe 0 = t - to, where to denotes the verticM
Time-br~,q~,~e~Q77Fime-,Scak~A~Mysis
250
asymptote of ~he group delay (i.e,, the curve on which the distribmion is localized). Then we obtain
{O)V = ~1 (0~ + 02) (a.123)
1
,,,
.... ~ (07~ + 0~')
(3,,24)
1
log 0~: = ~ (log 0,~ + log 0,~) log ~,/~7=
1 ~ (log ~.q + log <,)
(3.t25) .
Thus the three rules correspond to as many ways of computing a midpoint after performing a nonlinear transfonnaiion of the coordinates. Furthe> more, in the limit of narrowband signals the other t.wo midpoints tend to the usual arithmetic mean, which is characteristic of ~he Wigner-V{He distribution. :~2 Similar to the Wignel>Vitle distribution, we can interwet these midpoint, rules ~ symmetries considered in a modified geonmt.ry (owing to the nonlinear transformation). Conversely, we can d e v i ~ an explicit construetion of a distribution ba;sed on a given midpoine ruie. This procedure starts fl'om rewriting the operator, w h o ~ expectation value is the Wigne>Ville distribution (el. Eq. (3.114)), according to
In this form, the central operator
of a
is the o ~ r a t o r freq~jential parity associated with the idea of an arithmetic mean. Its a,:tion is deflated by
Chapter 3 Issues of Inteu~retat, ion
251
We can thus think of substituting another unitary operator for it, which is ~usso('.iated wit:h the idea of a ge
Art immediate computation shows that the action of the global operator, which can be derived as in Eq. (3.114), is given by
= -~ X
e~-~[('~/~!~ U(~).
ttenee, the corresponding time-frequency representation has the fbrm
\
This expression, except for a fac/~or 2, is identical to t h e defisfition of the Unterberger distribution in its "paxssive" fbrm (of. ~ b l e 2.3). AnNogously, it is conceivable to start from the rule of Eq. (3.122), which is connected with the D-distribution. Then an opera,:or can be defined by
and we find in this erase that
f e• i2rct[* i~, ~ i2~tf' \'/,,
1
= 7 D~,,(t:,,,).
7'im('-Freqae~cy/7~me-ScaIe A ~u~,lysis
252
3.2.3. R e d u c t i o n of t h e Interferences
In the previous section we noticed the importance of the interference terms in a Wigne>Vi/le distribution. 'Fheir sig~ifica~ce materializes in a 2-.fold mamler. First, their mm~ber is large, because N ij~teracting components generate N ( N - 1)/2 cross-lerms. Second, their ,size may exceed the signal terms by a factor 2, as two components of equal amplitude in the model equal;ion (3.9t) yield ma~ II.,,,:,(t,,,)t = 2 m a x ! U % (t, " ) t .
Conseque~My, it is worthwhile to dwell on t:he problem of their reduction; this wil~ otlce more relier t.o the mtalysis of the mechanism of their creation. Analytic signal. Because the number of (:ross-terms grows as the number of interacting eomponent;s of the signal im:reases, a first; step to avoiding an excessive proliferation of these terms is a best possible reduction of the redundancy in the representation of the sig~ml. Viewed fi'om a fi'equential perspective, t,he Wigner-VitIe dist;ributioT~ can be written as (3.131) Tlle mnnber of created interference terms depends on the numl)er of components in the spectrum X(t/). For a reat-,,~d~led signal ~he Hermitian symmetry of the spectrum is the reason for an excess of half the mmlbe~ of frequency~components, namely those that are located at negative frequencies. As an example, we consider a signal which is the smr~ of three (positive) frequencies. In its real representation i~ produces [i0,eei~ cross,-terms, ~i1~e of which have positive frequencies. Because the rmgative frequencies of the signal stand only for a repetition of che same inh~rma.tion of the p()sitive olms, it is certainty profitable to omit them completely, This can be achieved by passing to t:he analytic signal (of. Eq. (1.22)). 3..~Then only the physicNly important components with positive frequencies interfere with each other: In the preceding e:;ample the number of cross-terms is ~hus reduced to thr(:a instead of 1)i~Je, and this enharmes greatly the reMability of the diagram (cf. Fig. 3.12). "aqgner-Ville and atomic decompositions, A second approa,ch to reduce the int,erference terms makes use of atomic deeomposit~ioI~s, t;~)r this purt)ose let us begin wit,h the geIieraI resulL that the Wigner~Ville distribution of a multicomponent signal
:c(t)
= E
a~
x,(t)
(3.132)
253
. napa~r 3 L s u ~ of.fnterpre~ation
Figure 3.12.
Wigner-Ville: real signal vs° analytic signal.
The Wigner~Ville distribution of a real-valued signM (left) contains interference terms, which are created by the interaction of components with positive and negative fiequencies. These terms disappear, if the distribution is computed based on the corresponding m~alytic signal (right,), sim;e the components of the signal with negative t~'equencies are omitted~ is given by .;\
N - 1
N
Re {,~, a; w:,.,,:,:~.(t, ,)} . ~=;l
~t::l
It:' " l~q, 1
An obvious way to suppress the crosso-terms is just l~y- retaining the first sum of the above expresskm, which corresponds to the signal terms, and omitting the second sum, in which the eross-t, erms ~re collected. Untbrtunately, while the :r,(t) generally correspond to an objective decomposition of the signal into physical components, the observer em~ only dispose of their sum (Eq. (3.~32)) as a whole. This renders a separation of the two series impossible, tlowever, there is a trick by repla~ing the objective (and inaccessible) decomposition into pt~vsi(al constituenis with an atomic decomposition into m a t h e m a t i c a l components° Hence, given a certain b ~ i s of flmctions g,,(t), the anMyzed signal can be written as N
x(t) = E ( x , n=
I
~,~} ,~,,~(t) .
(3.133)
FJme-Freque~qr/'TJme-Sc~deAnaJ3:sis
254
Then a '%/eared" form of the Wigner-Ville distributkm, which suppresses the interactions of the different atoms of the decomposition, is given by :~5 N
This clever method is naturally limited by tim (t>,cessary) choice of {.he b~sis functions of the decompositiom When a sufficient a priori knowledge about the signM is unavailable~ this problem can be remedied by a iearning procedure that uses a large dictionary of admissible bacses. Smoothing, We expIained in Subs¢~:t.ion 3.2.2 that on{ of the characte> isties of the interference terms is ttmir usciltaml;vstructure. This contr~sts with the signM terms, which are more regular. This opposite behavior sugges{:s the use of a smoothing operation as an apwopriate tool for reducing the size of the interii.,rence t;erms. Such a smoothing can either be e:xpressed in the time-freqummy plaice by means of a convolution with a function [I(t, ,/), or equivalently; 1~.. a u~ighting or multiplication in the ambig~.fity plane by a f'unetion f(~, r), whk:h is the Fourier transtorm of II([, l~). In this regard, the reduet.ion of the interfere~ees results from replax:ing the Wigner-Ville distribution with a smoothed version
I12
112
We there~r~: recover the general definition of the distributions in Cohen's class (Eqs. (3.9)-(ads)). It amounts to a new geometric imerpretation of this class. Therefbre, the inherent problem of reducing vhe interferences boils down to a good &oice for the weighv function. The ~olution to this problem rel{es on the g~ome~ry of the interfbrences (explored in the previous Subsection). In fact, a.s the velocit.y of the o~:illations is bigger for intera,:ting eompo~mnts that are farther apart from ead~ other, the r~luired smoothing needs only a shorter wiIMow in such situat, ion. E x p r ~ in terms of the ambiguities, a big distance of the components of the signai manifi?sts itself by a localization of the significant cross-ambiguities far awgg: from the origin, and thus from the autoambiguities of the signal. We only want to keep the latter ones, which are the true images of r.he signM t.erms. In this c ~ it suNees to choc,se a weight flmction w,i~h a large support, which in return corresponds to a smoothing by a small window in the time-freqummy plaim. This very generM interpretation of f({, r) as a weight Nne~ion states that, we should choose f so that it suppresses the eross-ambigui V terms
Chapter 3 Issues of" II~terprct:af, ion
255
(which are natm:ally eccentric) and preserves the autoambiguities (concen-tra£ed in a neighborhood of the origin) to their best. possibtc integrity. It is thus justified to think of .f({, r) ~s being maximal at the origin and hayil~g a support, which is determined by the time-frequency positioIm of the interacting terms, a(~ Coupled smoothing. A first example for the reduction of interferences by a fixed smoothbJg is realized by the spectrogram. Denoting its window function by h(t), as before, we (:an write it; as &.(t,,,)
=
,,)
.
In case of a, smooth window function, the a~ssociated parameter function =
has a global lowpass dmrat:ter. This ascertains an effective, reduction of the cross-terms that are created t-ty components with a time-frequency distance, which puts them outside the region of influence of Alz(~, r). Here we recover another example for the restrictions that apply to the spectrogram: Providing a better reduction of interferences in one directiol~ of the plane (time or frequenw) causes a loss in {.he same property relative to the other direction, This can be regarded ~ a direct consequence of the inv~,riant vohmie prq)erty of the ambiguity function, which signifies that J~f IAt,(~,r)t2d~d~ = alfJr i 4~7(t,~,)dt&~ ~9 " .... E h2 .
(a.ias)
Combined with the requirement. IA,~(~,r)i ~ IAh(O,O)l = & , this property implies that, the reduction of the support of the ambiguity flmction in any direction1 is coupled with an enlargement of the support in the perpendicular direction. As an illustration let; us talkie the Gaussian window (with unit energy) fl(t) = (2c~)1/'~*e. . . .
t~ .
(a.la6
[ts ambiguity flmction is given by .Ah({,r) = e x p
-...~ {' + a t ~
.
(a.la7)
Tim,'~. b ) e'qu~*~c3.?;"f'ime.Scate A na(Fds
256
............................. 7)>>
i~ ! ..........................
I ..........................................................
] i Figure 3,1a,
Spectrogram of two atoms.
The spectrogram of a signal, vchich is the superposition of fwo a~oms, consists of two smooth contributicms (related to the atoms) and a crossterm (relative to ~heir inter~cti~m). As for the Wigner-Ville distril:mtion, the latter is located at the midpoint of the stndght [iue com~ecting t}m time-frequency centers of the given atoms. Its amplitude, however, decreases rapidly when the overlap of ~;he "signaF'-components g~ts smaller. Tim figure illustrates this behavior tbr identical pairs of atoms and difi'~re~,t time-frequency distance>~ For fixed c~, the present couplfllg of the smoothing in the time- and fie° queucy-direetions is clearly underhned by the fa{t that
{ (
(3.t38)
Even though a spect..rogram shows a significant reduction of t;he er,>> t:erms e~s compared to a Wigne>Ville dist:ribution, it is not completely exempt from ttl~se terms, ar Indeed, we can immediately see that,
s,~.,,(~,,)
=:= &(t,,,)
+ ~%(< ~)v 2Re {~:~(~,,.,) t(,; (t,,.,)}
.
Hence a cr~xss4erm &~es generally exist. This is not surprising after all: tt is related to the classical observation mmrte when measuring an intensity in the presence of interfering ph;~sica.I waves. Nevertheless, this eross.-term only arises when the short-t.ime Fkmrier transforms of the interat'ting components overlap, Its importan(e de{ rea,~s rapidly, if the distance between t,he component.s grows (el. Fig, 3.13). S e p a r a b l e s m o o t h i n g . The use of the spectrogram as a smoothed version of ,~.he Wigne>Ville distribution faces two m a j o r drawbacks. T h e first is e}m kxss in most theoretical properticas that constit.u~.e the advaut.ages of the
Chapter :~ l~sue,~ of Int(,rpr('.t~tion
257
Wigner.~Viite transfbrm. The second drawback of the spectrogram results from the restrict, ions of Heise~fl)erg;-Gabor type~ wifich linfit i~.s smoothing capability. Heuristically, we can say lha~ the time-fl'equeney smoothing by a, spectrogram is based on only one "degree of freedom," as it employs a unique short-time window h(t): If the smoot,hing in time is of order 5t(h), by which we denot.e the width of h(t:), then the smoothing in frequency must be of order 1/4t(h). If we take both dimensions of the time-frequency plane into consideration, we can use a smoodfing with two "degrees of freedom," one related to time and one related to frequency. This yields an improvement over the specl, rogram, ~ t3.r ~us the tbrementioned second disa~tvantag~ is concerned, A natural way to proceed is to employ a s~:pa;rable parameter funct,ion
It is equivNent to a, time-fl'equency smoothing by 1I(,.,.)
=
:;(t) H'(- ,,,),
which is controlled in time (by 9) mad in f~eqnency (by H ~) fnde.pendentl3.', r h . corresponding represent a.t ion is called smoothe:d ps'eudo-Vi.xgn~,r-¥file dNtribution :~s and has the tk~rm sPw.(t,
(3.139)
.) '
-C
::2::
Note that if no temporal smoothing is used, that is, 9(t) = 5(t), the original definit.ion equation (3,74) of t.t~e pseudo-Wigner-Vitle distribution is recovered by putting
h:(r)=t,:
r
h
(")
.
[)lrthermore, if we also let, the window flmction h tend to the const~mt t, the resulting distribution tends to the Wigner-..Ville distribution wi{h no smoothing applied. The use of separable functions permits a continuous mid independent control (in time and frequency) of tile employed smoothing. This contrasts with the spectrogram, which is based on a smoothing function, whose timeflequency concentration carmot be pushed beyond the lower bound of the
7 7me-b }'eque,~??/Time-Scale A ~:ag'sis
258
Heisenberg-Gabor uncertainty principle. In fa(t. when we return to the example (Eq. (3.136}) of the Gaussian window~ we find lls(t, ~,) = u),(t, :,) = 2 e x p
......2 :
a.t".
(3.140)
<:i
for the spectrogram. If we measure the ext.ensioa of the timeofrequeney smoodfing by lhe "duration-bandwidth p r o d u c t ' t~T(I1) = 4u
/f
lI(t, u 'I dt&ei t ~ Y-.....
o
,
(3.t41)
i
we obtain
~fr(n:~) ,~ 1
(a.:42)
for the spect;rogram. This result holds regardless of the (~msen p a r ~ n e t e r o; that is, it is not affected by the size of the short-time window. However, in the separable (use ~mi hing hinders us from using a smoo{.hing of the type
2
27r
+ -g
(3.1,13)
with e~i3 < 1, In this ca.~e we ob{;ain 1;:T(I [spw ) ...... v'%fi,
(a.M4)
and t.hks enables ~ts to pass contimzously fl'om the Wigne>Ville d[stribution (oi3 = 0) to the spectrogram (aft = :) (cf. Fig. 3.14), While the spectrogram is confronted with an unavoidable trade-off betweet: tim time- a:~d frequency-resolution of the anMysis, this compromise is now located soIn(> where in between the joint time-frequency resolution of t,he signal terms and the size of the interference terms. ~lhe ac~uaI choice of a pair of flmctions for a separable smoothit/g can be guided by t~I~ knowledge about the geometry of the interfermmes of I.he Wigne>Vitte distribution. If we consider, for instancG the pseudoWgner-Ville distribution (L'q. (3.74}), which opera:tes by a a~Iely Dequentim smoothing, it becomes (~ear that i~, reduces maiuly the osci[lations in the fi'eque::cpdirecth)n. Such oscillations result from the interaction of compoImnts that are separated in time. Hence, we can regard their reduction equally as being gained from ~,he introduction of a short-time
(Yb~ptc'r 3 Issues of h~t:e~'l~re~;af:io~i
O
,
O
'259
1
~
t
t
e%, ?~:?
F i g u r e 3.14.
i
Tra~sition between Wigner-Ville and spectrogrmn.
The spectrogram is a smoothed version of the Wigner-Vilte distribution, which employs a time- and fl'equeney-smoothing. By the use of a separable smoothing flmction, one ca.n obtain a continuous tra.t~sil.ion between the Wigner-Ville trm~sform (no smoothing, teft figure) and the spectrogram (smoothing with respect to a Gabor eell~ righl figure). The transition depicts the resulting trade-.off between the joint resolution of the analysis and the size of the interference terms. window, which induces a temporal separation of the correspondi~g components (el. Fig. 3.15). Only those interferences persist that are created by the il~teraction of components that are %een" through the window simultaueous/y. Alternatively, the use of a temporal smoothi~g essentially ~m~ounts to the reduction of interfbrences which oscillate in the time-direction. As we know, such oscillations result from (,he interaction of distinct spectrN components, and the velocity of the oscillations is deternfined by their fleque~tiaI distance. It is, thereby, quite easy to use this property as a guidelh~e for choosing the duration of the time-smoothing properly (ef. Fig. 3,16). While the use of a separaMe smoothing allows better controI over the reduction of the cross-terms, it ge~mrNly atfects certain theoretical properties. However, we should retain the important observation that the introduction of a smoothing in one dil~,ction can only influence the theoretical properties related to this variable. This is another advantage over the spectrogram, a~s the tatter disturbs the properties relative to both ~<ariables. As we already saw beibre, the conservation of the time-support of a signN is not affected in the pseudo-Wigner-Vilte case by the introduction of a shorttime window. Likewise, the property of the correct marginal distribution in time will be preseiw-ed if (('£ ~l~ble 2.4) fpw(~, 0) .... 1 , Note that this condition is automatically satisiied, provided that we use a
"l'im~ -l~?3:,quoncy/; Fime-fcede An~Vsis
260
?
f . . . . . . .
F i g u r e 3,15.
I
J
Pseudc.Wigner-Ville
The p~eudo-Wigner-Vi|h, distribution :.~es a short--time window that performs a ffequentiM smoothing of the Wigner-Ville distributiom Only the cross-alarms created by components that. overlap with the window simul~m~eously persist. The tlgure depicts two pseudo-Wigner-Ville dis-. tributions of the same signaI, but computed with two different windows. Their respective width is indicated by the horizontal line segme:,t.
....~!
~!i:i •
{ ~
i•
!i~!!
~!!;
{.............................. F i g u r e 3.16.
Smoothedp~e,:d~Wigner-Ville.
F h e short-~ ime window of the pseudc,-Wigner-Vitle disLribntion leads ~o a reduction of the i::terferences of time..separated compo~ents. In a dual maturer, the nse of a stx':ond smoothh:g in {lie time-direction amounts to a reduction of the interferer~ces of frequency~separated eompo::ents. The figure illustrates this by showing two computed distributio:Ls of the same siglla}, which employ !he same shorl-time window, but use a diiferent smoothing al,)ng the time ::Lvis (left: no smoothing; right: smoottd:~g with a duration indica~txt by the horizontal ti~m segmeni;).
Chapter 3 Is,s'ucs of: h~tc:,~pr'ctation
26t
normalized window t~ with ih(0)l
1.
Moreover, the instantaneous frequency can be recovered as the local center of gravity ff (el. ~lS~ble 2.4)
t,*
" { il;
}
=0.
This condition is guaa:anteed if (,he window flmetion is either real-valued or ha8 a Iltaxiinun:t at, t.he origin. Of course, we can collect a set of Mternative conditions for the case of a pure time-smoothing, However, we must note that the mentioned properties are destroyed when the separable smoothing incorporates both direction,s. ConsequEntly, in c~se a separt~ble smoothing displays a not;ice-,, able practical improvement over the spectrogram, it is affeEted by the same loss in theoretical properties (although on a smMler some). Joint smoothing. By giving up the idea of separability, we can introduce other types of smoothing, so as to preserve some of the desirable theoretical properties and reduce the cross-terms at the same time. If, {br insta~me, we wish to guagantee the correct, margilml distributions of the modified repre,~ sentatiom its parameter function must meet the condition (el. 'I~flfie 2.4) f(~, 0) = f(0, r) =-- 1 . If WE want the same parameter function to yield a reduction of certain cross-terms, it must be decaying away t?om th.e origin so as to diminish the size of the cross-ambiguity of distinct components. A simpD way to achieve this double objective is to impose the special reFIT1
f({, r) .... :#(gr)
(3. 145)
on *he parameter function, where g~(x) is decreasing and c7~(0) == [. The eo~z'esponding time>frequEncy distributions are generically called reduced interference distributions. :~ Their parameter funEtions of the type from Eq. (3.145) display a cross-like shape: The cross sections for ~ = 0 and r = 0 are constant, while the other contourlines ({r =const) form a net of hyperbolas. (~boi=WiNams distribution. A simple example (cf. Fig. 3.17) is obtaiimd, when we choose a Gaussian for our flmction ~, hence f~'(~' T ) = exp (
' ( < r ) 2) . 2~.1 ~.
(3,14(~)
Time-Yrequ<'r~c77 2!)'m(-Scale Amdysis
262
¢#.......................................
....5 t .-5
q
-'L5
o
-5
0
F i g u r e 3~17~ Parameter function of the
Chc/i-Wil!bmls distribution,
The parameter funetion of the Cho~-Williams distribu~ion ill the fiequency-time domain (or ambiguity plane) is a Gaussian relative to t M product of the two ~lriables~ with a bee parameter (~ denoting its spreeu.I. Its graph has the form of a %ross," because its co,ttourlines build a net, of hyperbol~s a~,d tim cross s~'tions along the two ~ e s are constant. Its properties permit compatibility with the marginal distributions toge{ her with a certain reduction of the interference terms. This reductiorl is more pronounced ~br smMler vahms of a. The figure presents three exampDs of such fmtctions with parameter ~Mues a :- 1 (top), a = 5 (cemer), and a ......25 (bott, om), r~pectivety
Chapge~ 3 Issues of Interpretation
263
Its variance a2 is used to fix its spread, "~;e thus obtain lira J..(~,r) ..... 1 , and this renders once more the WignerYVille distribution as a limiting case (with infinite variarme). The tormal application of the definition equation (3.1.4i) of the duration-bandwidth product of a smoothing kernel leads to 1
sr(fo)
(a,~,r)
= -. G
Hence, the smoothing gets more significant and the anticipated reduction of the interferences improves when we use smaller values of m Note that., strictly spe&ing, the Fore'let transform of Eq. (3.146) does not define a "true" smoothing kernel in the sense of a lowpass filter. The representation associated with the parameterization equation (3.146) is (:ailed Choi:Williams distribution. It has the tbrm 40
, ,
."
~G "
cn..,(t.,)..~
...2,,.2(.s....t)°~ir;: r
; '+7T
; ,
7" .-.~ .-"'"'a.,a~.
~2
(a.l,,lS) The ChoY-Williams distributkm acts more like a reduetJon of the size of the interfb.renees, combined with a reJocatimL rather than a true deletiou. This occurs ~s a necessary counterpart of the requirement of the correct marginN distributions. It caa better be explNned by a simple exmnple. Consider a signal composed of tv¢o pure frequencies (:u4.(~)
its Wigner-Ville distribution has the tbrm
W,,(t,~4 = ~0.-t.~)+ ;q~.-~.2)+ 2cos[27r0.~- z.2)t] ~(z.-a +~.2) (3.15o) while a straightforwa.rd computation leads to (a. 151 )
G
2~2
U
t/t ~
11~
264
"1'imeJ"req~ mT/Timc-5< ale A~atys'is
-:; £ ~ S a £ ~ .; g a k .5 =
F i g u r e 3.18. Pseud~..Wigne>Ville and Cho~.Williams distribution of two pm'e frequencies. If the anaiy~ed signal is composed of two pure fr<~uencies, the pseudoWigneroVille distribution (left) yields two ridges with constant amplitude located at both fi:equenci~, a~M a~ ~scilla.th~g er~:~s-term (of twice the amplitude) b~ bet,ween both frequencies. This interference structure is reduced by a Cho?-Williams distributio~ (right). The ca~,e considered he,-e (~* = 1.) shows that the Chdi-Wiliiams distribution acts more t~ke a wtocation than a reduction of the interferences. This is needed tbr prt,serving ~he corr~x"t marginal dis~ritmtions.
lit b o t h cases the cross-term is centered rotund the midpoh:~t of the two interacting frequencies. It t.acillat.es i~J. the time-direetio~i mid its period is im:ersely proporti{maI to the frequentiat distance. T h e differmme betwee~ the two dist;ribut:ions results fl:om the introductio, t of a fi~,ite p a r a m e t e r a+ tt brings a b o u t a transformation of *>he normalized l)irae distributim-J (responsible R}r ~,he pert~( i, localization of the (:~s~;~s-..term in the Wigne>Ville S e ]\ i l l t : o a ( ,"~ (:aa a u ~ ' S t a"l i with s o m e n t m z e r o sprea(1 aIt(t unit a r e a . A s l l i a l I value of u leads to a better reduction of ).l~e maximal absolute vahm of the im, erfere~me term. tto~v,~ver, it a £ o widens its support, which (measured by t,he WpicM exteixsioI~ of Ga~lssial~s) attains the value ),~ ~,et/2c~(eft Fig. 3A8). Let tts dwell on the com~ectior~ a i t h the correct; marginal di:stributicms o~ce agmn. As a,lready explai~md, when we use smaller values of the tmrameter cr the i~terference term is reduced in size. It is also %pread" in ~;he freq~mlmy~direct, km~ so t h a t the resul~ of its integration parallel t.o tim frequency axis remNics constant. Obviously, ~his behavior is lmcessary for the ccmservatio1~ of the cr~rrect margirJal distribution in time, ~Ls it involves the c o n t r i b u t k m s of the interference term.
(]hapter 3 Issuos of Inte~?')retatiou
265
Le~ us next return ~o tile interpretatiolL of the reduction of tim interf(wences in tim ambiguity plane. It ca~ easily be seen that the eNeacy of the Chd/-Willian~s distribution depends crucially on [he ha{ ure of the analyzed signal. By construction, it is most powerful if the cross-ambiguities are contained in the regions of the plane, where the attenuation by the weight function is maximal. On the other hand, {he condition of correct marginM distributions requires the cross sect.ions of the weight function at { = 0 and r = 0 to be constant. This hinders the Chdi-Wil/iams distribution from reducing the il~terference of terms thai correspond to a cross-ambiguity, which is concentrated near one of the coordinate axes. (This ceLse occurs when the int,eract, ing components have distinct ti:equencies, but are synchronous in time, or the other way m:ouud, when they are separa*ed in time, but belong to the same frequen~T band.) Conversel> the elIicacy of the disWibution is best, when the cros>ambiguities are located near one of the diagonals of ehe plane (of. Fig. 3.19). In this case the reduction is even more accentuated, when the interacting components are far apart,
Born-dordan distribution. The compatibility with marginal distributions w ~ only one conceivable restr~'dnt. We can add others and express them by further admissibility conditions on ~ in the setting of parameterizations of the type from Eq. (3. t45). tabr example, the conditions providing the instantaneous frequency and the group delay as local centers of gravity impose
'!~ (o> = o , Obviously, the Cho'i-Williams distribution meets this condition. However, the same distributkm does not, meet other desirable criteria such a~s the conservation of temporal support. ~s seen before (cf. Table 2.4), this corresponds to the requirement
F(t, r) = 0 f~,r an
It~/rl > 1 / 2 .
If we denof~e [~v q>(y) the I%urier tra.nsfkwm of the function ~(x) in Eq, (3.145), we can establish t;he general relation
Hence, the condition
+(v) = o ,
M > 1/2,
Time-lQ°equem'\~v'Time-S~ ale Al~a.]ysis
266
Z:
?i'i
Figure 3.19. atoms.
Wigner-Ville and ChoY-Witliams distribution of ~bur
The figure shows two examples that demonstrate how large a reduction of interferences by a Cho'/-Williams dista'ibution (right) can be as compared with the Wigner-Ville distribution (left). In both ca:se~, the analyzed signal is composed of four atoms, without (top) a~M with (bottom) overlapping time- or freque~cy support.s. Evidently, the first situation prov-es more favorable for an analysis by the Cho/-Wiltiams distributiom
is sufficient for the conservation of temporal support. The shnp[est way to fulfil1 this condkion wkhJr~ the c l ~ s of distributions iI~ Eq. (3.145) is given by the cardim-d ,si~e functio~. ~k:~ thus let
f(~ v)== svA.y.~. , i T ~7rF~
(3.152)
267
C h a p t e r 3 lszsu(~s o f l n t e r p r ( ' t a t i o n
whidi defiiJes the B o r n - J o r d a n d i s t r i b u t h ) n ,H
/~,i,: t. i,') =
1
....
x
s "F
r
)( :zr~
s
t]
2.
d,s c -~2r~' d r .
For the example of [:]q. (3. 149) (with ut < z,:z) this distribution yR,lcts a similar reduction of t h e cI~)ss-term a.s the Cho~i-Williains dist.dbution (3.151), as we obtai~l
sJ~.(t.~,)=~(,, ...... <) + ~(-- ~,2) .I + 2,o<>,(,,,-.--~:)~ii,-;i-:,.-~[I,. ,<(,,).
(3.is4)
We should also note that the frequential support of" the cross-term, in this special ca;s< is resl,ricted to tile im,erwd [l.q, u~]. This results from the fact that: the Born-Jordmi distribution abo gua.ra~ltees tile conservation of tile f~equency support in the wide sense (of. % b t e 2.4). Zhao-Atl:~s-Ma~ks distribuOon. Next, w e allow a deviatimi Dora the class of parameterization equatiori (3. 145), As a generalization of Eq. (3.152) we can consider every function of the form
f(e, ~) = ho(r) l~l
sin 7c4r
'~7
where h0 may be any apodizacimi fune¢'iom This gives rise to {;he Zh~u> ktlas-Ma~s
d i s t r i b u t i o n 42
ZAM:~:(t,~/) =
h~(r) .
x
s + r~. :r* s .... .[ ds
e_.-~2~''~ dr,
(3.~ss) which is also ca,lied a distribution with "eonmshaped kernel." This nomenclature reDrs to the support of the parameter function in the dine-time domaim A comparison of equations (a.lsa) and (a.laa) yields that, the Zhax>Atlas>Ma.rks distribution is nothing but a, smoodmd version of t.he Born-Jordan distribut;ion, the smoothing taken along {;he frequency axis. Due to the conservation of the temporal support of the signaJ, it cannot preserve the frequentiat support as well Accordingly, the property of a {orrect margimd distribution in frequency is no longer sar.isfied, unless we require that h o ( r ) = t / } r l , which brii~gs ~> back to the Born-Jordan case. Pur~hermore, we have to impose *:he condition Iim h o ( r ) ir = 1 for preserving Kite corre{ t marginal distribut.ion in time (and tbr making it an energy distribution in the s e n ~ of' Eq. (2.55)!).
'I~im~,-£Te,quencx/Time-Scale k ~xa/ysis
268
Remark.
By putting
h'(v) .... h,~(v) tr , we see that dae reta$ion between the Zhao-Atlas-Marlcs and the BornJordan distribution is the same as between the p~.'udo-Wigner-ViI~e and the ordinary Wigner-Ville distribution. Hence, some evident computationN advant.ag~s cm~ be gained from tlm l~se of a fm~ction hc,(r) of compact support. Then the evaluation of the distribution equation (3.155) is b a ~ d only on those values of the product
r that correspond to points (s, r) of a lTni~e cone (or "butterfly" area) in the time-delay ptane.
GenexviIizations.
There are many special e~mes a,s well as generalizations of the reduced interference distributions, which ai~ defined by a parameter function of Eq. (3.145) Vpe. As their main ingrediem~ is a decreasing univariat.e function, we can emptoy the whole range of apodization, functions of speetra:l analysis (Hamming, Harming, Kaiser-Besset, etc.). Moreover, we can draw our inspiration from the techniques of fih;er design, insofar as t.he r~sulting parameter function s*'~rves as a lowpass filter in t.he ambiguity plane (Bu~terworth, Chebyshev filter, etc.). Such techniques can be helpful tbr gaining control over the transition between "passband" and "a.ttenuated band" of the fih,er. In this regm'd, one can define a Buta:~rworth distributh:m by cho(ising the weight function 1
. . = . :l +. (~V,r/,./'e . . ,,,: f~,,r~ ~%' -
,
N =
1~2. . . . .
A p(>sible generalization of the Chc;f-Williams distribution is given by
2 r • *El
t v12~}
Even more generally, we can introduce an ao'mmetrical behavior of the time and frequency variables of the weight function, If we use the substitution
Chapter 3 Issues of Interpretation
269
in any of the previous definitions, the variances and exponents can be controlled for ea.ch w~.riable independently~ ~3 I%om a practical point of view, however, these generalizations run into some diNculty: It is not clear how the corresponding distribution ('.an be computed directly, that is, wid~out. involving the ambiguity function.
Rema.Hc ~lb bring the discussion of joint smoothing to an end, iet, us note that some const, raints are incompatible with the idea of a reduction of interferences, if we stick to parameterizations of the type of Eq. (3.145). This is evident for tim unit~rity of the distribution (or Moyal's formula). Indeed, t~he latter requires the weight function to be unimodular (cf. Table 2,4), and this certainly rules out m~y idea. of an attenuation in the ambiguity plane. Such examples are the s~Wigner distributions for which C
(el %~ble 2,1). On the other hand, this definition implies that every function of Eq. (3.145) can be written as
/?
p(~-~) :
(3. t56)
Hence, it (:an be viewed as a superposition of s-Wigner parameterizations. This hmi: proper{y carries over to the associated (reduced interference) distributions, which attain the form 44
[, ~{ :-x~
c:~. (t, .: ~) : *
--
l
~(,~) w::.'~(< .) ds .
(3.15r)
x .
V~riable a n d / o r a d a p t e d smoothing. For a.ll the forementioned ca:ses, which are the spectrogram, the pseudc~Wigne>Viile distribution and the "reduced interference" distributions, the emph)yed smoothing has two properties in common that make it too restrictive in some situations: first,, the smoothing is ~xed, that is, the action is the same at every point of the time-fi'equency plane; and second, it. prk)ritizes fixed dir~:tio~s, namely the thne- or fi'equency-direction of the plane.
Afline smoothing,
Regardin~ the first point, we can devise some variations of the foregoing forms. Let us return, for instaime, to tim spirit of I<% (3./0), wMch transforms the Wigner-Ville distribution into a time-scale distribution by an a.ffine smoothing. This leads to an equivalent form of the tmual scalogrmn
I~/+::<
i<,.(t,a)12 : -o
_~, *(*) h*
(a£) :!.
ds
(3.~5S)
'7ime- b}'equ,~:~tiqy/Time-Seale Ana(~,s]s
270
bT mea:~s of the repr~,sentation [7:,:(t,a.)l 2 =::
{U,.(s,[) il), - - 7 / - , a ~
dsd( ,
(3-t59)
As a simplifiea~,ion let us choose a Morlet wavelet R~r our function h.(t), which is defined by h,,,, (t) = ( 2 a ) li~ e ~"'~' d > ' ' ~
.
A straightfbrward calculation yields the ibrmuta [1/~,,,, ~ , , ' 4
=
2 exp
{ ( ,5~
)}.
c~
0>:60) By means of the fbrmM identification Jy .... uo/a; one ea,n saiv that ~he eomputa:tion of the seMogram equation (3.159) is Mcah'y ba.sed on the same smoothing as the spectrogram equation (3.70}. Here, however, the v,4dt~h of the smc~)thing window is linked ~,o the bcale and thus to the analyzed frequency. Let us dermte by At and Au the c<mst<+mtwidths {in time and frequeney, respectively) of the smoothing window h of the spectrogram. Then we infer from Eq. (3.160), that the scalogram operages with yawing widths according to the rule At(,,)
= ff:!~}
a,~d
U
a,,(,,)
......... /.si~
,
Hence, these local quantities depend on the anal3;zed fl'equency, as we. already observed in Eq: ( 2 , 3 0 ) . VxZe *bus realize dmt the scalogram behaves like a spectrogram with a large time-windc,w (arm consequent|y a shor~, frequency-window) at low frequencies. Conversely, at high frequeno cies it beheecc,s like a spectrogram with a shor~ time-window (and a large frequency-window), It is dear that ~his w,riat.ion of the resolution of the analysis does not: aff?,ct the joint degree of the smoothing, as the product of both widths yields a constant value At(u) Au(u) = At A u . As ah'eady presented, we can similarly reduce the effect of this tLvpe of smoothing, while maintaining its a ~ n e character, by having recourse t.o separable pazameterizations, rhis defines the so-called a n n e smoothed pseudo- Wigner- Ville distriSution ~,s
A,SPW~(t,a)=:!-
h 2 9 1l 2
--~
:r s + 7
:r* s --~ '2 d s d r .
Chapter 3 Iss~,~cso["h~t.crpre~ation
O O 0
'
Figure 3,20,
•
27I
•
Transition between Wigne>Vitle and scalogrm~.
The scalogram in it.s time-frequency form is a twice (mffinely}smoothed version of the Wigne>Ville distribution. By" mea~s of a sepm'able smoothing we ca~} devise a continuous tral~sition between the Wigner~ Ville distribution (leR, no smoothing) a~d the scalogram (right, smoothing by a Gabor atom). The trai~sition illustrates the era.de-off between the joint resolution of the analysis and the significance of the crossterms. It permits a smooth transition between the Wigner-Ville distribution and the scalogram (in an exact manner, if both h and g are Gaussians, and approximately otherwise) (el. Fig. 3.20).
Sig~iM-dependent smoothing. The "coustant-Q" property of t,he afline smoothing materializes in a modification of the usual smoothing, which better fits to signals that are wideband at high frequencies and narrowband at low ~equencies. More generMly, in the case of an arbitrary signal ,:~ can say that a smoothing is "good," if it is adapted to the structure of the signal (ii~ a possibly local way). Let us also consider the example of a signal that is composed of two parallel linear chirps (Fig. 3.21). In this case, the Wigne>ViHe distribution displays a diagonM structure, which does not support any of the direc{ional smoothing methods with a preferably "rectangular" or "crossdilm" shape. In fa~t, these latter smoothing operations axe confronted with an inevitable trade-off between the reduction of the interferences and the loss in the resolution of the signal t.erms. Hence, one can only observe their poor efficacy. In view of the same directional preference of the signal terms and the cross-terms in the Wigner-Vitle distribution, it. would certainly be nmch better to use a directionnl smoothing along the straight line of the instantaneous fl'equency. However, this raises the pra~:ticM problem of finding an automatic adaptation to the signal. 4(~ Wittmut dwelling on this issue in detail, we are satisfied with al~ in(tication that there exist local and global solutions to the problem of adaptation. Ie~'om a globM point of view, the easiest approach is to find a weight
F i g u r e 3.21,
Limits of fixed smoothing,
In this example t.lm analyzed signal is the superposition of two Iinear and parallel fiequ~:m:y modul~,ti~ms. The resulting \Vigner-Viile distribution is shown in the upper left, the spectrogram (~ptimizcd with respect to the slope of the modulation) in the upper right conmr. The figures in the bottom row represent lhe smoot h~;d p,~;u&~-Wig~erVilte (left.) and ChogWiltiams distribution (right). This shows tha~, the methods of tixed smoothb~g (~'rectang,~lar" or ";cross-like") are inefficient in slJch a (~diagonM'} situation, be~:ause they <'anm.,t I'uriSsh a reduction of the cros~terms withom affecting the signal terms too much.
flmctio~t in the ambig~fity pbme that depends on the signal. T h e usual i~lterpretat~}on of the interfer~mces a~ ~ross-ambiguities associates them with a~:~ off-c~m~er posithm in the ambigui~.y plane. In order to ~,~duce them. we can impose an immedia~.e and x,aturaI requirement on the searched weight ftme¢ion t h a t it be (real positive and) radiMly d~ere~sing, By the use of
Chapter 3 t.ss'u(,sof [ut<'rpretatiori
273
polar coordi~m, les, we can express this property in terms c~f the inequality
f'(r~ ~,osO, r~ sin 0) ~ f(r~ cos0, r~ sin 0) for a.uy al~gle 0 and radii r~ ~ r > Given such a form, we thus search for the best we[ght flmction ghat solves the optimiza£ion problem
);,pt(g, r) = m~x .][)f I](<, r ) . ' L ( ~ , r)l ~ d~ dr
(3.1(i2)
under the aecompaa~ying condition t2t } 1t'2
where a is fixed beforehand° In far't, for a given upper bound ol~ the volume of f the property of being ra.diNly decreasing penalizes the eccentric contributions of IA,.(~, r)t 2 in the integral on the right-hand side of Eq. (3.t62). An inclusion of these contributions would result in a 'qoss" in pmts of the volume of f , which are allocated t;o ~.he "empty" angular regiolls between the auto- and the crosso~orrelations. Thus ~he optimization preDrs those
JJ'lC:~,,(r, <; f)~,(r ........t, ~ K(t,
,J: II) =
,,) I~& ct~
(a.l(,a)
~:'~
( ~.(r ~' f)u(r....~-t,~ -~')1
"l'in~(,-.b'r~que~c3 /?~im~>Sca]e A na]3"sis where w(& re) is a time-frequency fl.mc~ion of lowpass-type that exprt~ses the local character of the evaluat~ion. TI~e locally ba~t sn c~.?thing is then defined by IL,:,~ (t, ~,) = arg max .t((~ l,; []} ti
Thks optimiza~.ion problem can ~wtualty be solved, if the smoothing is pa... rameterized in a simple w W (ibr example: by a Gatt~sial: ~,ime-frequency timer*on with unit energy, whose directionality and eccentricity may change). t I o w e ~ r , it is als<~ true that such a procedure is difficult to implement and expensive as far as its computation time is concerned. " I m a g e " approaches. If we regard the result of the compu:,atio~| of a joint representation as an image, we can at lasr~ think of the reduction of the cross-terms as methods of image processi~:g, in contrast to a:n image in the usual sense, however, a timed}equenc 5 represm:~ation possesses a very high degree of structured inibrmation (as can be deduced, R~r instance, from the formula for the inner interferences, Eq. (3.i03)). Th~:~e underlying structures must be taken into account in every post-processing of the image. We only wish to indicate that such methods exist. For an in-depth study we ret~r the reader to other sources. ~7 3.2.4. UsePalness o f t h e Interibrences In gm:era:, the interference terms reduce t.he readabitit;y of a t.im~:-4requency diagram. Therefore it is often desirable ~o get, rid of them. ltowever, i~. is also true that the same terms constitute a.n integral part of the dis~r[bueion, a.nd they carry atom information that ca:n be useflfl in certain c ~ e s or from a certain perspective. Unitarity. ~W,~ementioned before that the presence of the cross-terms is ~he price to pa5~ for gaining the ,mitarity of the dislribution. (Recall that the parame¢er function must be unimodular, which is incompatible with the idea of ~,educing the interferences by an apodization.) The important f~ature of this property (Eq. (2.95)) is that we cau subst, itute the joint inner product in the time-frequency plane f~r the usual inner product (in time or frequency). As a consequence of this fact, which will be studi~xl in more detail in Chapter 4, the comparison of vwo signMs by" means of tSeir timefrequency "signatures" r~:quires the consideration of tim complete unitary repre~mtation i~dudir~g the inu'rtb,'(v~( e terms. The unitarity of a distribution always implies its im:ertibility; that is, it is possible to reconstruct, the signal from its representation exactly, apart from a pure pha.se, Hence, tile inl.eH}~rel~ce terms are essential, as they are needed for a complete reconstruction.
Chapt~:r 3 lSsu¢.'s ot Ir~terlm,tatiou
275
i i¸ Figure a.:a2,
~
Pseudo~Wigner~Ville and phase displagement,
The (pseudo-)Wig,mr-Ville distrihutkm is sensitive to a posMble phase displacement between the diff'erent components of a signal, It codes it, in the corresp(mdirJg cros>terrrl. The figure preseI~ts {,he pe~mdoWigner-Ville distributions of two swperimposed tmre frequencies with (right) or without (left.) a phase displacement at the ~nstant t = O. The displacement materializes in ~i. time-shift of the oscillating structure of ~he interference. P h a s e information. The invertibility of a repr(~ent.ati(m (wi~h crossterms) signifies that it contains all inffmnation about all components of the signal. This includes the modulus as welI as the phase of the signal Unlike the spectr(Nram, which neglects the ph0~se informa~tion of the short-time Fonrier transtbrm ~V taking its squared absolute value, the Wigner-Ville distribution is a real quantity that codes these data in its cros~.-terms. We have already seel~ such a codling of t,he phe~se in the example of Eq. (3.149), where the superposition of two pm'e frequencies was eonsid~ ered. The a~ssociated Wigner-Ville dist.ribution (Eq. (3.150)) shows a direct sensitivit,y to a phase displacement of. the sinusoidal oscillations at time t. .... 0 (of, Fig. 3.22). In that example, the eros,'s-term itself acquir~xs a proper physicai meaning: ig can be regarded ~ a signature of the beat frequenqy' resulting from the coexistence of two frequencies, Indeed, the instanta:neous power of the signal of Eq. (3.149) is given by
i;,:(t){ e = 2 [1
cos2.,(<
Hence: it is governed t g fluct, ua~ions that have a lmlger period if the t~wo frequencies are closer to each other, Owing to the correct marginal di> t.ribution in time, this value must coincide with the sum of aIt amplitudes of the signal, terms and the eros>terms at this insta.nt. Hmme, i~ is the
The*~'J~}~qu~*qy/t~ime-Sca~c A~atrsis
276
in{erfere~ce term that expresses ~he existence of the beak frequency and sta.nds for its realizatio~ i~l the thne-fl~tueImy plane. Remark. Let us ~ote thag the phenomet~on of the beat frequeacy tail oIfly be perceived phvsically, if the frequencies are very cIose to each other. In a time-ti'equency setting, this can be explained in terms of a smoothh~g of the distri.but:io~l. For this purpose let us consider a post-processi~ g fih.er, which performs a~ lowpass filtering relative to a non,zero time ir~terval. Then on13 those in~,erf~rence terms subsist~ which corr(~pond to components with a small/}'equency distance: It must be smaller than tile reciprocal of the response time of the filter (playing the rote of tim interval of the integra~thm). For these components the effbct of the imerference cau be perfectly understood as aa i~tert~r(mce to~e in a physical sel~se. I;i~r other componengs, which are significantly farther a.part, the sm{x~thing destroys any coherence a~d leads to a them-frequency picture, which is simply the superposit;ion of the speetrM li~ms wi~.hout ineerax'tioa. A secor~d example, which explMns how the phase mt¢ers a Wigner Ville distribution, is an eventmally existing phase jump of a spectrM tine~ Le~ us consider tile model d an undamped pure freque~my, which has a phrase disx:ontinuity at the origin; hence :
~,
* - , ( ) .+ e ~-'~",' ~ ~ u l - t )
(a.ls,l)
Here U(t) denotes tile normalized HeaTdside step flmct.ion, a~ betbre. An immediate computation of the Wigner-Ville distributkm yields the result uq:(t, .) = cos(~
- ~)
~ ( . - ,,.)
sin( ~¢~--.4 :#2) ~os {4r~(- ,,,0)Ill ~' :2--J .... P~'~. .....................
We can certainly verily' that
holds in case of a perfect, alignment of the phases at the origin, and this corresponds to the Wig~m>VilIe distribution of the pure continuous fl'equency. At the breakpoint t = 0 Eq. (:3.165) turas into
It thus produces a Dirac distributiou competing with a hyperbola, and both terms come with a factor depm~ding on the phase displacement ~ - , ' ~ 2 . Its
(3~apter 3 tss~es of lnr~erpretation
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
277
.
I....................................... == ==
[
/
,! t
F i g u r e 3.23.
Psendo-Wiglmr-Ville and phase jump.
[~br a signal that consis~s of o,~ly one pure frequency, the (pseudo-) Wigner-Vitle distribution hab a characteristic [orm ~mar an eventua! phase jump. Tim figure illustrate~ this f~tct for various va.tues of ~he phase ,imlip betweem 0 altd 2;~, graph has a peak, whose amplitude and position telt how much the phase displacement ~i - :~2 differs from ~. In fhet, if ~I ~2 is ck~ser to ~r, then the peak is more accentuated; moreover, it is k~cated a~ a tower (higher) frequency than z,o, if g~ - ~2 is smalIer (larger, respectively) tha,n ~r. This theoretical behavior is depicted in Fig. 3.2a. ~,Vhat is a c o m p o n e n t ? We have approached the pix)blem of the creation of interl~rences by considering general models of signals, w}fich are iormed by the linear superposition of compollents that i~teract~ tlowever, we never defined what this notion of components really means, Although it is dift~cult to give a definite answer to this ques~i(m (what is a component?), it st;itl de,~rves some special h!vesgigation,
C(mstructive intertbrences. In order ~.o illuminate this issue, w~ should first recall that the distinction between % n e t " and "out;er" interferences wa~s rather arbitraw. 'lhis is wily the idea of decomposing a ~ime-fr(~luency distribution into "sig~ml" {.erms related to the components of the signal and ;;interference" terms rela,ted t,o their intera,'tion rut,s into di~cu}ties, Let us illustrate this problem by reierring to t.he example of Eq. (3.164) once more. When there is a phase j u m p at t = 0, the separation of the signal into
't 'im ~:>~:?"eq u e n q~,/'1 'i ~n e- Sca]e A ~ a .[}'sis
2 78
two components has a. physical meaning; however, if there is no p h ~ e j u m p at all, tile splitting of the signal and sticking back tog<'ther at tim instant t~ = 0 is totally arbitrary. I,et t)~s think over what happens to the W i t h e > ViHe distribution in this borderline ca:~e, gach virtual "component" creates a distribution starting or terminating at t: .= 0. Then the blterference term, which is superimpo~>d on the individual distributions, eause~,~ the representation equation (3,166) to be continuous at this instant and look like a single component again. In the opposite ease, whe~x~ a nonzero phrase j u m p occurs a.t t = 0, the interference term becomes asymmetrical and does not yiekt tile same perfect fit of the two parts. This resuh.s in the discontinuiV of the rewesentation at t = 0~ In this sense, the mechanism of the creation of (inner) interferences renders {.he notion of" a component meaningful. More g~ner.~ ally, due to Eq. (3. I03) of inner intertk, rences, the value of the Wigiler-Ville distribution in one point results from a "holographic" construction, which brings the values at all other points into play~ ~A'2~can thus say in a few words that a comportment N a constructive interference of the Wignm'-Ville
distribution, h~fi)rmation logon. Another way to give a. meaningful notion of a component relates to the definition of a logem in G a b o r ' s sense; by this we denote a,n information "quantum" in the plane. Provided that we are able to define a measure for the information conte~ts in the plane, a component can thus be associated with a bit of inlbrmation. Unfortunately, ~s we know by now, a time°iYequency representation is susceptible to having negative values, in generah Hence, it cannot be associated with a probability density in a straightforward manner, which wouht enable us to compute information contents in Shannon's sense. However, there exists a solution t h a t employs a definition due {;o R6nyi. 4~ As a substitute of Shannon's information measure the quantiV
._1 =
f.
log2 ~l" C'~ ,'5*It.. l.,: . J). .dt du a
c~ > I
.
!'12
is proposed. Recent studies have shown that this ~pproach allows us to "count" the ~mmber of constituent Iogons of a signal, especially when <x ~ 3.
Independence. Finally, a third point of view relies on ~ stoeh~stic ira,e > pretati(m. S u p p o ~ we analyze a random signM a:(~) t h a t admits a decomposition N
(Thapter 3 Issuers of tntcx~)rr,tation
27!)
where the a: (t) are deterministic signals (not necessarily orthogona[) and the a~ are stochastically in&~pend~ut random coefficients. Under t_tle~ a,ss~IHlptioIts w e t a i l conclude that
E{a,,~z;~}=O
if n ¢ m .
Hence, we can immediately deduce, due to Eq. (3,92), that N
(a.l{:;s) This shows tha.t a random signal, whidl is the superposition of stoeha,stically independent eompo~lents, has a Wigner-Ville spectrum which la.cks any ~ross-terms. This can be compared with the method of "atomic" tiecomposition discussed earlier (see E% (3.132)); here, a simplified Wigne> Vitle distribution is obtained by taking dm stun of die dist,ributions d tile elements of the decomposigion, prm~ided that t,hose are stochastically independent. 4t~ This last assumption contrasts with Eq. (3.132) where the atoms v:ere fixed ab initio. However, this new approach requires the availability of as many linear combinations as there are components iii the signal.
3.2.5. Statistical Estimation of the Wigner-Ville Spectrum We already expta,ined the importance of the Wigner-Vilte spectrum of nonstationary random signals in Subsection 2.42i. Let us first recall its definition w,(t,.)=
+
,
which is based on the autocovariar~ce r:~(tl,t~) d the anMyzed random process,
Suppose we dispose of a discrete~time observat, ion of the process. This is most often the case in practice and can be obtain¢
r~.[n + A', n - k] e -~i'~4" ,
Iul < 1/.1.
(3.170)
Now we face the problem of estimating this quant, ity, especially when only one realization of finite duration is assumed to be available. An estimator %r the Wigner-Ville spectrum can be based oil the Fourier trans%rrn
l'ime-1~)'equ('~~c37~TJme-ScaIe A,~aI* :sis
28,,.t
of an estim~tor tk~ the eovariance. Under certain co~ditions on tile raxtdom pro(:ess~ which describe the locally stationary behavior, we can thus devise a definition of a g,~meral class of estimators. They can be compared with the representations in Cohen's ela;ss of a single (certa,i~) reMization of the protess, There are severM motivations for defining such a unified conceptual framework: First, it pros some new light on Cohen's el;~ss; then it Mlows consideration of the properties of t~he different estimators from a very g~meral perspective; and finally, it includes ghe treatment of degerministie and raw,dora signals in one and the same settling. -,o A s s u m p t i o n s . Suppose an est~im~m)r for the Wigner-Vil/e spectrum is givet~ l~. tim Fourier ~ransform 4-y.
where '~j is an estimator fbr the autocovaria~me. In the stationary case, where only o~e realization of x(t) is availahle, a comm(mly used assuml> tion is tile e~godic hypothe,sis, by which we ca~ substitute a temporal mea~~ va.tue of the given data for the (inact:essible) s~,ochastic e-¢pectation value of the theorm:icaI eovariance, Unfortunate|y, we ca,mot tbI|c_r~'~,the same approach in the nonstationary case, in general, u~dess some additional conditions are iIlel;, Let us therefbre confirm ourselx~s to the class of quasi~ st,atioxm W signals, whirl1 al~ characterized by stochastic properties t}mt are steady enough, so that ~he instant.and:ms covarianee at any ins{;ant mah' be appro>:imated by a IoeM timeoaverage, This eonfbrms to the interpretation that a quasi-s~atiouary signN with instantane~ms em~.riemce r):)~, k, n k~ admits a tangential s~:ationaw signal at each instant n. T h e covariances -v~, of the given signal and %:,z of the tangential stationary signal are supposed to satisf};
<
2 ,.I
Here e(T) is a measure for tim gNned apN:oximation relative to a time.~ int,erval of length T~ For fixed e, the time of the stationa~ T behavior T~ is defined as the larg2:vst value of T %r which Eq. (3.I72) holds~ When we work with quasi-stationa,ry signals, we can thus consider the average over ghe t,ime i,~ter\v~l 7 as an estimator for the instantaneous eo~v~ria,nce. Two N r t h e r assumptions will t)e used in the (ourse of the following a~alysis: (i) T h e signals are given ht their arc@tic form, so as to avoid all problems of spectral redundancy and to bestow a physical meanhlg on the imtions of instantaneous amplitude and frequency; a~d
(ii) they are zero-meal~ (~;aussian processes.
(,bap~,~r 3 /~s~.s of In~:erpr~ation
2811
Classes of estimators. Assuming a quasi-s~ationary character of the signal, a na.turaI class of estimators for the co'~,riance c~m be defined by
where Fin, k] is a free parameter fimction. Its %rm a,r~d duration determine the way tff averaging the products x[m + t:i)z~ [m - k]. If we insert this expression into Eq. (3,t71), we obtNn a general cla.ss of estimators for the Wiguer-Ville speetrmn by X;
f-
Ilence, we reeo~,~r a discrete-time version of Eq. (3.I 1). Vvb can thc, retbre conclude that a general form of estimators 1)~r tim Wigner-Ville sp~:trum of a random signal (with only one given realization) is provided by tile represent~tions in Cohen's class of the observed reMization, Remark, Cohen's class defilms est, imators based on a smoo~hing, wifich is independent of tffe fl'equency. Other generN schemes tbr tim esgimation d l,he ~variance may be based on the a n n e class~ for exa,mple. We confirm ourseh~ss ~o tlle study of the ibrmnentimmd claims here,. The second elass of estimators witl be used tater~ in Subsection 4,2,1, wilere it is emplo~ed for a spectral estima~;ion of " l / f - n o i ~ < processes. The complete set of results, which map ~he properties of l:im parameter flmction into associated properties of the representation (ef. Chapter 2), cml be used here bv an adaptation to the stoeimstic context. Moreover, tile stoch;;~tic properties of the estimator equatSon (3. t73) can be understood as a function of tile paramegerization F' (or one of its equivalent forms) as well. In return, we are enabled to devNe parameterizat~ions~ and thereby estimal;ons, thag satisfy some a priori imposed conditions. Let us blvestigate the stochastic properties of first and ~.,cond order. We begin with a consideration of ~he general tbrm of ~he estimators, prior go specialization in some of t,he know~ ca~.s, It is sometimes more convenient to work with the equivalent forms of F. By analogy with Sub.el,ion 3,t .1 we denote them by llj~, ~,) = 2
t@t, 2k] e i'l~4~: ,
,
(:3.174)
(3.i7s)
t'ime-f;>equency/Time-&':aleAna~(y.sis
282
•
=
,
( , . I ~6"}
which are discrete-time versions of the definitions given there. Biaus. B,~.sed on the general tbrm of the estimator equation (3.173), we can immediately compnte the expectation value
=2
+::~ E
t.. :'<,
=
+~ E
1
/11/!4
'ftl E :~ -- X
/,i
f÷l/2
F[,r,-n, 2k]2>,2
~
'
4~-:
~
k ;-..a;
and we thus obtain
~.-cv / , ~ 1/~t
~ .... r -
~---~
.
""
'
7" '
This result, is evideIitly comparable with t h a t of Eq. (3,9). In t,he context of statistical estimation, ig meaas that, in general, the estimators of Cohen's (:lass are in t,ime and frequelmy. The bias is controih~d by the p a r a m e t e r flmction in the time-frequer~cy form (and it grows with the "extension" of this parameteriz~tion). 'I-b ha;~ a correct normalization of the estimator, we must further impose the condition
bia,sed
+'~ /+~/'tH[n,u) du= 1, which (:an be rewritten in a, simpler form ~-~s
f(o, oi =
(airs)
Chapter 3 [ssues of hlte~)Dretation
283
Variance. F o r the determination of the stochastic properties of second order we must compute the covariance of the estimator al two fixed t,im~> h'equency posil&ms. This yields the expression
"4 cx~:
+
:x>
+,.xJ
q <~
Under the given hypot, heses w e can further siniplify the righbhand side. In fact, by tim ~ s u m p t i o n of a Gaussian process we can express the momeng of fourth order, whicb appears in the co~v.wiance of the cross-product, by a sum of products of the second-order moments. The ~ s u m p t i o n of an analytic signal allows us to reduce this sum to only one term, owing to the fact that. reM and imaginary parts of an analytic signal are uncorrelated. This leads to coy
{x[m.t +/q]*
[ m l -. k l i ,
z[m~ +/~,:~] x
= ,.[,,,,~ + <,
tin2 .... k~]}
,,,~ + k~] ,.:[,.,, - k~,
,.,,,~
-
a:~].
By using the additional assmnp{;ions on quasi-stationary signals, we can employ the approximations 17"I @ '#7'2
2
]
ke
that are valid on the useful interval defined by F. If we next introduce the power spectrum ['~,~ (~J) of the tanger~tial stationary process at the moment 'rq =: (nl + n2)/2, we obtain ~ our final result (after some quite laborious computations)
2 "~
2
[[+i/2
This shows that under giveIl hypotheses the gelmral estimators of C(> hen's class are correlated in time and frequency. Let us add one further
7 imp,,-FTc,que~
28t
assumption, namely, that the radius of the local correlatio~ of die tangential stationary process is small as compared ~o the time of the stationary behavior of die signM. (This amounts to the assumption that the Wigne> VilIe spectrum is }ocMl3.' wideband retati~e to the frequency-band of the analysis.) Then we can derive an approximation of the variance of the estimator fl'om the. previous expression for the eovarianee. If we leg
s'(c<-:
f ( ( , t ~ ) f * ( ( , m . .....k] ,
we obtaht
(3. l SO)
..r
I h~," shows tlmt the variance depends o~t th~ frequency. Examples. All distributions i1~ Cohen's class can be viewed a~s estimators tbr the Wigne>Ville spectrum. As a first example let ns consider the specrrc~,'ram. In the context of statistical esthnation, ~t is identical wi~h the short-time period(~Tam of classical spectral estimation, as it is defined by 12
S,:[rt, ~,)
:
~.. :,[~,}i,."7," -
~'"'"'
'4 ~',
It can also be obtained by choosing the Wigner-ViIle distribution of the ~}u:,rt-tbne window h[-nI m~ the iime-frequencv ~ .iJlt HI'n, ~/) {br the esdiitaJ;or equalion (3.173). Consequem.Iy, the bias in Eq. (3.1.77} attains the
~\~ dins recover once. more the classical trade~off between the time aud frequency behavior, which materiMizcs here in the form of bias. As far as die correlation of dm estimated values is concerned, the N~proximate ibrm of Eq. (3A79) leads to
(3.1s2) Here die discrete-time ambiguity N n c d o n is defined by
Chal)ter 3 lSsuas of tnterpreta, tion
2~;5
This signifies tt~a¢ the approximately m~correlated beha~'ior is subdued to a time-fl:equency compromise as well. With a eorrecl; normalization of the window (a,ceording to Eq. (3.178) it, m~kst have uni~ energy), ' ~ can obtain the approximation of the variance
,,a.~ {s~, [..,,J)} ~-: w'~[.,~, .).
(3.1~.:I)
This follows fl'om assimilat, ing the Wigner-Ville spectrum locally (in time) t.o the spectral del~sity of t,he tangent, ial stationary process. In fact, this result is of the same type gm the clas~sical r<%ult ii~ sta,t{onary spectral a,nalysis, which sta.tes that the variance of a periodogra,m yields the square of the value to be est:imat~t (which is the power spec{rum). Consequently; and by a,talogy with the stationary case, an improvetlt~ent ealt be achieved, if we replace the raw periodogram with a smoothed or aver~tged ~ersion, 5~ This amoungs to the iI~troduG,ion of estimators of the type
Their associated p~{rameter funet.ion has the form -~ ~ . . . # , tel .,r(~1, 2~.1 =: ot~)A,¢L-~1
.
In this ca,se, the approximation of the variance in Eq. (3. f8 .~) becomes + ::~
rat
~. W ; i', ")
Z
t :Z
gI"'~'i A,,
071,,,,1 / "
(a,.~s6)
Ilence, if g['mI is a rectangular filter with ?4 taps and properly normalized so that [
()
:~
the [actor in gq. (3.186) is of order 1/M. We can thus derive the approxiIIt&[ioIf
Note tha£ eniargii~g Ihe horizon M of the smoothing amounts to a., redu¢t.ion of the variance of t;he estimat;or. But then ~:he bias in the timedirection inereaeses owing t~o Eq. (3.18t).
Fime-I'}'equenq}'/;Tim~>Scafe Anal3:sis
286
Let; us reconsider the g#:~mmtric point of view (this time in a sett;ing), which was develop~t before and rome t_ts gain greater In order to replace the spectrogram, which ha~ "one degl~e of with a smoothed pseud(>Wigner-Ville distribution having %wo freedom," it suffices (,o pu(
s~ocha,stic flexibility, freedom," degrees of
(3.188)
ni,~,.) = ~4[,-,,])~<),[0,~}. The corresponding estimator is defined by
(a.tsg)
As could be expect~.~.t from the general setting,, t:he b i ~ is also,a s", £pa~ rable quantity with respect to the two variables, b e e a u ~ we have + C-,~;
E {spu.i,.[,~..:>} =
E (3,190)
Hence, we can conclude in t~hepseud(>Wigne>Ville cao~ ( i.e., no smoothing in t,he gime-direet,iou is t>ed), that the bb~s of the estima¢or att)~ets only the frequency-direct:ion; this beeonms clear by examining the relation
j/,
,+ ~/ 4
E { Pw,~.I,,, .)} =
w~.[,, ~) w,,I0. ~ - , . ) < .
(a. ~9.t)
The advantages of the separability also materiNize in the properties of second order. Due to the generN result d Eq. (3.179) and the specific form of the sep~raMe parametcrization (Eq. (3.188)), we obtain
,:ov {sPw,:[.<. < ) , sPw;[~,.~, .'~)t
f+~/2 ~-~ -'),]-t ~/'2 Wh[O,~, - ~q ) ll:), [0, t~ - z.,:~)C~..,, in, In d'~[s exprcxssion we Imt
f"(2~, k] = !C(¢)[ 2 .
n2,~:f")&,.
287
( :hapt(.~ ~ 3 Lssu,es o f 17~terpreta, tion
If we a~ssume, a,s in the forementioned discussion, that h is wide enough, we obtain the approximate form
~ ~ F~,,~ i [~Z~ -" 1~2~ (//l 4" ~-'2)/2;,
d --1/2
It follov,~s that only t,he correlation in the frequency-direction enters t,his expression, The ur~:orrelated behavior in time can be understood as a result of the separation of the considered, insta~lts, whose distance is at k~a.st as large as the duration of the smoothing~ Hence, under the given hypotheses it emerges that
lni -n21
> M
~}
Iz, ......
> Vm
J (3.
92)
Here M and N denote the widths of the windows g and h, respectively. For such windows an approximate decorrelation of the estimated valu~ at distinct points in the plane is guacanteed, i£"their distance is large enough. Conversely, for two given points in the plane an approximate decorrelation can be achieved, if the time-frequency °extel~ion" of the smoothing is ch(xsen to be suNcient:ly smM1 evm~tuMly smaller tha.n for a spectrogram. Finally, we assume that the normalization condition of Eq. (3A78) is satisfied and the window functions 9 and h are of re(:tangula.r shape (of length M and N, where N is supposed to be big). This leads to an approximation of tlm vaz-iance by (3.193) If no smoothing is performed, we gain the result (3+194) Hence, we ca~l rewrite Eq, (3.193) in the c ~ function of len~h M ~s
,J)}
7;7
I77 ~
-- J<~
of a rectangular window
M )
Chapter 4 2~me@~'equeney as a t~radigm 4.3.2. M a x i m u m
;/45
L i k e l i h o o d E s t i m a t o r s for G a u s s i a n P r o c e s s e s
Turning t,o the original formulation (Eq. (4.62)) of the problem~ we no~ supw~se that x(t) is a Gaussian random process, so that E {x(t)} = ,(~),
~.~,(t,,~) = E {~.,.:(t) ~-. - ,U:)]
[~.(~) - #(s)] * } .
(4.66)
It ix known that the detection problem under
whidl is
More precisely, we recall t.ha~ x(t) admits a decom-
doubi~7orthogond; ttm.t is, tile relations # , , j ~ } = ,~,~ &,,,,
E {Ix,, ..- , - d [z,~
/ are verified, where A, and p~.~(t) are the eigenvalu(;-s and eigenfunctions, respectively, of the autocovariance of z(t). ttence, they at*e defin~M by the int.egral equation
f ,.:,,(t,.~)~,,,(s)ds = A,~.~(t),
~:c (r),
(7} The coe~cients x,,, y,,, and #, of the decompositions d x(t), y(t), and #(Q, respectively, axe the projections o~tto this basis of eigenflmetions ( ~ xn = (x, ~ ) , e~c.). 2Ne optimal det(x~.tor (in the s~use d a maz~:imum likelihood estimator) is obtNned t~. a comparison of the decision statistics A(y) = A~,(y) + Ad(y) to a thr~hoht; in this definition we put --
~ ¢1
.
.
.
.
@(Y) = 7o ,=0 A,~ q % (14,67) J
X
T~==0
•
(hal)~
3 Issues of InteJj)ret;~.tffon
289
ttenee, we infer
0 1I%, [m, . ) =
,
4P2[n~ sin4rc~'rr~
- 1 / 2 *~ ~, < 0 , 0 < ly < 1/4
7r7/~,
4t?[n] sin47c(t/2
~)m
1/4 < ~ < 1/2
V~h~further derive fl'om Eqs. (3.184) and (3.I94) that var {,5'~[n, -)} ~ 4p~'[~I ,
0 :< z~ < + 1 / 2 ,
(3.197)
alld
/ 4p2[n] 4t~ var {.PH~,.[,,, r,)} ,~ [ @2[.,,] 4(1/2 - ,,)
,
0 < ~ < ~/4, (3,198) 1/4 ~ ,, < 1/2
Therefbre, in contrast to the spectrogram, the (nonsmoothed) pseudoWigner-ViHe estimator has a. variance that depends on the frequency; this is illustrated in Fig. 3.24, where the reducing effect of a time-smoothing on the variance is also shown.
§3.3. A b o u t the Positivity The existence of nega:tive va.lues of the distribution sets limits on the analogy betweer~ a time-frequency representation and a. probability density function. This also renders its interpretation more delicate. Evident problems arise when we attempt to bestow the rank of a local energy densi V on a negative value. Another import, am observation to be made concerns the rather vagtm justification of a point:wise interpretation of a tim(~ frequency distribution, even if it happens to be positive everywhere. In spite of these restrictions, it is inter(~sting to encompass the relations between t;ime-fi'equeney representaf:ions and positivity ~s well ~s some of their conseqtlence8,
3.3.1. S o m e P r o b l e m s Caused by the N o n p o s i t i v i t y The limitations caused by the nonpositivity ha:re a local m~ture in the first place. '['his means that a negative value of the representation ~t one point renders any direct pt~;sicM interpretation as an energy density at this particular point impossible.
AnM~,~sis
290
0.5 (/
- 0 . 5 -,
F i g u r e 3.24.
Wigner-Ville spectrum and i(:s estimatiom
The Wigner-Vitle specl;rum of a whige r~oi~e (defined as the ensemble a:verage of the Wigner-Ville distributions) is constant theoretically, This can be obser',.ed in the left figure, which shows ti~e average of I00 computed Wigner-Vi[le distributio~s of i~dependent rea.lizations, In the case, where only one realizatior~ is available, the pseudo-Wig~erVille distribution (cet~ter) yields an e~imator with a big variation, as expect.ed~ An elJhanced estimator is obtained by employing an additionaI smoothing in the time-direct;ioa (rigt:~t). T h e y can a ~ o have a m o r e global nature, in~)far as the a~ver~ge quantities of a distribution with negative values can lose their sp(x:ifie ilteailtllg, If we wish to interpret a representation p~:(t~z,') a,~ a joinf proba, bili~y density of the pair (t, ~), we should require, at least, t h a t its marginal dist, fibut, ions be identical with the individual probability den~s'itie;s in t and z.~ (or the a priori densir;ies) ,:,.(t) = i:c(t)l :~ and
p:,.(~,) = {X(r,) 2.
(3,199)
W i t h i n the s~:~tting of C o h e n ' s clasps, we iaave s~,n t h a t s i m N e conditions on the p a r a m e t e r i z a t i o n guarant;ee this p r o p e r t y (eft Eq. (2.67)). C a r r y i n g on this analogy, we eau also conceive definitions of conditionM probability deu.sitie~ p:~.(t I J~) and p~,(r/i t) (using h a y e s ' form.ula.) a.ccord~ ing to
re(t, . ) = p:~(t i - )
r~(.)= p~,(-{t)0:,.(~) •
(3.200)
W h e n we stick to t,,,ou n s class, some~ additional conditions (of. Eqs, (2.99) and (2,101)) turn out to be sufficient %r an identification of the
Ch,',l)ter 3 ls;s'ue.~'of 1nt~rpretati{m
291
asso( iated conditiomd expectation vntues with the instantaneous frequency m~d tim group delay. RecM1 that under these hypod~eses we ha.ve
S {,~ it}
,.,,%..(,~ it)a,
:
(3.201) ~ z 6 : , , ( t , u ; f ) d u = .~,.(t) ,
t:r(t)]::
S {t!u } =
tp:~,(tI u)dt 7~0
(3.202)
= IW~,,-)[2
.~ t<,~,(<,J;f)dt = t , + J ) .
All conditions together (which are verified by the Wigner-Ville distribution, in particular) seem to provide a satisfactory (though not pointwise) interpretation, insofar ms they suggest to define a localization of t,he representation around its local centers of gravity. It is therefbre tempting to proceed wit& the s t o c h ~ t i c analogy and to define a dispersion around these averages by means of conclitkmaI variance.s. 52 Let us consider the rule of the instantaneous fi'equency, for example. Then the expression
~ ( . It) = E {,~ l t} - (S {,. i~}) ~ .......ix(0f
(3.203)
J._,.,..
~'2c~ (t, ~'; f) d~, -
~,~(t)
is a suitable candidate for such a dispersion. UnibrtunateIy, simple computations in the Wigner-Ville case show that this "pseudo-variance" has the v a h e
1
d ~log]:r I t ,
and this need not be nonnegative at NI. The same happens Nr the "pseudovariance" associated with the Itihm~zek distributiml..An analogous eMculus leads to
4~
P:(t)l dr2 (t).
We can observe that in both cases a locally positive curvature of the envelope of the signad may lead, again locally, to a negative "pseudo-vmiance."
Time-bb'clu<'ucv/Tim¢-S(ale
292
Aw,il).~'is
Ibis, of course, rules out any legitimacy i~/terms of a disperskm round a mean vMue. One cai~ show more ge~mr~dly ihat the "pseudo-variance" of an arbitrary distributio~J, i~ Cohe~'s class (with totted; margimd distributionis and local first-order moments) ca~ be writ.rel~ 8 . s 1 d2 log i:r] (t)
j/'"
I
D:~l,:' (3.2o4)
If we cow,sider, in particular, a paramd:erization d "produeg"-Vpe as in Eq. (3.145), we obtain
(3.205)
~.~
+ (1
(t~ri-i)1 d2t~t:l dr2 ( 0
7r2dt2(O) .
Such a quantity coBdition c,:.~
{'a]i
1
"
•
be ~ c e r t a i n e d to be m)n~egative if it; meets die d2.~
..........~ (0) = ~
cl.r~
.
,
(3.2{}6) ,
Under this constrNt~t ~.l~ecorresponding "pse~do-varia.]:~ce" attains the -value
(\'--;V dh:% :t i ..... (t))
., G;(~, it)=
(3.2{17)
,
floweret, the t)regohig condition is llOt sat:isfied by most~ eIa~ssic~d para~mterizations (i~ f~,:'t, it corrc,spo~l(N to a p o s i t i v e curvature at the origin of the ambiguity phme, whkh is in(xmsiste~t wi~h the usual idea. of a m a y iron u<(q~g~,tat the origins). Nevertheless, o~m cam. propose examples of ad hoe constructions such as the "differeuce of {wo (-;aussians" ...." ,
,-
~
"
'~
'"#
(3,2{)8)
(which actuatly correspo~tds to the differencP of two Choi-WiIliams distributio~s}, subject to the restri{ tion 1
1+
'( 1+ '1
Thinkilig more gtobaly~ w~ can use the quailtity
a](f ) =
/(Jt"z;-
( . { t , t/; f ) dt d~
t¢?
(3.209)
Chapt. r 3 lss'ucs~()f Interpretation
293
as a mee~sm'e lot the joint dispersion of a representation. Again t.he Itcmposi!ive chara.cter of C~,(I, ~; f ) can lead to a uegative value i~ certain cases. Indeed, we know from Eqs. (3.44) and (3.4~d that dx I~ dt t ~ ~,}:i-.(t)
,.-(i~ ~
t
O~f
( ( \7
(3210) A lower bound for the first term in bra.ekets will be derived in Subsect,iotL 4.t.i. D)r the time being, we are sa,tisfied with the considera, tion of Gaussian signals, for which a direct (omputation yields t:he result
I f+~>t, e
dz(t) 2
:{
ttence, f;he condition 04 f (0,0) > 0~2 0r2
-~r 2
is necessary tbr the positivity of Eq. (3.210). tlbr the family of parameterizations of product tyt)e l;his condil, ion simply turns into d ~ (0} re2 ct-i~-' ' > ..........
and in case of the Chdi-Witliams distribution (Eq. (3.146)) it is equivMent to a ~ l.
3.3.2. Positivity by the Signal An immediaf;e issue, which arises in conjunction with a t~ed parameterization, is to find all signals whose distribution is nommgative everywhere. example. Answering this question in fhI1 generality is too difficult. However, one can easily find one class of signals with a positi',e distribution. Let us consider the simplified tbrm of a linear chirp as given by An
z(t) = e - ~ .......~
.
(3.211)
l'hne-Krequenq!~ /T'ime-ScMe Amd3~'is
294
Straightforward :ompmations show that its Wigner-Ville distribution has the % r m =
-,,,~ -
> t),
(3.2t2)
where we put
~'~ =
(v% t,
.¢~; ~,//~),
A =
,
~ =
(3,2:3)
Hence, the exmnples of Eq. (3.211) provide a class of ~<~b,~tionsm the posed problcn: %r the Wigner-ViIle distribution. As a generalization the i~sult remains true for all distributions in Cohen's (:lass and tile a n n e class, which are derived by a smoothing of the form of Eqs. (3.9) or (3.10) employing a positive time-frequency functkm I](t, z~). Hudson's Theorem. 'ili~ be of the form of a lihear chirp is suNcient for a signM to have a positive Wigner-Ville distribution. The n~_x~essitvof d:is condition fbtlows from a theol~m: of tludson, r,,~ The idea of its proof is to introduce the signal
z,.~.,(t.) = ~.xp ~-:(<w + ~,t ' ( [ ~
e)."I
which depends on arbitrary complex numbers a, b, and c, wk.h the only assumption that l:t.e {(~} > 0. A generalization of Eq. (3.212) shows that the Wigner-Vilte distribution of such a. signal is nonnegative everywhere (in Nct, it is the exponentiN of a quadratic form). An apNication of Moyal's formula (Eq. (2.95)) gives
I (:c
'
", }~
/tfw~.(t,z,) ~I'~.:..(t, t,) dt dt.,
This is a function of the complex variable z, If the signal x(t) has a WignerVille distribution, which is positive everywhere, the so-defined i'unetion has no zeros. Co:~equent.ly, the analytic funct,ion F ( z ) = e ~v2 ( x , a,:~;,
(3.21.4)
has no zeros as well. Fhrthermore, an applica.tion of the Cauelky-Schwarz inequali V yields }Ft:)l2 < ~
G
Hence, F(z) is a zero.free entire function of exponentiM gype at. most 2. It therefore suttees to imx)ke a theorem of Hadama.rd in order to azcertain the fact, t,ha~ F(z) is the exponential of a quadratic form of z, If we finally let z = i2m,, we ca:: deduce from Eq. (3~214) that the unknown signM x(t) itseK is the exponential of a quadratic ibrm of the variable l, and this comNetes the prooL
C]mpt(-.~r 3 L~snes (ff Interpret, at, ion
295
R a n d o m signals and positive spectra. The posiiivity is the exceptional case for deterministic signals; without being the rule, it cal~ be lbund much more frequently fbr r~mdom signals. Due to Eq. (3.]68), especially, every independent linear combination of (deterministic or random) signals with positive Wigner-Ville spectra has a positive spectrum as well. Moreover, there are plenty of examples of random signaN with a positive spectrum. 55 The positivity of the Wigner~Ville spectrum is evidently assured for the (a]reaMy tinge) class of weakly sta.tiom~iv signals, because their spectrum coincides with tbe power spectrum density. This is a nmmegative quantity, by definition. This fi~a.ture extends easily to the class of signals, which arc local~, sta.tionnry in the sense of Eq. (1.44), as well as to a number of more particular ca;ses. Let us consider, for example, the (mziformfv modula.ted) diserete-tinm signal x['n] = e[n.] (@~.] be[l~ .... 2j)
with
E {e[nle[mI} = 8,,.,~ .
Then a strNghtfl)rw~:d cMculation leads to
Itence, the conditions
are sufficient for gaining the positivity of the Wigner-Ville spectrum. An analogous analysis can be carried out for the thmily of tim(~dependent MA(1) process(as of the form
:~[,,-,} = ~[,~1 +
hi,
21 c[,,. 21.
In this case we obtain
and this leads to the sufficient condition
2 tb[n + 1]! ~ 1 +
b'~[r~,~
for the positivity: This restra,int is always s~tisfied in the st~ntionary c~se, where b[n 1i = bin,l = b. It also holds {br some special choices such as b[,~l = ~ ..... u [ . ? ,
~,. > 0 ,
A n:=flysis
296 or
In all of" these special cases, the positiv[ty is c:osely linked to the structure of the signals. All of them stay close to the stationary e a ~ iJt certain w~vys. It is important i.o note, however, ~,ha~. the qua.si-stath);rm.W properties are not needed, in generN, in order tbr tim Wigner-Ville spectrum to be positive: as an example we may refer to the Brownian motion, whose spectrum is positive (eft Eq, (2,164)).
3,3,3. Positivlty by t h e D i s t r i b u t i o n A second question in connectiou with the posi~ivity eon(x~rns the distributions and their capability of a{eaining only nommgative vatues regardless of the analyzed signal. If such solutions exist (we haa~e already encountered a f~w, such as the spectrogram or the scalogram), it is important to find o m about the trade-off emerging from their reatizati(m (i:: theory or practice). Positive distributions. We mentio:md in Subseed(m 2.3.3 that there are no obstacles to gaining a nonnegative distribution with correct marginal vahms, if we accept repre~ntations outside Cohen's class, or if we te~ the parameter function of Cohen's cla~ss depend on the given signat. The simplest solution is given by ~'m,")
>5 },.(;,
=
.X(") e -
(3m,5)
Cohen; Zapa.rovamW, and Poa?h showed that this solution can be ext~en(ted ~o a Iarger class of distributio~> having d:te ibrm 5t~ czp:.(~,.)
with
=
/
: l:~:(t, ~:-,7 .) e .t
lX(,,)i~ . . [1 .:. cr(c,(t). . . '~(.))}
I~ :=~
(a.2t6)
[' }x (~oh d(,
and
I~ the' la~t expression, It(a, ~3) is an arbitrary p<~itive flmctiom whose integra~ over the square 0 < a, f3 < 1 ~Nuals !. The consta:~t e, by which the correlation term .r(a, :~} is multiplied, mu:~t be chosen so that
1
l
..................................................<...7.C <
~nax V(~' n)]
-
-
rain [,.(o, J~)t "
C/~apter 3 j(ss~es of lhl.erpretation
297
At first sight, this soh~thm is satisfa.ctoW ht~}tax as it, enables us to overcome the cons{:raiil~>of Wigner's theorem. But it also hmorporates certain disadvantages, which raise some doubts about its usefulness. 5r I~ fa,:t, making the positivity compatible with the marginaI distributions cause,s a loss in other import, ant theoretk:al properties such as t,he a~tNnment of the iImtanta.neous freqtmnW or the group delay. More<~'er, there is no on~>toone correspondence between the given representation and the signM. In order to see (his, it suffices to observe that any positive dNtribution, ~ , definition, depends only on the squa'ed absolute va,lues :r(t){2 and }X(u){ 2 of the signM and its Fourier transform. Hence, such distributions cam~ot distinguish betwx~.en sigTm.ls that differ only by their phases. This happens, in particular, for two li,mar chirps whose slopes of the modulation have the same absolute value, but a di~ierent sign. The positive distributions of such signMs coincide, no matter if the modulation is increasing or decremsing. This is unacceptable for methods that are used in order to determine the rule of the modulation, whi(.h is unknown a priori, Positive smoothing. The close relationship bet':,,~.en smoothing a~M positiviV can be regarded as a conseque~me of our analysis of the reduction of the interference terms in Subsection 3,2,3, In fac% ~he nega~&,e values of a 1~presentafion (the Wigne>Ville distribution, for exa,mple) are due mainly to the cros>terms and their oscillatory nature. A smool~hing opera~,ion is therefore likely to reduce the importance of tim negative values. This observation applies to both outer and immr interferences. The simpIest, approagh to gain. positiviV by a transformation of a l~o~positive distribution (positive smoothing) is based on Moyal's formula (Eq. (2.95)}. ~3&. ~ re(all for the reader's (:onvenien('e, tha! .
.
.
.
[/<<,.e;j:)c>,e;j'>d,<= S .
¢ :x;
*.
[2
t
holds for every m~itaw distribution; that is, its parameter fimctiou f is mHmodub~.r (of. Eq. (2.97)). By putting
and using the eovarian.ce with respect to arbitral" time-i~equelmy shifts (which is verified by every repre,'~mtati<m in Cohen's chtss), we can ilrunediatety deri~x~
(3.2~r)
298
JSme-Freque~tC~ /7'ime-Scale AtlaJ}5sis
Hence, we obt,ail~ a noImega,ti,:e dist:ribu~ion sta.r~;ing fi~om a1W tmitary representation of the gi\'~,~n sigual :r(~) and empk~ying a certMn smoothing operation. The snmo~hing kernel is the same type of distribm;ion of an a r N t r a r y signM h. (This observation exteads~ of cottrse, to any linear combination of functions of ~;his Vpe.) Remark. A result of the same Vpe can cergainly be obtained for the affine ciass, for which the ~reguhrizagion" of a un~itary distribution yields a ~alogram. This result, which was rediscovered several times, se pl~:es the spectrogram (and the scalogram) in the midst of the arguments about p~itivity. h particular, let us look at t.he c ~ e of the Wigner-Ville distribution again. I~.ecali tha~t its s~lpport cannot be limited to an arbitrarily smMt sub~et d the time~ii'eque~lcy plane. Equation (3.217) suggests ~:ha~. the smoothing, which is applied for gailfing positivity, should occupy a region that is at least, ehe size of the minimal timed'reque~my (:eli of the Heiseliberg-Gabor uncertMnW principle. Different results become effbctive in thN sense. The first is r d a t e d ~;o the smoothing by Gaussians. tt yields t.he t~sertion that, ].
B T > 71~i ~
[[
/
Hu'Ms'~)e I(2
. . . . . . . .
x2 * , ) - 2
.~
~ 2 ,'-> ~ ? a,
at
>
0 .
(3.21.8)
Let us demonstrate ghis equi~dence in more detail. The condition is certainly su~eient. Indeed, if B T := I / 4 ~ and owing to Eq. (3,I38), we ha;ve e -[(~/~J~÷s~;~"~t = ll),(t,~)
with
h(t) = (8~rT~) ~/'~e ~ (~i~'~)~
If we now assume that B T > 1/4~r, ~hen with tim same fimction h(t) and
13~ ~= 13 v/ .1....-......(1/4gBT:; :,.d~::~}~ . . . . . . . . .
it fbllows thag
Hence, for any [3T ~ 1/4~, we can derive the rda~,ion
,
x
-
*v 2 ~ B
:-"
I he t ~ t expression is positiw ~.~ a. convolution (in ffequer~c'¢) of a positive distribution (a spectrogram) a~ut a G a ~ s i a n .
Chap..e~" 3 Issues of h)terpretation
299
Remm4c The smoothing by Gaussians yields positive distribu{ions a<s soon as the duration-bandwidth product exceeds the lower bound of the lleisenberg~Gabor uncertainty principle. However, a smoothb~g with an arbitrary window function that has the same effective time-frequency support as such a Gaussian need not flu'nish posidvit,y, We will l~eturn to this issue in Subsection 4.1,2. The condition BT k 1/4.7r is also necessary for positivity. But t.he proof of this fa.ct is more involved. In order to study the behavior of the smoothed representation in gq, (&218), we first make use of the fact that the Wigner-Ville distribution is inva.riant under time-frequency shifts, Then we can restrict our attention to ~;he singb ~erm E,r(B,T) ....
W~.(t,u)e
~o/~l ..-;'2~
dt;dt,.
(3.2t9)
Due to attother inva~rian(e of the Wigner-ViIle distributiorl with respect to changes of some, o~m can easily see that this quantity depends on the duradon-bmldwidth product BT, but m-,t on B and T independently. We thus find the equivalent rela.tion
E ~ ( B T ) = / / " Wy(a., b}e "(a~+' b~2)'/]17'
do, dO ,
1,1
by introdu(ing the auxiliary signal (with a dummy variable a) v(a) = ( T / B ) ~7~ x((TID)~/2~,)
•
An expansion of y(a) into a series of It@mfitian functions %',,(a) yields @ ;N-
z~,.(B;O .= ~ a,.,(B;r) l e , 'v,d i~ n, =; 0
and A.(BT) = ( - 1 ) "
f
+ :>2
L, <2, ) e<~+d,./4~<~>. d'r .
(3,220)
Here L.(r) is tile rtth ][mffuerre polynomial Hence, we in}r that the smoothed ff)rm in Eq. (3.219) is assured to be nonnegative~ if and. only if all A,(BT) are nonnegative. An explicit cah,,ulation of the values in Eq. (3.220) gives
A,,(,~T)
=
4~e,
'r
-~ :l. k \ 7 1
and this leads to the necessary condition B T ~ t/4~r,
'
(a.22~)
+lTme-b )+,:+q++++e'nqy/TP+rJe-55::+g~,le A :+mi3,'sis
300
A stochastic interpretation, Fhe (eventually positive) smoothing of a time-fr~tuency distribution admits a simple stochastic interpretation, For this purpose let us consider a randcma signal z(~), which is induced by d~e ]itterfl,g of a known deterministi~ signal xd(t) il~ time and frequency. Then the (random) Wigne>Vifle distribution of :r(t) can be written as
wiuere (% ~) represents a pair of random variabIes associated with a p r o ~ abiii V densko< flmction G(r,~), As a, consequence: the Wigner-Ville N)ectrum of x(t), defined in Eq. (2,158), he~s the fbrm
(3.22~)
""
which is just a smoothed version of the Wigner-Ville distribution of xd(t). The smoothing kernel is nothing but the probability density of the jitter, In the case of Gaussia,n fiuctua.tions, the previously established result (ff positiv[~y shows that a sufficient degq'e~,of disorder mqcertains the positivity of the Wigner~ViBe spectrum. Although it was introduced b~sed on a speciM example, this fa~:t is rather general in nature, A second example pointing in the same direction concerns combinations of the type % i g u a / + white noise"
On the assumption that the noise b(t) has zero mean axed variance cr2, we obtain ~V~(~, v) = W , ( t , ~,) + ¢2 .
(:3,223)
Furthermore, the inequMib
{~t;(t,.){ 5 2E, can ea,'Jly be verified for any finite energy signat s(t), find that the condition
(a.224) Consequently, we
E~< i ~r2
-
2
is sufficiem ~br the poskivity of E% (3.223). The term on the left-haled side of tile k~regoing inequality can be regarded ~_ma sigmJ-to-noi~ ratio (SNR). We thus recover the fbrementioned interpretation, which states that positiviV is gNned ~om a minimal degree of disorder.
Chapter 3 lssue~ of b~t~?rpretatiozi
301
Chapter 3 Notes
3.1
See atso Hlawa~.sch (1991; I992), 3,1.1.
2 The classical spectral estimation (corrdogram, periodogram, etc.) is cov~ ered iI~ many books. Possible refi~rences are, for example, Kunt (1984), Kay (1987), or Marple (1987}. A first int;roduction to time-frequency methods considered from this angle is given in Schm:,der aa~d Atal (1962). :-~The concept of cyctostationary signals wa~sespeeiMly explored by Gardner and collabor~tors. Their investigation is l~sed on a %ignaI" perspective. An overview can be found in Gardner (1988}. We should Mso mention the work by Hurd (1969), who explicitly us~t the frequency-fl'equency inter~ pretation given here. 3.1.2.
~ The main work devotc.:i to the flmctional calculus of operators in correct> tion with correspondence ~ules is cont,ained hi the literature on quant, mn mechanics. The first approach thal; establishes the link to the notion of joint representations waLs gb;en by Cohen (t966), There are more receat pz~pers on this subject su& t:~sthe one by Springborg (f983). 5 See, for instance, Cohen-Tamloudji, Diu, and Lalo(~(1973). t; f h i s interpretation is borrowed from Cohen (1.970). 7 A detmled interpretation of the correspondence rules from an angle of pseudo-d~ffhrential calculus is presented in the monogg'aph by Folland (1989) (wieh the needed mathematical rigor). The %Veyl symbol" owes its name to the work by Weyl (1928), who proposed the respective correspondence rule. Some examples of appticatio~Ls of' this concept, to signM theo~3~ are explain~ in the work by Kozek (1992). s The notion of the "twisted product" (or "left product:" or ';*-product") is discussed in FoHand (1989) ~md Bayen et al. (1978a,b). u This is the correspondence rule that w;?~ contained in the original article by Born and J o N a h (1925). ~0 The exau@e of the dilation operator is distressed in L. Cohen (t992), for instance. There it leads to an introduction of t.he Meltin transfom~. A mark of its appearance can aNo be tbund (as far ~ i;he Weyl symbol is concerned) in Flandrin and Escudi~ (1981a) am] t~).)ltand (1989).
302
l'h~e-Freq~l~:,~q~"/Thn~-S~ ale Am@:sis
1~ ] h e transition reh.~t[on mentioned her~. and the way of its prooL are taken &ore Flandrin and Eseudi6 ([98[a), A similar relation appeared in Altes (197]). l~ There is no unique definition of ~ wideband a,mbiguity function (or qn compreasion')~ see the discussio~l ill Shenoy (1991). The symmetrical ibrm of E% (3.53) is due to Altes (1990), The forms mentioned here do not, exhm~st the infinity of mathematically ¢onceivable delinitions, which are discussed in Shenoy (1991), Papandreom Hlawatsch, a:nd BoudreauxBarrels ( 1 9 9 3 ) o r Cohen (l~D3). Considered from an m~gle of physical significance, thm,~' should exisv a most appropria.t,e form in tile foIlowing sense: The a,mbiguity fnnction measures the correlation of a copy of an emitted sigmd and an echo, which has undergone the combined actiot> of a d e l W and a Doppler efI~ct; hence, the mosl, appropria*e model ~br the ambiguity function should correspond to a unique modeling of the echo. This approach, which is d[scl~s~ad in Mamode (t981), for inst.ance, tends to fa~or E% (3,50}. It corresponds to a faetorization of the type "propmgat[on emit.ter-target, dilation/compression, propagathm targel-en~itter " which is used for the echo. Another discussion of tliis iasue of modeling of echoes for s o n a r / r M a r is contained in Biahut, Miller, and Wilcox (1991}. l:; For a different evpproach 1.o the same problem, see L. Cohen (1992). ~a The basic properties of the Mdlin tra,ns%rm are described in Braeewetl (1978), for example. Ovar[ez (1992) oN~rs a. very compelling discussion about; ils application in signal processing. la See Marlin)vie (1986), l,~ The quoted definition of the Q-distribution is taken from Altes (1990). Let us note that the lransformation can be defined in two different ways: e l m of them is based on the signal and the other one empl~zvs its spectrum, The latt;er possibility sex,*ms to be more natural, as it does not require the introduetkm of an (arbitrary) point that plays the role of tile temporal origin. A further g*meraliza.tion of these concepts of scale-invariant, Wigner distributions to random signals can be found i~ Flandrin (t990). This work also eon~.ains examples, which explain the use of these coneeNs for the description of self-similar (slochastic) proo~sses. ~r The hyperbolic class was introduced by Papandreou, Hlawatsch, and Boudmaux-BarteN (1992; 1993),
3.2.1. s The com',ept a,nd ~erminotogy of i,he pseudo-Wigner(Ville) d[stribmion is due to Cl~msen and Mecktenbri:iuker {1980a),
Chal)ter 3 l,ssuos of Iut.erl)rcCatio~
3(}3
l~ qhis example essentially follows the analysis found in Flandrin aud Escudid (1984). 2o This minimal width in frequency is also called Sto~y bsnd, refi~rring to the discussion in Storey (1953). It equally determines the best achiew~ble resolution for the separation of two parNleI linear chirps. 21 The idea of reassignment was introduced to the spectrogram Rv Kodera, Gendrin, and de Villedary (i976; 1978). Hence, it is a rather old idea, and we can contemplate the reasons why it almost felt into oblivion in spite of its promising results. In the original publications its introduction was b~sed on both t;he Rihax:zek distribution and the method of stationary phase. This is difIhrem fl'om our presentation here, which uses the Wigne> Ville distribution. The reintroduction of the reassignment met.hod in this new form by Auger and Handrin (1995) is not limited to the spectr(Nram alone. It can be used with any distribution of "window-type" in the bilinea.r cla~sses, and this treacly extends its possible impact. 22 The concept of ~;he differential spectral analysis was introduced in Gibiat et nl. (1982), It w~us mostly applied to audio-acoustic applications for music. 2:t The discretization of the Wigner-Ville distribution h~s been addressed in very little of the existing literature. Nevertheless, current understanding exceeds our (deliberately) elementary discussion by far. The definition equation (3.87) is the sin@lest and most natural one. It goes back to Clae~en mtd Mecklenbr/:iuker (t980b). In t.he same publication the use of the analytic signal is proposed in order to avoid spectral alie~sing. A mot,~ exhaustive discussion can be ffmnd in the x~x}rksof Peyrin and Prost (1986) or O'Neill (1997). 3.2.2.
>~ The discussion of the interference terms of the \Vigner-Ville distribution and the mechaNisnl of their creation relies on tile results of Flandrin and Escudi6 (1981b), Flandrin (1.984; 1987), Hlawatsch (1984), Flandrin and ttlawatsdl (1987), and Hlawatseh and Flandrin (1998). ~s LookiNg at the intert~rences from all angle of ambiguities w ~ proposed in Flandrin (1984). 2~ The distinction of inner and outer interferences w~s introduced by ttlz~,s~f.tseh (1984). ~r See Janssen (1982). 28 It seems that, historically, the first use of the approximation by the method of stationary phase (tbr investigating the structure of the WignerVille distribution) is due to Berry (1977). There it was presented in the
304
T~mc'-,~'r~qm,l~c~\/'Tin~e-Sc~leAmffysis
context of semiclassica.| medlanics. Similar approa¢hes in a "signal" [}-amework can be found in Flandrin and Es(:udi6 (} 98I b) and Ja,nssen (1982), and later i~ Flandrin and l{b~watsch (1987), Flandrin (i987), a~Jd Hlawat.sch and Flandrin (1998). ~ Berry (1977) was the lirst; t.o propos,~~ the use of catastrophe theory for a description and classification of the possible geometric features in a Wigner~Ville ~.ransf})rm. ~k~r a general description of cataastrophe theory one should consult: tile fim(tamen~.at book by Thorn (1972), The special application that we have in mind here is a characteriza, tion of singularities stemming from oscillatory integrals. This part is very clearly expla,ined i~l the book by Post;on and Stewart (1978). 30 The connect.ion between the Wigner distribution and the parii;y operator was made evident by Grossmann (1976) and Royer (1977). 3.~ This argument is borrowed from Ova.rtez (I992). 32 A more comprehensive treatment of generalized midpoint rules of afline dist.ributions can be found in Flandrin and Gonqaiv?~s (1996). 33 This approach, which brings us back Io tile IJnterberger distribution, was proposed by Grossmmm and Escudi6 (1991). V,~;~should mention that Paul (1985) proposed another construction, which also relies on the idea, of a 'hnidpoint" in a modified geornet:ry, tlowever, it leads to a different definition. 3.2.3.
~* Boashasb (1982) (fl~rmer Bouacha(he) was one of the first to realize the usefulness of ~he analytic signal in this context.. 35 Such a form was proposed by Qian and Morris (1992). It was based on a G~d)or decomposition. A generdizat.ion by means of (Gaussian) atoms, which are indexed tV time, frequency, and scMe~ was first discussed by Mallat and Zhang (1993). at~ The general philosophy of reducing tile interf/erences by worki~lg in the ambiguity plane first appears iu Flmadrin (1984). Since then }t has served as a guideline fbr most of the considered methods. In ttiawatsch et M. (1995) one can find a comparison of diff~rent methods, which are based on this approach. 37 We refbr to Get:Ldrin ~md de Vil/edary (1979) and the more recent work by Williams and Jeong (11992), where tt~e existence of intert}rences in the very nature of a spectrogram is exposed. A similar discussion with respect to the scMogram is contNned in K M a m b e and Boudreaux-Bartels (1992).
Chapter 3 &sues of luterpretatiorl
305
:~s We explicitly introduced the smoothed pseudo-Wigner-ViIle dist, ribution for the reduction of interf~.~rences in F|andrin (1984). It ~dso appeared shortly before in Ma.rtir~ and Fla~drin (1.983), where its advantages for tim sta~t~istical estimat, ion were emphasized (we will come ba.ck to t.his in Subsection 3.2.5), and earlier in Escudi6 and Flaudrin (1980), where a purely ff)rmaI point of view of separable parameterizations was considered. Olin should Nso note that a definition of the same Vpe was introduced in para.lle} by Jacobson aa~d Wechsler (t 983), who used the name of composite Wither distribution. a, The concept of "Reduced Interference Distributions" (RID) is explicitly described in Williams and Jeong (1992). It formalizes the program tha* was sketched in Ftandrin (1984). 40 See Cho'f a i d Williams (1989). '*~ As w e stated earlier, the Born-Jordan distribution is implicidy rooi,ed in Born and Jordan (1925),/mr its explicit expression first appears in Cohen (1966). It wa.s rarely used, however, and only a ti.~wexamples of its application along with a justification of its introduction in terms of "reduced interference distribution" are eont:ained in Handrin (1984), t2 See Zhao, Atlas, and Marks (1990). ~a This generalizatioIi was proposed, discussed, and illustrated by Papandreou a.lld Boudreaux-Ba.r{ets (1992). :l~ See Hlawatsch and Flandrin (1998), tot example. *~' The ternfinology of a.qine slx~oothed pseu&>Wig~mr-Wlle distribution w~s introduced in Ftandrirl aald Rioul (1990) and Rioul and Flandrin (1992). In these refe~xmces one can also find an explanation of the contfimous tra~lsition between spectr(Nrams and s(alograms via tim Wigner-Ville distribution and its modified versions, which employ a separable smoothing. '*~ The idea of a directional smoothing, whidl automatically adapts to the special structure of l,he signM, was proposed in Flandrin (1984). But there was no efficient algorithmic solution included. This latter point, was im~stigated in Andrieux et g. (1987) or Riley (t 989), but the most successful and efficient approaches were developed by Jones and Parks (1990) and Baraniuk and Jones (1993). 47 One can find a survey of "image" nlethods used t:br the post-processing of t.imc-frequency distributions in Auger (1991).
306
Tim~'-b?eq~e~c37:Time-Se~1o A~m]ysis
3.2.4.
~:~ The [l¢~yi m~:~asm'eof it~f~rmatioL~ is d~:.fined iH Rdnyi (1961). lilts appli(:ati(m for estimating t~he dimer~si(n~ of a sig~tal il~ t~he ~ime-fl'equeney p|ane was proposed in Williams, Brown. and Hero (1991). 4~ This approach is d,,m to Duvaut and 3orand (1991), We re%r to Comon (199!.) fl)r a general presentation of the pri~miples of an a n @ s i s by stochastically independei~L comptments. 3.2.5.
~'~)This whole paragrap}~ re(apituial.es the results in FIaDdrin aIid Martin (t983b; I984; 1998), FlaD_dri1~ (1987; i989a), mM Martin and Flandfin (1983; 1985b). :~ See No~e 2, this chapter. 3.3.1.
z2 The idea of me~suring a iocal dispersio~-~ by the value of a eondMona/ variance of a joint, distribution wa.s introduced in Flandrin (t982). The subsequent analysis is also ~a,ken from this source. 53 The i s u e of the (no~}positivity of the "variance" of a joint, distributi(m w~s copiously discussed by CoheT~ and Lee (1988). Another use[ul reference is the work by Poletti (1993). 3.3.2.
~>l The Theorem of Hudsort was proved ir~ Hudson (t 974). A generalization is given in .lanssen (1984a) 5:, The quoted examples of random signals with a nonnegative Wigner-Vitle spectrum are most.ly taken from Flm~drin (1986a). 3.3.3.
~(~ The posit.ire distributio~s were first introduced in the context of quan~ t~mi mechanics by Cohen and Zaparovanny (1980). La,t.er they were considered in signal theory by Cohen, and Poseh (1985). Int,er¢~ has been revived in recent years, The main impac~ came l~om the design of efficient methods for their construction under certain constraints (Loughlin, Pitt(m, and A~l~s, 1994). 5r This question was raised by Aires (1984) and initiated a vivid discussion (see Oanssen (1987) and the response t~; Cohen).
Ch~pter 3 I,s,~ues of h~t~:'~7)retation
307
5~ As Lax as g~ining positivity by a. "regularization '~ of the Wigner-Ville distribution is concerned, we reff'r to Bopp (1956), Kuryshkin (I972; 1973), Srinivas mid Wolf (1975), O'ConnelI and Wigner (I981), Ja, mlssis et ai. (1982), ~md P. Bertrand et at. (i983), among or.hers. More specifically, de Bruijn (1967) m~' have been the first to formalize the not.ion of a positive (Gaussia.n) smool;hing.
Chapter 4 Time-Frequency as a Paradigm
I h e time-fl'equency representations do molx~ than oflbr an arsenaJt of adaptive methods for nons~al, ioIm.W signals: They manitbs~ a, new pa,radigm. This last &apt, er attempts to i}he~trate by some typical examples, how an explicitly joint descripeioi1 cal~ lea,t to a ~lew vision of several problems in sigrm,1 analysis and sigIml processing, and how it amounts to finding solm;ions tha,~ have "natural" blt.erpretations, Sectkm 4.1 deals with the first of these issues, which is related to the questions of Chapter I. It is eoncen~ed with ~he joint localization of a. signM in time and i}equency. More precisely, we introduce several definitions of nfixed descriptions {based on the represm~tations i~ Cohen's class), which reflect some of the inherent limitations. First, in Subsectior~ 4.1.t, we ex,amitm different tbrms in which ~he Heisenberg-Gabor uncertainly principle carries over ~o bi}inear distributions. ThN leads to new t.yp~s of timefrequency inequalities regarding the minimaI spread of a distribm, ion in the plane. Second, we follow the idea of a maximal e~ergy (oncen~xa.tion in a. fixed time-fl'equency region in Subsection 4.t,2. This is related to the problem of Slepian-Pollak-La~dau. The joint, perspective amounts to maximizing the integral of a time...frequeIlcy distribmion over a given bounded domNn. This leads to a m?w eigenvalue problem. W( are able to find its explicit solution for tile special case, when the Wignm:oVille distribution is considered on el}ipsoidal regions. We further discuss some genera.lizatioias and conjectures. Finalty, Sub~ction 4.1.3 deb,:ribes or.her possibIe time-fl-eque~m:, inequati~;ies, One construct:ion is based on the use of different norms to quantii~, the local character of the Wigner-Viile distribu~iom A second approach relies on arguments of the method of stationary phase, It te~uis to a bet.ter description of the geometry of ~i~e Rihaczek distribution of frequency-moduS:~ed signals. 309
31 ()
7'i.~e-t:r,"quel,qv/ l'ime-Scale A ~*aLvs*is
Some problems of signal arz~@sia are addres~d ill Section 4,2. In eazh case we gain adva~tage of the two variables of tim de~:ription for estimating the seaix'hed characteristics of the signal, l?be first example (studied in Subsection 4,2.1) dean with the spectral e:-;timation of sta{ionary random signaN and its time-fl'equeney interprelation, \;~.%show, in particula,L h~gv the "constant-Q" paving of the plane, whidl is associated with the timescNe representatior~, lea& to a.n efficient spectral amdysis of "l/f~noise" processes. More generatly, we can use the same approach for selgsimilar stochastic processes. Next, we discuss several issues in cormection with the characterization of explicitly nonstmionary quantiti(~ in Subsection 4.2.2. Among these are measures fi)r the dNtance from the stationary (::~sx~and estimators for the corresponding modulations by means of thne-frextuency tools. We also dea~ with {;t~ estimation of local or evolutionary singularities u.sing time some tools. In each ca~e the velT nature of the representational space (timefrequency or thne-sc~de plane) reveals the underlying (local) struct, ure of the signal. In the last section, Section 4.3, we focus on applications of tim~> Kequeney tools going beyond the purpose of anat3=sis, Here we investigate how eer~.ain reference patterns in the transtk)rmed plane can be defined in order to formalize statistical deciNon problems (such a:s detection, or cIa> sification). The first possibility (in Subsection 4,3A) consists of emNoying a known (deterministic) reference and defining a matched time-frequency filter, This gives rise to a ~ew interpretatim~ of the optimaliV of some di~ tributions (such as t~he Wigner-Ville or Bertrand distribution), which relies on their unitarity, Then we investigate the elassieM binar3 detection problem, hypothesizing a Gaussian random process, in Subsection 4,3.2, It turns out that the unit,arity of the representation is the key go finding optimal solutioas in the sense of a tnaximmn likelihood estimation. Finally, we dwell on im~erpretations of ~:h<se solutions in Subsection 4,3,3. They are intimately associated with dle formulation of several problems of signM processing in a ~ime-frequency context, such ~ the detection av a noisy output channel, the tolerance to the Doppler effect, or the matched filtering with an incompletely known reference. The chosen perspective enables us to collect a whole family of subopt, imal receptors into one class (which again turns out to be Cohen's class). Their performance caI:t be controlled directly by the choice of ~clmsmoot,hing function of a joim, representation,
Cha:l)t~r 4 'l"ime-Frequezicyas a
}~¢adigm
311
!i4.1. Localization As seen in Chapter I, there are flmdamental obstructio~s to a.n m'bitrary loea,lizatiou of a signal in both time and frequency. A mixed description in the time-fl'equeney plane cannot be exempt fl'om an analogous restraint. But; by its mere structure it shouid offer a new vision of these limitations, which is better suited to the posed problem. We illustrate this by consi& ering several examples in the present sectiom At this point we focus on the two major issues of mi1~imal t,ime-li'eque~w ,st:)l~ad~md maximal e,~e~gy co~,centration, limitiI~g ourselves to the representations in Cohen's class aztd some I.ime-.l?:equency va.riants of" the corresponding Heisenberg~Gabor and Slepim>Pollg~k-Landau problems.
4.1.1, Heisenberg-Gabor Revisited The primal restriction of the time~frequerLcy localization is eertainIy expressed by the Heisenberg-(]~d)or inequality (gq, (1A)). It, tells that the product of the time- mid frequency extension of a signa.l, as measured by the second-order moments of the instantaneous power (temporal energy distribution) and the spectral energy density (frequentiM energy distribution) is bounded from below by a strictly posii, ive constant, Intuitively, a similar inequality should exist tbr a, time4'requen~y energy dkstribution, provided that we have suital)le mea~s to measure a joint dispersion in the time-frequency plane. Example 1.
As a first, measure one can propose the quantity ~
,~" ( ~ + T2~e2) <.(t,v;f) d~&/, fT~ a
where T is an arbitrary nonzero duration and f(~, r) denotes tile (frequencytime) parameter function of the distribution C:~,(t,/J;f ) i~l Cohen's class, (As in Subsection 1.1.2, we always assume that the a,nalyzed signal, and ttnm i*;s distribution, have zero mean.) A straightfbrwa.rd computation based on definition equation (4,1) ]e~u]s to At ~ zX,,(f) ...... 7rT , + r
2
')
+
where we put
1 [1 a2 (o,o)+!r a2f, o)l
(4.2)
312
;I'ime-~q°eq~m~qy/Tim,.,-Scale A ~ a t 3 s i s
The general retation
-T-,T + I " Az,- =
-.,, T A -
i 2 A t A~, > 2 At: A~/
and ai, application of Eq, (I.4) imply 1
A~:(f) Z : ~ 7 + ~ : ( f ) .
(4.a)
If f({, r) is real and a,.tains a locat maximum at the origit~, we infer from Eq. (4.2) that (%:(f) >" 0. Hence, the timc-frequency extension as defined by F:% (4.t) is miTfimai for a.tl pa.rameter functions that sa.tisfly f((,O)=f(O,r)=l.
As the Wigne>Vil/e distribution (fw ({, r) = l ) obviously ",~rifies t.bese reiations, we find ~hat, d:~-(fw) = 0 and A~,:(fv,') = - ~ .
~
+ 7 ' e , 2 I4%:(&,,)dtd,x ;> 2-7 "
The same proper W remains true for the s-Wiglier and ~he Chdi-Williams distributions. On the or.her hand, we obtain for a spectrogram wigh window h ( t ) (and associated parame~:er function fs(~5, r ) : = A;(~, r)) t
This impli(~ A:,(fs) = ~
!~,;?~, T2tfi
S:,:(t,*,)dt&,
> .... 7r "
(4,:5)
Therefore, ~,heminimal time-frequm~cy spread of a spectrogram is twice ;{s big ~ the one of a Wigner-Ville distribution. V}2~can regard ~.Ms result as a consequence of the regularization equ~tio~ (3.70), (ttm to which a spectrogram CO1Tesponds to the smoo~;hing of the Wigner-Vitle distribution of the signM by the same type of distribut, ion of the window. This amounts to mi accmnulatiox:t of the individual extensions.
C;V~aptcr 4 Time-f}'oquez~(3~ avs a t ~ r a d i g : m
3[ 3
Rema~k 1. The time-frequency inequality of Eq. (4,4) was established by an explicit application of the usual tteisenberg-Gabor inequality. The lower bound is therefbre attained fbr Gaussian signals. As a corollary, the timefrequency inequality of E% (4.5) for the spectrogram turns into ~m equMity when both the signal and the short-time windcrw are Gaussians. 2 Remark 2. By Eq. (4.2) one could formally reduce the lower bound (I/2rr) in Eq. (4.4) by tnking a parameter flmction f(~,.r) that has a, local minimum at the origin of the plane. This wouId make both second derivatives in Eq. (4.2) negative. However, the physical meaning of such a parameterization becomes questionable. Example 2. I)efinition equation (4.t) presents one way to measure the time-frequency extension. There are, of course~ other possibilities. We can use, for instance, t,he qumltity
,•ft
2 u '~ C:,:(t, z~; f ) d t d * ,
(4.6)
,
Ii 2
which was already mentioned in Chapter 3 (see Eq. (3.209)). The computa,tion in Eq. (::I.2:10) showed that
dt (t) " dt ....... ~,2
4rc2 0 C " O f
When we proceed a~s in Subsection 1.2.1, we can. further prove that tile first term in brackets is bounded fl'om below. Indeed, the introduction of the internmdiary quantity ! =
x~:(t.)dt
t .....
-:h2
a~d integration by parts yield the result ], 2,2+
z = {t I:,:(t)I~l !i~ - zc~-
........=
t ~:(t)
dz~
~
(t,] ( i t .
Hence, for all signals tha,t decrease suNciently h s t (especially those with finite energy) we have Re {I} ..-
i% 2
Consequently, we cal~ write :~ = (Re {I}) :~ < tll 2 <
t~ ~(
dt .
i:r:(t)t2dt
3t 4
Tim~,-Fb~quenqr/Time-Scak~ Ana!ysis
by an a.pplication of the Cauchy-Schwarz inequality. This leads to
1 /~×'t, e
}dx
}2dt
> 1
(4.r)
It, should be noted that this |ower bound cannot, be attained by any finite energy signal. In particular, we obtain for a Gaussian g(t}
.........
1 i +'c t 2
.,?.(
~)
3
dt = ~i .
Hence, the time-frequency equality
{bllows for all distributions of a Gaussian g that us~ a paraaneter flmetion f(~, r) with
04f 0~Or2
(0,0) > - r r 2 •
~.'~ thus find a~a identiea~ result for the tim~>frequency extension of a Gau~siaa ~s in Fh1. (1.4), if the respective partial derivative of the p a r a m e t e r flmetion vanishes at the origin. This truly happens to be the ( : ~ for the Wigne>-Vil}e distribution, It is worthwhile to ment, ion, however, that in c~me of a 1~sual paranieter tunetion ( i.e, maximaI a;t die origin) the righthand side of Eq. (4,8) may be hess t h a n 1/4re, and even zero can be attNned. This causes some evident probtems concerniug the interpretation of this nlea,sllre.
On the other hand, if we consider the spectrogxani of a signal z(t) with a window h(t), the mere structm'e of the ambiguity mnctions leads to the dispersion ')
•
9
,
'?
-
cV(/s) = ~;: ( f w ) + ~ ( f w ) .
(4.9)
As a consequence of the positive character of the spectrogram we can now refine t,he lower bound in Eq, (4.7). Indeed, t:or every finite energy signal z(t) we obtain
I /+:~t2 Zdx(t) 2 at
L:,~ _ ~
t
~ 5 "
(~l,i0)
Moreover, in case we analyze a Gaussima signal with a Gaussiari window~ the minimal time-fYequenw extension in the sense of gq. (4.6) attains ~he bound 4; i
~o(L~) = v 2 v 7 =
¢'ff~,,(fw).
Chapter ,I Time-I~)'equenqv as a t~,r~:dfgm
315
4.1.2. Energy Concentration The inequalities of Heisenberg-Gabor type characterize a first family of time-frequency restrictions by installing certain measures tor the spread of a time-fl'equency function. A second type of restrictions concerns the fact that the totM energy of a signaJ cannot be totNly concentrated on finite intervals both in time and frequency, even if we allow the supports to be arbitrarily large. In this regard the time-freq~mncy approaeh consists of defining a measure 5or the portion of the totat energy of a time-frequency distribution, which is allocated to a hounded domNn. This measure should operate directly in the plane and should permit a determination of the distributions with the best possible energy concentration. P r o b l e m formulation. The classical problem (el. Subsection 1,1,3) of the energT concentration of a signal is posed in "two times one dimension." Given a fixed duration, for example, we seek a signal that concentrates as much as possible in a certain frequency band, In this way we define implicitly a time-frequency "~'cta.ngle," to which the signal allocates most of its energy. Hence, a joint time-h'equency t)erspective should certmnly offer an adequate and more generM setting for this problem. It should be adequate in the sense that, the time- and frequen W variables appear in the formulation of the problem; and being more general means that the copiousness of the time-frequency plane should be incorporated by passing to "one times two dimensions," which eliminates tt~e restrictions to rectangular domains, :~ Let us begin with the basic property of an energy' distribution, which assures that P P g';~. .... ll/C.,(g,,,;f)dtd,,.
(4.11)
I{, 2
Then an appropriate definition of the partial egmrgy is given by
E~:(D; f) =
/]"<,.(,~,.; f)dt & ,
(4.12)
D
where D rnag be any subset of the timedh"equency plane. The new problem thus consists of finding the (largest possibD) energy portion of a given type of distribution when restricted to this set D. In a more r e l ~ e d version, we can replace the strict concentration on D, as expressed kty the chara.cteristic function of this set, with a (possibly smooth) function G(t, v) whose effective support is equivalent to D. Then the problem is d~m~ged into tim evaluation of
E;,:(D;f) =
j]" O(t,~,,)C,(t,~,;f)dt&,.
(4.13)
316
Time-I'~'eqt~en( ;~;/Time-Scate Aliatysis
Due to Eq. (3.27) the so-de~ilied partia~ e*mr>2~' is nothing but the expectation vatue of the operator, which is associated with the flm{tion G(t, v) b 3' means of the (orresponde~ice rule b~o~ed on f . Computiug tim m a x i m N value of the partial energ3.' is tJiereby equi~dent to finding the signal that belongs to the largest eigew,'ah~e of this "proje( t.hm" onto D.
The general eigenvalue equation.
The projection operator relative to the se(: D, in its completely general form, has die kernel (eL Eq. (3.29)) - t (s,t) =
F
- 8 , t .......s q N L t .......s)d0
with
?(t,
r ) :=
¢(t,~,)
~. . . . .
The corresponding eigenvalue equat[on reads as
f+'~' ~./.(t, s) z(s)ds = ),~r(t).
(4.14)
Every signal that constitutes an eigenflmcdori t,o the lalTtc.st eigenvalue k,,,~ yieMs the best possible energy eoncentra,tion of the distribution C,.(t, v; f) to the domain D. This is easily inferred from {he relation
E~.(D; f) <
Q -~D1;tx
-
It is too diNeult~ of course, to solve the eigenvabm problem (Eq. (4.14)) in the most general cax~e (any dist.ribmio~ and arbi{~ratT domain), ttowever, we shaJl develop explicit solutions to the foregoing problem for some special c~ses. R e s t r i c t i o n to ellipsoidal domains. Let us consider the cla.~<~of functions G(t, I/), which are eliiptically symmetric in the sense
where H can be aay flmction subject to the condition
tH(.~)t"~ ~ & < + x
317
Chapter 4 TimeJ'~'{~$u~'nO' a~ a Paradig~n
for all b > 0. In {his ca:s< ~lle eigenvalue equadon for the Wigner-VilIe distribution can simply be written as ?+ ~\~ 2 , t -
/:(s)d.s = Xr(¢) .
(4.I5)
This renders an explicit computation of the solutions possible. In fact, by introducing the re
one can show
tha,~ ~
E,(D; f w ) ~:~ 2..,t-' . v " ' 1)" ~,
c=" RB~ (r] L , ( 2 r ) d r
,i<]7, <'-'-.:)'[z[
r I : (~
In the preceding ff~rrnula~ ~,~:,'employ the Hermitian fimctions
~>.(t)
-: (2 '~- ~'%,!)~,,,'2
H, (,/2:~
t) e -~'~'
defined in terms of the IIermit,ian polynomiaE B , ( t ) = (~ ~.)'~ e "~
(e -~'~+) ,
aFtd the Laguerre polynomials i S;,,O,')=
,%-[ e
',,~,t) (e~<')"
It follows thai: tile }Iermitian functions are eigenfmtctions of Eq- (4.15) with corresponding e[genvalues ÷:x.
An(BT) = ( - . ) l '~
0
,:~" Ri~ (r) L,,(2r)&" .
(f.16)
Moreover, these values only depend on tim (effective) dm'atiouO)audwidth product B T of the domain D. Next, we appeal to t,he g(u:~erat,ing timer, ion of t,he I, aguerre polynomiaJs
,,/'s-..(,O ~<=0
.... 1 - ~, e x P , i . / - - - -
1 =--: 't,u'
"
Thr~e-b?'equenqL/Time-o~.~caJe Amil ysis
31~
Hence, the eigem'a[ues in Eq, (4.16) sat~ist~y
As, by constructiom this sum ha~s the same order of magnitude as B T , t.he l ~ t equation conforms to the interpretation that the number of predominant eigenvalues close to t is direcdy proportional to the durationbandwidth product BT. In case of a strict confineItmnt to the ellipse, whet~ 141 ,,_~
1
,
0<s<
0
,
otherwise,
U(.~) =
(4,18)
the sum of Eq. (4.17) has the precise value
E
:
Br,
One can further show by some iilore involved ana,lysfs t h a t A,~(BT) = 1 - e .... ml/~
(-I)
ak L k ( ~ B T ) ,
(4,19)
k ~:: 0
witt~a~::2for0
A,~(BT) = 1 ,
which was to be expex:ted due to Eqs. (4.11) and (4.12). A ~-~cond and tess trivial obsem~don is A, ( B T ) £~ Ampex= A0(BT') = 1.
e ~w72 < 1 ,
(4.20)
and the maximal eigenvalue is obt, Nned for the Gaussian signN, Indeed, b,.sed on Eq. (4.19) and the identi<¢ L,~(r) dr = L,~(s) .... L . + l ( s )
(4.2!)
( ? h a . p t e r .1 T i m t ~ F r e q u e n c y
~.s a.
Paradigm
319
i/
{).8
/ /
0~6
/ /
/
/
/
/
/
/ I
0.4
/
" /,
/ 7"
/ / ,/
,,
5/
/
/
/
/
f~ "
Y/
./
///ff , e x\
~0,2
/
/
1/
'
,/
./
,/
/I
// ~.'~e /
/
//
/
0,2
/"
~4 ," '
/
.
,
,/
e
"X
~\ /
/ \,
/
'\ ,
/
.......
~:;~
-OA
0
5
10
15
20
nBT
Figure 4.1.
EigenvMues (Eq. (4.I9)).
Each of these different eigenvalues, a.s a function of the product BT, measures the energy portio~ of a Wig.er-Ville distribution of a Hermitiaa function, which is strictly contained in an ellipse. for the Laguerre polynomials, we obtain the relation
~,~,(BT) = , \ , , _ ~ ( B T ) + (-1) '~ ~-"~'~/~[L,,~ ~(~.ST)-. *.,(~:e:V)]. A recursive application of {iris equality gives
A.r,(BT) : A0(BT) + e "'~*~:'/2~ ( - 1 )
~. ILl,: ~( ~ B T )
........L , ~ ( r c B T ) ]
.
k; 1
Using Eq. (4.21) once more yields the result
5\,~(BT) = Ao(BT)
e ~uz¢t'/2
.......1)~:La(t a0
k k =: 1
dt ,
n 2 1 .
3
If we finally realize tim{, the sum in brackets is nonnegative 5 tbr every n, we obtain the first inequality in Eq. (4.20)
.xT,.(m') < ),0(B~').
320
T i~,~(-t~?'eq~zenqv/'Thne-Sc, de A~,al3,sis
Equality is at, rained fi:~rthe function £;~0(a}. that is. fbr the Gaussial,. Hence, the pari.ial energy of a Wigne>Vil/e distributio~ cont, ained in an el.lipsoidat domain is maximal fbr a Gaussian signal. The second inequality in Eq. (4.20) foNows from Eq, (4.19) ~md the definition of the Laguerre p(.@nomials, which imply f.
A,ffBT) :: l -
~t~,:Fi'2
and, therefore, 1 .
Ao(BT)
As ai~other ;,2 1/2. IntuitG'e!v,,. we could be tempted to expect that t;he result will always be posiliw~, ttowew~r, this conclusion is false, because for evew finite value of B T , no maZter how large, {,here exists an index ~? so thai, A , ( B T ) < 0. This contradicts our naive belief and ascertains that the smoot.hing of a Wigne>Ville distribution by a flmction, whose (eff~mtive) support in the plane exceeds the Heisenberg-Gabor limit, n ~ d not furnish, in general, a positive distribution. ~; On the other hand, t.he anticipated conclusion about the positivity, which proved tMse in the situation of a strict confinement,, becomes true, if the smoothing kernel is a Gaussian. This is related to the results of Subsectkm 3.3.2. In f~:~, when we replace the characteristic flmction in Eq, (4.18) with a Gaussian H(s)
=
e
,
we find
.OIlSeqUelltI} A~, ( BII') n=O
=
9~ r c B l-"
.
Chapter 4 {Iime-b?~eq~encv as a Pazadigm
i
32t
I
~
0,4 =:
f
~i ~'t~
~
~'~
0,2
-0,2
- 0 . 4 '......... ~ .................. 0 2
4
6
8
I0
4xBT
Figure 4.2.
Eigenva]ues (Eq. (4.22)).
Each eigenvMue is drawn as a fmmtion of BT. It mea~ures the enexgy portion of the Wigner-Ville distribution of a Hermitian flmction., which is co~,tah~ed ii~ aI~ ellipsoidM domain associated with a Gaussian smoothing. The result of Eq. (4.~2) further leads to
B T ~ 4~
~
A,~(BT) ~ 0
tbr ever?"
A graphical illustr,~tion of this fact is given in Fig. 4.2, wllere we draw the graphs of t;he eigenvalues of Eq. (4.22) as functions of the product B T . As in the ctu~e of a strict restriction to an ellipsoidal doma.in, the maximal energy concei~t.ration with respect to a Gaussian weight fimction is obtained tbr a Gaussian signM. In this ca.t~" it attains tile upper bound
A,jI3T) ~ A,,.~ = ),o(Br) =
4rrBT
4rrBT + 1
(4.23)
Remark t. One can easily observe that in Fig. 4.2 tile eigenvalues with odd indices are neg~tive for all values of ti~e duration~basldwidth product below the threshold B T = 1/4rr. We can realize by a simple computat.ion
322
l'im~:,-l+c~queu
that the to~al extrema of Eq. (~I.22} in the int,:rval 0 "~ B T < 1/'l~r (ibr evell and odd r~) a~re
Helmc~ the negative eigenvMue wit;h the }argest absolute vMue is )~. [~s mhfimM vMue of - 1 / S corresponds to a duration-bandwidth product B T = 1/12w, which is one-third of the lower bomad in the Heiserfl)erg-Gabor m~ce~"~,ainty relatiom R e m a r k 2. bk)r the same case of a Gaussian weight flmction, t;he Gaussian sigI~at can be shown to be an eigenfunction of' the equa.tio~
;~s well. This is the eigem,atue equation mssociated with an s & \ l g m r di> tribution. Its Mrgest eigem, alue has the lorm .
r~
e
4,v B T ao(B7'; s) = 4~rBT[1 + (s/2~BT)~'] ~/~ + 1 '
from which we inf)r
x0(B:I: s ) ~ 5 0 ( ~ ; 0). Hence, eomong the whole (:lass of s-WigIter disgribut.io~s the Wig'lter-Ville case (which corr~=~pon~ to the value s = 0) yields a~ maximM energy conc e ~ r a t i o n relative to a given e~fipsoida! domMm This resuk presra:~ts a n.ew )ust, ification of gtle mi~fimal dispersion, which is characteristic of the Wigl~er-Vitte distributiom : Remark 3. The pr¢~ent.ed approach, wNeh was restriemd to etlips~% eonld possibly be geI~eralized to other domains, tlowe~x~r, it turns out tha~ finding exact, solutions to the eigenvMue problem bee,m~es rat.her difficult. Aa a p w o x i m a t e sMution can be mot.ivated l U the observ-ation, that maximizing Eq. (4.13) is equiv~dem to minimizing the distance d (G,C~,) =
ff[
( ; ( L @ -'~ (:~(t,~';f)l 2dtd~/ •
This tra.nsfers the {onsktered problem t.o the synthesis of a signal which ha.s a Wigner-Ville distribution with a best least squares fit to a given time-frequency fm~ction, s
Chapter 4 Tin~e-I';requenqlv a.s a I~zad.i)gm
323
I n t e r p r e t a t i o n s and conjecture. The idea of met~uring tile energy concentration by integra.ting a time4Yequency distributiot~ over a bounded subsel~ of the plane is confronted with the ki~own diNculty, that the distribution ma\y attain negative values. In Net, if we would regard a time-frequel~cy distribut, ion (such as Wigner°Vitte) as an arbitrary function subject to the mere condition of Eq. (4.11), there is no reason why it should not have large values outside the domain of integration. We can easily devise flmetions that have arbitrarily large values wi~h alternating signs and whose contribution to Eq. (4.11) is negligible. Moreover, there exists no obstruct:ion, in general, against a local integration providiI N gret~tter vMues than the total energy of th.e signM, because the excess could possibly be eount.erbManeed by negative ~,lues outside the domain of integration. However, it seems that all this does not happen lbr the Wigner-Ville distribution, on convex domNns at least. We want to justify this by a purely geometric argument, which is based on the structure of the inter~)~r~ ences in a Wigner-Ville distribution as it was discussed in Subsection 3.2.2. Suppose, in N,et, the integration of a Wigner-Ville distribution over a convex regi(m yields a result that is arbitrarily close to the total energy of the signM. Assume fl~rther that the distribut.ion has non-negligible w~lues outside this domain. Due to the formula of inner intert~rences (Eq. (3.103)), the ~alue at each point results fl~om the interaction of contributions, which are symmetricMly distributed about this point. Consequently, ~s the domain of the integration is supposed to be convex, the non-negligible values outside the domain can onIy result from an interaction of pairs of two contributions, at least., one of which is Mso located outside the domNn. On the other hand, the same pri~ciple of interferences together with the almost perfect: conformity of total and partial ener~2y imply that N1 signifiemlt contributions of the signal lie inside the convex dommn. This contrMicts the previous hypothesis. Without having tile quality of a mathematicM proof, the forementioned re~zsoning supports the conjecture that
W~,(t ~/) dt dv =:<"E,,
E~,(D; ,]\,v) =
(4,24)
D
for every convex domain D.
4.1.3. Other Time-Frequency Inequalities Apart fl'om the generalization of the Heisenberg-Gabor and Landau~SlepianPollak approaches, the localization of the energy of a signal to a bounded domain in the tim~>frequency plane carl be measured by other means, each having its own interpretation as a time-frequency inequality.
324
Analysis
L~-norms. sith'e t.o tile L~,-norm of (Eq. (2.95))
A m e a s u r e for the eoncentr~:io~t in t;he plane, which is sen-, existe~tce of large values in a s~aa]] s u p p o r t , is given h 3 the the Wigner-Ville d i s t r i b u d o m RecNI thai Moyaf's formula implies ,
I 2
...... ,<:,
(4.2,5)
R~
It therefore suggests itself to consider tt~e more genera! q u a n t i t y
(4.2(~) 1{
First, we can show by a simple a,rgument that this p - n o r m is finite, tndee& let us write I1,~ ,.it, t/)i II'I',.(t, ~){ "~ .~dtd~,
[IUr(t, ~/)I" dtd,/ ..... I{~
t{2
a,nd ug~ the H6tder inequality, which gives P We next e m p l o y tile genera[ r e h t i o n s Iiln I tI*r/; " ' :: max writ, ->-
~}l <...... 2~%=
&I~d
I i.i; }
A recursive application of the (:mimate by H6Mer s inequality b a d s to
and this gives
(,:l,sr) T h e foregoing estimate is not sharp. Lieb proved t h a t a more a c c u r a t e estlma~:e "' ' is furnished try ~ n > '2,
(~,2s) 1 !~
'~ < '2,
Chal:)/+>r 4 Tim+'-t+b:,qu+>sTcy++s :++ParadiNn:+
+~2,+~
The bounds on the right-hand side are act;uaily a.t.taim?d lor Gaussian signals (aad only t7)r then@ These inequatit;ies supply a new Jbrm of a t,iine-frequeney uncert, aint,y, which renders an. arbitrary concentration of tile Wigner-Vit/e distribution of a finite energy signal impossible. Localization and stationary phase, AI last, let us consider another form of the localization, which was also mentioned in Subse(¢ion 3.2.2. It, is re[at;ed to frequen(y- or phase-modulated signals. V~,~ have already seen that; certain pairs of the t.ype "distxibueion/phase rule" permit a perS+x:t, localization; that is, the distribugion behames like a Dirac mass localized to a curve i.1~ the tim~fl:equency plane, provided that this curve is infinitely exl;ended. For dealing with more realistic modds of signals it looks ral, her na~m'al to measure this property of localization 1)y means of a dispersion of tlle distribution about the cons~-" 'tiered ~ : time-fl'equencv, curvc~. This must be compazed with the total+ spread of the signal in the time-frequency plane as measured by Eq. (4+6), for example, tlen(e, we arrive a.t the deti~tition ,.,. (f)
= ~S {.7.'_~L,:,,), cr.,(f)
0~, 2 9 )
~tS a itew i;llea.~t!re for l:he loc+.flizatjoIb wheTt(,
<.~(,fl,<,:) =
ii;i V{,,
i;
........
,a(l)i~c,(t.,~
f)m.d,..
(4.:~0)
According to this definition a p.erfbct, localiza.tion (i.e., the ideal energy conc(:,ntrat:ion on the curve of the instantaneous frequency) corresponds t,o ~( a. vanishing measure of I);q. (4.2J). (We should note that dual definit,ions with respect to the group delay are possible as well.) As we infbr fi'om previous discussions, the Wigner-Ville distribution is perfectly localized t,o a linear instantalmous frequency, provided that this line is extended infinit:dy. Let us nexe consider the model signal (of.
Eq,
(3.211))
:r(t)
~ ~ ( ........iJl~:;
which is (ffoser t+o the physical rea.lib. A simple computation yields the result p,,(f,,v)
...... t~(I/;l')
~ -, 2] °~i:~
,~7'=
t t...- ~,:~)
This implies lira
::x
I3"1'--,+
p..(fw) .... '
0.
Hence: t.he localization improves, when the vMue of the duration-ba,r~.dwidth product: gets larger.
lime-Vr<'qu<,nqy/TJme-Sc<-deAnai~~i~
326
Remark I. q h e same situatkm can be expres,~ed difl)rent[y, when we appeal to the explk:it form of Eq. (3.2t2) that is ~ssociated wid~ the pre~:eding model. Because the contourthies of Eq. (3~2t2) define ellipses, we can use as a measure for l~he conce~tra:tion in the plane the eccentriei V e(BT) of t h e ~ ellipses. The latter ix given tV
•
l+2o4-vl
a = ( B T ) 2-,1:--.~
{...4~ J '
,
Consequendy, w e have tim
Renmrk 2. The concentration on a curve in the dme-frequen(.~," plane can be regarded a.s t,he analogue of the "classical" Iimit c)f quantum mechanics (existence of localized trajectories in p h a ~ space). The reciprocal of the duration-bandwidth prodnct ibrmally plays the role of the Planck ~onst~mt h~ this context. The previous r ~ u t t has a rather general significance, although it was based on a special e~.~se: The localizatkm of a time-fi'equency representation of a modulated signal tends to be enhanced, when we elflarge ~he durationbandwidth product. Here the notion of the duration bandwidth product must be unders~.ood in a wider sense. The importaut point to be required is t.ha~ the approximatio~ based on the method of stationary phase be admissible. Then we can develop other relation;~ to express the same property of concentration in the plane in a new fi~rm. Let us begin with tim approximation of the spectrum of a sigual by the method of st,a£ionary pha~se, ~ it, w~s laid out in Subsection 1.2.2. Given an (analytic) signal in its generic form
we suppose that the law of the insta,ntaneous frequency r%(t) is strictly monotone, tf the oseitb_tions of the phase are fast as compared to the variatiom~ of the modulus, the spectrum of ~r(f} admits the approximation
dlJ,,
J7 ,,~4th
(L)i a:~.{t~) exp{'~ ~I~C(~C)} !/2
....
327
Chapter 4 77me-b'requeucy as a Paradigm
Nex~ we introduce the Le~gendre trm~s'Dorm to of a (monotonic) flm<:tion ~(t) by d';'( t £{,p}(1/) = vt~ - ~(~;,) , ~/= dt" ~'" ' Then we immediately obtain :I. , ; ( v ) = 2N
.~{
1 } ~.g~,. (,i) +
1. d,/v ~ sgn ~ ( t s )
,
Vr(t~) = v .
(4.31)
This implies, under the given hypotheses, that. the phases of the signal alld its spectrum are linked to each other by a Legetldre transform. The involutive character of this operation permits a new justification of the fact, that the illstantaneous frequency and the group deIay are inverse Nnctions. Another con~quence is obtained by making use of }}_)u/it's inequality in connection with the Legendre transform, which, gives f ._ tlJ ~(t)
£{~}0/)
"t<
tv
if ~(t) is convex, if ~(t) is concave,
with equalit,y for t = ts. As far as Lhe Ilihaezek distribution (cf. Table 2.1) is eonterned, we infer from (431) that
arg{R~.(t, z+)}
rc
if lJa:(t) is in(reasing,
.....4rr < # .~
if v:,:(t) is decreasing.
(4.32)
EquMity holds for all points on the curve of the instanta.neous frequency v = v~(t). Th.is furnishes a,nother form of a time-fl'equency inequality.
§4.2. Signal Analysis Speaking of "good" properties of a representation draws its motivation from the goal of gaining a ~ood anMysis tool, no matt.er if this is expressed explicitly or implicitly. Such a tool shouM enable us to l~:~vealthe m~dertyi~g stru(ture of a signal and to estimate its characteristics. Viewed from this perspective and regarding most signaIs in our environment (in bot.h their own nature and our modes of perception), the time-frequency approach offers a nat.ural nomenclature for tlmir description and tiros facilitates their anNysis.
4.2.1. Time-]~¥equency, Time-Scale, and Spectral Analysis Although *~he time-frcque~cy methods (in a. wider sense) m'e designed to deal with nonstationary signals, they also t)rovide a natm.al setting for the spectral estimation of stai, ionary signals and signMs with stationary increments.
328
'1~ime @}'equenqv/'f ~mc-Scah~ Ana]y,~is
Paving and marginal distributions. The most classical procedures of nonparametric spe~ tral esI;itilation are ba~ed on the? smoothing or averaging of a periodogram, ~! This term is used fbr d~e santo transform as a @ect:rog~'a.m, since its operation can be understood as a %quared absolute value of tim Fourier trans/brm of (itMependent) consecutive segments (ff the signal." Therefore, tile spectral estimation by an averaged periodogram acquires a natural time-frequency interpretatiom The way of partithming the signal prior to taking its Fburier transfbrm implicitly defilms a pav:ing of the time-frequency plmm, The l i m ~ r e ~ h l t i o n of t,he corresponding ceils is fixed by the short-time window, and the frequency~resohttion is dete:rmined by tim specteum of this window. From a. point of view of timo-frequency analysis, taking the timc~ averagv of a periodogra,m is the same as finding the marginal distribution of the spectrogram. We can dearly realize the bia:s-va,rianee trade-o~i which is inherent in this type of speei;ra.1 estimation. Indeed~ one can only reduce the variance of the c~tima~ion by increasing the maN)er of (independent) mean ~ I u ~ , which in turn results in reducing the ~ength of tile short-tinm win&hr. At the same time, the bias is aggravated by (3~70),
(-I)
lira
t / f/ ;1:; ,/~//'1:,.(~)H)~(s ....... t, ~ ......t,) ds d ( dt (7} t~2
This shows timt tile resulting estimator yields a smoothed version of the searched power spectrum FrO~) as its expectation v~lue. The paving a~soeia.ted with the considered procedure has a unifbrm and rccta.ngular geometry, all cells having the same size axed shape. This geometric structure renders the spectraI estimation by an averaged perio&Nram suited for spectral densities with a constant rate of tNange in the Dequency direction. Due to the interpretation of a spectral estimation in terms of marginM distributiorcs of a time...frequeney representation, we can rela_x the restraint of identieN time-frequency (:ells. This is useflfl for ml adaptation to more
C?laptei 4 Tinle-I'}'equenqy as a Fa,ra*hgm -
.
)
-
,
•
329
general va,riations of the pov,<;r spectrum density; .As an example, let us consider the case of a paving with constant quality tb.ctor, as it arises in connect.ion with a time-scale analysis. In this case the frequencyq'esohlt.ion diminishes, when we turn t.o high frequen(ies, whiD I,he time resolution improves a,t the same rate. Hence, the trade-off between b i ~ and variance is unevenly distributed over the frequencies, hideed, at high frequencies the constant-O paving provides a large number of segments, which (generally) leads to a smaller variance; however, this is only achievable by the simultaneous loss in the frequency resolution. Conversely, the frequency-resolution is good a~t low frequencies, while the ax~ailability of few independent segnients results in a bigger variance. Owing to the structure of this paving, the spectrM estimation based on a scalogram fits to those cases where the power spectrum is g~;erned [V a law of the type of a power of the £-equency, such as the "t/f-~oise" signals. T h e example of ' q / f - n o i s e . " i2 The notion of " l / f - n o i s e " processes is commonly used in a very geueral way. It encompasses all processes, which h~tve a density spectrum of the forni cr 2
: i;;[;:;
(4.aa)
in a large frequency range, especially at; low frequencies, i:~ The problem of spectrM estimation of such processes reduces to the estiniation of the exponent ce, This is often realized by a linear regression in a "tog-log" some, ~ts w e know log P , ( v ) = log cr'e - ~: log Iv! . (4.34) We want to estimate E% (4.33) by a marginal distribution of a repre~ sentation in Cohen's class, that. is, F~(,)= "
lira v
(,%:(t,v:f)dt,
--1
o+,,x; T
(4,35)
'
(T) Provided that the process is stationary (which is needed in order that k:q. (4,aa) is well defined), we obtain
7d;
wtieIz, we use the notation V>from Subsection 3.1,1 for the parameterization. Hence, we obserw~ that the power spectrum density is affected by a bias factor that depends on the analyzed h'equency. This bias carries (~er to the
330
Fimr-b)~,quc, r,qv/~hIVme-b):ate AnalvsLs
estimation of the exponent ~, if we il~sert the estimator of E% (4.35) imo the Iinear regression model of Eq. (4.34), This consideration applies especially to the spectrogram (and its estimation by an averaged periodogram), beta,use in case of ~ shorb, time window k(t} we have '~/,,(0,~) = IH(~£)t~Le~ us next e m p b y a repre~ntation of the a n n e c}a~s (used in its time-f?equeney lbrm, Eq. (2.64)) ff~ order to derive the estimator
P~:(,.,)=
~im T-,,~+:>.
:1
~',~(t,,,:f)dt
~
'
'
(4.a< "
Then a computation, which is similar to the preceding one, provides s
=
el-
This shows that the bias (factor) becomes independent of tim frequency. Inserting this new estimator ff~to the regression model of Eq. (4,34) leads to an unbiased est,ima,tor for the exponent a,, t h e time-fr(~:tuency interpretation of the sp~:tra,/estimation thus justifies the conchlsion tha~ the procedures based (m cons~:ant-() repr(~en(;atffons (especia|ly the scMograms) are well suited for the study of "t/f-noise" processes. There is another obvious situation, in which the time-scale analyses are naturally motivated and reveai very clearly the structure of the signai. This is t~he case fbr processes with an underlying .scaling operation as expressed, for example, by a sell%imilarity, Analysis of self-similar processes.
By defit~ition, a stochastic process is sd£simitar ~ (with exponent H), if it verifies the relation
{:;-(l~,)} = {t."s(t)} for aI1 k > 0, where the equaiffy is to be understood in distribution. The simplest example of a, ~:qf-similar stoehastic process iv a l~a.ct:ionM Brownian motion 15 Bt¢ (t), which represent.s a (zero mean) Gaussian process with an autoeovariance of the form r ~ - , ( t , s ) = ~Icr" ~ [tti e l l + sl 2 H - ~ t - s f e*I
,
O
1.
(4.37)
The structure of the covariance renders its nonstationa~y character visible. It is self-similar with exponent H due to the identity r<,, (kt, ~s) = ~Y~ rf3. (< s) = r(A:-~.)(t, s) .
Chapter t 5/~me-P}'equ(uqV a.s a Paradigm
331
It actuaily represents a generMization of the ordinary Brownian motion, which corresponds to the special case H = I./2. Note that in this case Eq. (4.37) coincides with the known result rs,,~(t,s)
'~ "
-- cr~I
t.,s > 0 .
While being a. nonstationary process, the fractional Brownian motion has the property that its increments are stationary: Indeed, if we put gsB,~(t, r ) = B . ( t
+ r ) .......t e . ( t )
,
we find by ~m application of Eq. (4.37) E {aBH(t,r)aBH(t,O)}
IlrI ' 2H + IO[2H - - i t = 2!:~r ~ ~'
01 ~t] .
This expression is independent of the considered instant t. Furthermore, if we take ,~vmmetric increments (i.e., so t h a t r = - 0 ) , the va.lue of the normalized autocorrelation is
E {hBH(t,r)~SBH(t,...r)} .= 2.,n_ l _ t . This yields a co~lstant value for every time-horizon of the increment (no m a t t e r how large). In case of the ordinary Brownian mot.ion ( H = 1/2) tile correlation between disjoint increments is zero. The other ca~es reflect a long~range correlation of the process, which can have either positive (persistence) or negative (antipersistence) sign depending on H being greater or less than 1/2, respectively. In the first e~L~e, the trajectories of the process become smoother when H gets closer to 1. Com'crseIy, they become more and more irregular when H approa,ches 0. This behavior manifests itself in a ti'acta.1 dimenskm, ~; which can be shown to be equal to D = H - 2. This dimension traverses the interval [t,2] when H passes from 1 to 0. The existence of long-range correlations, which decay slowly (typically by a rational power of the time-distance rather than exponentiMly ~s is the ca.se for the A R M A processes), is also responsible for the eve~tual spectral divergence at the zero frequency (by a power of ~,). tlence, it; fits into the generM picture of a '~l/f- noise" process. The anMysis of self-similar processes, including the fractional BrowMan motion as their prototype, runs into several di[Iiculties. These can be reduced by assuming an adaptive viewpoint and introducing the timefrequency or time-soMe context explicitly. For example, it is diNcult a pI~ori to attribute a mea.ning to the notion of the power spectrum of a fractional Brownian motion, because we are dealing with a n o n s t a t i o n a w
Time-lq'eque~qV/'Time-,%:ale At~t)cWiS
332
process. New.¢rtheless, it oft.en serves a:~ a model fl:,r ~he ' t/f-~mise" from a practical perspective. This disjunctio~ can b~:, taken care of aaturadty by using a ti:me-freq~ency cont.ext and following the ideas of spectral estima, tion already developed here. tn tS.~ct, we ca~ make use of t.he stationary increments of the fractional Brow~fim~ motion ~ogether with the gen.erN ~dmissibility condition equation (2.26) of a wavelet (~/,(t) h~v,,ing zero mean). Ther~ the exp~ssion for the covafiance between two points of the time, soMe plane (a > 0) can be simplified to
(4.:~8) This r~ul.t, is both simple and importam. It indicates tha~; a wavelet transforln r~,k(~ a fractionM Brownia~ motion st:atioua~y, t7 Hence, one ca.n define a scMe
,q
S
"F
X;
This function reflects the self-similarity of the process, a~s we further obtain
Moreover, td~e existence of a staUor~aO' a.utocorrelat.io~ gives rise to a scale-dependent power spectrum, which is defii~ed t:9~the f;burier ~;ransfbrm I~,'~t~ ( . , a} =
")t'.H i v, t.1) e ....... d r o~2
By the obsep:ation that V ~ * (a~) is tile freque~lcy response of the "wa~xdet filter" at scaling level a, this relation describes ~he spectral behavior of the fr~mtional Brownian motion as a "I/f-noise" pro(~ess, in which ihe frequency variable is raised to the usually assigned power 2H + 1, As aI~other simpIe coilsequer/ce of Eq. (4,38) we obtaht ~
2-
(12H+1
Chapter 4 Timc-Freque~c;~ ~aos a I:5m~.digm
333
where
This leads to a possible way of est:imating H via
log E { I7>,~ (t, ~)t e } = (2H + l) log,,, + log ~,%(u).
(4.4~)
The use of the scalogram thus admits a natural and particularly wellsuited description of the structure of the fraztional Br(nvnian motion. Tim stationary increments find their counterpart in the stationary behavior of the reptvsentation o n each s(aling level; loosely" speaMng, the "wavelet, filter" acts like a differentiator. The setgsimilarity materializes in the homogeneity of the expectation value of the scalogram relative to the scale para,meter, as we infer from Eq. (4.40)
The flexibility of the w a ~ l e t analysis amounts to a further advantage regarding the estinmtion of the exponent (in the spectral or the self-similm" form) of the process, t/'or this purpose we can use Eq. (4.41) t(gether with an empirical variance. The wavelet analysis not only leads to the stationary behavior on each scale of the initially nonstat, ionary process, but it also reduces its long-range dependencies. We can give a quMitative explanation of this fact by rewriting Eq. (4.39) as
~l~.(r,a) = C ~r~a ~ ~
.-~.
t~(t')liz, i 2 . + l e~,.(w,~,) d,e.
Hence, we observe that the behavior of the autocorrelation at infinity :is determined t.¢ the behavior of the ratio {~(~)t~/IvI 2H+l at zero. Con°. sequently, the frequency response of the wavelet; at. the zero fi'equency can counIerbatance the divergence of the ' q / f - s p e c t r m n " of the process. The determining f~'tor in. control of this behavior is the ~mmber of vanishing momer, ts (or degr~'~ of "cancellation") of the analyzing wavelet. More precisely, for a wa;~qet with a.~ most R vanishing moments the ratio ]~P(~)I2/]~} 2x¢ ~' behaves like near the origin. Hence, the condition f~ ;~ H + :1/2 must be satisfied in order that we can expect a reduction of tim hmg.-range dependencies in tim transl2)rm of the signal. This requires the waecelet to have at least two vanishing moments, a priori, given that we limit ourselves to the cases 0 < H < 1. Provided that R is large enough, indeed, the asymptotic dece4~ of the autocovariance is !rl 2(H~t0. Hence, the correlation between adjacent segments of the scalogram becomes less important for wavelets with a Iarge number of vanishing moments, is
334 4.2.2. N o n s t a t i o n a r y
Tim('-Frcqt~e~,
We have alre~My explained, why purely temporal or fl-equentia} methods are often unsatisfl~c{ory for the analysis of nonst;ationary signals. In such situat.ions the time-frequency methods deploy their fuli capability, Moreover, they can serve to d~Nne new quar~tities t.ha~ m:e well-suited for nonsgati,:mary siruat~ions and allow simple physical interpretatkms. D i s t a n c e from the s t a t i o n a r y case. F h e notion of stationary signMs in the wider sense (or weakly stationary signals) is a well
According to t:he definitions of Subsect.km 2.4.3, a signal is (weakly) stationary if and only if i{.s Wigne>Ville spectrmn coin{ides with the power spe~,trum at each instant; that is, we have
W, (t, ,~,) = 1'., (,.). t:t)r a fixed Dequency *~ aa~d a time-holizon T, a nat.ural measure ior tim distance from the statkmary ca:~e can therefore be defined by
(4.42)
This quantity is zero if a,nd only if tile signal is weakly sta,tionary, and it incree~ses as we depart from t~his situation. The me~sm'e of Eq. (4A2) can obviously be ~sed by itself (e.g., for problenm of chssification). However, it can also serve to find an optimal time-smoothing for the estimatffon of the Wigne>Ville spectrum. 1~, Intuitively, a,u optimal smoothing should be more pronotlneed for proee,~;es that are a.hnost stationary; then one can expect a reduced variance of the estimation without; the penNizing effect of a~ enlarg,~,d bias. Hence, we shoukt be ~d~le to find an optimal smoothing by an adaptatiou to the iocN time of the stationary behavior. A pc~ssible solution can be fotmd in ~he design of an adapted piecewise stationary model for the considered signal. The underlying idea comprises three consecutive steps, namely: (i) Start fl'om an es~ffmated spectrum without, smoothing;
C'hap~er 4 Time-~iwquency as a Pe,.r~digm
335
(ii) Estimate the insta,nts at which the stationary properties change; and finally (iii) Replace the, nonsmoothed estimator with the sequence of averaged estimators on e~d~ interval between two such instant& Let us coI~ider the situation of a discrete4ime estimation with an fffitial estimator PW~.[n,*e), n = i , . . , N , of pseudo-Wigne>Ville type, Then the first problem consist.s in finding, for each selected ~equency 7J, a pa.rtition of the interval [t, N into p blocks of leng{hs N, so theft P
ES(~ = N . The smoothed estimator is then defined by t.he sequence P
1 I'V :~[i, z:'t = "--
~ ?~a:z,;g ~.
PIt~,. [~, I/) ,
i = 1, 2 . . . . , p,
(4A3)
@t
where
{
j= I
The N~ are the estimated times of t,he stationary behavior, and ;% are the estimated points in time where w~ pI:L~-;stl'om one stationary segment to tile next. By intuitiml, a model using f~rwer segments than needed ( i.e,, a value of p whMl is too small) results in a s[gtiificant bias and a reduced variance, Conversely, if the xmlue o f p is chosen too l~rge, the bbe~ becomes small but. the variance increasaes, This suppm'ts tim idea of &oosing p according to a comprom.ise of bia~-varim,ce type, which is the result of a minimization of the total mean square error (MSE). (Note t,hat ~br an a p~'iori fixed number of segmem;s the opt, imal partition, in the s.rose of a minimal MSE, can be constructed by a dynamic programming technique, because the problem resembles the approximation of a curve by a step Nnction. 2o) 15k~r each stationaw part the analyticN Wigner-Vitte spectrum coin. cides with the power spectrum F . . . (v) of the ta,ngential stationary process~ Therefore, we can compute the distance betw(~en the smoothed ~xstimator on this segment and the po,a~r spc-ctrum k~v taking the sum of squares of dleir differences. ~l~king the different variances of t.he estimators on different pieces into account (which results from a possibly unequal partition), a proper choice tbr a confidence measure of the estimator in Eq. (4.43) is P 1 *a~lP(l/) = E T~° ~--1
~< E r ~ = n i _ i-F1
[PlI;~:[i'~)-F:r,'~(l/)i 2
(4.44)
77me- b)~¢'que~(y/7hne-HcMe .4 na/ysis
336
Here I.) d t,~n o t .e~ s the xariance of the estimator in E~.~((4.43). In order t<) minimize t;he expecta.tiou ~due of Eq. (4.44) one can use the statistics (of Akaike type 2.t )
d~(~,:X]
......:v . +2p,
{4.45)
f =: 1
with
"
Ni
[ r W , [<,~) -. n w , . [ i , ~ ) ] ~ .
(4,46)
Note that Eq, (4.46) is a ]is(:~et_-tm-l,. a.nalogue of Eq, (4.42). We dins see that the distance from the s~ationary case, as defined here, occurs a.s a natural ingredient in the solution of the underlying optimization problem D e m o d u l a t i o n . The demodulation of signals is certainly among the first and most important applicati(ms of time-i~xiuency representations. Concerning the quantitative and quMitati',~, aspects of a representation, its succe~ in appli(ations depends not only on its dleoretica} capability of rendering the rules of the modular.ions acce,-.sible, but also requires it. to lay these rules %)en in the time-Kequency plane to the best possible degree.
Example 1. When. we consider a freqtmncy-modulated signa.t, we expect that a representation displays a "ridge" that mark:~ the cm've of the instanganeous frequency, t¢~ fLxample 2. The group delay carries the useflfi information in case of a dispersive propagation of different wavelengths. ThN notion describes a situation, wtmre the synchronous frequency compcm(nts of an iuitial pulse become more and more asynchronous by a dispersion during their propagation, In a t.ime-frequency terminology~ the different fi:equen(ies propagate with different (gTOUp) velocities; this maps a straight line, wlfich ks parallel to the tYequency axis initiMly, into a curve describing the g w u p delay. A time-fl'equency repr~entation should display ghe information about the law of this type of dispersion directly. This appea.rs M1 the better as the ~ t i m a t i o n of the group delay gets more accurate and the localization to its curve is m o r e pI~nOUllced, 2:~ Let us recall the two conditions
f(~, o) = 1
(4.47)
o-~ (~' o) = 0
(1.48)
Chapter 4 Thn<,-Freq,t~:nqy as a tS~radigm
337
from 'I?:~b~e 2.4 in Chapter 2, They were hnposed on the parame{.er tim(> tion of a representation in Cohen's class, in order that the instantaneous amplitude a:~(t) and frequency v~,(t) of a sigtmt (in i~;s anNyt}e ft~rm) can be derived from the marginN distributim~
(4.49) and the local center of gravity
ai~.(t) .......:~:, ,C:,.(t, v; j') du = v~(t)
(4.50)
of the representation, respectively. Obviously, the Wigner-Ville dist.ribution satisfies both conditions (Eqs. (4.47) and (4.48)). It is worthwhile to retain that; the same conditions hold tbr the mc)ving methods such a.s Neudo-Wigner-Ville (3.74), provided t}mt the employed short-time window is real and even, for instm~(x?. More generally, if only the relation (4.48) is satisfied (which can be the c~:~ for a smoothed pseud~-Wigner-Ville distribution or a speelrogram), we Obtain
#o(t; f ) =
]iq-:~
C,: (t, ~,; f )
dv
(4,5t)
/,,t, {" g -
. . . .
N
F(t
-
s,0)
,
a T' , , { , s ) d s
This sig~fffies physically that the local moments of a, time-frequency repre~ sentation give a,ccess to the modular.ion Iaws of aa(t) and V~,(t), if F(t, O) resembles a paise as compared to ~tle rate of d~ange of t h e ~ two functions, For example, whe~ we apply a rectangular smoothing of duration At to a representation of a signal gEar has a constant instantan(~ms amplitude during this short period, we can easily see that Eq, (4.52) ~urns into #l(t;f)
1 arg{;4t + ~ t / 2 ) } - a~'~{x(t At
2~
-
~t/2)}
TimeoI'i'~-quency/"!l'ime-S{ ale AnatsMs
338
He~}e< the local first-order moment pr,~ides ar~ estimate %r the ii~stantaneous f'reque~cy; which is obtained by replaci~tg the &~rivaxiv~' i~ its original definition (E% (1.21)) wi~h a finite difl;,rence, As t,tm difference is s3'mmettic about the instant t the estimate yields the true xv~lue, if the instantaneous frequency admits a linear approximation relative to tint time horizon At of the smoothing. Within the precision of the e~ssumed approximations, this manifests {he possibiliIy of finding die structm~e of d~e modulations of the signal by an inspection of the ridge:s of a time-frequency representation. L o l l singularities. [,et us recaB that the time-fi'eguenqr representations were constructed mainly for giving prominence to the £requential properties of a. signal changing in time. Accordingly, the time-scale representations offi~r a ha.tufa1 para, tigm tbr the description of properties with a scaJest.ructure that may change in time as aeli. ~\~ dealt, with such an example in connection with the self-similarity of a. stochastic process in Subsection 4.2. f. ~Ihis w o p e r t y w~;~ reflected by a corresponding rule [br the progression of tile variance of the w a ~ l e t coef|~eients over alI scMes (cf. Eq. (4.40)). We can attach a ~econd interpretation to this restdt, which te}!s that the ex'olution of the size of the wavelet coefficients (or the seatogram) a c r ~ s the scales provides a measure for the (global or local) H61der regulariD ~of the signal (or the trNeetories of the stochastic process). ~l In fact, let us consider a wavelet transform T~(t,a) of a signal x(t). Oiveu an admissible wavelet "~9(t), we can use Ihe vanishing momen.t condition and derive
T A t , a) --= ( ,
'*" ~J:'
/f+~v •
= a
~,'~
.i':(8) ,.)"
( S -- t )
7~
d8
.~'tt)l~.,*
T
%.
[.(~)-
ds.
Suppose the amdyzed signal is mfitbrmly H61der continuous with exponm~t 0 < I1 < 1~ that is~ we ~ssume that
(4,53) If the wavelet is such that I,~,(t) is absolutely imegrable, we can immediately
Chapter ,I Time-,Ereque~cT a.s a t.~radigm
339
verify
L , ( t , a ) ~ ClaI '~/~
l;~-tl H *
:it
ds
--(x~,
(4.54)
iti H IO(t)idt" {aiH'~'2
(7 ~Y.2
O(ial"'~, ,'2)
=
and the upper bound is mfiform with respect to t. Therefol,~, the regularity of the signal is reflected bY the behavior of the wavelet, transform for small scMing parameters. The ~onverse implication (:at: Mso be p r o ~ n . 25 B e c a u ~ the trajectories of a fra,::tionM BrownMn mot;ion have the umf:orm regularity H, tile rule for its variance (Eq. (4.40)) fits to the p,~ceding Eq. (4.54). As a function of scale and time, the wavelet, transform enables us to derive ~ t i m a t e s for the focM regularity of a signM as welt. N>r this let l~s aasume that at a given instant t~ there exists a bound of the type Ix(t0 t-- r) - ~'(t0)[ _< C {r I~~(~'~ ,
0< H < t .
(4.55)
Then we can wrige
:&.(t,a)=.
:a
~"~ '
[z(t,~+r)
-
ztto)],,,~
and }~y employing the i~mquality (4.55) ~
I%:(t,a) ~ Cl
c
I~-t"<'''',
'
r-(-to) =
dr
obtain the result :-(t-t0)
:>':<~ ~ :l~>(t):dr. I~,I""~'>~/2 +
a
dr
dr-:t - .o
(4.56) 'i,'~ can thus say that the small scales of t.he :va~let transform depict the regularity of the signal arr~mged by a time-docalization, whirl: refers to a "cone of influence" centered at the considered insgant G, Again the convince is Mso true: a sufficient decay of die wavelet coefficients inside this cone of influence leMs to estimates ibr the locN regutariV of the signal.
34{1
; N m e - f ) ' e q u e t w y / T i m ~ > S c s ~ l e kt:~atysis
R e m a r k 1, ~,\,): ax~umed that: ihe regularity irtdex H satisfies 0 < H < 1, There occurs no problem wittx the (x)nsidera t ion of larger HStder exponems, if we impose a c(mdi~ion of higher-order vanishing mouients on the v:aa'elet insteax] of ~,he usual admissibility condition (zero mean). In Net. iet I~s suppose that _
C~-I~t
~
where P , ( r ) is a suitable polynomiaI of degree , , 'lTbeI:~we obtain the same upper bound ~ks in Eq. (4.56), provided tlmt t a ('(t)dt = 0,
]~: ,~:::0 . . . . . r~.
This last condition signifies that the wavelet is orthogo,m.1 to all potyno:mia~s of degree less than or equaI to -t~. Remark 2, Tim H61der exponent of a singularity iliads a simple trar~slation in a wavelet tra,usibrm or a ~'alogram in terms of the behavior for smalt scaling parameters. Howew:,r, the same infbrmation can be provided by other bitinear distributions in the affine class as well: esp,x:ially the ("active") Unterberger distribu~:io~ (see T;~ble 2.3) yields comparN~le results. If we associate the situation in Eq. (4.55) with the asymptotic spectral decay
the definitio~l of the "active" U~terbe~ger distribution can easily be seen to yield the app~oximatioll U.(t,a)
.... l(~i ~;~H~','~a(t
....... t~)
~
~ .......... 0.
Hence, it: is governed by a rule depending oa the scale, which contains the ~xponent of the singub~rity in a perfecgl3 localized mamter. It completely ignores the notion of the cone of influence, which retli,rs to a neighborhood of the singularhy and was stiil present in Eq. (4.56). However, as a typical %ature of bilinear transformations of th~ signal the kmalization is affccted by the existence of possible interlbrenee terms, if two (or more) sbIguIarities coexist close to each otfmr. :-'~ E v o l u t i o n a r y singularities. Remark 2 (precedii~g 1his d~scussion) suggests that we can use the bithiear affine dist, ributions, rLot;(rely for a local (s~atie) estimatiol~ of singularities~ t~ut even ior put.suing ew~tutiom~ry sit> gularities. As a guidel}ne we consider the analogy between time-frequency and time-scale representations~ or more specially, between instantaneous
Chai)ter 4 Tim(~l~'equenqy as a th~ra,di,gm
3ti
frequency and local singularity, The upper bound (Eq, (4.54)) for the scalogram tells that a ';good" time-scale representation should feature the relation f~:,:(t., a; f ) ~ ta}2Hu)+i , a --~ 0, (4.57) at every point t, at which the signal has a H61der exponent H(t). In t~his ca~se we obtain
H(t)
... l ,
,~,:~
:
,/!]
9;:,:(t, ~t; f)
(4.58)
e - ~ " da
where k is a positive number, which is feasible in the sense t.ha£ Eq, (4.57) is valid on the usefifl domain of the integration. We thus obtMn a result, which is comparable with Eq. (450) in the following sense: a IocM first-order moment of a (,a:cighted) bilinear distribution yields a local characteristic of the signM (in this case the HSlder exponent in the place of the instmltaneous frequency), Let us now try to find a condition on a time-scale distributior~, in order that the relation of Eq. (4.57) is valid. For the sake of simplicity, we a~sume that the analyzed sig~al is real. Vt~ start from the situation i~ Eq. (4.55), which implies [x(t + ~) - :~:(t)] ~ I~l ~"(~:' (4.5~}) :-,
ibr small increments T. This gives, of course, 1[ ', ] x ( t ) z ( s ) ~ 5 x~(t) + ::(s) - I t - ,I ~"(') Hence, for small scMing parameters a we obtain B:.(t,a;
:)
ff
[x2(t-+ a ~ ) + x2(t ,~. a0)-, Ia(T-~ 0)12H(t ~ r ) ]
X ~'~
,--~ ~ c(t:) + IaI ~H(~)+~ ,
-
,7"--0
dTdO
b-42H(o f ( o , r) d r
}tere we used the notations of Subsection 3.1.1 and
. (,~.60)
342
Time-f?'eq~e~!O'/Time-Scale A~alvsis
In the general ce~se, this ~ast quax~tity contribu~es to a bi~,~b(wi{h respect to ~he expected rule fi'om Eq. (4.57)}. A suitable choice for ehe t~imescale d~stribu~ion can amfihitate t,hks con~ribttt.ion. In t~ct., we should use a parameterization tha,t satisfies the condition ::7
"~~'; ( r +0
/i
-
-,r-O
)
e ~'~e~drdO
(4.6~)
A straightibrward argmnent, proves that the scalogram meets this criterion, e~s the usual admissibility condition of Eq. (2,26) gives
Remark. We ha~? f(0, r) = T,;,(-r, 1) for a seaiogram. Hence, b3q. (&60) yields a k)cat result thai; is compatible with the global mean value of Eq. (4.40). The preceding condition in Eq. (4.61) does not. limit the solutions }ust to the scalogran~s, It is fulfilDd by all a ~ n e Wigner distributio~s a~s well, see Eq. (2.63), while the separable dist,ributions (of. ~l:i~ble 2.3) with G ( g ) H ( { / 2 ) ~ 0 provide approximate soIutions. §4.3. Decision Statistics The theory of decision statistics (detectiom estimation, cb~ssifi~ation) is a well-studied prob¿em ill signal processing, with abundant literature and well-tried solu~,ions av~dlable, % Nevertheless, ~he time-frequency approach offers a way* to reconsider this issue from a special perspective that better reflects ~he nature of the analyzed sigr~als in certafi~ cases. Intuitively, a detection or estimatio~ it~ the time-frequency plmm should result in "recovering a structure of a known form" (or signa~:,ure). This leads to the following pair of problems: (i) G i ~ n the optimaI strategy (matched fikering, m~:imum likelihood, etc.) for a temporal or frequentiM approa~ch, how can we find aa equivalent form in ~l~e time-ffequelmy plane that amounts to a simple physical interpret at ion ?
Chapt~er 4 Time-D'eq,zem-3," as a t~,radigm
343
(ii) Hove can one formalize tile empirical notion of comparing two timefrequency structures in order to deterrni~te their optimal or suboptimal character? This section addresses a :first approach to these questions within the fl'amework of energy distributions. 2~ The detection problem, which we shall consider here, is given in terms of the (cb~ssicat) bilmry ~est of hypot~heses
{ Ho : y(t) = b(t) H~ :
:,j(t) = x(t) + b(t).
~.~\% use the notations y(t) tbr the known observation, which is restricted to a finite time-interval (T), b(t) fbr a (complex and stationary) Oaussian white noise with E {t,(t)} .... 0 ,
E {b(t)b*(~)} .......~0 6(t
~),
and x(t) for the (complex) signal to be detected.
4.3.1. Matched Time-~-equeney Filtering Let us first consider the case where the signal in question x(t) is degermiuistic. A conceivable approach to constructing a matched tJme-t)'equenc3 ~ til~er proceeds by maximizi~ag a contrast flmction (or SNR) based on a suitable time-frequency representation. Within the framework of Cohen's class, the detection problem from Eq. (4.62) can be written as H0 :
C,j(t, v f ) = C*~(Lv; f )
H, :
C~j(t,z,)) = C:r+~,(,v:f)
By analogy with the cb~sicM theory of matched filtering, a. (timefrequency) filter fbr the detection is defined by A(y,f) =
G(t l / ) C ~ ( t , v ; f ) d t d v .
(4.63)
In this formula the function G(t, u) represents a time-fl'equency template that we ha~'e to determine, so that detection becomes optimal. Hence, we should try to maximize a contrast fimction relative to the null hypothesis
344
t'ime~9~'equen£~/Time-Sca]~, Amdysis
and its alternative. A naturaI choice of such a flmct;ion is the output SNR given by the expression:
SNR(C, f) :
IE{A(y,f)
H ~ } - E { a ( y , f ) IH0} I
[var {A(y,f) lHo} ]',./2
When we empkr¢ the IYe(iuemy-time parameterization of Cohen's class, an application of the Cauehy-Schwarz inequality to the correlat.ive detector of Eq. (4~63) leads to ::~'
,~ l[l[ !f (&
,
SNR.(C, f ) < SNn.(C:,., j) =
.
l.f(~,'r)I * l: ~t.,~,r)l°d~ d'r)
It follows that the SNR is maximal, ff the impulse response of the timeq~requency filter is identical to the ~ime-frequency distribution of the signal x(t) in questiom This renders the empirical notion of matched time,frequency filtering meaningful. After we have just %und the optimal solution for fixed represent, ation, we ca~ move o~ward to a secotld level of optimiza:tion, which consists in finding the best representa.tion in Cohetfs class. Another application of the Cmmhy-Schwarz ir~equatit3 to the right-hand side of t~. (4.64) gives the final relation a
max SNR(G,f)= SNR(C~:, f)5~ E':, (; 2~0
.
(4.65)
The la~%inequality turns imo an equMity for atl unb~ary dist;ribudons. RecMI that t h e ~ are chara~:terized by a unimoduIm: parameter function in the frequency-time domain (i.e., so that kf(~, r)} = 1). Under this assumption the maximal SNR k~ exactly the same ~s tbr the 1hatched filtering emnbined with an envelope detection. This furnishes a new interpretation d Moyal's formula (Eq. (2.95)). The spectrograms are ruled out by this optimality condition, and this explains why a time@equeney det(~:tion based on spectrograms requires auxiliary procedures of deconvolut.ion. 31
Chapter 4 2~me@~'equeney as a t~radigm 4.3.2. M a x i m u m
;/45
L i k e l i h o o d E s t i m a t o r s for G a u s s i a n P r o c e s s e s
Turning t,o the original formulation (Eq. (4.62)) of the problem~ we no~ supw~se that x(t) is a Gaussian random process, so that E {x(t)} = ,(~),
~.~,(t,,~) = E {~.,.:(t) ~-. - ,U:)]
[~.(~) - #(s)] * } .
(4.66)
It ix known that the detection problem under
whidl is
More precisely, we recall t.ha~ x(t) admits a decom-
doubi~7orthogond; ttm.t is, tile relations # , , j ~ } = ,~,~ &,,,,
E {Ix,, ..- , - d [z,~
/ are verified, where A, and p~.~(t) are the eigenvalu(;-s and eigenfunctions, respectively, of the autocovariance of z(t). ttence, they at*e defin~M by the int.egral equation
f ,.:,,(t,.~)~,,,(s)ds = A,~.~(t),
~:c (r),
(7} The coe~cients x,,, y,,, and #, of the decompositions d x(t), y(t), and #(Q, respectively, axe the projections o~tto this basis of eigenflmetions ( ~ xn = (x, ~ ) , e~c.). 2Ne optimal det(x~.tor (in the s~use d a maz~:imum likelihood estimator) is obtNned t~. a comparison of the decision statistics A(y) = A~,(y) + Ad(y) to a thr~hoht; in this definition we put --
~ ¢1
.
.
.
.
@(Y) = 7o ,=0 A,~ q % (14,67) J
X
T~==0
•
Chapter 4 2~me@~'equeney as a t~radigm 4.3.2. M a x i m u m
;/45
L i k e l i h o o d E s t i m a t o r s for G a u s s i a n P r o c e s s e s
Turning t,o the original formulation (Eq. (4.62)) of the problem~ we no~ supw~se that x(t) is a Gaussian random process, so that E {x(t)} = ,(~),
~.~,(t,,~) = E {~.,.:(t) ~-. - ,U:)]
[~.(~) - #(s)] * } .
(4.66)
It ix known that the detection problem under
whidl is
More precisely, we recall t.ha~ x(t) admits a decom-
doubi~7orthogond; ttm.t is, tile relations # , , j ~ } = ,~,~ &,,,,
E {Ix,, ..- , - d [z,~
/ are verified, where A, and p~.~(t) are the eigenvalu(;-s and eigenfunctions, respectively, of the autocovariance of z(t). ttence, they at*e defin~M by the int.egral equation
f ,.:,,(t,.~)~,,,(s)ds = A,~.~(t),
~:c (r),
(7} The coe~cients x,,, y,,, and #, of the decompositions d x(t), y(t), and #(Q, respectively, axe the projections o~tto this basis of eigenflmetions ( ~ xn = (x, ~ ) , e~c.). 2Ne optimal det(x~.tor (in the s~use d a maz~:imum likelihood estimator) is obtNned t~. a comparison of the decision statistics A(y) = A~,(y) + Ad(y) to a thr~hoht; in this definition we put --
~ ¢1
.
.
.
.
@(Y) = 7o ,=0 A,~ q % (14,67) J
X
T~==0
•
Chapt(,r 4 77meot%'¢'queu(Ta:s a Ibradigm
347
4.3.3. S o m e E x a m p l e s So far we has,e dealt with equivalent time-frequency fbrnmhtions of the classical strategies. Now we will briefly demonstrate by some simple examities , how the explicit time-frequency setting of decision statistics can reconcile the optimality of the solution with the physical interpretatkm of the results, a3 Rayleigh channel, The first example s t w s close to the situation described in Subsection 4.a.1. Suppose the signal under consideration has the form
* ( 0 = . J:,~(t),
(4,69)
where xd(t) is a known deterministic signal and a is a (complex) Gmlssian random variable so that
E{a}=O,
E{la[ u} = c / e .
This situation (called a Rayleigh channd) tbrmally enters the framework of the model equation (4.66) by letting
#(t) = 0~
<,(t, s) :
~2z,~(~)<,(.~)
(4.r0)
Hence, the decision statistics in Eq. (4.68) reduces to
A(v)
:
ap(y) = V0 :t +
(~o/~2E, o,)
~I~(t, ,~) ~ x.~(< -) dt (h,. (v')
(4.n) V~fe are thus looking a£ a correlative time-frequency structure, which makes immediate use. of i;tle Wigne>Ville distribution of the known refe> ence as a template {br tim observation. ~br a high SNR ( i.e., for %/rr2£~,., << 1) we obtain the asymptotic result
%
This is compatible with the purely deterministic case in Eq. (4,65). Detection of chirps and Doppler tolerance. Let; us next: consider the case of linear chirps, The time-frequency formulation of Eq. (4.71) may present an advantage over the classical strategy of matched filtering, This results mainly from the localization properties of the Wigner-Vilte distribution, a~s it generally provides the most localized pattern in the p h n e (in the sense of the discussion of Section 4.1, for instance). Let x(t) (or xd(t))
Tinic-t~?'~,qt,enc~?c/Time-ScMe A,mlysis
348
be defined tzy the idealized model of a linear chirp The perfect localization of its Wig~er-Vilie distribution to the cmwe of its (linear) i,~stamaimous frequency ~Z,:(t.) Mlows us to write
fr
u~5(t, ,.,) u~. (t ,,) dt d,., =
/"7
I.~;(t, ,<.(t)) dt.
This redtmes the detection pro(edure to a simple pat]J intreat'at in the plane, c~4 Due to tile covariam e properties of the distribution, this resuk can be extended to an es~.imatio~l procedure wRh respect to a temporal displacement. It/ fact, suppc~se the signM
~r(t) = :<(t
r)
in known apart from the delay r. The~l an immediate choice fbr the estimator of the delay is given by tQ~(t + r. ~:~:~(t)) dt .
:.= arg max T
J -- X,
Under the stone perspective, but with an observatimx affected by a (wideband) Dopp/er cttbc< an adequate time-frequency formulatkm yields a rather simple interpretation of the optimal ~stimator for the deh~. As both estimat.ors of tim Doppler rate and the delay are not decoupied, in general the delay estimator is bi~ed, if the Doppler rate is unknown. Consequently, a naturally arising task is to design signa}s that tolerat, e a Doppler ei~bct. Thin means that an arbitrary (unknown) Dopl)[er rate should leave the estimatio~ of the delay m lbiased. Such signa}s exist: They are charax:terized by a l~yperbdie it~tantaneous frequency or group delay. ~5 The time-fl'equel~.cy approach leads to a very simple gexmmtr}c justificatiou of" this fact. Recall that a Doppler effect with a rate 'J? corresponds to a trar, siormation in the time~fl'equency plane according to
(t, .)
---,
(,?/~, . / ~ } )
.
(4.72)
This imp}ies that under certain conditions an unbiased estimation of the delay (:an be devised by adopting the simplified s t r a t e ~ ~ :? = arg max
i f @ :x~
p~(t 4 r, IJ~.~(t)) dt
which is again based on a pazh integral These conditions on t}te distribution pu(t, u) are ~ Mlows:
Chapter 4
7.~me-~;~'equenqy ~s.~a.
f~radigm
34!)
(i) It is cow.u'iant by traf~slado~s;
(ii) Ir is unitary (for ascertaining the (mth'nality); and (]ii) It is localized to a curve that remains im'ariai~t under the transfi)rmation equation (4.72). As the curve in tile last; condition is a hyperbola, the eompatiNlity with these three restrictions leaves only one possible solution, ~m~mty the Bertrand distribution (see Table 2.4 and Suba~ction 2.3.2), This furnishes a purely geometrieN solution to the posed problem of Doppler tobran.ce in a tim(.~frequency setting.
Locally optimal detection.
The ease of a high SNR was already mentioned here. The opposite cr.se, where the random signal has srnaI[ energy in comparison with the noise, leads to a time-frequency solution (called IocMIy optimM), which can again be endowed with a satisfactory physical h~terpretatiom According to the hypothesis of a small SNR, the eigenvalues of the autocovariance veri~}~ the relation At,
ibr all 'n. Hence, we have
,,.=0 A,~ + ?o The l ~ t equality follows ~om
V¢~:(t, t/) = E {Wig(t, r~)}
+
+ 5
=
¸
(u,,).
% ,~}~
' "
?0
Auatysis
35{}
No~e t.hat the double s~mt in the precedi~.g eqm.~ti~ va,nishes due to the s~ochastic independence of th(~ coefficie~ts ;n th.e Kartm~en-Lobve decom.position, Provided that the sigm~l has zero memo, it; follows ~hat the detector (Eq. (L68)) reduces to
(4 7:~)
A(t~) = AA~.~) =
(7)
This shows that the reCUrrencesignature is ~:~othing bat the Wigner-Ville spectrum of the I)roce~s under {:onsideratio~. [Note that this argument cart be used to recover the same asymptotic solution (Eq. (4.7111) for the model of Eq. (4.69).] In the p**~sent e~:, of a small SNt:I, we must therefore use an expectatio~ ~/~lue as our time-freque~cy template, The joint timefrequency approach yieMs a simple (onstruction of this template (e:s a~ ensemble a,verage of the distribuf.ions of the ~f{~renee). This is simpler, fbr ez~ample, than defini~g the notio~ of an a~x~raged reference based on the
T i m e - f r e q u e n c y j i t t e r . The forementioned solution to the locally optimal detection problem can be fl~rther iIlustra~ed, when we invoke the model of Eq. (4.69) once mo~, and let it be affected by a time-fl'equency jitter; that is, we consider the rai~dom process
where (t, ~) is ~ pair of ra~ndom variables with ;~ joint probability density G(~-, () (ulrich is sup~ p_. s e d • to be eve~0. As a eoi~seq~e:nce of d:~e propert, ies of the Wigner-~ille distribution we find
w ~ ( ~ , ~) = ~ {t~! ~ w , , (~ - T , . -
= E {Ial 2}
~)}
I.v,.,(t ....... r,~J ......{) C ( r . { ) d T d (
(4.74)
Iten{Te, the correspondil~g tim(~frequency detect:or b~s the fbrm
l.l'~(t,~,)C:~,,,(t,~hg)dtd~.,.
A(y) = Ap(?l)= 2? [(
x. 7)
(4.75)
(T/mpter 4 Thne-t;}'eqtlenc?¢ as" a, Paradigm
351
As compared with Eq. (4.73), we see thai, the jilter introduces a smootMng of the reference signature by the corresponding probability density. This nmlehes with our intuition: as f~.r as the known reference is affected by a rmMom shift. (in time a.nd freque.ncy), the procedure of Eq. (4.75) simply "thickens" the reference relative to the joint probability densiW of the shift. This enables the wider reference to "catch" the eventual displa~'ement. A broader class of time-frequency receptors. The preceding example of jittering in the time-fi'equency plane suggests comparing the Wigner-.-Ville distribution of the observation lj(g) wid* a modified (smoothed) version of the reference signature. 'Fhis ma;v be done for the purpose of a mere generalization or for expecting a gain in the robustness of the detector. Let us note that the same expression in Eq. (4.75) can be rewritten and newly interpreted I)3, use of the conmmtativity of the convohltion; we can l,hus regard it as a correlation between the reference and a smoothed distribution of the observation a~:cording {o
a(:v,
C)
:
f -H,'~/ -~:x)
d~(t, ' " ,, 9) W,.~, (t, z~) dt & , , ,.
£tt
Viewed from a pra~Mcal poi~t of view, this struci;ure is quite natural, insofar as a smoothing of the Wigne>Ville distribution of an observation is often commendable, be it ibr reasons of reducing the amount of the observed data or for the sake of readability or estimation. Moreover, this new perspective can be especially interesting in case of a weLMocalized reference. 'I'hen we can use a decision statisties based on t.he path integral
,/4(:~(t,z:~(t),g)d,t:.
A(~A G) = *
x.
IlegardLess of its interpretation (smoothing of the reference or of the observation), Gq. (4.76) may serve to define a general cLa~ssof receptors that are parameterized by an arbitrary smoothing function. In fact, this e l ~ s offers a great flexibility for applications, as it provides a unified framework for a whole f3,mily of solutions of different complexity a.nd performance. It also yields a new interpretation of Cohen's class, thus addfitg to the illustrations of its capability. As a justification of the definition in Eq. (4.76) it may be wise to consider some Limiting {ases, The principal members of this class are listed in Table 4.1. They correspond to either none or total smoothing in tinm a n d / o r frequency.
7'ime~ b}'~'q~te~(:5?."Time-Scate A ~;t~_d3:sis
352
Table 4,1 Time-frequency receptors related to Eq.. (4.76}: some limid~g ca~!~s
c(t,~,)
A(g, G)
,~(~),~(~)
v(t) :,::~(t) dr!
T y p e of detector
matched filter + enw~ope detection
J/¢i
tY(t)12 iXd(t)t 2 dt
( .]f..]~ lr(~,) ~ I.x,~(,~)l ~ d~.
int;ensity eorretator
power spectrum col'relator
energy detector
From this taMe it emerges that w~tly different configurations are ineluded in one and the same fl'amework, such. as the matched filtering eomMned with the envelope detection (semicoherem ~x~ceptor), or the purely eilerget, ie detection (incoherent receptor). Furthermore, it. permits a smooth tra.n,sitican from ot:~? extreme cm~e to another. This can be achie~x~t, for instance, tU mnploying a s e p t ' a b l e smoothing in very much the sanie way as it wa~ used in Subsection 3.2.3. (There we obtained a. smooth transition between the Wigaer-Vitle distrib~ldon and the spectrogram,) The introduction of a smoothing leads to suboptimal receptors. They show a, bsser performance allan the respective nonsmoothed solutions. The deteriora,tion can be quantitied in terms of the normMized SNR:
,o(C) =
1 E {a(g, G) ,'t~ } .--~E { A ( y , a ) ! H,,} i (f~':,.,/?, ) [ w,,- { A (y, C) U,, } 1'/2
It: can be shown that Ff
H,., tl~:~, (t, z.) c,:~, (t..; g) dt dz. l
Ills
1,,'2 < 1 .
(4,77)
Chapter 4 Time-Erequency ~s a t-'ea.radigm
353
Subject to some nominal conditions, this property of suboptima.lity comes with an increased robustne,ss, when the reference is knowIJ, b3completely. l~7or the detection of a chirp, for example, the smoothing allows formalizing of the notion of a time-frequency macgin or tolerance, within which the chirp is Mlowed to be. This yields a comparable peribrmance for the detection of all real chirps that are included in this model. Once agMn the time-frequency approaa:h is no doubt the most natural way of defining an average ref>rence for this case, 37
Tim~'-}'}~eq~e~cy/Time-Scate A ~m]ysis
354
Chapter 4 Notes
4.1.1.
t This men,sure fbr lhe time4i:equen(:y extension was proposed by Claasen and Mecklenbr:guker (1980a) i,l connectioI~ wb.h the Wigne>Ville distribution. Its generalization to Cohen's class was studied in Fla~drh~ (1987). There is a diffi~rent approach Io this problem in Jm~ssen (199I). 4.1.2.
:~ It seems that the formulation of the problem of energy concentration of joint time-~equency distributions goes back to Fhmdrin (1988a). Later work, for example, that by Ramanat:han and %piwala (I992), follows the same ktea (ba.sed on bilinear dis~.ributions and the Wigner-ViHe case. in particular). A Nmilar point of view is used fbr linear decomposir~ions (of Gabor or wavelet v p e ) in Daubechies (1988b) a~d Daubechies a~d Paul (I988). \¥e can finally observe that the idea of a projection ont:o a ~Amefrequency domain is connected with ~he question of time-frequency anNysis and synthesis of certain spaces of signa.is. This subject is intensively im,estigated in the monograph }g ttlawatsch (1998}. ; M~:×stresutts presented here are taker~ from Flandrin (I988a). They rely in part on other arguments given in 3anssel~ (t98I). .5 See the work by aans~'n (I981). ~~This fact was first emph;~sized by 3ansseI~ and Ctaasml (1985). r Another way to (/tmracterize the optimality of the Wigne>Vilie distribution among M1 :s-Wither distributions with respect to a minimal spread in the pbme is provided by 3anssen (1982). Ite uses a simi}ar measure u.s in Eq. (4.1), which involves die sq~za~w of the distribution. s In spite of its impor~an(:e, we do not deal wi~h this syntheMsproblem. It is clear that the dif~culty stems from the Net thav a general time-i:requency flmction need not support any rdation to a representation, of a signal ( e.g., Wigne>Ville type). Tim most natural approach consists of solving a proG lem of best approximation by admissible Nl~ctions; that is; we consider only those time,ions that I~present the time-frequency distributim~ of a signal. Here ~,~, may work with any reasomdfle norm as a mee~sure fi~r the distance from the given flmction. The first solution pointing in this direction w~_~s proposed t g Boudreaux-Bartels (19S3) and Boudreaux-Bartels and Parka (198~i). They considered ~..he Wigne>Ville distribution and used a quadratic distance }~etveeeJl time-Dequency funct.ions. The underlying t.ask
Chatot~?r 4 ~l ime.-F}~el:~( r~qv as a P a r a d i g m
355
w ~ shown to be related to an eigenva]ue problem. An ex~en.ded and more general anaIysis~ including other unitary a~ld nommitary distributions, was carried out taier by IIlawatsch (1988) and ttlawatsch and Krattenttialer (1986; 1992; t998). 4.1.3,
v Here we refk.r to ~,he work of Lieb (1990)~ Some preliminary results of this type af)peared in work by Price and t.tot~tetter (1965). Ext,ensioim wi~h speciM emphasis on entropy and related i n e q u d i t i ~ can be found in x<~rk by Janssen (t998). J.0 The definitior~ of *:he t,eger~dre transi;arm, as ig is used here, can be lound in the book by Arnol'd (1976). 4,2,!.,
~i See aiso Chapter 3, Note 2. ~ This examp}e is taken from Abry, Gon~alvhs, and Ftandrin (1995}, which includes a finer analysis of the ~timat~ion than givt?n here. ~:~ ['he '~.1/f-noise . . processes . .have been ~tu~h,.d c "~ exhaustively. This is true for their physical nature, on the one hand, and als() for mathematical problems concerning their modeling. A~ introduction to this subject caa be found in Kesimer (1982), tk~r example. t A simple and heuristicat approach io the notion of self-similar stochastic proc~s~s in the form that we use here is in work [tY Feder (1988). I/etF~rences to more rigorous a.nd/or ( omprehensiv~ deseriptioi~s can be found in the cumulative bibliography compiled by Taqqu (t986). 15 The definition of the fractional Brownish motion with exponent H (whidl stands for Hurst) is usually attributed to Mandelbrot; and van Ness (}968). An earlier source might be the work of Kohnogorov (see Yaglom,
198r). *~ A defiaition of the not.ion of t.he ffactal dimension, with special empha.~is on the problem considered here, is given in Feder (I988), A more in-depth approach is contained in Falconer (D)90). lr The fi,wt paper in which this property is ~tablished is by Flandrin (1989b). Even more generally, the wavelet transfl)rm of every random process with stationary incremen|s turns out to be stationm T ( M ~ r y , I!;)93). ls V~e mentioned only some ~spects of seifsimilar st;ocha~stic processes tinked to the con{ilmous wavelet transform. There exist, other r~utts, especialIy for the discrete case (Wornell, 1990; Fhmdrin, 1992; and Tewfik and Kiln. 1992). Most of them are contained ill the survey, articles }~;,, Flandrin (1994) and Wornell (1993).
356
t~irr~e'-f?'~'quem:y /TISm~*-Seah ' Am~lysis
4.2.2.
l~ This measure w~s i1~troduced by Martin i~ connection w}th a classifica~,ion problem for certain signals in biology (Martirl, 1984). I~s applicatioI~ tbr gNniI~g an optimal smoot.hing is discussed in more detail by Martin and Flandrin (1985a). 20 See. for example, tlellman and Roth (1969}. 2~ The mentioned relation to a strat.egy of Akaike type refers mainly to its i~tterpreLat;ion by Clergeot (1982). 22 A typicM example t).~r this sit.uation is the a,nalysis of simple sonar signals, such as the emitted chirps of a bat, s~:~ Flandrin (1988c). 2a This application of time-fr{:utuency methods to ~ dispersive propagazior~ wais especially studied for underwater acoustics, el. l~'landrin, Sag~?loli et at. (1986), Zakharia et al. (1988), and Sessareg'o et d. (1990). 2~ The usefulness of the wavelet transform in signal analysis, in particular for the determim~.tion of the singularities, was described }~ Ma}lat and tlwang (1992). Their work t,~iies partly on the flmdamentN r~uies of Holsctmeider and Tchamitehian (1989), and 5 atI~,rd (1989). ea Sec ttm work by Holschneider and Td~amitchian (1989). 2'~ This point is addressed in Flandrin and Gongalvees (1994), for example. 27 See Gongalvbs and Fla.ndri~:t (1992) and Ftandrin m~d Go~{i.Mv6s (1994). 4,3.
2s A ciaasica.1 rei)~rellce concerning the statistical decision theory i~ signal processing is tile book by Whalen (1971). It cm~ Mso serve a:~ a useful in~roduetio~t to more advanced books such as the one by ' v ~ Trees (I 968). 29 We do not consider the solutions ~hat rely on linear decomp(~sit, iorks (similarly to the localization problem in Section 4.1}, The reader m~y consult the literature for gMning more insight into ~Ns area, for example, Friedlander and Porat (1989) or bbwler and Sibul (1991). There e x i t precursors of the bilinear approt~:h followed here in the work of Ftaska (1976) and Alt:es (1980). However, it gain~t its veritable importance, after the work of Kumar and Carroll 11984) and Kay and Boudreaux~Bari;els (1985} ~ppeared. They were tim first, to make explicit use of the WignerV ilte distribution in this context, Hence, they prepared the ground for the more general approaches based on Cohen's class (or the a.~ne class) in Flandrin (1986b; 1988b; 1989c), Saye<
Chapa~r ~ Time-Frcqt~el~cy as a P#m~digm
:{57
4.3.1.
:~o Comp~,,re work l~}"lqandrin (1989c). :~1 Such protedures were hnplememed in the original approach by Aires (t980), who used the correlations of spectrograms. 4.3.2.
32 The classical ~,~ution describexl here is borrmved fl'om Vaa~ rDees (1968). 4.3.3.
a3 This paragraph consists essentially of examples Laken from Iqmtdrin (mSSb). 3~ This property w~.~ exploited for the first, time tK Kay and BoudreauxBartets (1985). It was used, for exanNe, in Flandrin (1986b; 1988c) tbr the construe|ion of a simphfied detector, which matches the structure of bat "chirps,' A g/meratization to the problem of detecting gravitational waves can be found in Chassande-Mottin and Flandrin (t998), :~5 The issue of the tolerance ~o tim Doppler effect is ad(fi~'ssed in Attes and Tittebam~ (1970), Altes (197t), Rihaczek (1969}, or Mamode (1981), ~br instance. Its time-frequency iIiterpretation given here can be timnd iIl Flandrin and Gon~alv~s (1994). Note tha,t a graphical (~;hus qualitative) description was ah'eady contained in Fhmdrin (1986c). :~'~This strategy was used with considerable profit for a detection problem of faalts in a thermal e~ghle, see Chiollaz, Flandri~, aud Gache (I987), 3r 1he robus~:.m'.,ss of tim~-f'requermy recepto~ is discussed in Flandrin (1989c). Some cow,crete examples supportfi~g ~,he given i~terpretation can be fbund in Ftandrin, Web(r, and Zaktm,ria (t989),
Bibliography
Abry, P. (1997). ()~fddette:s et Turbulences, Paris: Diderot Editeur. Abry, P., Gon(jaiv/~s. P., and Fl~mdrin, P~ (1995). Wavelets, Spectrum Analysis and 1 i f Processes, ii~ l.~.hvelets and St:atistics. Leca~re Notes in Statistics, 103, A. Al~go~iadis et al., eds., New "~Swk: Springr~r-V~r|a.g, 15 29. Acktx.~vd, M. tL (1970) Short.4ime spectra and time-fr~,quency ex~ergiv dis~xibueions~ & Acoust. Soc. Amer. 50, 1229 1231. Akay, M. t~.(l.). (1997), Time-fk'equency ~md g~):~v~4etsin Bkmledic~J Ng!la/ Proce:s.~ing, Piscataway: IEEE Press. ¢
Allen, J. B. and Rahiner, L. t{. (I977), A nmfied approach to STFT analysis and synthesis, Pro{'. IEEE, 65, 1558 156.£ Alb:,s, t/. A. (1971). Methods of wide-band signal design fix radar mM sonar sys~;ems, Ph.D. Disserl;ation, Univ. of R.oche~ter, Rochester (NY), Altes, It. A. (1973). Some im'ariance propert;ies of the wideband ambiguity functiorl, 3. Acoust. S'oc. bner., 53, 1154 1t60. Altes, It. A. (1980). I)e~ection, estimatkm mid classification with spectrograms, & Aco~s*. Soc. Amer.: 67, t232-1246. Alte,s, R. A~ (1984). Non-uniqueil~ss of a m.~w time-fi'equency siglial representation, O|~INCON Tc,ch. Memo. ] M-322, t,a Jolla (CA}. Alter, R. A. (1985). Overlapping windows ~md signal representation in the dme-fi'equency plm~e, in CoE Nt. (~\rI{S SrstOIJes So~ar AO'iens Ang maux: 7}'Mtement ct A natyse des Signaux, Ly(m, FYa~tce, pp. i 7.1 -17.47. Ali;es, I/. A. (19N)). Wide-.band, proportional ba~ldwidt}i Wigner-Vilte analysis, IE_EE 7~ans, Acoust:., Speech and Sigr~Fd Proc., 38, 1005 1012. Altes, t2.. A. and Titlebaum: E. L. (1970). Bat signMs as optimally Doppler tolerant w~t:veforms, d. Acoust. Soc. Amcr, 48. t 0 1 4 1020, Amblard, P. O. and Lacoume, Jo L. (t992). Const.ruction of fourth order Cohen's (la~ss: a, deductive aq~proadi, iIl [EEE..SP Int. ~;{ymp. on 7'im~ N'equency and Timc-S'c~-d~: Analysis, VictoIia (BC), Cmmda, 257-.-260. 359
Andrieux, J. C . F(dx. M. R . Mot~rgu~'s G.~ Bertrand. P . lzrar, B., and Nguyen, V. T. (]987). Optimum smoothing of the Wigner-.Ville dis~zibtltLiOtl+ IEEE Trans+ Aco~Jst., Speech ;rod Si+gnM Proc+, 35, 764 769+ A rn ot'd, V. I+ (1976)+ Mdthodes ffla t hdm atiq aes de la M(;caniq ue Cta:~siqTle, Moscow: Mir. Auger, F~ (1991). Repr6sentations temps h.&:tuence des signaux non-statiom,aires: synthbse et; contribution, Thhse de Doctoral% Univ. de Nantes. Auger~ F. and l)oncarli, C- (t992). Quetques commentM~s sur des repr& bentat&:ms t:emps~fr6quence propo~es r6cemment, 7)'~,,i~,ement, du Sigw& 9, 3 25. Auger, F and FIandrin, P~ (I995), Improving the r(adability of timefreque~cy and time-scale represent, at}ons by the re~ssignment; me~hod, IEEE TrmJs. SignM Pn)c., 4a, ~068 1089. Baliam R. (1981). i n pri~lcipe d'incertitude fort, en thdorie du sigaal ou en m6canique quandque~ C. R. Acad, S(:. f~ris, fi&*ie It, 292, 1357-~1362. Baraniuk, R, G. and Jones, D. L. (1993). A signal-dependent time-fYequency representation: optimal ken~eI deMgn, IEEE ~l?;ms. Signal Proc, 41~ t 589....1602. Bm~s, .l. {t945). Les loner.ions M(~at.oires et, bur interpr&ation m6cauique, Revue ScientiL;que, 836me Annde. 3 20, Basseville, M , Flandrin, P., and Martin, N. (eds.)~ (1992). Signaux nol~stadommires, analyse temps-fr6quence et segmentatiom Fiches descript,ives d'algorithmes, :[}'aitemet~t du Signai, 9, Snppl. t.o No. 1, B~,stimms, M. 3, (1978). Fhe Wiggler distribution fimcdon applied to optical signM and systems, Opt. Commm<, 25, 26- 30. Bwsdaans, ?\I, J. {1980). Gabor's expansion of a signM into Gaussian elementary signMs, l:'roe. IEh;E, 68; 538 -539. Bt~stiaatts, M. J. (198I), A sampling d~eorem fbr the complex speet~Jgram and Gabor's expa,nsion of a signa.1 in Gaussian elenmnta,ry signMs, Opt:. E~N. 20, 594 -598. Ba]~taglia. F. (1979). Some extensions of the evolutionary spectra} analysis of a stoch~sdc process, B~di, [h~io~e 3lJtemat.k~ Balimla, 16-B, 1154 1166.
Ba%'em F., Flato, M, H'onsda[, C., Lichnerowicz, A, and Sternheilner, Do (1978a.). Deformatio~ theory and quantization L Defi)rmation of symplecdc structures, Ann. Plt y:*., 11 I, 61 -t 10. Bayen, F.. FIa~o, M., FronsdaL C , Lichnerowicz, A., and St;ernheimer, D, (1978b). Deformation theory and quantization II, Physical applications, Ann. Ptzv:s, 111, 111 151.
Biblio~rapfo'
361
Bedrosian, E. (i963). A product theorem for [{il}:,~rt transtbrms, Proc. IEEE, 51. 868 869. Bellman, R, and Rod~; R. (1969). Curve fitting by segmented straight lines, J. Amer. Stat. Assoc., 64, t079 -1084. Benedetto, J, a. (1990). Uncertainty Principle htequalities and Specvrum Estima,tion, in Recent Advauces in I%uri~r knal3:sis and its Appticatkms, J. S. Byrnes and J. I?. Byrnes, eds., Dordrecht: Kluwer, 143 182. Berr> M. V. (1977). Selniclassiea,l mech,mics in phase spa*e: a. study of Wigner's funclkm, Phil. Tra,s. Roy. Soc, A 287, 237 271. Bertrand, J. and Bertrand, P. ([987). A tomographie approach to Wigner's function, l~bund, of Physics, 17, 397 405, Bertrand, J. and Bertrand, P. ([988). Time-frequency representation;s of broad-band signals, in IEEE lnt. Cone on kcoust., Spe~ch and Sig= hal Proc. IC~ASSP-~'~8', New York (NY), 2196 2199 (also Combes, J. M., Orossmann, A., and Tchamitchian, Ph. (eds.). (t989). 1,15.~,v~:4ets, 75me-Fr'equenqv Methods and Phase-Space, Berlin: Springer-'vi:~rlag, 164- 171). Bertr~md, J. and Bert.rand, P. (1992a)~ Afl:ine Time-Frequ:ency Distributions, in 77me-l;~reque.ncT Signal Anat>sis Methods and Applications, B. Boashash, ed, Melbourne: I,ongma.n-Cheshire, / 1 8 140. Bertrand, ,l. and Bertrand, P. (1992b). A elaass of Mfi:ne Wigner flmctions with extended covariarme requirements J. Math. Phys., 33, 2515 2527. Bertrand, P. (t983). Representations des signmlx dans le plan temps-ft'& quence, Rech. A&o,sp, 1983-1, 1 12, Ber~;rand, P., Izrar, B., Nguyen, V. T., and bi~ix~ M. R. (t983). Obtaining noI>negative qua.ntum distribution functions, Phys. Left., 94, 415 ~17. Btahut, R. E., Miller, Jr., W., and Wilcox, C. H. (1991). Radar mM Sonar, Part I, IMA, Vol. 32, New 55~rk: Springer~Verta.g. Blam>Lapierre, A. and Piciubono, B. (1955). Remarques sur la notion de spectre instmttand de puissance, Pub. Sci, Univ. d'Alger B, 1, 2 32, Blmm-Lapierre, A. a,nd Pieinbono, B. (1981). Fonetiol~s At&toire,s, Paris: Masson, Boashash, B. (1992a), Estimatiltg a~d interpretilg the instmltaneous frequency of a signal. Part I: l~Smda.mentals, Proc. 1EEE, 80, 520 538. Boashash, B. (1992b). Estimating and interpreting the instanta,neous frequency of a signM. Part II: Algorithms and Applications, Proc. IEEE, 80, 540 568. Boashash, B. (ed.). (1992c). Tim~-b}'equenqy Signal Ana!ysis Methods and Applications, Melbourne: Longman-Cheshire.
362
"1im~,-F}vq{aea,cy
13oaMmsh, B. and I¢istic. B. (I992), Pot5 nomial ~\Vigtxe>.Vil/edist:ribu~io~ls and time-.varyi~g higher-order spectra, in IEEE-~5;P lut. ~g}a~tp orJ TimeFreis , \:'i(toria (BC), Caltada 31 34. I~onnet, (E (I968). Consid~ratiolis sur la repr~sema~,iou et t'anMyse harmonique des signaux d6terministes ou al~atoires, Ann. Tdl&:om., 23: 62 86. Bopp, F. (1956). La m(k:mtique quan~ique est-etle uue m~<:aniq~mstatistique particu}i6re? Am~, h~st. II~;uri Poiucar& XV~ 81 1 I2. Borchi, E. and Petosi, G. (1980). A quantum medmnical derivation of the o'O~lp a} uncertainty relation for ~eat sig)~ats in communicatiou th<~_.)ry, .... r..~ Siguat Proc., 2, 289- 292. Borm M and Jolda~, P. (1925). Zm" Quantenmechanik, Zk~it. t~ Pttys, a4, 858 888. Bouachag:he, B. (1982), Repr~sentatkm ~emp>frdquence application ~ la mesure de Eabsorption du sous-sol, Th~,se Doct.-Ing, INPG, Grenoble. Bouacha,:he, B, Escudid, B., add Koma~itsch, J. M. (1979). Sur la possibiii~;d d'miliser ~a reprdsentation col~joiI~teen temps et ii'~quenee daI~s l'anatyse des signaux moduh;s emis e~ vibrosismique, in 7~,me Coil. GRET,.~I, ~ Nice,, pp, 121/1 121/6 Boudreaux-Ba~rte}s, G. F. (I983). Thne-Kequency signal proces~sing aigorkhms: analysis and Wnthesis usir)g Wigner distributions, Ph.D. Disserbation, Rice Univ. Houston (TX). Boudreaux-Bartets, G. F. {i1996}. blixed Time-Dequeney SignaI Trmlsforma~;itms, in The 'I~'aI~s~n'u~s a~M Applk:atio~s ttarMbook, A. D. Poulafikas, ed, Boca Raton: CRC Pt~ess, Boudreaux~Bartels, G. F. and Parks, T. W. (1986). Time-varying filtering and signal est.imatiott using Wigner distribution synthesis techniques, tI~37JE 2}'at~s. AcoHst., Speech aud Sig~ml Pro<, 34, 442 451.
I{{OtlVC%M. (t992). X)°air,ement des Sigr, a~x pour k:,s ,5):stt,mes So~,ar, Pea'N: ma.<Jqon. Bracewelt, R. N. (1978). The Dmrier J~'ar>'~)rm aud its Appiicatious, 2nd ed., New York: McGraw-Hill. BrilIouin, L. (1959). La Scieuce e~ N. Th&>rie de I'I±il'orma@)u, Paris: Gauthiers-Vitlars. Brornar~, H. (1981). The i~tstantaneous freque~cy ~' a Gaussia~ signah tim one-dimensionN density flmetio~, IEEE 7ra~s. Acoust., Speech mM 5iguat Proc., 29, 108. 111. Bur(., P. and AdeI~)m E. (i983). The Laplax:ian pyramid as a compact image code. IEEE Trans. (?)mm., 31,482-540.
Bibliography
363
Carsoi~, J. and Fry, T. (1937), Variable frequency electric circuit, theory with applic~:~tion to the theory of fl'equency modulation Bell Syst, ~lb~d~.d.~ 16, 5t3--540, Cha~sande-Mo|;tin, E. and Flandrin, P. (1998). On the. thne-kequency detection of cilirps, Appl. Comp. Harm. Anal., 5. Chiolla,;~, M., Flandrin, It, and Gache, N. (1987). Utilisalion de la repr6sentation de Wigner-Ville comme outil de diagnostic des d6Nuts de fonctionnement des moteurs thermiques, in 11brae Coll. GRE21SI, Nice, 579 582. ChN, H. I. and Williams, W. J. (1989). hnproved time-frequency representation of multicomponent signals using exponential kernels, IEEE 2}'a.,)s. Acoust., Speech and Signad Pwc., 37, 862.871.. Chui, C. K. (1992). An Ntroduction to ~;Dwelets, New York: AeMemic Press. Claa~en, T. A. C. M. and Mecklenbrfiuker, W. F. G. (1980a). The Wither distribution a tool for time-fl'equency signM mlalysis. Par~: I: Continuous-Time Signals, Pbilips J. Res., a5, 217 250. Claasen, T. A. C. M. and Me&Ienbr/iuker, W. F. G. (1980b). The Wither distribution a tool for time-frequency siguN anMysis. Part II: DiscreteTime Signals, Philip,s 3. Res., 35, 276 300, Claa~sen, T. A. C. M. and Mecklenbrguker, W. F. G. (1980c). The Wigner distribution .......a tool for time-frequency signal analysis. Part III: Relations with Other Time-Frequency Sig~al Transformations, Philips d. Res., 85, 372 389. Claasen, T. A. C. M. and Meeklenbr~uker, W. F. C. (198I). Time-frequency analysis by means of the Wigner distribution, in IEE~~ Int. Cont: on Aeoust., Speech and SignM Proc, ICASSP-81, Atlanta (GA), 69 72. Clergeot, H. (1982). Estimation du spectre d'un signal aI6atoire Gaussien par le crit6re du maximum de vraisemblanee ou du maximum de pro}> abilit6 a posteriori, Th6se Doer. Etat ~i,sSe. Phys., Orsay. Cohen, A. (1992). Ondelettes et 'i}'aitement Num&'ique du Signal, Paris: Ma.sson. Cohen, L. (1966). Generalized phase-spa,:e distribution hmetions, a. Math. Phys., 7, 781 786. Cohen, L. (t970). Hamiltonian operators via Feynman path integrals, a. Mh~t;h. Pious., 11, 32963297. Cohen, L. (1984). Distributions in signal theory, in IEEE Int. Con£ on Acoust., Sp~eh and Signal Proc, ICASSP-84, San Diego (CA), pp, 41B,1,1 41B.1.4.
3(~::t
'I i m~- b req ~~e~(v/'~l 'im e- Sea l(, ,4 ~~a]3'sis
Cohen, L, {.~989). Tim<-freq,te1,,:}~ distribmio~ls 9i~
a review~ P~oc. IEEE. 77
981.
Cohere I,. (~992). Scale a~l im.'etse i¥~quency represent.atio~s, iY~ A F I T / A ~ O ~ R g'hve/ets ~:~i~rk~shop, Day,.on (()t)I~ 118- t29. Cohen, L. (t 993). The scale representatiort, IEEE if?arts. Signal Proc., 41, 3275 --3'29'2. Coh,,n, L, 1995), "~Tme-I}'equenc,," knMvsis, Engtewood Cliffs: Prent:iceHaIL (,oh ' e.n., 1.,. and Lee., C . (1988). h~st m-~(m~eo~s quantities, standard deviation and cross-terms, i~ Proc. SP,[E Conf. tdv. Atg. and Arch. fbr Sig:nM Proc. ItI, 975, San Diego (CA), 186 208. Cohere L. m~d Posch. T. E. (I985). Positive t:ime-fl'equeuey dis~;ribution Nnctioas, IEEE Trans. Acoust, Spo~ch and ,SSgaal Proc., 3a, 31 37. Cohen, L. and Zaparo~anns, Y I. (1980). Po.*it:ive quam;um joit~t disl;ribut:ions, .L Math. Ptzvs, 21, 794 796, (x)herl l amt( udjI, C., Diu, B., and La,Io~},F. 197:f) M&aniqu~; Quantique, Paris: lIermamL Comon, P. (199 I), | n d e p e n d e ~ compo~lent aualysis, in lnt. SignaJ Pmc. 14})rkshop on [Bgher Order St~tt:ist:ics, Chamrousse: 111 12(l. (opsoIl, T. E. (1967). As vmpt:otit Exl)an;~Jons, Cambridge: Cambndg.. )ltlV, Press. Cowling M. G. a~nd Price~ 3. F. (}981). B~mdwidth versus time cone{ ntrae'( '" "o" ; )~:m}i-\~cy} ' ' 2~ t~ion: t;he H..lsenbeto-I inequalit}:, SIAM ,L Math, AnaL, 15, t51 165. C.rant.r, H- (!971). Structured and Statistical Prob}ems for a Class of S~,ochastw Proe(~ses, The t u . t b. S. Wilks I,eetm:e a.t Print< ton University, il~ Random Processes: Muttiptieir.y The:~:~ryand C2monicaI Decomposit~ons, A. gphremMes and J. B. Thoma~s, eds,, ~Strottdsbm'~ . g'" Doaden. _. , Hutchinson & Ross, htc., 305 34.~1. d'A/essandro, C. (1989). [lepr(?semation du signal de parole par une somme tie fonc~ions ele~.~e~mane,:."s ~ nese de l)octorat~ _.niv. Paris 6. d'A less~mdro, C. arid Demars, C. (1992). Repr6sem, at;ions ~emps-fl'(,quence du sig~:~al de parole, 7}'aitement du SignM, 9, 153 173. Daubechies, I. (1988a}. Orthouormat bases of comp~el:ly supported wavelets, Comm. in Pure and AppL Matt~.. 41,909 996. Daubechies, 1. (I988[@ Time-frequency [ocaliza~km operators: a geometric phase spa~:e a,pproaeh, [EEE T}'s:ms. lul)~ Theory, 3 4 : t i 0 5 612. Da.ul::,echies~ I. (1992). 2bn Lectures on ~:~:?~vetets,Philadeiphia: SIAM.
.Bibli<:>gl'apto~
365
Daubechies, I,, Grossmami, A~ amd Meyer, Y, (L985), Painless non-orthogo~mI expansions, d. Math. P/o;s'., 27, 1271 1283, Da.ubechies, I., aaifh.rd, S,, and aourn6, J. L. (199t). A simple Wilson o> thonormal basis with exponential decay, SIAM J. Math. AnaL, 22,554 573, errmmn p. 878. Daubechies, I. and Pa.ul, T. (1988). Timeqi'equency locMiza(ion operators-g geometric phase space approach: [I. The use of dila.tions, Nverse Prob1eros, 4, 661 680. De Bruijn, N. G. (1967). UneertainV Principles in Fourier Analysis, in l)lequatities, O. Shisha, ed., New York: Aca,demic Press, 57 71. Donoho, D. L. and Stark, P. B. (t989). Uncertainty principles and signal re(overy, SIAM .L AppL Math., 49, 906 93I. I)orize, C. and Villemoes, L. F. (t991). Optimizing time-frequency resolution of orthonormal way(tiers, in IEEE Nt. Con£ on Acoust., Speedi and HigilM Pro~, I(TASSP.-91, ~Ibron¢o, 2029 -2032. Duv~mt, P. (199t). 71"a.it~,m~nt du Sig:nat- Concel)ts et Applications, Paris: Henn6s. Duvaut, P. and Jorand, C. (I99t). Analyse en eomposantes ind6pendantes d'un noyau multilindaire, Application g une nouvelle repr6senta,tion temps-fr6quence de signaux rion Gaussiens in 13&me Coll. GRETSI, Juan, les-Pins, 61 -64. Dyrn, H. a,nd MeKean, H. P. (1972), /??Jurier Series and Ntegrals, New York: Academic Press. Eseudi6, B. and Fhmdrin, P. (1980). Sur quelques propri6t6s de ta reprG senta.tion conjointe et de la fimction d'ambig~fft6 des signaux d'6nergie Iii~ie, C. R. Ao'~d, So. P~ris, Sdrie A, 291, 171 t74. Eseudi6, B. and Grda, J. (1976). Sur une formulatiou g&16rale de la repr& senta¢ion en temps el; frdquence dans l'analyse des signaux d'&lergie time, C. R. Acad, Sc. Paris, S&ie A, 283, 1049 q051. EseudiG B. mid Gr6a, J. (1977). Repr6sentation hilbertienne et reprdsenration conjointe en tenips et fl'fiquence des signaux d'6nergie finie, in O~me Coil. (;I:LETSI, Nice, pp. 5/1 5/6. gsteban, D. and GMand, (L (1977). Applh:ation of quMrature mirror filters to split-band voice coding schenies, in IEEE Int. C,m£ on Acotzst., Speech and Signd Proc. ICASSP-77, Hardbrd (C'[), 19/- t95. Falconer, K, (1990). th'acta.t Ge~m~etry MathematieM ~?)undatious and Applications, New York: J. Wiley & Sons. I¢>der, 3. (1988). b}'actaN, New York: Plenum Press. Feichtinger, H. G. and Strohmer, T. (eds.). (1998). Gabor AnM/~is and Atgorithnls: ~lNeow and Application, B~~ston: Birkhgnser.
366
l'ime*D'equene~'/Time-ScMe Analysis
Flanagan, J. (1972). Speech Analysis, S)nthc~is aud Perception, New York: Springer-Verlag. Flandrin, P. (~982). Repr(~en~,ations des signaux dans Ie pbm temps~.~ff6quence, Tht~se Doct.-Ing., INPG~ Grenoble. FlandriI~, P. (198.1)~ Some %atures of time-kequency representations of multi-component signais, in IEEE hm Con/: on Acoust., Speech and Signd Proc. [(\4SSP~84, Sa.n Diego (CA), pp. 41.BA.1 41B.4A. Flandrin, P. (1986a), On the positivily of the Wigne>Ville spectrmu, SignM Proc., 11, t87 I89. Flandrin, P. (1986b). On detectim>estimation procedures in the time-fl'equency plane~ in 1NEE Int Cone on Aeoust,, Spee& ant/SignM Pwc. ICASSP-86, ~Ibkyo, 23312334. Flandrin, P. (1986c). Time-frequency interpretation of matched filtering, in IEEE Academia Sinica Workshop on Acoust, Spee& arid S~g~lal Pro< ~%SSP-86, Beijing, 287-290. Flandril~, > (1987). Repr&en{,ations temps-fr&tuence des signaux no~> sta.t,ionnaires, Th~se Doer. Etat bs Sc, Phys., Orenob~e. Flandrin, P. (1988a). Maximum signal ener2,, concentration in the timefrequency plane, in IEt'3£:Int. Con£ on Acoust., Speoe,~ and Signd Proc. ICASS,~88, New York (NY}, 21762179. Ftandrin, P. (1988b). A time-fl'equency formulation of optimum deI~ec~ion, IEEE Trans. Acoust, Speech and S~g'nM Proc., 36, 13771384. Flandrin, P. (1988c). Time-Frequency Processing of Ba.t Sonar SignMs in Animal Sonar- Proc(::~e;esand Performance, K E. NachtigaJ1 and P. W, B. Moore, eds, New York: Plenum Press, 797 802. Ftandrin, P. (1989a)~ Time-Dependent Spe(:t~ra fbr Nonstationary Stoeha;stic Processes, in Time and FTrequenqy Representation of Signals mid Systems. CISM Coarses and Leetm'es No. 309, G. Longo and B. Pieinbono, eds, Wien: Springer-Verlag, 69 124, Flandrin, P, (1989b). On the spectrum of t}actional Browniaxl. umtions, IEEE 2}Tans. Nfo. Theory, 35, 197 I99. Flandrin, P. (1989c). SignM detection in the time-frequency plane, in 1N4C l'I,t?rkshop on Ad~wnc~:M Information Processing in Automatic Contrd AlPACa89, Nam% 131134. Flandrin, P, (1990). Scale-Inw~riant Wigtier Spec{;ra and Self-Similarity, in Signal P r o ~ i n g V Theories and Applications, L. 2"brres et aL, eds., Amsterdam: North-HoI[and, 149-152. Flandrin, P. (1991 ). Sur une elasse gdn6r~de d'extensions a~nes de Ia distribution de Wigne>Ville, in 13brae Coil. GRETSI, Juan-les-Pins, 1720.
l~ibliography
367
Handrin, P. (1992). Wavelet amdysis and synthesis of fr~,c~iol~alBrownian motion, I~2E ~)'aiJs. hdb. TheoO', 38, 910 917. Flandrin, P, (1994). Wa:velets arid Self-Similar Processes, in Wm,etet~' aud their Applications, NATO ASI Series C, 442, J. S. Byrnes et M., eds., DordrechJ~; Khtwer, 121 142. Flandrin, P. and Eseudid, B. (t980). Time and fl'equency representat.km of finite energy signals: a physical property as a result of a Hilbertian condition, Signal Proc, 2, 93 100. Fland:rin, P. and Escudi6, B. (1981a). Sur une relation int6grale entre/es ibnctions d'ambigN% en translation et en compression, Rapp. Int. 1CPt TS-8110, Lyon. Flandri~, P. and Eseudi(!, B. (1981b). G(~om~trie des t}mctions d'ambigtfft~ et des repr(}sentations co~jointes de Villa: l'approetm de la th~orie des catastrophes, in 8brae Coll. GR.ETSI, Nice, 69 74. Flandrin, P. and Escudit~, B. (1982). Sur la localisation des reprdsentations conjointes dans le plan temps-frdquenee, C, R. Acad. So. Paris, S(~Ne I, 295, 475 478. Flandrin, P. and Escudid, B. (1984). An interpretation of the pseudoWigner-Ville distribution, Sigual Proc., 6, 27.. 36. Flandri~, P. and Go~4alv~s, P. (1994). l/Yore~¥~velets to Time-ScNe Energy Distributions, in Recent Ad~mces in ~}~w~.letAnalysis, L. L. Schumaker and G. Webb, eds,, Stm Diego: AcMemic Press, 309 334. Flandri~, P. and Gon(~alv~s, P. (1996). Geometry of arlene time-frequency distributions, AppL Comp. H~rm. AnaL, 3, :10--39. Fbmdrin, P. a,nd Hlawatseh, F. (I.987). Signal Representa.tion Geometry and Catastrophes in the Time-Frequency Plane, iI~ M~.~thematics in Signal ProcessbN, T. S. Durrani et al., eds., Oxibrd: Claxendon Press, 3 14. Flandrin, P. a.nd Martin, W. ([983a). Sur les conditions physiques a~'ssurant l'unicit6 de ta representation de Wigner-Ville comme repr~,sentation temps.4r(Squenee, in 9?~meCoIL GRET,qI, Nice, 43-49. Flandrin, P. and Marl;in, W. (1983b). Pseudo-Wigner estimators for the analysis of non-stationary processes, in IEE[?2 ASSP ~)ectrum Est. }~'brkshop H, Tampa (FL), 181 185. Fiandrin, P. and Martin, W. (1984). A General Chess of Estimators for the Wigner-Ville Spectrum of Non-stationary Processes, in AnMy~sis and @timization of ,~stems. Lecture Notes in Control and hfformation Sciences, 62, A. Bensoussan arm J. JL. Lions, eds., Berlin: SpringerVerlag, 1523.
368
'l ime-b}'cquenc3\/' l'ime-Scak? Analysis
Fbmdrh~., P. a~<:l 51ar~in~ W~ ( ~998}. The Wif.~;~er-Vit}eSpectrum of Nonstationary l/andom Signals, iil t~h~~ I~ i~ner Distribution Fho>ry and Applications ir~ Si~4na[ t>roces~h,;~" \V. F, G, Mecklenbrthlker aim F. Hia.watsch~ eds., Amsterdam: Eb~vier, 212 267. Ftandrin, K at~.l [~iottl O+ (I.990). AtI/i~e smoodling of the WigIler-Vilfe distributiom in IEEE bit. Contl on Acoust., Speeeh aim Signal Prot. ICASSP-90, Albuquerque (NM), 2455-2458. Flandrin, P., Sagetoli, ,J., Sessare~o, J. P,, and Zakharia, M. (1986). Application of time-frequency analysis to the characterization of surface waves on elastic targets, Aco~s~. Lett., t0, 23-28. Fla,ndrin, P., Weber, P., and Zakharia, M. (1989). A~Nyse temN-fr&tuence de r6ponses hnpulsiomtelIes de cibles, Rapport Interne ICP[ TS-8912, Lyom FI~ska~ M, 1). (1976). Cross-correlation of st~ort~-time spectral histories, & Acoust. Soc. Amer+, 59, 3;Sl 388. I:ollai~d, G. B. (1989). H~.rmonic Analysis in Phase 5pace, Ann. of Math. Steadies No~ 122, Princeton: Princeton Univ. Press. FoHand, G. B. and Sitaram, A. (I997). The m~certainty principle: a mat.hema.tical sm~,'ey, J. fburier Anal AppL. 3+ 207 238. tJ;k~wler, M. L. and Sibul, L, H. (I991). A unified formulation for detection using time:freq~mncy and ~iln.e.-s~:a|emethods, in Asilomar C o J . ozt Sig., ,~s'~.. a~d Comp., Pacifi< Grove (CA), 637 642. FriedlmMer, B. and Porat, B. (i989). Detection of l:ra/i:siel.,~ signNs by the Gabor representa~iol~, IEEE 7r;ms. Acoust., Speech and Sigtnal Pr<>c,, 37, 169- t80+
Fuchs, W. H. 3. (1954). O~l rife magnitude of F;mrier transforms, in Proc. hit. Cong~'ess ofM~th., Amsterdam, 106 [07. Gabor, D. (1946). Theory of communieatiom d. tEE, 93, 429 457. Ga.mbardeHa, G. (1968). Time-scaling and short-time spectral analysis, d. Acous~.. See, Amen, 44, i745 I747. Gardner, W. A. (t988). Statistical Speea'al Ana(~sis A Nor~-12robabNstic Theo~T, Engtewood Cliff:s: Prentice-Halt. Gasquet, C. and Witomski, P. (t990). AnaIyse de Fomqer et AppBcations .... Fil~rage, Calcul Nume~rique et Onddem~s. Paris: Masson. Gendrin, R.. and de Villedary, C. (t979). [ nambig~lous determinaIAou of fine structures in nmt~icompouent time-varying signats~ Ann. ]Mecom., 35, 122130. Gendrin, R. and RoberL P. (1982}. ~[?~mpsde gr<mpe et targeur de bande de signaux modul6s simultandment e~ amplitude et. en fr6quem-:e, Ami. TehOeomm., 37, 289=297.
Bibliogr;~ph~"
369
Gibia,t V. Wu, F . Perio, P. and Cimintret~iL S. (1982). Analyse spe(trMe difl?,re~ti~lle (A.S~D.), C!. R. Acad. So. P~ms, S¢!rie 1I, 294, 633 636, Gon%'alv~s, P. aim Flandrin, P. (]992), Scaling exponent es¢ill~latiorl froI?l time-scale energy dislributions, ill [fZb~ hit. Cant cm Aco~lst,, ,~peecJt ~nd Sig~M Proc. ICASSP-92, Sa,n FYatlcisco (CA), pp. V.157 V~160. Grace, O. D. (I98I). Inst;anganeous power spect.ra, d. Aconst. &~e, Amer., 69, 191-198. Grenier, Y. (t983). Time-dependent ARMA modeiling of non-stat, ionary signals, IEEE ~/}'ans. Acoust., Speech ~md Sig'x~d Proc, 31, 899911. Grenier, Y. (1984). Mod~}lisation de signaux non-stalionnMres, Thi~,seDoct. Etat ~s So. Phys., Univ. Paris~Sud, Orsagv. Grenier, Y. (1986). Mod~les AI]MA g coefficients d4pendants du temps: estimateurs et applicat;ions, Tl'~dtemc,~lt du Sig;~)al, 3. 219. 233. Grenier, Y. (1987). Parametric Time-Frequency Representations, in 2i~M1.ement du N g l d / Sig~d Proc(~ssing, J. L. La,com~te et M, eds., Les Houches, Sessio~ XIN, Amsterdam: North-lloltaud, an9 397. Grossmann, A. (I976). Parity operator and quant;izal.ion of ~-Nnetions, (k~mmun. M~th. Phys., 48, 191194. Grossmami, A. and Escudi6, B. (1991). Une repr6sentation bilin~aire en temps et echelle des signamx d'~nergie finie, ir~ 18tmie CoIL GRETSI, Juan-les-Pins, 33 ;/6. Grossmann, A. and Morlet, J. (1984). Decomposition of ttardy time|ions into square integrable wa,veiets of constant, shape, SIAM ,1. M~th+ AnaL, 15, 723 ~.736. Grossmam~, A , Morlet, J., and l~a.ul, T, (1985). ~Ii'ansforms associated to square integrable group represent.ai:ions I: generM results, d. Math. Pl!vs., 26, 2473--2479. Grossmaml, A., MorteL J., and Paul, T. (1986). Tra~lstbrms a.ssociated to square integrable group representations II: exmnples, Ann. Inst. Herin Poincar(i, 45, 293 309, Hammond, J. K. and Harrison, IL Fo (1985). Wigner-Ville and evotutio~ary spectra tot covariance equiv~.lent non-stationary random processes, in IEEE I~t. Con£ o~ Acoust., Speech ;~l;~d:~ig1~N Proc, ICASSP-8:5, Tampa (FL), 1025 1027. Hell, C. E. a~M WahmL D. F. (1989). Contimaous and discrete wavelet transfi~rms, SIAM Rev., 31,628-666. Helstr6m, C. W~ (1966). An expansion of a signal in Gaussiaz~ elementary signals, IEEE 2}tans./n~b. Theory, 12, 81 82.
1]ippenstiel, H. D, and d~ ()liv~ira. P. M, (1990). Timt~varyfilg spe~:tra,! estintaliori using ~,h~-:i~s~a~ )a~)~,~>uspouer 5p(i.'trmn (IPSL IEEE ~l}'a~s. Aco~,st, Sp~.'och a~Id Si~md Proc. 38. 1752 1759. Hlav,~a.t~sch, F. (Iq84). b~terfcre)~ce terms [t~ tim \Vigner distribution, in I~i~. ('ont~ ou Digital Sig~mt Proc,, b'loren(e. 363 367. Hla.wa.tsch, F. {1985). "Fra~sformation. inversion ~ud conversion of bitinear signal representations, i~ IEEE .h~L C2~£ ol~ Acous~:.. Speech and Sigtmt Proc. lCASSP-85, 23mpa (FL), I029 [032. Jib,watt:h, F. (1986). Unil;ary 'I ime-~Nequenty Signal Hepresentations, in Sigmd th'+>ce,ssix~gIIl Theori~s aJ~d Apptica~.io~s, I. T. Young eta],, eds., Amst(:rdam: Nor~h-H~)I~and, 33 36. Hb~watsch, F. (1988). A study of bilfimar time-frequency sigmd represe~> rations wigh applications ~.o fime-h~qt~ency signal synthesis, Doctoral Thesis, TU Wien, Viemm° Hlawatsch, F. (199t). Duati W and classification of/)ilinear time-~?equency sigmfl re:presen~,alions. 1EEE 2}'a~s~ Si24~al Pro(:.. 39, 1564 1574. Hlawatsch, F. (t992). Regularity and mfi~arit3 of bilfi~ear thr)e-freque~cy sig~aI rept\ese~tations, IEE'E ~l~at~s. h~b. Tkoow, 38. 82 94:. Hlawat:sch; F. (1998). Time-f~}'eque~cy An,trsis and ;%~wwhesis of L/near Sig~at S/mc~,~: Time-N>que~C~ Fil~(,r,~, Sig~ml l)~.~ecti(m ~md Estitna, tio1~, and Ra~g~'-Doppler Estim~riom Bostot~: KIuwer. Hlawatsch, F. az~d Boudreaux-Bartels~ G. F, (t992). Linear and quadratic time-frequency siguM represemafion.b. IE~2E Sigma] P~oc. ~'~1~gazh~e, 21 67. ft|a~vat;sO), F. and Flandrin, I>, (!998), The h~ert?'rence Structure of the Wigner Distribution and Heta)ed Time.-Frequency Signal tlepre~.senta-. t[ons, in The WiNlier Disfribut:i<m TJteo~;v a~d Applic~'~ti<ms i~ S~g~at Processing, W. F. (1. Mecklenbr~iuker and F. Hla:,,vatsch, eds., Amsterdam: Elsevier, 59-t 33. Hlawatscb, F, anti Krattenihaier, W. (1986), Signal Synthesis Kom Unitary Time.F'requency SignaI Rel)resentations, in Sig';~mIProces:~ing I l l 'lSeories a~M AppIica*:io~s. I. T. Young (~t M., eds., Amsterdam: North-, ttolland, 3740. tttawatsch, F. arid Krattenthaler, W. (1992) Bitinear siguM synthesis, IEEE 7~ai~s. Sig'uaI Proc,, 40, 352 363, |{[avva~sch, F, and Krattentha~er, W. (199g). Signal Synt;hesis Algorithms tbr Bilinear Time-I~Yequency Sigl~at tlepresentatio~s: i~i The Wig,wr Distributkm Theo*W az,(I Appli( ations i~ Si;;~aI Processilg, W. F. G. Me&lenbrguker and F. IIlawatsch, eds., Amsterdam: gl~:>vier, 135 209.
Bibiiograpt~ ~
371.
H/await,sob, F., Manick~ml, 'F. G., lJrbanke, R. L., and Jones, W. (1995). Smoothed pseudo-Wigner distribution, Choi-Willians distribution, and conc-kernel representatiori: anlbiguity-domain analysis and experimentaI comparis(m, SfgnM Proc., 43, 149-168. Holschneider, M. and Tchamitchian, Ph, (1989). R6gularitd Locale de la bkmct:ion 'Non Diffdrent;iable' de I:/,iema~m~in Les Ondelettes en i989. Lecture Notes in M~thematics, 1438, P, G, Lemarid, ¢d., Berlin: Springer-Verlag, 102-124, HudsolL R. L. (1974). When is the Wigner qua;si-probability density non negative? Rep. Math, Phys., 6, 249252. Hurd, H. L. (1969). All inv~tigation of periodically eorrel~ted stochastic processes, Ph.D. Dissertation, Duke Univ., Durham (NC). Ivanovic, I. D. (1983). Note on a new interpretation of the scalar product in Hilbert space, Phys. Lett., 99A, 161-162. Jacobson, L. and V~reehster, It. (1983). The composite pseudo Wigner distribution (CPWD): ~ computable and versatile approximation to the Wigner distribution (WD), in IEEF l~:~t. C(mf. on Acoust,, ,~peecb and SignM Pwc. lC~ASSP-83, Boston (MA), 254-256. Jaffard, S. (1989), Exposants de H61der en des points donn~s et coetticiants d'ondelettes, C. R. Acad. Sc. P~tris, Serie I, 308, 7981. JarLssen, A. J. E. M, (1981). Positivity of weighted Wigner distributions, S I A M & Math. AnM., 12, 752758. Ja~sen, A. J. E. M. (t982). On the locus and spread of pseudo-density functions in tim time-fl'equency plmw, Philips" J. Res., 37, 79-110. Janssen, A. J. E. M. (1984a). A note on Itudson's Theorem about functions with non-negative Wigner distributions, SIAM ,L Math. A~M., 15, 170 i76. Janssen, A. ,1. E. M. (1984b). Positivity properties of phase-plane distribution functions, & Math. Phys., 25, 2240 2252. Janssen, A. J, E. M. (1985). Bilinear phase-plane distribution flmctions and positivity, .1. Math. Phys., 26, 1986 4994. Janssen, A. J. E. M. (I987). A note on 'positive time-frequency distributions', IEK~ ~D'ans. Acoust., Spcech and SignM Proc., 35, 701703. Janssen, A. J. E. M. (1988a). Positivity of tinm-fl'equency distribution timetions, SignM Proc., 14, 243-252. Janssen, A. J. E. M. (1988b). The Zak trarasform: a signal transform for sampled time-continuous signals, Philips J: Res., 43, 2369. Janssen, A. J. E. M. (1991). Optimali V property of tim Gaussian window spectrogratm, IEEE Trans. SignM Proe., 39, 202-204.
372
Tim~-[q'eq~e~c~ /Time-Sca~e A ~mdysis
Janssen, A, J, E, M. ([998), P¢~siti;ity and Spread of Bi}inear Time-Erequeney I)istributi(ms, i~ ~lhe ~,~/g1~erDi:~tribution - Theory," ;rod Appli(:atio~s in Sig'~d Ib'oeessi~g, W, t,'. G, Mecktenbrfiuker al~d F. Hlawatsch, eds., Amst;erdam: Elsevier, [ 58. Jm~ssen, A. J. E. M. and Claasen, T. A. (L M. (1985). On positivity of time-flequency distributions, I£L~JE {I}',ms. Aco~st., Speech m~d Sigud Proc., 33, 1.029 1.032. Jamtssis, A, D., Leodaris, A., |:-'atargiae;, N., Philippakis, T., Philippakis, P., and Vlachos, K. (1982). ApI)licat,i(ms of the positive-definite Wigner distribution function, Let:t:. N~ov. Cim, 34, 433-437. Jensen, It. E., Hoholdt, T., arid Justesen, J. (1988). Double ser}es represent:ation of bounded signals, IEEE 7}'a~s, Intb, Theory, 34, 613-624. Johamlesma, P., Aertsen, A., Cranen, B., and van Erning, L. (t981). The phonochrome: a coherent spectro-t.emporal representation of sound, H?:~a,riug Rr~s., 5, 123.145. Jones, D. L. and Parks~ T. W. (1990). A high resolution data-Maptive time-fl'eq~eney representable)n, IEEI~ 7k~ms. Acoust., ,~peech arid Siguad Proc,, 38, 2t27 2135. Kadambe, S. and Boudre~mx-BarteIs, G. F. (t992). A comparison of the exisi;ence of ;cross-terms' in the WiEner distribution and the squared magnit.ude of the waJ,'elel; trm~sibrm and the short-time ~kmrier trar~sform, .~JL>2 ~Ii'a~J~s,Ng~a.1 I:h'oc., 39, 2~198 2517. Kay, I. and Silverman, H. A. (1957). (h~ the uncertainty relation fl~r re~d sig~mls, .h~tbrm. Col~trot, 1, 64 75. Kay, S, M. (1987). M~der~ Spectr~d Est:imatio~, E~glewood Cliff~: Prentice° Hall. Kay, S. M. a.nd Boudreaux-Bart.els, G. F. (1985). Or~ the optimNity of the Wiggler distribut, ion tier deteceion, in 11;~; Int. Cou£ ou Acoust., Speech a~d Sig~)a+t Proc. 1CASSP-85, Tampa (FL), 1017 1020. Kelly, E. and Wishner, B, P. (1965). Matched-fil~er theory for high-velocity, accelerating targets, IEEE Tra~ls. MiL Elec.tr, 9, 56 69. Keshner, M. S. (1982). 1 / f noise, Pro<:. IEEE, 70, 212 218. Kodera, K., de Villedary, C., and Gendrin, R. (1976). A ~mw method fbr ~he mm~erk:at analysis of non-stationary sig~mIs, Phys. Earth aud Pla~. Mt., 12, 142 150. Kodera, K., Gendrin, R , a~d de Villedary, C. (1978). Analysis of timevarying signals with smalI tJT v~tues, liEh;E ~I}'al/s. Acousr., Speech and Sig~mt Proc., 26, 64- 76. Koenig, I~., Dunn, It. K., and Lacy, L. Y, (1946). The sound spect, rograph, •L Aco~.~st. Soe. Amer.. 18, 19 49.
Bibliogrwfl~y
373
KSrner, T. W. (I 988). t~burfer Analysis, Cambridge: Cambridge Univ. Press. Kozek, W. (t992). Time-Kequency sigmd processing based on the WignerWeyl Kamework, SJgnM Proc.~ 29, 77-92. Kronland-Martinet, R., Morlet, ,1., and Grossmamh A, (1987). Analysis of som~d patterns through wavelet trm,sforms, Int. J. Pattern Recogrt. trod Arti£ h~tell., 1,273 30t. Krfiger ,L (L and Poffyn, A. (1976). Quantum mechanics in phase spauce, I. Unicity of the Wigner distribution functkm, Pto,'sica, 85A, 8.4 100. Ku:mar, B. V, K. V. and Carroll, C. W. (1984). Performa.nce of Wigner distributkm flmetion ba~ed detection methods, Opt. Eng., :13, 732--737. Kmit, M. (1984). Traitemerit Num&ique des Signnux, Paris: Dunod. Kuryshkin, V. V. (1972). l,a m6canique qmmtique a:vec une fonction nonn4gative de distribution dans l'espace des phases, Ann. lnst..blcm5 1~ghwar(5, XVII, 81- 95. Kurysbkin, V. V. (1973). Some wobiems of quantum mecha.nies possessing a non-negative phase-space distribution %netion, Int. J. Theor. Phys, 7, 451 466. Lacoume, J. L. and Koflrmn, W, (1975). Etude des signaux ~lon st,a~ionnaires par la reprdsentation en temps et en D~!quence, Ann, ~'~dJfi!C()llllll,, 30, 231 23& Lamel, L. (1988). b))rmNizing knowledge used in spectrogram reading: a~oustical and perceptual evidence from stops, Ph.D. Dissertation, MIT, Cambridge (MA). Landau, H. J. a.nd Pollak, It. O. (t961). Prolate spheroidN w a ~ flmctions, Fourier analysis and uncertah~ty II, Bell S)'st.. Tech. d., 40, 65 84. I,e Chevalier, F. (t989). Prin(~ipes de Tra.itemem, de~ SiglJaux Radar et Sonar, Paris: Masson. Lemarid, P. G- (ed.). (1989). Les Onddettcs en 1989. Lecture Note,s' in M;tthmnath:~, 1438, Berlin: Springer-Vertag. Levin, M. J. (1967). lnstaataneous spectra and ambiguity flmctions, IEEE 2bms. In~~. Theory, 13, 95 97. Levshin, A., Pisarenko, Vo f'., and Pogrebinsky, G. A. (1972). On a fl'equency-time analysis of oseillations, Ann. Geophys., 28, 211~-2i& Lgvy-Leblond, J, M. (1973). Les in6galitds de Heisenberg, J. Phys., 14, Suppl. (educational insert), 15.-22. Lieb, E. H. (1990). Integral bounds for radar ambiguity flmetions and Wignet distribntions, ,I. Math. Phys., 31, 594599. Lobve, M. (1962). Probability 7J~eory, Amsterdaxn: D. van Nostrand Co.
374
l im(~-t~)~¢q~(~n~.~v/Time-ScMe AnMy,si,~
Longo, G. and Picinbo~lo, B. (eds@ {i989). Time and t@equency Repr( ,sent:~tion ofSig~mls ~md ~Sy,stems. CISM Courses and Lect~re~ No. 309, Wien: Sprmger-\,:erlag. LoughIin, P. ted.), (1996). Special Issue on Time-F~equ(ncy ,4nals~is, Proe. IEEE, 84(9). Loughlin, P., Pitton, J , and Atlas. L. (1994). Construction of positive timefrequency distributions, IEEE Trz~l~s. SignM Proc, 42, 2697-2705. L~mes, R. M, (1968). On the concept of the spectrum for nonstationary processes, J. Roy. Star. Soc. B, 30, I 30. MNlat, S. G. (1989a). A theory iSr multire~oIution signM decomposition: the wavelet rewesentation; tEEE Trans. l~a.ttern AnM. ~md Machine fnteII., 11,674 693. MalIat, S, G. (1989b). Multiresotution N~proximations and wavelet orthonormN bases of L2(~,), ~I}','ins. Amer. Math. Soc, 315, 6988. MMIat, S. G. (1998}. A I,:~;)welet lbur of SignM Procassing, San Diego: Academic Press. MMlat, S. G. and Hwang, W. 1-,. (1992). Singularil,y detection aM processing with wavelets, IEEI~ 'D'ans. Ira%. INeo;iv, 38, 6t7.-643. MMtat, S. G. and Zh~mg, Z. (1993). Mat,ching pursuits with time-fre
"
r
Bibli,~raph3
375
Martim W. and Flandrin, P~ (t983). Analysis of Non-stationary Processes: Short;-Time I"eriodograms ~'~rsus a Psendo-Wigner Estimator, in Signat Processing 1l -l'5eorie,s and Applications, H. Sc}~iissler, ed., Ams[erdam: NorCh-Hotland, 455 45& Mart, i~, W. and Flandrin, P. (1985a). Detection of changes of signa} struct:ure by using l;he Wignm:-Vitle speet.rmn, SignM Proc., 8, 215 233. Martin, W. and Flandrin, P. (1985b). Wigner-Ville spectrN analysis of nonstationary processes, IEEE 7}'a.ns. Acoust., Speech and Signal Proc., 33, 1461 t470. Masry, E. (1993). The we2,,elet transform of stochastic processes wil;h sl:a,,,, tionary increment:s and it.s applicatkm eo fractkmal Brownian mo/,ioll, lt;!EE 7bms. Nfo. TheoI), g9, 260 264. Matz, G. and ttlawatsch, F. (1996). Time-frequency tbrnmlation and design of optimal detect,ors, ir~ lf/~EENSP Nt. ~ymp. on Tim~>f}'equenqy and Time-Sc~fle knMvsis, Pa.ris, 21:{-..2t6. Mat.z, G., Hlawatsch, F., and Kozek, W. {1997). Generalized evolutkm... ary spectra~l analysis and tim Weyl spectrum of noustationary random processes, IEEE Tra.ns. Sig,_in,flProc., 45, 1520 1534. Mecklenbr}inker, W. F, C, (1987), A 3,~torial on Non-parmiletric Bilinear 'lPimed~¥equency Signal Representa.tions, ir~ 'lbfitemcnt du Signal / Signat Processing, J. L. Lae(mme (~t aL, eds., Les Houches, Session Xl?~:, AmsterdaJn: Nori;h-~HoIIand, 27? 336. Mecklet~brguker, W. F~ G. and Hlawatsch, F. (eds.). (t998). The Wigncr Dist.ributiou Theory and Applications in Sigmat Processing, Amsterdam: Elsevier. Mc}tard, G. (1978). Propri~:;tds du spectre 6votut;if d'un processus uon-st;at:io~lnaire, Ann. lnst. llem'i t%incarc; B, XIV, 411-424. Meyer, Y. (1990). OudeJ~,ttes ct Opdrateurs L Onddette.s, Paris: Hermanm Meyer, Y. ([993a). ~41tvelets: Ab_~orithms a.nd Applications, Transl. by R. D. I/yah, Philadelphia: SIAM, Meyer, Y, (1993b). Onddett~:s et A]gorithmes Colieurrelm~, Paris: Herm~,lIlll.
Miehaut. F. (1992). Mdthodc,s Adaptatives pour le Si~4nd, Paris: ttermt~s. Montgomery, L. K. and Reed, I. S. (1967). A generNization of the GaborHelstr6m transform, IEEE ~tbms. Intb. Theory, 13, 344 345. Mortet, J. {t982). Sampling Theory and \¥aa,e Propagation, in. NATO ASI Series, I.ssues in Acoustic Sigml/N:~ge Processing and Rec
376
Tim~.-~:}~qu(,uq;;J'/'im~*-Sc;~]e .~.md3',sis
Mourgu*,s, G. (~987). U~_ilisation du lbrm~disme de Wiggler en mdcanique statistique classiqu(; et eri traiteme~t des sig~laux (ertail~s, 'l'h~:?se 1)oet. Etar. ~:,sSo. Phys, Univ. C[ermo~t [l, Moyal, J~ E. (1949). Quantum mecha.nics as a statist, icat theory, Proc. C;~mbridge Phil. Soc., 45, 99 124. Mugur-SchSchter, M, {1977). The. Qmmt~ma Mechanical One-sys~.em D~r~ malism~ Joint Probabilities a~ld Locali~)~ in Quantum M~chanics, a It~d[ Cen~;m2~, Later, J Leite Lopes and M. Paty~ ~ts., Dordr(~cht: D. l~eMel, 107-446. Nawab, S. H. and Quatieri, T. F. (t988). Short;-Time Nmrier Transtbrm, h:~ Ad~'anced 2hi)its i~ Sit,tat Processing, J. S. Lira and A. V. Oppenheim, eds., Englewood CliffS: Pi~mice-IlalL 289 337. Nikias, C. L. aim Petroputu, A. P. (1993). tfigher-(.)rd~.~" Sl)ect~'~ A~laI~.'si~ A ?';(}m'/n~r 5ignd Proc{~s.~ing f?'~mle~ork, Engiewood Cliffs: PrenticeHall. O'Com~etl, R. F. and Wig~mr, E. P. (1981). Qua~ltum, mechanica.1 distribution flmctions: conditions for unktueness. Plt~. Lett., 83A, 145-1.48. O'NeitI, J, C. (1997), Shil~-cow~riai~t time~fr~quency distributions of discrete sig~als, Ph.D. Dissertation, Univ. of Michigan. Ova~riez, J. Ph. {1992). La transformation de ?delli~: un outil pour l'a~~alyse des sigr~aux h large bande, Th~?se, Univ Paris 6. Pagc~, C. H. (1952). lnsta~tmmous power spectra,, d. Appt. t'f~,.~:, 23, 103 106, Papa~dreou, A. and Boudreaux-Bartds, C;. F. (1992). Distributions for time-frequency analysis: a generalization of Choi-Williams and the Butterworth distributior~, i~t [EEE b~t. Cotdi on Aco~st., Spet~h and Si~'mf Pro(:. lCL4SSP~92, San ["Yancisco (CA), pp. V . I 8 I V.184. Papandre~m, A., ttla,za.tsch, F., and B(mdreauxd~artels, G. F. ([992). A ratified framework for the Bertrand distribution a~ld the Alws distribution: the new hyperbolic class of quadratic time-fl'equency distributions, i~ IEI~;'E-SP Nt. S}'mp. o~~ Tfm(,-I~eque~c~ ' and Time-S(:~de Am~tysis, Victoria (BC), Canada, 27 30. Papandreou, A., Hl~cwa~sch, F , and Boudreaux-Bariels, O. F. (1993). The hyperbolic ela;ss of quadratic time-frequency represent, ations Part I: constant~Q warping, the hyperbolic parMigm, properties, and merebers, tEEE 21}',ms. Signal Pr~c., 41, 3425-3444. Papo~flis, A. (1977). Sig'~a/Analysis, New Yc,rk: McCraw-ttill. Paul, T. (1985)~ Ondelettes et mdcaniqtm quantique, Thbse Doct. Etat ~s So. Phys.~ l.Ji~iv. Aix-Mar~¢ilte IL
Hibliogr¢q~hy
377
Pearl, J. (1973). Time, frequency, sequency, and their uncerlainty relations, IEEE Trans. InYo. (l'heoW 19, 225 229. Peyrin, F and Prost, R. (1986), A unified definition for the discrete-time~ discrete-frequency and discrete~time/frequeney Wigner distributkms, IEF2E 2}'ans. Acoust., Speech and SignM Proc, 34, 858 867. Pieinbono, B. (1977). El&nents de ~I5&;fie du Signal, Paris: Dunod. Picinbono, B. (1997). On instamaneous amplitude and phase of signals, IEEE Trans. Sig~ial Proc., 45, 552 560. Pici~lbono, B, and Martin, W. (1983). Representation des signaux par amplitude et phase instantandes, Ann. T~q~comm., 38, 179190. Pimonow, L. (1962). Vibrations en Rdgime 7}ansitoire, Paris: Dunod. Poletti, M. A. (t993). The deveIopment of instantaneous bandwidth via local sig~m,1expm:lsion, SignM Proc., 31, 273 281. Portnoff, M. R. (t981), Time-fl'equency representations of digitN signals and systems b ~ e d on short-time l%urier anNysis, II£tr;E [Ibms, Aeoust., Speech and Signal Proc., 28, 55o 69. Post<m, T. and Stewart, I. (1978). Catastr<sflie Theow and its Applications, Loudom Pitman. Pott;er, I~. K., Kopp, G. A., and Green, H. C. (1947). Visible Speech, New ~ r k : D. wm Nostra,nd. Price, R. and Hofstet~ter, E. M. (1965). Bounds on the volume and height. distribution of the ambiguity fimction, ~Ef:2h; I}'ans. Infb. [iH~eory, 11, 207- 214. Priesfley, M, B. (1965). Evohltionary spectra and non-stationary processes, J, Roy. Star. S'oc. B, 27, 204 237. Priestley, M. B. (1981). SpecaM An@sis and Time Series, New ~(~rk: Academic Press. Priestle?x M. B, (t988), Non-Line,at and Non-Stationary Time Series AnMysis, New York: AcMemic Press. Qian, S. and Morris, J. M. (1992), Wigner distribution decomposition and cross-terms deleted representation, Signal Proc., 27, 125-144. tlanmnathan, J. a.nd Topiwaia,, K ([992). Tinm-f?equency IoeNization operat;ors, in IEEI:).~ P ~Nt:. Syrup. on (ISme-l+eq~mncv and 2'line-Scale Ana(vsis, Viei.oria (BC), Canada, t 5 5 158. Ildnyi, A. (t961). On measures of entropy mid information, in Proc. 4th BerkeIw Syrup. on Mat.h. Star. and Proba., 1, 547.561. Riesz, F. and Sz.-Nagy, B. (t955). 1~hnctional AnaIysis, New ~%rk: Ungar. Rihaczek, A. W, (t968). Signal energy distribution in time and fl:equency, IEEE ~lYans. Int~. Theory, 14, 369-374.
378
l ?mc-t:)~..q~ency/'Thn<>S'cMo A t~alysis
Rihaczek, A. W, (1969), Principle,, of fli#b,Resolution Radar, New York: McOraw-tlill. Riley, M. D. (1989). Speer'h TJme-th'equenc~ Rqn'esentat~ions I)ordreeh~,: Kluwer. R%ul, O. a:~ld F[andrin, P. (1992). ~Iime-sca.le energy dis~ribugions: a gen..~ eral class extending waveiet transforms. IIt@)E 7/hns. SJgr~at Proc., 40, t746- ] 757. Rciyer, A. (1977). Wigner fun(tion as expectation value of a parity operator, Phys, Rev+ A, 15, 449 ,150+ Ru~eri, G+ J+ (1971)+ On phase-space description of quantum mechanits, l)r(:~g T~Jeof PtO:,., 46, 1703 1712. Sayeed, A. M. and Jon(~, D. L. (1995), Optimal detection using generNized bilinear time-frequency and time-scale ~epre,~en~a~ions, IEEE Jbms. Signal Proc., 43, 2872- 2883. Schrmder, M. R. and A~.al, B, S. (t962). Generalized shorb..time power spec{ra and ~mto~correta.tion fimctions, J. Acot~st. Soe. Amen, 34, 1679 1683, S<ssarego, J. P., Sageloti, J., Degoul, P., FIandrin, P., a~d Zaktmria, M. (I990). Analyse temps-frdquenee de signaux en milieux dispersit~ appliea~ion 5. l~tude des ondes de lamb, J. Acoust., 3, 273 280, Shenw, R. G. (199t). Group rept<~senta~[ons and optimal recovery in signal modeling, Ph.D. Disserta,tion, Cornelt Uuiv., Ithaca (NY), Silvenna.n, R. A. (t957). Locally stationary random proe~?:sses, IRE 7}'ans, tli~o. Theow, 3, 182 187. Slepian, D. and D~ltak, It. O. (196i), Prolate spheroidal wave f*mctio:ns, Fburier analysis and uneert, aintT I, BdI Syst.. 7bch. ,L, 40, 43 64. Smith, M. J. T, and BarnwelI, T. P, (1986). Exact reeonsI;rueihm techniques for tree-structured subband coders, IEEE Tral~s. Acous*.., Speech and Sig=M P,>c., 34, 434441. Speiser, J, M. (1967). Wide-band ambiguffy functions, IEEE I>a.,ks. In&. 7Neory, 13, 122 123. Springborg, M- (t983). Phase space flinctions and correspondence rules, J. Phys. A, 16, 535 542. Srinivas, M. D. and Wolf, E. (1975). Some nonclassical features of phasespace representat, ions of quantum mechanics, Y'h,:~. Rev. D, 11, 1477 ..... i485. Storey, L. R. O. (1953). An investigation of whistling at,mospheri(s, l~il. 7hns. Rq}: SZm. A, 246, 113 141.
BibfiogJow)hy
379
Strang, G, and Nguyen, T. (1996). ~A':avclets and t~ilter [hmks, Bost;on: V,bllesley-Canbridge Press. 'lhqqm M. S. (1986). A Bibliographical Guide to SelgSimilar Processes and Long Ralge Dependence, in Dependence in Probability and Statistics, E. Eberlein and M, S. Taqqu, eds., Basel: Birkh~iuser, 137- 162. Tewfik, A. H. and Kim, M. (1992). Correlation structure of the discrete wavelet coefl?.eient,s of fl'actional Brownian motions, IEEE 2bms. Nfo. 23eo~iy, 38, 904 909. Thorn, R. (t972). StaNlit(~ Structurette et Morphogen}~se, New kk:,rk: Benjamin. Tjostheim, D. (1976). SpeOrM generating operators tbr non-stationary processes, Adv. App1. Prob~, 8, 831 846, Torr~sani, B. (1995). Analyse Continue par Onddettes, Paris: Inter,_gditions / CNRS Editions. Unterberger, A. (1984). The ealcuIus of pseudo-diflbrential operators of Fuchs type, Comm. in l~5~rt. DiK Eq., 9, 1179 123(3. Vaidya,tmthan, P. P. (t993). Multimte Systems and Sig~aJ Pn)cessh~g, Englewood ClffN: Prentice-HM1. Vakman, D~ Y. (1968). Sophisticated Signals ~md the UncertMnt,y Principle in Radar, Berlin: SpringerA:erlag. Van der Pot, B. (1946). The fundamentM principles of frequency modula~ t.iom Proc, h~E, 93, 153 158. Van 'iYees, H. L. (1968), Detection, Estimation and Modulation Theory, New York: J. Wiley & Sons. Vetterli, M. (1986)~ Filter banks Mtowing perfect reconstruction, Sit, m1 Proc., 10, 219 244. Vetterti, M. and Kovacevic, J. (1995), Wavelets and Subband Coding', Englewood Cliff:s: Prentice-Halt. Ville, J. (1948). Th6orie e~; applications de la notion de signal analytique, (Nbles et 2hnsm., 2 A, 61 74. Ville, J. A. and Bouzitat, J, (1955)o Sur un type de signaux pratiquement born6s en temps et e~ &dquence, (Nbk,s et 7}"~msm., 9 A, 293-303. Ville, J. A. and BouzitaL J. (1957). Note sur un signal de dur6e finie et d'dnergie filtrfe maximum, C~Nes et Transm., 11 A, 102 127. Weyl, H. (1928). Gruppentheorie und Quantenmechamk, Leipzig: S. HirzeL Whalen, A. D. (1971). Detection of Signals in NoNe, New York: Academic Press. Wither, E. P. (1932). On t:tte quantum correct.ion tbr thermodynamic equilibrium, Ph3:s, Rev,, 40, 749--759.
380
Time-l: }'cqucr~cy/'!I'in~o-Scale A n~fiys:is
Wigner~ E. P~ (I971). quantmu-Mect~micat Dist~ribut.ion l?tmct,io~:ts Revisited, iIi Pe:rspec[iv~:s ill Q~la~fl:,~lu~ 'tJ~e!ors" , ~¥. Yourgrau and A. ~m der Merwe, eds., New York: D~}ver. Williams, W. J., Brown, M, and Hero, h~ (1991). U~icertafitty, information, and time-frequency distributions, Proc. SPIE, 1556, 144-[56. Williams, W. J. and 3eong, J. (1992). Reduced [merference Timc-FYequency Distributions, ht Tim~:yf}'eqnenqy Signal Amt1?.ad;s-....Methods and AI)plications, B. Boashash, ed., Melbourne: LongmaI~.-Clmshire, 74 97. Woodward, P. M. (1953). Prob~bili~y and hfformation "I]moO~wif.h Apt~ti... cation to Radar, London: Pergmnon Press. Wornell, G. W. (1990). A Karhunen-Lobv~-tike expansion fc,r I / f processes via wavelets, IEEE ~I~'~ms. Int}). Y3¢x~ry, 36, 859-861. Wornetl, G. W. (I993). V*q~velet,%ased represental,ions for the 1 / f family of ffa~:tM processes, Proc. I~;?E, 81, 1428 1450. Yaglom, A. M. (1987). Corrdation Theory of StationatT and Rtqat.ed Random Functions, 2 voL, New York: Sprlnger-Verlag. Z~kharia, M., Flandrin, P,, Sageloli, 3., and Sessarego, J. P. (1988), AnMyse temN-t?,}quence apptiqude g la earact6risatkm a~'oustique de cibl(~s, J. Aeoust., 1, t85-t88. Zhao, Y, Atlas, L. E. , ~md Marks, R. J. (1990). The use d (:one-shaped kerneIs fbr generalized time-frequency ~presentations of nonstationa,ry signals, IEEE ~/}ans. Acoust., Speech and Signal Proc, 38, t084-I09[. Zhu, Y. M., Peyrin, F., Ooutte, R., and Amiel, M. (1992). Anaty,~:~spectrale loeMe de I'image par transfbrma.tion de Wigner-Ville, ~/?aitem~nt du SignM, 9, 281-- 289. Zue, V. W. and Cole, R~ A, (1979). Experiments in spec~;rogram reading, in IEEE Int. Cont: on Ac(~ust., ,~)~ch al~d S#,~~M Proc. IC~4SSP-79, WeMfington (DC), i t 6 119.
Index
The uumbers in boldf~tce refer to the definition of the reievant term, or to its first appea.ra~ce in the text, Autocorrelation function fl'equential 191 local 40, 189 time-frequency 193 Autocovari~mce function 37 Basis continuous 57, 67 discrete 66 dual 75 overcomplete 69 Bertrand distribution 63, 115, 149, 180, 212, 249, 349 Bias 282, 329-330 Bias-variance trade~off 285, 288, 328, 334 Born-JoMan distribution 64, 111, 204-205, 265-267, 305 Browniall motion 168, 296, 331 fractional -motion 330-33t, 355 Bmterwor~h distribut&m 110, 268 Canonically coI0ugate 17 Catastrophe ....... of the Wigne>ViHe distribution 237, 304 cusp 241~242 elementary 238, 243 ibld 240
Ackroyd distribution 1i0-111, 180, 204~205 Adapt.ed smooehing 269 Adaptive method 45, 47, 181 Admissibility cot~ditiol~ for v.~w,'elets 77, 83, 133, 332, 340 Affim~ cla~ 108, 112-1t3, 132142, 150, 17~180, 281, 330, 340°342 ..... group 76, 107 smoo~hing 269-27t, 298 Airy Nnction 240 Ambiguit,y Nnction in Woodward's sense 100 invaria.m, votume property of 255 narrowband 100, 107, 191, t96-I97, 231-232 wideba~td 101, 17,9, 207, 211212, 302 Approximative dimension of a sig~ nM 24, 54 ARMA modet 158, 331 Atom 66 t:ime frequency 44, 53, 71,252 time,.,sca,le 77, 83 Auiocorrelation ftlllCtiOll 37, 4xI, 193 381
382
Tim~'-b) cq~aency/'Tim<~-Scale AnMysis
Causatit3 60, 119, 158, 173 Chirp 15, 32-33, 43, 47, 58, 102, I30, 2t7, 293-294, 347-348 ChoY-Wittiams distribution 111, 261 ..266 Cohen distribution 296, 3(N CohetFs class 62, t06, 111, 131132, 15I, 281, 35I Commutation reiation 17, 65, 196, 2t)4 Constant-Q 51, 78-79, 271,329330 Correlation ......function 99 long~range 331 rMius of 41 time-frequency 100. 187, 231, 346 Correkgram t90 Correspondence rule 64, 197-200, 301 Cram& d~:omposition 38, 152, 156, 17t Cross-term (see interfere~lce) D-distribution 115. 150, 249, 25t Density critical sampling 20, 56, 74 power spectrum 37, 44 probability 58, 117, 127, 197, 290 pseudo.~ 57 Detect,ion binary problem 97. 3 4 3 locally optimal 349 time-frequency receptors for 351-352
Differential spectral analysis 225, 3O3 Dispersion (see measure of supporK) Dispersive frequency propagation 336
t)i~s~ribution enel.~y 44, 53, lO{i, 116, 132 marginal 58, 62-63. 116, 133, 143, 145, 1.55, t58. 262, 337
l)oppler etfec~ 99-KKI, 207, 348 Doppler Iolerance 347 Energy content ra I;ion joint measure of 273, 3 1 5 measure of 18 Enscmbte a.verage 64, 151 Ergodic hypothesis 280 Expectat, ion value of mr operator 17, 195 Fi>urier duality 13, 71 scNe 208, 210 transfbrm 10, 46 short time transform 50, 56, 70~ 71-74, 80, 100, 177, 214 Fourier-Stieltjes 38 Fractal dimension 33t Frame 69, 76 b'requen(V dispersion 219 - o p e r a o r 17, 195, 198-199 beat 32, 35, 275-276 instantaneous 2&30~ 33, 36, 47, 59, 127, 146, 165, 223, 26I~ 29}: 337 mean 2 9 - 3 0
Gabor decomposit.ion 54-55, 75, 10f1, 177 Glauber-Jordan Lemma 196 Grenier spectrum 1 5 9 Gr,:mp deb\v 31, 33~ 36, 128, 139, 146. 222, 291 Heisenberg-Gabor uncertainty 14-18, 46, 80, 298, 311 Hermitian fimction 299, 317, 319, 32t
Index rlilbert ~;ra,Hsfbrm 28, 120, t65 tthlder regularity 338 Hurst exponent 355 Hyperbolic cbzss 212, 302 modulation 63, 211 Image inethod 274, 305 Independent component analysis 279 Inequality of concentrations 24, 46 Heisenberg-Gabor 14-i8, 46, 80, 298, 311 joint 311,324-325, 327 Instantaneous amplitude 28-30, 122, 15~, 337 - frequency 28-30, 33, 36, 47, 59, 127, 146, 165, 223, 261,291,337 spect;rmn 50, 53, t74 Interference ..... formula 232°234, 246 of the Wigner-Ville distribution 228 tone (see beat frequency) inner 232-,,234 outer 232-234 Invertibility 120, 274 Janssen's formula 233-234 Jitler 300, 350 Karhunen spectrmn 154 Karhunen-Lo~ve decomposition 39, 153, 345 Kernel .....of an operator 199 reproducing 68, 73, 10t Kodera method (see rea~ssign~ rneilt) Laguerre polynomial 317 Legendre trm~sfonn 327 Levin distribution 60, 111
383 t,itttcwood-Paley decompo8ithm 83 Localiza~ion ....to a curve in the t)tmte 130, 136-I39, 217, 22,1, 239240, 2dd, 247, 325, 348 energy -, in a domain t8, 315, 346 Lot}re condit, iot~ 39, 41 "Logon" 54, 57, 66, 278 M arinovie- Aires dist,ribution 211212 Maiched filtering conventional 99, 10l time-fl'equency 343 Mean spectrum 168 ensemble (see ensemble average) ge~er~lized '~tlue 199-200, 250 geometric 248, 25t Mellin scale 210-211 transfbrm 210, 302 ~\'| OlIlelt |;s
global of a distribution 202, 205, 292 local of a distribution 127128, 130, 140, 165, 222, 291,337, 341 ~mishing (see wavelet) MoyN's %rmula 126, 146, 181, 233, 297, 344, 346 Muttiresolution anMysis 84, 178 Noise impulsive 169 white 37, 156d57, 167, 288 t / f 329 Operator associated with a function 198
Operato~ of ~;ime~frequer~cy k~caliza~ tion 316 dilatio~ 108, I20, 206, 301 displac( merit 244 shift I04, 120, 195 truncation 19, 60 Page - distxibntion 60, 111, 148 ..... speu;trum 166, 172 Parametric modeling 44, 47, 158 Paving of the time-frequency piane 54-55, 328 Penrcey fun(tion 241 Periodogram averaged 285, 328, 330 short-time 284 Phrase • displacement. 275 information 221, 275 jump 276-277 method of stationary (see s~a-
tiollars' ) Priestley spectrum 62, 154-155, 18t Product duration--bandwhtth 15, 22, 34, 258, 299, 3l %322, 326 twisged 201; 3(11 Prolate spheroidal w~:e flmction 2 2 , 56, 17"4 Pyramidal algorithm 91, 93 Q-distribution 211,302 Quadrature --- filter 28 - mirror filter 87, 89~90, 96, 178 Quantmn mechanics I6, 46, 58, 64, t76, 19,1, 326 Rayleigh-ehannel 347 Reassignment method 222, 224~ 303
Ileduced i~terti~rence dis~,ribmioIi 261, 269, 305 t{6nyi h~t?~rmation 278, 306 t{esoluti(m of the idengity 68, 77 Rihaczek distributioi~ 60-(il, 65 111~ 175, 245-247, 291,327 s p e c t r u m 164 Robustness of time-frequency receptors 357 Sampliug time 19, 29, 46, 226-227 time-~}'equency 54-55, 73 Scale dyadic 81 Dmrier 208, 210 Meltin 210-~21t Scalil~g factor 76 function 85, 98 operator 108, 120, 206, 301 rule 330, 338, 340 Scedogram 113, 269-271,330 Self similar six,christie procesxs 330, 355 Sigual (see also stationary) analytic 28, 32-33; 1t4, 226, 252-253 baadlimited 12, i9, 22-24 harmonizable 39, 47, 160 molu)chromatic 27 multicomponent 31 252, 277 i,arrowbaad 32, 140 oscillatory 154, i70 unifbrmty modulated 41, 47, 155, 158, I67:295 with;band 62, t(tl, 1t4, t94 Sig~lal-to-m)is4:~ ratio (SNt{) 30(/, 343, 352 Sil~gularity of the Wigner-Ville distribution 238
h, dex
Singularity ew)lutional3 340 local 338 Sor~agram 51, 174 Spectrogram 52, 111, 174, 213, 22D225, 255-256, 284, 328 Spectrum evolutionary 43, 175 instantaneous 50, 53, 174 re(an 168, 328, 330 power den,sity 37, 44 Sta~,ionary ,, deterministic sig~ml 31,128 ..... in the wide sense (see weakly) cyclo- 191, 301 degree of non behavior 219 dist,ance f?'om the case 334 locally 40-4I, 47, 1(]7, 295 method of phe.se 34-36, 47, 234-235, 238, 303, 326 piecewise 158, 334 quasi- 42, 45, 280, 29(i tangential 280 t,ime of behavior 280, 334335 weakly 36, 47, 152, t56 Storey band 303 Support conserv~tion of 123-1.26, 1,45, 216, 259, 266 measure of 14, 15, 18, t95, 219, 258, 273, 29D293, 311, 3t3 Symmetry 214-215, 242-244 Theorem of Balian-Low 56, 74, 81 Bedrosian 30, 47 Hudson 294, 306 Wiener-Khinchin 37~ 161 -Wigner 143, i8i, 297 Time operator 17, I98-199
385 Thneq'requency at,ore 44, 53, 71,252 jitter 300, 350 shift 104, 1(16 synthesis 322, 354 Time-scale atom 77, 83 ]]estheim-Mdlmxt spectrum 157, 172 Uncertainty principle 14, lfi, see also Heisenberg~Gabor Unitarity 126, 139, 146, 269, 274, 344 Unterberger distributkm 115, t80, 248, 25t, 340 Wa ve/et admissibility condition of 77, 83, 133, 332, 34(1 Battle-Lemari6 94 eol~tinllOils tralisform 56, 76, 80, 332 Daubechies 95-96,974}8, i02 discrete transform 81 Haar 83, 94 Lit,tlewood-Paley 83 Mexican hat 81 Morlet 82 orthonormal .....hams 57, 83 regularity of 95 vanishing moments of 95, 333, 310 Weyl quantization 200, 202, 206 -. symbol 200, 301 Weyl-Heisenberg group 71 Wim~erd~6vy process 168 Wigner distribution afline 113,115, 142-I43, 149, 180 composite 305 s- 111, 124, 147-148, 188, 245, 322 scale-invariant 211
Wiguer-V i~l(~ specBrun~ 164. 20{}, 29{}, 3{)0, 350 positive spet'~ra 295 WigI~er-Ville distribution 58~ I{)6, t1I~ 113, ldS, 175, 187,200205, 213, 227, 266, 271,294 adapted smoo{hi~lg of 269, 3O5 a[Iilfe smoo~hed pseudo- 270, 305 discr{~t~-time 171, 225, 279~ 303 generalized (see ;~-\Vig~terdistribution) interDrences of 227-228 positive smoothing of 297, 320 pseudo- 215, 218, 260, 264, 275,277, 302, 335 smoothed pseu&}- 257, 259, 286 sm()olhing of 187, 25.~L 307, 31.2, 351-352 \\,~)td decompositioIl 156-157 Woodwind 100 Yomig's ira,quality 327 Zak transform 75, 177 Zhao-AtIa~Marks dist,ributio~ 112, 267
