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...
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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 (19701980) 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 timefrequency 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 Hmeti'equc~cy amd)~Js of signals. This am~lysis involves unfolding the signal in the timefrequency 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
TimcFrequencL/i[SmeScale 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 pseudodifferentia! 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 timevarying 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 wellsuited 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 Ville distribution (naxrowbaad) ambiguity ['unction Cohen's class, zparamet~er function f agine class, parameter function f timefrequency 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 HeisenbergGabor uncertainV principle (Subsection 1,1.2), which stipulates that a signN cannot be concentrated on arbitrarily small timefrequency regions. The second result (theory of SlepianPollakLandau, 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 timefrequency 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~imeli}'c~quonc_tg/TimeScah , Anal;~=~;is
10
!il.1. The TimeFrequency 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(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 timefrequency 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. HeisenbergGabor 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 IB~2, +B/2}, we would also have
(;hapter 1 Tim TimctCrcquency 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 timetYequency 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 socalled 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 CauchySchwarz 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 ThneFr~.que,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 righthand side holds, t~s both operators [ and P are selfadjoint. 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  .  47r
1.1.3. SlepianPollakLandau Theory As flmdamel~ta,1 ~s it is, the tteisenbergGabor 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')

JT~2
I~,(~.)l~zt: and
~::'x(B) '
a ,.,,,B/2
ix(~,)t2d~,.
(1.6)
C/mpt~er I T h e 't'imet;¥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
(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.imefrequency 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 imch)~equ~>ncy/'*l'imeScale :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 timeinterval. 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:iuicl;?'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 ,.Cfa.([lomt, the ratio ft(B, T) of the energy of the twicetruncated 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)
77met ?'equet~qv/'7:imeSca.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.ionbandwid~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 durationbandwidth 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 aplatieE k,, n~.{>
z.
2 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 timefrequency 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 flequency), what is the best possible concentra.tion in fl'equeney
(.ha~ ter I The ~l~meFrequency 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 timefl'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) 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 timefre(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 TimcF~('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 limeD'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 "modulusphaemean 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 zeromean.) 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}
:
~ :,,(0) .
(1.a~)
Moreover, it. is posi*:i,(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 FourierStiettjes 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 TimeFrequenqv 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 frequencyfrequency 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 Fouriertype relation
[12
This equation generalizes tile WienerKhinehin 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 ~ 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 quasistati(Tma(~,. §1.3. T o w a r d s
TimeFrequency:
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 timefrequency 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 timefrequency representation. This Nso Mlows us to ~ some further notations.
(,?hat)ter 1 The I~mel?rcquex~q}, Problem
43
1.3.1. Tile TimeFrequency Plane and Its Three Readings The wealth of the timefrequency 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 crossrelations. 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 timefrequency 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 frequencyl)yfrequency 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 timefrequency plane. In principle, tim two dimensions of the timefrequency plane Mlow us to draw a separating curve between the two components, while this cannot be achieved in a purely temporal or frequentiM setting. Timefrequency. 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 timefrequency 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 timefrequency 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 %linefrequency aoms," play the role of a b~sis of the decompositkm. Within the picture of a linear decomposition, the timefrequency representation is given by a {discrete or continuous} set of weights, each being associated with one atom. Moreover, th 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 timodservation. 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 "shorttime" horizon or t.he validation of the ;~sumptions of the quasistatkmaD ~ 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 quasistationarity, 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 quasistationary ceuse. They can be called joint methods, as they treat the time and the frequency symmetrically, although they are explicitly based on a timedepeIMence.
Adaptive and evolutionary, 1~ In the setting of a p~rametric modeling we can adopt two viewpoints ibr the introduction of a timedependence, 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 HeiserlbergGabor 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 timefreq~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 Fimebi'~,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 slmrttime Fburier transf?~rm or Gabor decomo position (in Subsection 2.2.2) a,nd the (timeseMe) 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 timet~equency and tile thnesca.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~;/TimeSc,~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 WignerVitle; 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 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 ~, = "'//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 timefrequency 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 welldefined 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 timefrequency), 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 timefrequency 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 shorttime spectral analysis). It was a slow process to move it closer to the more fimdamental concepts such. a~s energy distributions, the WignerVille 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 shorttime window') needs to be endowed with physical rank and interpretation.
l~imcb}'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 timet?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.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 ,(i.)d~ d#,.;(~:;\)
:I.~
Hence we find
It;:
where we have i:i~roduced :,he socal: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# matic waves. By w~ay of eou~r~st, the shorttime 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.
Shorttime ~))urier transh)rm: timefrequency atoms.
The azmlysis by the shorttime l?burier transform (Eq, (2A8)) can be regarded as a projection of the a~lalyzed sig~al onto "timefreqllency 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 shorttime 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 shorttime 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 shorttime 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 frequencyshifts. 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 shorttime Fourier transform is an admissible representation if the analysN
Timef?'cqueucy / TimeScale 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 shorttime 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 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 "timescale 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 (soid 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 leftinvarim~t measure is defined by 2~;
d# ~{t, a)• :
(it da ({2
Hence, we can rewrite the cosure 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/TimeSt 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 shorttime 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 constantQ 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,
= 5i,;?;
eonst;ant.
Although the wavelet transtbrm is a timescale representation in the first place, it also admits a ~;imefrequency 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 shorttime Fourier transtorm,
'Timeb}'equencv/TimeScMe Analysis
80
which ofli:~¢s an identical resolution at each point of the plane, Indeed, an anNysis with constantO fil~,ers has a firm frequencyresolution 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~Nuencyreso}ution. Whichever point in the timefrequency plane is considered, a tradeoff of Heisenberg type (cf. Eq. (1..4)) between time and frequencyresolution 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,ntQ 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 shorttime Fourier transform contain the same information, as both ~*~pres*ent the signal in a on.toone 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) O, ao>O,
Ch~:~pter 2 C/asses of Solutions
S]
l
I
0t ................................
+,:~
=IH(
One can show in the same way that

.
(..4,~)
This shows that the coeNcients of the approximation and the detail at a fixed resolution leel 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 =
/
~sa: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 durationbandwidth 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 crossambiguity 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 detectionc~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'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 timefrequency and timescale 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 timefrequen.cy representation must "tbHow" the signM, whe~ it is shifted in the pMne. The constrair, t can thus be written as
pVille 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). Timescale. The definition of a ge,mrat cla~s of bilinear timescale representatimls parallels the Umefrequency 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" ' ' '
:./~,) 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
Ddist, ribution
(~/~,o) 2
.+ ?
ei2rruT~
77m+>N'
=
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 ar,slations. Tile mos~ important covariance to be called ior refers to t,he translations in tile timefrequency 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 timefrequency 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 wellknown 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) 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'imeScale 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 sWigner 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
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 timetrequency 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 timefrequency 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 singlecomponent signal with frequency modulation, a '*good" timefr(: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/TimcScale 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~ ") ,frequency ca.~< We therefore list only some of its specific properties, Enor£?:
A timesca,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 timefrequency 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"imef~reque~cv3/Time'Scate
Analysis
Provided that th.e parameter fun~ tion satisfies ~9(0,~e) = 5(u .......u0)
.
.
.
'_2
.
I{? x
e~2:~d*Freq u(,tJqK/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). Notrm 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 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 byproduct. 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 .
TimeFreq ue,ic~y/'l'imcSeale 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 timefrequency 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 righthand 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 iirstorder 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. sWiguer. The sWigner distributions are unique soh.ltions of diti;~rent collections of constraints: (i) 'l'he srWign~" 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 sWigner distributions are the only timescale 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 sWigner distribution in its timescale form, where the scaie a is a~ssociated with the frequency ~, = *Jo/a.
TimVille 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 .firstorder 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 WignerVille 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 firstorder moment. This result is dual to the previous one and established analogously. (v) The W~gnerWlle 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 IV/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
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 timedependent function, a.nd this leads to lhe forma;l definition of a timedependent "power spectrum" by K , , ( t , ~,) = le(t, t')l 2 F~(zJ) . (2.14{1) \¥e thus dispose of a truly timedependem, 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*byDequency 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 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] 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~et+(?que~,.::y/limeScake 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~; 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 sWigner 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, XWignerVille s p e c t r u m . Within the chess of sWigner spectra, ~he task of finding conditions ibr the uniqueness of the WignerVille 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 WignerVille :~)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~erViIle spectrurrJ is the on.(y sWither spectrm~ whose local hrst~order momel:~t (center of gra,vity) in t.he frequenqV domain is the m = 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 WignerVille 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~>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
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+? The conditions on the density of t.he sampling lattice for the shorttime 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 timefrequency 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 "cycleoctave 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 BattleLemari6 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 timescde 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 ~imescale class with a tim~>fl'equency interpre{.ation (an issue to which we will crone back later),
180
Ti*~i(f)eq~e~lc3'//JimeScMe 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 BoudreauxBartels (1992), Auger (t991) and Boudrem~xBarrels (1996). IIistoricMly, the Aekroyd distribution waa proposed iIl Aekroyd (1970), die ,sWiggler 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 BoudreauxBartels (1992). in a sense the BornJordm~ 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 timefrequency" distributions is due to J. Bertrand and P. Bertrand, They were the first: to conslmet a family of i;imefrequency 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 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 timefrequency representation is a severe obstacle. It sometinms renders a l.horough interpretation of the crossterms 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 secondorder 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;/TimeScate 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.vailabe (timefrequem~y, t i m ~ gi~?, t?equencyfr(~quency, and fl'equen(;yt;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
Timefrequency. 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 WignerVilte 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 tirnefrequency plane,
Pz where a reference object (the WignerViHe 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 timefrequency 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 WignerVille distribution; that is, it might not con~spond m a bivariate filtering of towpass type in the timefrequency 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 WignerVille 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/TimeScafe A nalysis
Remai'k 3, Let us fina.llynote that the central role which the WignerViIle 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 sWither 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 timefrequency parameter fimcgion a~:cording to
1]( ~wab~e came, we can write
=
,.
r) e
~ ,
dr
wid~
~ (t,r}=ta11/2
..... g
~
x
s+
s8
ds.
J 2"*4
We thus see that the timehorizon 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~eequencyfrequency. The preceding de~::ription in the timetime sett.ing h ~ a naturM com~terpart of l)'equencyl}~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 timetime 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 socalDd Wclosgationa.O, signals, which are characterized by periodic stocha,;tic properties without being stationary. :~ l~¥equeneytime. The fburth ir,terpretation concerns the fix,quencytime 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 twodimensimrml ~ u r i e r t r a ~ J o r m of the WignerVille 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
_
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 ab, Anaivsis
198 by employing the defimtion o((,
r)
=
VilIe, Ackroyd, or BornJordan 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 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 soformed 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 timescale 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 WignerVilleW~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 'operatorsymbol' 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 WignerVille 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:sis
214
situa,tion is in force for t:[~e WignerViile 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 WignerVitle 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 WiglmrViIle 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 WignerViiIe 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~,¥ignerVille. 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 WignerViile 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 pseudoWignerVille distribution. ~s While it stays in the spirit of the WignerVille transform, it is a (movb~g) shorttime 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 pseudoWigm rViile 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 pseudoWigner~ Ville distribution using the preceding notion of a "local mirror image," which is nov., restricted to a shorttime neighborhood of the point of the evaluation. Indeed, we can e~sily see that (3.75)
Timeth'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 pseudoWignerVille, 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 WignerVitle 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 shorttime analysis (spe~.rogram or pseudoW~gnerVille) 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[gnerVllle 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 pseudoWignerVille 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 shorttime 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'smo(~llmd version of the WignerViiie distribution, we can write
L,(s; ~, ~,) &
s,,(t, x,) = x.
TJmeF}'~quer~qD/~l~meScaJe Analysi.s
222 where we let q :~
f
&(s;t,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 crossterm, 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 WignerVille 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 timofrequen( 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 timehequeiwy 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 WignerViIle 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~!erViile transfbrm of two atoms. If a sig~at is the superposition of two atoms, the Wigi~erVille 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 crossterm given IB,
L: :,,+(t,
.)
= 2
rain A~. (~, r) = ,~'~~~ (~, ~) + A~.~ (~, ~') + &~.,.~ (~, r) where Dora the Dmrier relation of Eq. (3,17) between the WignerVille 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 crossambiguity 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 crossterm 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 timefi'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 timefrequency dist.ance of the two signa;ls. Certainly, this result conIbrms to the interpretation of an ambiguit.y function in. terms of tinmfi'eqummy corrdatio.as: If two comp,> nents are locMized, to neighborhoods of tile points (tl, u~ ) and (t~, u~) in the timeiYequeney plane, the essentiN contribution to their crosscorrelation involves those shift, parameters, which put them in an overlay position; these correspol~d to tt~e time aad fi'equencyseparations At and Au between the two components. By combining the crossambiguity 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 crossterm 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/TimeSc?~.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 crosbterm 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 timefrequency 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 WignerVille 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 interibrences. We adopted both notions only tbr the sake of (xmvenience. ():m can observe that they overlap and complement each other by inspecting the socalled 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 WignerVilte 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
WignerVille and inner interferences,
}br a noMinear frequency modulation (which is of simmoidal type in the example), the WignerVille 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 WignerVille 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 ?/TimeScate 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}gnerVille distributions of the components a:(t) and y(t) are concentrated in regicms around (~'1,~/I) and (t2, ~.'2), ~heir crossterm 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 wellsuited 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 timefrequency 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 WignerVille 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 sidecondition
T
t,5:

,
236
'l h~wJq'~ quency/"FhneScale 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 WignerViile distribution ha~ noilnegiigible rabies at Mt points (t, ~) of the timefrequency 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~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 WignerVille 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¢=~(,,:) ~: ......+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 Ville and cusp singulariti~>s
The WignerVille 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
WignerVille and higherorder 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 WignerVille 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 WignerVille 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 WignerVille 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,
Timeb}~'qu~.~( q/ Fim,eScale 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 WignerVille distributiol~ must be perfecdy iocNized to linear chirps. Turning this argument upside down, we conclude ~.hat. the WiguerVille distribu{im~ can only be }ocalizcd to a. curve in the timefl'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 WignerVille distribmion as
t'~,tt,~ ) = 2
;r(2t. r)e il~:'~''ri.  c'(r)dr. •
..
rent ways tha£ only the WignerVille dis{,Hbugion admits a perfect 1ocaliza~.ion to a timew~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 timefrequency 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 crosstetras, Their geome{ric interpretation, however, differs va~stly from the corresponding principle of the WignerVille 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 timeflequency 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 crossterm 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 kuowedge 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 timefrequency representatMn (~e = ~eo/a) has the form
tt can easily be seeI~ that the crossterm ~>s~miated with two pure ffequencieL~ :rl(t)
=
.... t ,
e
is given by
= 2~,
t +
6(~,,
u~.) X ~
 >,
e e~'(~'
'/"!* ds.
This y~eds 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 Ddistributions (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,. (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,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 timefreqummy 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 WignerVille 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 crossambiguities 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 timefreqummy plaim. This very generM interpretation of f({, r) as a weight Nne~ion states that, we should choose f so that it suppresses the erossambigui V terms
Chapter 3 Issues of" II~terprct:af, ion
255
(which are natm:ally eccentric) and preserves the autoambiguities (concentra£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 timefrequency 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 crossterms that are created tty components with a timefrequency 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 WignerVille distril:mtion, the latter is located at the midpoint of the stndght [iue com~ecting t}m timefrequency 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 timefrequency distance>~ For fixed c~, the present couplfllg of the smoothing in the time and fie° queucydireetions 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 shortt.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 timefl'equeney smoothing by a, spectrogram is based on only one "degree of freedom," as it employs a unique shorttime 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 timefrequency 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, timefl'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'eudoVi.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 pseudoWignerVitle 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 7meb }'eque,~??/TimeScale A ~:ag'sis
258
HeisenbergGabor 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)
he normalized l)irae distributimJ (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 crossambiguities 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 ChdiWil/iams distribution from reducing the il~terference of terms thai correspond to a crossambiguity, 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,
Borndordan 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'iWilliams 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,
TimelQ°equem'\~v'TimeS~ ale Al~a.]ysis
266
Z:
?i'i
Figure 3.19. atoms.
WignerVille and ChoYWitliams 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 WignerVille 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 proves 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 cardimd ,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~)ssterm a.s the Cho~iWilliains dist.dbution (3.151), as we obtai~l
sJ~.(t.~,)=~(,, ...... :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<mstVitle and scalogrm~.
The scalogram in it.s timefrequency 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.deoff between the joint resolution of the analysis and the significance of the crossterms. It permits a smooth transition between the WignerVille distribution and the scalogram (in an exact manner, if both h and g are Gaussians, and approximately otherwise) (el. Fig. 3.20).
Sig~iMdependent smoothing. The "coustantQ" 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 tradeoff 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 crossterms in the WignerVitle 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 \VignerViile 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 ";crosslike") are inefficient in slJch a (~diagonM'} situation, be~:ause they 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
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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277
.
I....................................... == ==
[
/
,! t
F i g u r e 3.23.
PsendoWiglmrVille and phase jump.
[~br a signal that consis~s of o,~ly one pure frequency, the (pseudo) WignerVitle 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 ~imefr(~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 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, Unfortunatey, we ca,mot tbIc_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 quasis~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
~
'
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 timefrequer~cy form (and it grows with the "extension" of this parameteriz~tion). 'Ib 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 hen's class are correlated in time and frequency. Let us add one further
7 imp,,FTc,que~ VilIe spectrum is }ocMl3.' wideband retati~e to the frequencyband 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'(cWignerVille distribution having %wo freedom," it suffices (,o pu(
s~ocha,stic flexibility, freedom," degrees of
(3.188)
ni,~,.) = ~4[,,,])~ 0. A generalization of Eq. (3.212) shows that the WignerVilte 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 sodefined 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 CauelkySchwarz 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 WignerVille 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) diseretetinm 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 WignerVille 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.sistath);rm.W properties are not needed, in generN, in order tbr tim WignerVille 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 tradeoff 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 ~ ? 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 durationbandwidth 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 timefrequency 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 WignerVille distribution is inva.riant under timefrequency 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 WignerViIle distributiorl with respect to changes of some, o~m can easily see that this quantity depends on the duradonbmldwidth 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, ~ The discussion of the interference terms of the \VignerVille 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~eSc~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 BoudreauxBartels (1992).
Chapter 3 &sues of luterpretatiorl
305
:~s We explicitly introduced the smoothed pseudoWignerViIle dist, ribution for the reduction of interf~.~rences in Fandrin (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 BornJordan 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 BoudreauxBa.r{ets (1992). :l~ See Hlawatsch and Flandrin (1998), tot example. *~' The ternfinology of a.qine slx~oothed pseu&>Wig~mrWlle 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 WignerVille 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 postprocessing of t.imcfrequency distributions in Auger (1991).
306
Tim~'b?eq~e~c37:TimeSe~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 ~imefl'equeney pane 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 WignerVitle 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 WignerVille 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 TimeFrequency as a Paradigm
I h e timefl'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 HeisenbergGabor 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 timefl'equency region in Subsection 4.t,2. This is related to the problem of SlepianPollakLa~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 timefleque~m:, inequati~;ies, One construct:ion is based on the use of different norms to quantii~, the local character of the WignerViile 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 frequencymoduS:~ed signals. 309
31 ()
7'i.~et:r,"quel,qv/ l'imeScale 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 timefl'equeney interprelation, \;~.%show, in particula,L h~gv the "constantQ" 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 thnefrextuency 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 thnesc~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 timefrequency filter, This gives rise to a ~ew interpretatim~ of the optimaliV of some di~ tributions (such as t~he WignerVille 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 ~imefrequency 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"imeFrequezicyas 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 timefl'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,imeli'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~kLandau problems.
4.1.1, HeisenbergGabor 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 secondorder 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 timefrequency plane. Example 1.
As a first, measure one can propose the quantity ~
,~" ( ~ + T2~e2)  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 timefYequenw extension in the sense of gq. (4.6) attains ~he bound 4; i
~o(L~) = v 2 v 7 =
¢'ff~,,(fw).
Chapter ,I TimeI~)'equenqv as a t~,r~:dfgm
315
4.1.2. Energy Concentration The inequalities of HeisenbergGabor type characterize a first family of timefrequency restrictions by installing certain measures tor the spread of a timefl'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 timefreq~mncy approaeh consists of defining a measure 5or the portion of the totat energy of a timefrequency 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 timefrequency "~'cta.ngle," to which the signal allocates most of its energy. Hence, a joint timeh'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 timefrequency 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) =
/]"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)
r 4 Tim+'t+b:,qu+>sTcy++s :++ParadiNn:+
+~2,+~
The bounds on the righthand 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,iinefrequeney uncert, aint,y, which renders an. arbitrary concentration of tile WignerVit/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 phasemodulated 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 ~ : timefl'equencv, curvc~. This must be compazed with the total+ spread of the signal in the timefrequency 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(,
,~ (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 selfsimilm" 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 longrange 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~,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 pseudoWigne>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 WignerVitte 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 spcctrum 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 _ iF1
[PlI;~:[i'~)F:r,'~(l/)i 2
(4.44)
77me b)~¢'que~(y/7hneHcMe .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':(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 timedocalization, 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 higherorder 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 timefrequency and timescale 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" timescale 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 firstorder 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 timescale 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
b42H(o f ( o , r) d r
}tere we used the notations of Subsection 3.1.1 and
. (,~.60)
342
Timef?'eq~e~!O'/TimeScale 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

,rO
)
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 wellstudied prob¿em ill signal processing, with abundant literature and welltried solu~,ions av~dlable, % Nevertheless, ~he timefrequency 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 timefrequency 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 timeffequelmy plane that amounts to a simple physical interpret at ion ?
Chapt~er 4 TimeD'eq,zem3," 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 timeinterval (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 tJmet)'equenc3 ~ til~er proceeds by maximizi~ag a contrast flmction (or SNR) based on a suitable timefrequency 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 timefl'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£~/TimeSca]~, 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(iuemytime parameterization of Cohen's class, an application of the CauehySchwarz 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 ~imefrequency 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 CmmhySchwarz ir~equatit3 to the righthand 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 frequencytime 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 Ville distribution of the known refe> ence as a template {br tim observation. ~br a high SNR ( i.e., for %/rr2£~,., 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?."TimeScate A ~;t~_d3:sis
352
Table 4,1 Timefrequency 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 WigaerVitle 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 TimeErequency ~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 timefrequency 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 timefrequency approaa:h is no doubt the most natural way of defining an average ref>rence for this case, 37
Tim~'}'}~eq~e~cy/TimeScate 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 WignerViHe 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 timefrequency 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 :sWither 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 timei: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 timefrequency 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 BoudreauxBartels (19S3) and BoudreauxBartels and Parka (198~i). They considered ~..he Wigne>Ville distribution and used a quadratic distance }~etveeeJl timeDequency 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 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 phasespa,: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, ICASSP84, 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