on the locus and spread of pseudo-density functions … bound... · on the locus and spread of...

· Philips J. Res. 37, 79-110,1982 RI050

ON THE LOCUS AND SPREAD OF PSEUDO-DENSITYFUNCTIONS IN THE TIME-FREQUENCY PLANE

,I

by A. J. E. M. JANSSEN

AbstractThis paper compares various time-frequency pseudo-density functions usedin signal analysis with respect to spread. Among the members of Cohen'sclass of pseudo-density functions satisfying the finite support property aswell as Moyal's formula, the Wigner distribution is the most well-behavedone in the sense that it has the least amount of global spread around itscentre of gravity. The Wigner distribution does not perform significantlybetter globally than the real part of Rihaczek's function; it does, though, ifthe global criterion is replaced by a localone, especially for signalsf of theformf(t) = a(t) exp(21tiIP(t» where lP is a smooth real-valued function anda is a slowly varying positive function. We formulate a general principleaccording to which the various pseudo-density functions of f should beconcentrated around the curve (t, lP'(1», and we present a more detailedqualitative analysis of the behaviour of the Wigner distribution off aroundthis curve.

1. Introduetion and announcement of results

It has frequently been observed that a tool is needed for accurately de-scribing signals in time and frequency simultaneously. This is for example thecase for a signal whose spectral content varies rapidly in time. Then the usualmethods of Fourier analysis fail since the time windows needed are either toolong to follow the transient behaviour of the signal adequately or too short togive satisfactory resolution in frequency. In the past, several time-frequencytransformations have been proposed; well-known in this respect are Rihaczek'sfunction and its real part, and the Wigner distribution 20,23,25). These time-frequency transformations are supposed to yield density functions indicatinghow the signal energy is distributed over time and frequency. The interpreta-tion as density function is seriously hampered, though, by the fact that thetime-frequency functions considered take in general negative, and sometimeseven complex, values. This is the reason why we speak of time-frequencypseudo-density functions. We compare the various possibilities by verifyingwhether for certain test signals a particular time-frequency transformationyields density functions concentrated mainly in those regions in the time-fre-quency plane where, according to intuition or experience, the signal energy isexpected to be located. As test signals we consider in this paper what we maycall concentrated signals and signals of the type exp(2n iço(t» with ço a smooth,

Phllips Journolof Research Vol.37 No.3 1982 79

80 Philips Journalof Research Vol.37 No.3 1982

A. J. E. M. Janssen

real-valued function. The first type consists of signals that can be consideredas having their signal energy concentrated around a point in the plane (e.g.exp( -n (t - a)2 + 2n ibt». The signal energy of exp(2n itp(t» is, according tointuition, concentrated around the curve (t, tp'(t» of instantaneous frequencies.For the first kind of signals the relevant quantity to consider is a sort of spreadof the various density functions around the appropriate point. For the secondtype one may consider the spread of the density functions in frequency directionat time t around the instantaneous frequency tp'(t). This leads to what we call aglobal and a local notion of spread. Before wedescribe this in detail, though, weshall consider a general class of time-frequency pseudo-density functions.

In ref. 9, part Ill, the Wigner distribution is compared with several othertime-frequency pseudo-density functions, in particular with those introducedand studied by Cohen and Margenau 11,21). These density functions aredefined as follows. Let cp be a reasonably behaved complex-valued function oftwo real variables. Then for any reasonably behaved complex-valued functionf of one real variable one defines H),~) by 0)

H),~)(t, w) = JJJ exp( -2n i (Ot+ 7:W - eu» cp (e, 7:)

X f(u + h)f(u - h) de dr du (1)for t e IR, WEIR.It can be checked that

H~:}, Ta/(t,w) = H),~)(t + a, co), H~~},Rb/(t, w) = H),~)(t, co + b) (2)

for all a E IR, bE IR and all f.Here (Taf)(t) = f(t + a), (Rb/)(t) = exp( -2n ibt)f(t).

In order that an H),~) deserves to be called a time-frequency density func-tion, certain requirements must be made. In ref. 9, part Ill, sec. 3, onerequires that for all fJ H),~)(t, w) dw = If(t)12 (tE IR), (3)

J H),~)(t, w) dt = I ($f)(w) 12 (w E IR), (4)

JwH),~)(t,w)dw/J H),~)(t,w)dw = in Im :t lnf(t) (r e IR), (5)

JtH),~)(t,w)dt/J H),~)(t,w)dt= - 2~ Im d~ ln$f(w) (WEIR), (6)

f(t) = 0 (I ti > T) ~ H),~)(t, w) = 0 (I ti > T, cO E IR), (7)

($f)(w) = 0 (Iwl > Q) ~ H),~)(t, w) = 0 (t e IR, Iwl >Q). (8)

0) f denotes integration over the whole real line; our normalizations differ from those used inref. 9 in such a way that the Fourier inversion formula takes a completely symmetric form.

On the locus and spread of pseudo"density functions in the time-frequency plane

Here $fis the Fourier transform off, "and Tand Q are positive real numbers.Of course, proper assumptions on f and (/1 should be made so that the for-mulas (3)-(8) make sense.Conditions (3) and (4) are quite natural requirements; conditions (5) and (6)

have to do with the expected instantaneous frequency off and with the groupdelay of a linear time-invariant system with impulse-response f respectively;conditions (7) and (8) are called the finite support properties in ref. 9, part Ill.It can be shown (ref. 9, part Ill) that

(3) holds for all f if and only if 4>(0,0) = 1 for all 0,(4) holds for allfif and only if 4>(0,r) = 1 for all r,(5) holds for allfif and only if 4>(0,0) = 1,4>,(0,0) = 0 for all 0,(6) holds for allfif and only if 4>(0,r) = 1,4>0(0, r) = 0 for all r,(7) holds for allfand all T> 0 if and only if, for any r, (/1(., r) is entire with

4>(z,r) = O(exp(1t(lrl + e)lzl» for all e>O,(8) holds for allfand all Q > 0 if and only if, for any 0, (/1(0,·) is entire with

4>(O,z) = O(exp(1t(1Ol+ e)lzl» for all e >O.A further restriction, which is quite convenient for this paper and natural in

quantum mechanics (but perhaps not really necessary for signal analysis), isthe validity of Moyal's formula (ref. 8, Theorem 14.2 and 27.15),

II H),~)(t,w)H;,~)(t,w)dtdw = Ilf(t)g(t)dtI2 (9)

for all f and all g. We shall show that (9) holds for all f and g if and only if14>(0,r) I = 1 for all ° and all r, and also that validity of (3), (4), (7), (8) and(9) for all f and g implies that 4>is of the form 4>(0, r) = exp(21tia Or) withaelR, lal ~t

Requirements of a more restrictive type than (9) are (a) for allfand all g wehave

and (b) for all f and all g we haveH(~) - H(<J» *w H(~)

f·g,f·g - f,f g,g •

Here * and . denote ordinary convolution and multiplication, and *1 and *wdenote convolution in the time and frequency domain respectively. Althoughneither of these conditions are equivalent with validity of Moyal's formula forallfand all g, it can be shown that (3), (4), (7), (8) together with (a) or (b) alsoimplies that 4>is of the form just given. The proofs follow the same lines asthe one presented here for the case that (9) (instead of (a) or (b) is assumed tohold. Observe that (3) and (a) imply that for allf and g

II H),~)(t,w)H;,~(-t,w)dtdw = Ilf(t)g(-t)dtI2, (10)

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A. J. E. M. Janssen

and that (4) and (b) imply that for all j and g

11 H),~>(t, w) Hi,~>(t, -cv) dt dco = 1Ij(t) g(t) dtl2. (11)

The class of pseudo-density functions H),; obtained by taking

tP«(), T) = exp(2n ia ()T)

in (1) for all j contains e.g. the Wigner distribution (a = 0) and Rihaczek'sfunction 23) f(t) ($j) (w) exp(2n itw) (a = -!), but not the real part of thelatter which is considered sometimes (ref. 9, part Ill, (3.21».We demonstrate that, among the members of this restricted class, the

Wigner distribution has the least amount of spread in the sense that for any fand any (to, wo) E 1R2 the global spread at> (to, wo) of I H),; 12 around (to, wo),defined by

at> (to, wo) = II [(t - tO)2- (w - WO)2] I H),; (t, w) 12 dt dw (12)

is minimal for a = O. We consider (12) and not e.g.

II [(t - tO)2+ (w - WO)2]H),; (t, w) dt dw (13)

sinceH),; takes negative (even complex) values so that, by cancellations in theintegral in (13), we may get a wrong impression of the amount of spread.Also, it can be shown that (13) does not depend on a; it is therefore of no usefor comparing the H),; 'so ,The quantity of at> (to, wo) is not particularly useful for e.g. signalsj of the

form j(t) = a(t) exp(2n i IP(t», where lP is a real-valued, smooth phase func-tion and a is a positive slowly varying amplitude function. Intuition tells usthat a pseudo-density function for such an j should be concentrated aroundthe curve (t, lP'(t» (also cf. (5», while at> (to, wo) indicates how well H),; isconcentrated around (to, wo). A more useful notion for such signals is thelocal spread 0'),~~(wo) of I H),; 12 at time to around frequency wo, defined by

a),~~(wo)= 1(w - wo)21 H),; (to, w) 12 dco, (14)

in particular when Wo equals the instantaneous frequency lP'(to).Since the computations get rather involved for the general case, we shall

mainly consider the cases a = 0, a = -! and the case with ReH};]> insteadof H),; in (14). We shall show that the Wigner distribution performs morethan marginally better than the other pseudo-density functions, especiallywhen the curve (t, lP'(t» can be approximated satisfactorily by a straight linethrough (to, lP'(to».

Let us summarize the further content of this paper. In sec. 2 we considerthe pseudo-density functions of Cohen in the context where they arose, viz.

82 Philips Journalof Research Vol.37 No.3 1982

2. Time-frequency pseudo-density functions and quantum mechanics

The history of pseudo-density functions of two variables starts in 1932withthe paper by Wigner, see ref. 26. In this paper Wigner introduced what is nowcalled the Wigner distribution as a device that allows one to express quantummechanical expectation values in the same mathematical form as the averagesof classical statistical mechanics. This line of development was taken up againin the late forties notably by Groenewold 13) and MoyaI22). Since then manypapers have appeared on the role of the Wigner distribution and other pseudo-density functions in quantum mechanics; see e.g. the papers by Berry 5),Cohen and Margenau U,21) and the references therein. For a good under-standing it is necessary to indicate how pseudo-density functions arose inquantum mechanics.

In classical mechanics the motion of a particle on the realline is describedby specifying at each moment t the position q(t) and momentum p(t) of theparticle (to avoid confusion with the notion of time used in the introduetionwe note that it is more proper to identify the notions time and frequency ofsignal analysis with the notions position and momentum of quantum mech-anics). The particle moves according to the Hamilton equations

On the locus and spread of pseudo-density functions in the time-frequency plane

quantum mechanics. In sec. 3 we prove the claim made in connection withMoyal's formula (9) about the global spread. In sec. 4 we formulate a generalprinciple for locating time-frequency pseudo-density functions of functionsfthat satisfy an equation Tf = 0 with Ta self-adjoint operator. In sec. 5 weprove the assertions made above about the spread of the various pseudo-density functions, and in sec. 6 we present more details for the behaviour ofWigner distributions of functions f of the form f(t) = exp(21ti lp (t» along thecurve (t, lp '(tl). We have tried to organize the paper in such a way that thetheoretical part (secs 2, 3 and 4) and the more practical part (secs 5 and 6) canbe read more or less independently of each other. Finally a remark aboutrigor: we present things in a rather informal way, although much of the papercan be put into a rigorous form.

where H =H(q,p) is the Hamiltonian which we assume to be time-indepen-dent.

In quantum mechanics a particle is described by specifying at each momentt a so-called wave function, or state, IfIt(q) with f IlfIt(q) 1

2 dq = 1. Theprobability that the particle lies, at time t, between a and b equals

Phlllps Journalof Research Vol.37 No.3 1982 83

A. J. E. M. Janssen

b

f IlfIt(q) 12 dq,

a

and the probability that the momentum of the particle, at time I, has a valuebetween c and d equals

d

f 1 (~ IfIt)(P) 12 dp.

c

Here (f!7hf)(p) = h-i f exp( -21t iqp/ h)f(q) dq, and h is Planck's constant.The motion of the particle is described now by Schrödinger's equation

:1 IfIt(q) = i(~lfIt)(q),

where ~ is an operator acting on functions of q and that replaces in somesense the Hamiltonian H (we give more details below).

In classical mechanics an observable, a, is any function of q and p. Inquantum mechanics an observable is represented by an operator A; the role ofthe function values a(q,p) is taken over by the values assumed by (A 1fI, 1fI).Here (1fI, ((J) denotes the usual inner product f lfI(q) ({J(q) dq of two square inte-grable functions lfI and ({J.Thus, if lfI = IfIt and the observable a is representedby the operator A, (A 1fI, 1fI) is the expectation value of a for the particle attime I. The problem is to assign properly quantum mechanical operators tothe observables from classical mechanics. For observables depending on q orp only this is not so hard. For example, the quantum mechanical versionof the observable "position" a(q,p) = q is the operator Q defined by(Q 1fI)(q) = q lfI(q) (if IfIt is concentrated in a small interval around a point qr,then (Q 1fIr, IfIt) = f qllflt(q) 12 dq will also lie in that interval), and for theobservable "momentum" a(q,p) = p we have the operator P defined by(PIfI)(q) = h 1fI'(q)/21t i (if f!7hIfIt is concentrated in a small interval around apoint pr, then (P IfIt, IfIt) = f (h 1fIr'(q)/21t i) IfIt(q) dq =f p 1 (~ IfIt)(P) 12 dp willalso lie in that interval). Thus the operator A corresponding to a(q,p) = f(q)and a(q,p) = g(p) are given by

(AIfI)(q) = f(q) lfI(q) and (AIfI)(q) = ~-l(g. f!7h 1fI)(q)

respectively.To handle more general observables a correspondence principle between

functions of q and p and operators of L2(1R) (= the set of square integrablefunctions defined on IR) needs to be formulated. The first step in setting upsuch a principle consists in defining for any lfI EL 2 (IR) a kind of density func-tion H""",(q,p) that indicates the location of a particle described by lfI as far

84 Phlllps Journni of Research Vol.37 No.3 1982


as position and momentum are concerned. A rather obvious condition is thatH""", gives the "correct" values for the moments of functions of q and palone. More specifically, one requires that 24),

II a(q) H",,'I' (q, p) dq dp = I a(q) IIfI(q) 12 dq, . (15)

II b(p)H'I','I'(q,p)dqdp=1 b(p) 1 $"lfI)(p)12 dp (16)

for all sufficiently well-behaved functions a and b. Note that, apart from theoverhead bar, (15) and (16) coincide with the conditions (3) and (4) for H),j>.A fairly general way 0) to get functions H'I','I' such that (15) and (16) hold for

all lfI is by taking cp= lfI in

H&~~(q, p) = III exp( -2n i (eq + ip - eu)) ifJ (e, i)

X IfI(U + !rh) cp(u- !rh) de di duo (17)

Here ifJ is a reasonably behaved function of two variables with

ifJ(O,i) = ifJ(e,O) = 111,19).

The next step in setting up the correspondence principle is to associate witha function a of two variables a linear operator A by requiring that

(AIfI, 1fI) = II a(q,p) H'I','I'(q,p) dq dp (18)

for allIflEL2(1R). That is, instead of substituting a particular value (q,p) in a,one integrates a against the conjugate H'I','I' of the pseudo-density function.

The various choices of ifJ in (17) give rise to a great number of correspon-dence rules. The best known rule of these is Weyl's rule where one takesifJ == 1. One also uses ifJ(e, i) = exp( -n ie i) or <p(e, i) = cos ne i (calledstandard rule in ref. 2 and symmetrization rule in ref. 11 respectively).An important reason why Weyl's rule is popular is mathematical elegance.

A further important advantage is (quoting Hörmander in ref. 15) that in thecalculus based on Weyl's rule there is closer agreement between compositionof linear operators and pointwise multiplication of the functions they corres-pond to than in the calculus based on the rule with ifJ(e,i)_= exp(-niei).There is, furthermore, in Weyl's rule definitely a relation between the notionsof positivity of operators and positivity of the functions they correspond to.We refer to refs 7 and 16 where some results in this direction can be found.

0) The claim in ref. 21 that (17) gives all H""", satisfying (15) and (16) for all If! is not correct;for this to be the case the function <IJ should also depend on q and p.

Phllips Journol of Research Vol. 37 No.3 1982 85

3. Time-frequency pseudo-density functions and Moyal's formula

In this section we prove the claim made in connection with (9). Adoptingthe notation of the introduetion again we assume that we have a reasonablybehaved function c[J in (1) such that (3), (4), (7), (8) and (9) hold for all/ and g for which these formulas makes sense. It is already known thatc[J(e,O) = <Ji(0,T) = 1, and that c[J(e,·) and <Ji(.,T) are entire functions withcertain restrictions on growth for all e and all T because of (7) and (8).

We shall first show that 1 <Ji(e, T) 1 = 1 for all e and all T. To that end weapply (9) to / = zf; + W/2 and g. By using

H(<Il) - 1 12H(<Il) + - H(<Il) +Zfl+wf2, zfJ+wh - Z fJ,fJ zw h,h

+ WZ H)2~jl + 1 WI2 H)2~j2(19)

A. J. E. M. Janssen

and equating coefficients we get-----;-::-;---f f H)1~j2(t, cv)Hg:~)(t, cv) dtdcv = UI, g)U2, g) (20)

for all /1, /2 and g. Applying (20) with /1, /2 and g = zg, + Wg2 we get in asimilar way

f f H)1~j2 (t, cv) H;~~2(t, co) dt dco = UI, gl)U2, g2) (21)

for all/I, gd2 and s«.Define a mapping T: L2(1R2) ~ L 2(1R2) by

(TF)(e, T) = f exp(2n ie u)F(u + !T, u - h) duo (22)

It is not hard to see that T maps L2(1R2) onto L2(1R2), that it preserves innerproducts in L2(1R2), and that for any /1'/2

H)1~j2(t, cv) = f f exp( -2n i(e t + T co» (TUI Q9h» (e, T) c[J«(),T) de dr. (23)

Here UI Q9h)(t, s) = /1(t)/2(S). Now by Parseval's formula we can rewrite(21) as

f f 1 c[J(e, T) 12 TUI Q9 h) (e, T) T(gl ® g2) (e, T) de dr = U!' gl) (12, g2). (24)

Since T preserves inner products, we may write (24) as

If (IfIn) is a complete orthonormal system for L2(1R), then the propertiesof T ensure that «(/Jk,/) is a complete orthonormal system for L2(1R2). Here(/Jk,1= T(lfIk Q9 ifr/). Hence f f (I c[J(e, T) 12-1) (/Jk,/(e, T) (/Jn,m«(),T) de dr = 0 forall k,l,n,m, and this implies that 1 c[J(e, T) 12 -,1 = 0, Le. 1 c[J(e, T) 1 = 1 for alle and all T.'

86 PhillpsJournalof Research Vol.37 No.3 1982

I I I 1 - z/z" Ilog tP(B, z) = log IT" /1 - z Zn

+ cImz (26)


We must show now that tP(B, r) = exp(2n iaBr) for some aE IR, Ial ~ i.Fix BE IR.We know that tP(B,·) is an entire function, and that tP(B, z) =O(exp(n(1BI + e)lzl» for all e>O.Hence tP(B,·) is regular in the upper halfplane, continuous in the closed half plane, its zeros Zn in the upper half planehave no finite limit points,

lim in f(r-110gmax [z] = r, Imz ~ 0, I tP(B, z)l) < 00,r- co

and we also have! [log+ItP(B,z)11/(1 + z2)dz = 0< 00 since ItP(B,z)1 = 1 forall real z (log+a = max(O, log a) for a> 0). The conditions of theorem 6.5.4 inref. 6 are satisfied, whence we have the representation

in the closed upper half plane for some CEIR.The function

. 1 - Z/Zllg(z) = exp( -1 cl) IT" /

1 - Z z;

is regular in the closed upper half plane, and we have ItP(B,z)1 = Ig(z)1there. Hence, as tP(B, 0) = g(O) = 1 we must have tP(B, z) = g(z) in the closedhalf plane. But tP(B, .) is an entire function, therefore so is g. We concludethat g has no poles in the lower half plane, i.e. no zeros in the upper halfplane. This means that tP(B,z) = exp(-icz).

We have shown now that for every BE IRthere is a cl(B) EIRwith tP(B, r) =exp( -ic1(B)r). In addition, Icl(B)1 ~ n IBI. Similarly, for every r e IRthereis a c2(r)EIR with Ic2(r) I ,,;;;nlrl such that tP(B,r) = exp(-ic2(r)B). Weconclude easily that tP(B, r) = exp( -icBr) some CEIRwith Icl ,,;;;n , and thiscompletes the proof.The choice tP(B, r) = exp(2n iaBr) with aE IR, Ial ~ i leads to the pseudo-

density function H),~ which can be written in the convenient form

H),~(t, w) = ! exp( -2n i sw)f(t + s(!-a»f(t - sO + a» ds. (27)

It can easily be checked directly that Moyal's formula is satisfied. Hence, theconverse of the theorem proved in this section also holds. For a = I, -!(27)simplifies to exp( -2n itw)f(t) (PJf)(w) and exp(2n itw)f(t)($f)(w) respect-ively, and the choice a = 0 leads to the Wigner distribution.We finally note that the pseudo-density functions of (27) also satisfy

H).~(yt, y-1w) = H~~LYit, w) where (zyj)(t) = yif(yt) for y >O.

Phlllps Journalof Research Vol. 37 No.3 1982 87

A. J. E. M. Janssen

4. A general principle for locating time-frequency pseudo-density functions

We take in this section cJ>(B,r) = exp(21tiaBr) with aEIR, lal ~! so thatwe get the pseudo-density function H),a_)of (27). We shall formulate a prin-ciple according to which we can get a rough idea where theH),a_)'sof anfsatis-fying Tf = 0, with T being a self-adjoint operator are located. This principleis based on the fact that under the rule (18) of assigning linear operators tofunctions of two variables there is agreement to a certain extent between com-position of linear operators and pointwise multiplication of the functions theycorrespond to.If we take Hf,f = H),a_),the rule (18) can be reversed so as to assign func-

tions of two variables to linear operators. Let T be a linear operator. Wedefine the a-symbol p~a) of T by requiring that

(Tg,f) = II p~a)(t, w)H),~)(t, co) dtdw (28)

for all f and all g. It can be checked, by using the fact that Moyal' s formulahold for Hf, s» that p~a) is given by

p~a)(t, ol) = I exp(-21t isw) hT(t + (l- a)s, t - 0 + a)s) ds, (29)

where hT(u, v) = (T <5v)(u) is the kernel of T (<5vis the delta function at v). Ofcourse, in a rigorous approach (29) should be interpreted as an identity be-tween distributions (see ref. 12, where the case a = ° is treated rigorously).If T represents a linear system, hr is the impulse response function of the

system, and p~a) can be considered as a kind of time-dependent transmissionfunction. Indeed, if we take a = !we get

p~)(t, ol) = exp( -21t itw)(Tew)(t) = exp( -21t i tw)(Tew, <5t),

where ew is defined as ew(t) = exp(21tiw t). This is the transmission functionconsidered in ref. 10. And if we take a = -Iwe find

p~-l)(t, ol) = exp(21t i tw)(T<5t, ew) = exp(21ti tw)(<5t, T* ew) = p~2(t, co),

where T* is the adjoint operator, given by (T* j, g) = (J, Tg) for allf and g.Let us give some examples. Let T be the modulator given by (Tf)(t) =

V(t)f(t), where Vis a well-behaved function. It can be checked from (29) thatp~a)(t, Ol) = V(t) for all a. And if Trepresents a linear time-invariant systemwith transmission function g(w), then p~a)(t, co) = g(w) for all a.

The following example will be used later on. Define T by Tf = (j, (/J) If! forallf(projection operator). Then

88 Philips Journni er Research Vol.37 No.3 1982

Philips Journalof Research Vol. 37 No.3 1982. 89


p~U)(t, en) = H&~~(t,en) (30)

for all a. This follows easily from Moyal's formula.As already said, there is some agreement between composition of operators

T and multiplication of the p~U)'so This gives a method of finding approxima-• ç B(U) . t fH(U) p(a) d p(a) F di t (30) H(a)nons lor Tj,TfIn erms 0 nr> T an T'. or, accor mg 0 'Tj,Tfcan be considered as the a-symbol of the operator g-+ (g, Tf)Tf = (T*g,j)Tf.This operator is the composition of the operators T*, g--+ (g,j)f and T.Hence, an approximation for H~J?Tf is given by p~a! . H),cy . p~a), the productof the three a-symbols.The case a = 0 is especially convenient. For we have Hti = H~~}follows

from (26)), so that p~2 = p~O) (follows from (28) and the fact that (Tg,j) =(g, T* f)). Hence the approximation for H~~Tf found above can be written as1 p~O) 12 H),~ . This is a nice result considering the fact that p~O)has an interpre-tation as a time-dependent transmission function and H),~ is something likea time-frequency density function.To give an idea how this approximation result should be looked upon, we

give two examples where B(O) = 1 plO) 12 H(O).rr.rr T j,f(1) Let T be a modulator with modulation function V(t), and letf(t) = c5a(t),

the delta function at a. We know that p~O)(t, en) = V(t), and it can bechecked that H),~ (t, en) = c5a(t). Also, (Tf)(t) = V(t) = V(a) c5a(t) so thatH~~Tf(t,en) = !V(a)12oa(t) = Ip~O)(t,enWH),~(t,en).

(2) Let T be a linear time-invariant system with transmission function g, andlet f(t) = exn(Zn i b t). We know that p~O)(t, en) = g(en), and it can bechecked thatH),~(t,en) = ob(en). Also, (Tf)(t) = g(b)exp(21tibt), and asin example (1) it can be shown that H~~Tf(t, en) = 1 p~O)(t,en)12 H),~(t, en).

The approximation result can also be used to find a clue as to where theWigner distribution of a function f satisfying an equation Tf = 0 is located.We formulate the following principle. If T is a self-adjoint linear operator(T= T*) andfis a function such that Tf = 0, then H),cy has its main contribu-tion near the set {(l, en) 1 p~a) (t, en) = O}. This can be seen from the fact that(p(a»)2 H(a) is an approximation for H(a) = 0 so that H(a) can only haveT j,f rr; Tf j,fsignificant contributions in regions where p~a) is small. Indeed, far away fromthe set {(l,en) 1 p~a)(t,en) = O}we shall often find that H),cy is small, or thatH),cy is not necessarily small but rapidly oscillating, which in either case im-plies that integrals of H),cy over regions of sufficiently large area are small. Atthe end of section 6 we shall give more details for a special case.

In order to indicate in what sense the above formulated principle holds weshould say to what extent composition of linear operators and multiplicationof symbols agree. This is a quite involved subject to which a whole branch of

exp(-x/2) Ln(x) = (-l)n(2n)--lAi(u) + O(I/n), (33)

A. J. E. M. Janssen

mathematics, the calculus of pseudo-differential operators, is devoted. For de-tails we refer to refs 14 and 15 (ex= 0) and ref. 4 (ex= - ~).We stress again thefact that validity of the principle is more convincing for the case ex= 0 thanfor the case ex= -!; in fact, much of this paper is meant to verify this claim.

Now let us give some examples. Consider f(t) = exp(21ti<p(t», where <p is asmooth real-valued function. This f satisfies (l/(21ti)d/dt-<p'(t»f(t) = O.Hence Tf = 0, where T = 1/(21t i)d/dt-<p'. According to the examples givenearlier we have p~a) (t, w) = co - <p '(t) for all ex. Hence we may expect H),~ tobe concentrated near the curve (t, <p '(t» in the plane. This was verified byRihaczek 22) in case ex= ~,and we will verify it in the next sections, mainly forex= O. As a second example we consider the Hermite functions v«. These If/n'Ssatisfy «1/(21t i) d/ dt)2 + t2 - (n + !)I1t) If/n = O. Hence. the principle appliedwith T= «1/(21ti) d/dz)" + t2 - (n +~»)/1t (so that p~a) = w2 + t2 - (n + ~)/1t)andf = If/n gives that H~~~'IInshould be concentrated near the circle around theorigin with radius «n + !)I1t)!. The formulas

H~~~'IIn(t, w) = 2(-l)n exp( -21t (t2 + w2» Ln(41t (t2 + w2», (31)

H~~~n(t, w) = exp(21t i tw)( _i)n If/n(t) If/n(W) (32)

show that ex= 0 gives better results than ex= ~ (ref. 16, sec. 2; we havef!7lf/n = (_i)n If/n, and Z, is the nth Laguerre polynomial of order 0). Indeed, it canbe seen from the asymptotic properties of Laguerre polynomials (ref. 24, chap. 8)that H~~~'IIndecays quickly for (n + ~)/1t < t2 + w2 -+ 00, oscillates rapidly fort2 + w2 < (n + ~)/1t, and is positive near the circle t2 + w2 = (n + !)/1t. In fact,for the neighbourhood of the circle t2 + w2 = (n + !)/1t we can use the follow-ing precise result (ref. 24, chap. 8; the relation between the Airy function A usedthere and our Ai used in sec. 6 is A(z) = 3--l1tA i( -3--l z)

where u is given by x = 4n + 2 + 2(2n)i u (the O-term holds uniformly in anybounded set of u's). We refer to ref. 5 where cases with more general Hamil-tonians than t2 + w2 are considered.

To illustrate the principle for ex= 0°) we have included pictures of the Wignerdistributions of f(t) = exp(21t i<p(t» with <p(t) = at, bt", ct", dt" and off(t) = If/n(t) (nth Hermite function). We make the following observations, Theprinciple holds exactly for f(t) = exp(21t i<p(t» where <p(t) = at.bt". We also seethat the Wigner distributions of f(t) = exp(21t i<p(t», where <p(t) = ctï.dt' areindeed concentrated near the curve (t, <p '(t», and that they decay fast at the

0) Coloured displays of Rihaczek's density function for several functions f can be found inref. 19.

90 Philips Journul of Research Vol.37 No.3 1982

Philips Journalof Research Vol. 37 NO.3 1982 91


convex side of the curve and oscillate rapidly at the concave side of the curve.A similar thing can be said about the Wigner distribution of f(t) = Ij/n(t)which oscillates rapidly inside the curve t2 + w2 = (n + ~)/rr.. In sec. 6 weconsider the Wigner distribution of exp(2rr. icp(t» in detail; it turns out that thetypical behaviour just observed in the examples (fast decay vs. rapid oscil-lations) is a more general feature.

In figs 1-4 we have used the pseudo-Wigner distribution (see ref. 9, part II),

Fig. 1. Wigner distribution of f(t) = exp(IOrr i t) for (t, w) E [0,1] X [0,10].

A. J. E. M. Janssen

Fig. 2. Wigner distribution of J(t) = expf lOrriz'') for (I,W)E [0,1] X [0,10].

92 Philips Journalof Research Vol. 37 NO.3 1982


Fig. 3. Wigner distribution of JU) = exp(IOn if3) for (/, w) E [ -I, I] X [-t IS].

Philips Journalof Research Vol. 37 No.3 1982 93

A. J. E. M. Janssen

cot __

Fig. 4. Wigner distribution of J(t) ~ exp(5n i (4) for (I, w) E [-1, IJ X [ -5,6tJ.

94 Philips Journalof Research VoL 37 No. 3 1982


Fig. 5. Wigner distribution of J(I) = 1/16(1), the 6th Hermite function, for (t,w) E [ -2,2) x [ -2,2).

i.e. the Wigner distribution of a windowed version of the signal involved; forfig. 5 we have used the explicit formula (31).

5. Global and local spread for time-frequency pseudo-density functions

Let f be a sufficiently well-behaved complex-valued function defined on IR,and let exE IR, 1 ex 1 ~!. If (to, wo) E 1R2 then we call

a;a> (to, wo) = ij [(t - to)2 + (w - WO)2] 1 H),a; (t, w) 12 dt dw (34)

the global spread of 1 H),}> 12 around the point (to, wo), and we call

a ),~~(wo) = f (w - WO)2 1 H),a; (to, w) 12 dw, (35)

the local spread of 1 H),a; 12 at time to around frequency wo.

We shall show now that for any f and any (to, wo) E 1R2 the global spreada;a> (to, wo) is minimal for ex = O. It will also follow that the optimal per-formance of ex = 0 is most apparent when we take for to and Wo the centres ofgravity of f and 87f, defined respectively by

Philips Journalof Research Vol.37 No.3 1982 95

A. J. E. M. Janssen

to = 1 t If(t)12 dtIIlf(t)12 dt,

Wo = 1 wl ($f)(w)12 dwlII ($f)(w)12 dw.

(36)

(37)

To that end we evaluate the first few moments of I H),~ 12• This can be doneby using Mayal's formula (cf. sec. 3, in particular (20», Parseval's formulaand the identities

t Hl'(t, w) = !Ht}lf(t, w) + !H),{l,(t, w)

+ als exp( -21t is w)f(t + s (! - a»f(t - sG + a» ds, (38)

to H),cy (t, w) = (! - a) H},J,~(t, w) + (! + a) H),a;,f(t, co), (~9)

which follow easily from the definition of H),cy. Here we have

(Qf)(t) = tf(t), (P f)(t) = f'(t)/21t i.

Weget

IIIH),~(t,w)12dtdw = Ilf~4, (40)11tIH),~(t,w)12dtdw = Ilf~2(QJ,f), (41)

11wIH),~(t,w)12dtdw = Ui2(PJ,f), (42)

11 t2IH),~(t, w)12 dzdco= 1 +24a211Qfl1211fl12 + 1 -24a21 (QJ,f)12, (43)

11 w2IH),~(t, w)12 dtdw = 1+24a2IPf~ 211f~2 + 1 -24a2 I (PJ,f)12, (44)

11tco IH),~(t, w) 12dtdw = 1 \4a2

(QJ,f)(PJ,f) + 1 -24a2 UP Re(PJ, Qf) .

. (45)Compare ref. 8, sec. 14. We further note that (QJ,f) = (I, Qf) and(P J,f) = (I, P f) = (Q$J, $f) are real numbers.

Before we consider the global spread of I H),cy 12 we observe that the centreof gravity of H),CY, i.e. the point (a,b) given by

a = Il ! IH),~(t, w) 12dt dwlII IH),cy (t, w)12 dt dw (46)and

b = 11 wIH),~(t, co) 12 dt dwl! IIH),CY(t, wW dt dor, (47)

coincides with (to, wo), where to and Wo are given by (36) and (37) respectively.This follows easily from (41) and (42).

Using (43) and (44) we get

11 (t2 + (2)IH),~(t, W)12 dtdw

1 +24a2

OQf~2 + ~Pfi2)llfI12 + 1 -24a2

(I(QJ,f)12 + I (PJ,f)12).(48)

96 Phllips Journalof Research Vol.37 No,3 1982


Clearly, I(QJ,f) 12~ IQfP Ifi 2, I(PJ,f) 12~ 11Pfl12 UI12 by Schwarz' in-equality, and hence (48) is minimal for a = O.This proves the claim made atthe beginning of this section for (to, wo) = O.

The general case can be treated by noting that

(49)

(cf. (2)). Now (48) can be applied with TroRwofinstead off, andthis completesthe proof.We have proved, more precisely, that

a?> (to, wo) = O'Jo>(to, wo) + 2a2

+ 0Qfd 211fd2 + 11Pfd 211f1112- I(Q!1.!1) 12- ICPf1.!1) 12), (50)

where f1 = TroRwof. We can assume, without loss of generality, that the cen-tres of gravity of'j' and 81fequal zero. This implies that (QJ,f) = (P1,f) = O.Now it can be shown that the expression in (50) between parentheses is inde-pendent of to and Wo so that it equals 11Qf11211!112+ 11Pf~2U112.This showsthat for any a E IRthe ratio 0';0>(to, wo)/ ar> (to, wo) is minimal (as a functionof to and wo) at the centre of gravity of H),~(for O';o>(to,wo) is minimal atthat point).

A time-frequency pseudo-density function which is not of the form H),~with a E IR, Ia I~! is the real part of Rihaczek's function ReH),j>. The ex-pression (48), with ReH),"'}> instead of H),~, gets a bit complicated, and it isnot easy to see whether it is always larger than the corresponding expression forthe Wigner distribution. We have calculated j'j (t2 + (.02) IH!~!(t, cv)12dtdcv .and !!(t2+w2)IReH:,i>(t,w)12dtdw for the first two Hermite functionsg = 2t exp( -n t2) and g = 21n-l t exp( -n t2) (the centre of gravity is the originfor both cases). We found 1/4n, (1 + 1/(2V2))/4n for the zeroth Hermitefunction and l/n2, (1 + 9/32n2 V2))/n2 for the first Hermite function. Hence,the Wigner distribution performs for certain functions only marginally bet-ter 0) than the real part of Rihaczek's function if we use the global spreadaround the centre of gravity as our criterion. It will turn out, however, thatthe Wigner distribution performs definitely better when the local spread isused with signalsfofthe form j'(r) = exp(2niqJ(t)).

Let us consider the local spread O'),~! defined by (35) in more detail. Wewould like to evaluate 0'),~!with f(t) = exp(2n iqJ(t)), where qJ is a smoothreal-valued function. This is not possible since H),a_f is for thesef's in generala distribution, and the definition of 0'),~!involves the square of H),a_f . Instead

0) It is more proper, though, to compare the ratiosff (t2 + co2) IH(t, coWdtdw/ fflH(t, coWdtdw. h H H(O) d R RH)WIt = g,B an e g,g. , .

PhllIps Journalof Research Vol.37 No.3 1982 97

A. J. E. M. Janssen

we consider f's of the form f(t) = exp(2n itp(t) - 1t e t2), and we study thebehaviour of (1},~!as e ~ o. According to the general principle stated in sec. 4,we expect (1},~!(cvo)to be small for CVoclose to the instantaneous frequencytp'(to). This expectation is supported by formula (5) showing that the averagefrequency at time t equals tp'(t). It was also noted by Rihaczek in ref. 23 thatH},"j) is concentrated around the curve (t, tp'(t» in the sense that integrals ofH},"j) over regions far away from that curve are negligible.

We shall evaluate the first few local moments of 1H},~ 12 for a = 0, -~ andof 1ReH},"j) 12withf(t) = exp(2n itp(t) -n e t2). We can do that for a = 0, -!by using formula (39) together with

__"..:c:----f H)l~~l(t,cv) H)2~~2(t,cv) dcv

= f f1(t + s(~ - a»f2(t + sO - a» g1(t - s(! + a» g2(t - s(~ + a» ds (51)

which holds for allf}, fz, g}, g2 by Parseval's formula. We get, after using theidentity (P f)(t) = (tp '(t) + ie t)f(t), for a = °

f 1H),~ (t, cv) 12dcv = e-i exp( -4n e t2), (52)

f cv1H},~ (t, cv) 12dcv = exp( -4n e t2) f exp( -4n e S2)X (tp'(t + s) + (/J'(t - s) ds, (53)

f cv21H),~u. cv) 12dco = exp( -4n e t2) { ;~

+ f [tp'(t + s) + tp'(t - sW exp( -4n e S2) ds} . (54)

And for a = -~ we get

fIH},"j)(t,cv)12dcv = If(t)121IfI12, (55)

f cvIH},"j)(t,cv)12dcv = If(t)12(Pf,f), (56)

f cv2IH},"j)(t, cv)12dco = If(t)121IpfP, (57)

(these formulas hold for generalf; the formulas for generalfand general a areunfortunately much more complicated).To calculate the local moments for ReH),"j) we write

1Re H},"j) (t, cv) 12 = ~IH),"j) (t, cv) 12+ ~Re (j2(t)«$f)(CV»2 exp(4n i t cv»,

and we use the convolution theorem, the formula Q$= $P and (55)-(57) toget

fIReH},"j)(t, cv)12dcv = Hlf(t)12 UP + Rej2(t)(f*f)(2t)}, (58)

f cv 1Re H),"j)(t, cv)12dcv = HIJ(t)12(P f,f) + Ref2(t)(P f*f)(2t)}, (59)

f cv21ReH},"j)(t, cv) 12dcv = H If(t) 1211P fi 2 + Ref2(t) (Pr- P f) (2t)}. (60)

98 Phllips Journalof Research Vol.37 No.3 1982


Here (fl *h)(s) denotes the convolution f fl (s - x) f2 (x) dx of fl and 12 at thepoint s.

The formulas (52)-(60) are especially interesting for the case that the instan-taneous frequency (/J'(t) is equal to zero, for then (54), (57), (60) give thespread around the instantaneous frequency. The following table givesformulas (52)-(60) for f(t) = exp(21ti (/J(t) - 1te t2) omitting terms of order el,and with t = 0, (/J'(0) = (/J(O)= 0.

TABLE I

IH(OlI2 IHHll2J,f J,f IReHHll2J,f

WO e-1 (2e)-1wl !exp( -41t e S2) lPó(s) cts !exp(- 21t e S2) lP '(s) ctsw2 Hexp( -41t e S2)(IPÓ(S»2 cts !exp( - 21t sS2)(1P '(S»2 cts

H{2e)-1 +!exp( -21t es2) coszn lPe(S) cts}!!exp( -21t es2)1P:(s) COS21t lPe(S) ctsif exp( - 21t e S2)(IPÓ(S»2 cts +

-!exp( - 21t e S2)1P'(s) lP '( -s) sin21t lPe(S) cts

Here (/Je(S)= (/J(s) + (/J(-s) and (/Jo(s)= (/J(s) - (/J(-s).The formulas (52)-(60) and the table give rise to the following remarks (we

ignore terms of order ei).(1) In order to compare the Wigner distribution to (the real part of) Rihaczek's

function we normalize the expressions (53), (54), (56), (57), (59), (60).Thus we consider

R/I> == R/I> (t, e) = f w 1H;(t, w) 12 dw If 1H;(t, w) 12 dw, (61)

R/2> == R/2>(t, e) = f w21 H;(t, w) 12 dwl f 1H1(t, w)12 dw, (62)

for i = 1,2,3, where HI = H},~, H2 = H},;>, Ha = R.eH},J>. Observe that(according to (25)-(27» RJI> and RJ2> are independent of t. Hence H},J>spreads uniformly around the curve (t, (/J'(t».

(2) Assume that (/J~(s)= (/J'(s) + (/J'( -s) is bounded, and that

LI := lim T.!.!(/J~(s)dsT- 00 0

exists. By a Tauberian theorem in ref. 17, chap. 4, sec. 2 we know that00

lim lel f exp( -1t e S2) (/J~(s)dse!O 0

exists and equals LI. It follows (since (/J~is even) from the table that

limRfl>(O, e) = limRJI>(O, e) = LI.dO dOSimilarly, if 1 T

MI := lim 2T f «(/J~(S»2dsT- 00 -T

Phlllps Journal of Research Vol.37 No.3 1982 99

100 Phlllps Journalof Research Vol.37 No.3 1982

A. J. E. M. Janssen

exists, then

and if

exists, thenlimR?)(O,e) = M2.elO

Hence, if rp~ is small compared to tp", H),j will have much less spreadthan H},"j).

(3) Consider the case that rp(t) = la t2(a E IR). Then it can be checked thatH),j (t, (1) = 150«(1) - at) and that H},"j) (t, (1) = (iati exp(-1t ia-1«(1) - at)2)or 150«(1) according as a =1= 0 or a = O.Hence H},~ performs considerablybetter in this case. More generally, assume that rp '(t) and rp '( -t) have op-posite sign if t> 0 and stay away from zero if l r+ 00. Define for 15>0

cous, = öi !exp( -1t 15S2) (rp~(S»2 ds.-co

Since I rp~(s) I ~ I rp '(s) I we see from the table that

R12)(0, e)/Rá2)(0, e) ::::;;L(4e)/4L(2e).

As to R~2) the situation is slightly more complicated. By assumption,rp:(s) = rp '(s) - rp '( -s) stays away from zero if s >O.Hence

1 T2T _£ cos 21trpe(S) ds-» 0

if T-- 00 since cos 21trpeoscillates rapidly, and this implies that

j exp( -21t e S2) cos 21trpe(S) ds = o(e-1)-co

as e .i0 by the Tauberian theorem in ref. 17, chap. 4, sec. 2. As

rp'(s) rp '( -s) sin" 1trpe(S) ~ 0,

we see from the table that R?)(0,e)/R~2)(0,e) = (! + 0(1»L(4e)/L(2e)as e lO. If e.g. limL(ö) exists, =/= 0 then R12) is at least four times as small

alOas R12) and at least twice as small as R~2) (asymptotically). Usually,though, R12) will be much smaller than R~2) since the integral involvingrp '(s) rp '( -s) sin" rpe(S) will be a negative number of large magnitude.

(4) The case where rp '(s) and rp '( -s) have the same sign for all s ;» 0 can beanalyzed in a similar way. We find now, under proper assumptions, thatR?) is not larger than RJ2) or R~2) (asymptotically).

A second way to get an idea as to how far the various pseudo-density func-tions spread around the curve (t, rp '(t» is the stationary phase method. We have



H),~(t, co)= f exp( -2n i cos+ 2n i(tp (t + is) - tp(t -is») ds, (63)

H),]> (t, ev) = f exp( -2n i cos + 2n i(tp (t + s) - I/)(t») ds (64)

(these integrals mayor may not converge as oscillatory integrals). If t is fixed,the integral in (63) has stationary points for all co in the set

U(I/)'(t+ u) + tp'(t- U»IUE IR}:= El(t),

and the integral in (64) has stationary points for all to in the set

{tp'(u)luelR}:= E2(t).

Note that El(t) C E2(t) and that E2 is independent of t. Usually, E2 will besignificantly larger than El (t).

In the figs 6 and 7 we have drawn portions of two smooth curves (t, tp'(r)

tco

Figs 6 and 7. Portion of critical curve (I, rp '(I» for the Wigner distribution of f(t) = exp(21ti rp(t»;the shaded region shows the points (I,w) for which the integral (63) has stationary points.

A. J. E. M. Janssen

together with a portion of the set of (t, w)'s (shaded region) for which theoscillatory integral (63) defining the Wigner distribution of exp(2n iip(t» hasstationary points. The pictures are slightly misleading since the regions shownare obtained by using only a finite piece of the curves. The corresponding setsfor the integral (64) would consist of the strip

Fig. 8. Wigner distribution of J(I) = exp(IOn i I + ~ni sin x I) for (I, w) E [-~, ~jx [0, IDj.

102 Philips Journalof Research Vol. 37 NO.3 1982



{(t, co) Imin lp 'CS) ::;;; co ::;;;max lp'(S) J.s s

Especially fig. 7 (also cf. table I and point (3) following table I) shows theadvantage of the Wigner distribution H),~ over (the real part of) Rihaczek'sfunction.We have also included fig. 8 showing the pseudo-Wigner distribution of an

FM signalf(t) = exp(2n ia t + n ifJ sinn y t) which was also considered in ref. 9.The curve (t, lp '(t)) = (t,a + in y fJ cosn y t) has some interesting features. If kis an integer and t = (k + !)y-1 then the set E2(t) consists of the single point(t,a), and if t = kly then E2(t) is the set {(t, a + in y fJ a) I - 1~ a::;;; 1J. In-deed, H),~ «k + i) y-1, co) is concentrated completely at the point co = a as canbe seen from ref. 9, (27) and fig. 1. This does not contradiet the theoremproved in sec. 6 about Wigner distributions concentrated on curves.

6. More details for the Wigner distribution of exp(21t ilp(t))

We consider in this section the behaviour of the Wigner distributions ofsignals f of the form f(t) = exp(2n icp(t)) along the curve (t, lp '(t)) in moredetail. Such an analysis is facilitated quite a bit by the fact that the Wignerdistribution, compared to other pseudo-density functions, has a particularlypleasant behaviour under certain area-preserving transformations (symplectictransformations) ofthe time-frequency plane. Using these transformations wecan rotate, shift or dilate the Wigner distribution of a function in any positionor direction we want; the result is always the Wigner distribution of a functionwhich is determined by the initial function and the particular transformation.

Following ref. 8, sec. 27.3 we can describe any symplectic transformationby six real numbers all, a12, a13, a21, a22, a23with a11a22 - a12a21= 1: the point(t,co) is mapped onto (t', co ') = (all t + a12co+ a13, a21t + a22co+ a23). Togetherwith this transformation we consider the matrix

-i a21allo

(65)

In ref. 8, sec. 27.3 an inner product preserving operator has been associatedwith every symplectic transformation in such a way that multiplication of thematrices corresponds to composition of the operators (apart from a factor ofmodulus unity which is of no importance here). Denoting the operator cor-responding to A in (65) by rA we have by ref. 8, sec. 27.12.2

(FAf, r,g) = (f, g), (66)

H(O) (t' ') - H(O) (t )rsr, r: g , co - f,g ,co (67)

A. J. E. M. Janssen

for all f and g and all tand ca «t', to ') has been defined above).The following table gives the elementary symplectic transformations to-

gether with the associated operators; from this table all symplectic trans-formations and associated operators can be obtained by multiplication ofmatrices and composition of operators as indicated above. In this tablea,b,a,fl,e are real and À. is positive.

TABLE II

Symplectic transformation

f -+ exp( -1t i a b - 21ti b t)f(t + a)f-+)Jf().. t)T(O)$;$-1

Ir+ f(-t)f -+ exp( -1t i fJt2)f(t).r -+ (i a)-l f expm i a-I(z - t)2)f(z) dz

Associated operator

(t', w') = (t - a, ca - b)(t',w') = ()..-lt,ÀW)(t', w') = (t cos 0 + ea sin 0, ca cos 0 - t sin ())(t', w') = (w, -t); (t', w') = (-w, t)(t', w') = (-t, -w)(t', w') = (t, ca - fJ t)(t', w') = (t - aw, w)

The operator T(e) needs some further explanation. If e is a multiple of n,then T(e) is, apart from a sign, the identity operator (e = 2kn) or the operatorgiven by f- f( -t) (e = (2k + l)n). If e is not a multiple of n , then T(e) isgiven by

(T(e)f)(t) = (i sin e)-l f expC~ie «Z2 + t2) cos e - 2zt») f(z) dz (68)

with some choice for the square root.H we take, e.g. f (t) = 1, then it can be verified that

(T(e)f) (t) = (cos e)-i exp( -n i t2 tan e).

Since the Wigner distribution of f equals oo(w) we see from (67) that theWigner distribution of (coset*exp(-nit2tane) is a delta function coneen-trated on the line {À. (-sine, case) IÀ. E IR}.We have the following consequence of the table. If we have anf of the form

f(t) = exp(2n icp(t» whose Wigner distribution we want to consider near apoint (to, wo), we may assume that cp'(to) = cp"(to) = O. For otherwise we con-sider fl(t) = exp(2n i(cp(t) - cp'(toHt - to) - !cp"(toHt - tO)2» which has thesame form as f and for which we have

H)l~~l(t, w) = H),~(t, co + cp'(to) + tcp"(to».

To get some further insight into the behaviour of Wigner distributionsaround certain curves we consider the following question. What can we sayabout functions whose Wigner distributions are completely concentrated on

104 Phillps Journalof Research Vol.37 No.3 1982

Phlllps Journni of Research Vol.37 No.3 1982 105


a curve (or set of curves)? It was shown in ref. 18 that, under rather severerestrictions on the curve, only chirps, c)-functions and exponentials haveWigner distributions concentrated on a curve. If one relaxes the assumptionson the curve then, as a rule, one can say that at all points of the curve wherethe radius of curvature is finite the Wigner distribution cannot have "mass" .

Let us give a not too rigorous proof of this statement for a rather generalcase (for a rigorous treatment we need to consider the Wigner distribution as ageneralized function). Let f be a function whose Wigner distribution is con-centrated on a smooth curve C in the plane. That is, we have a smooth func-tion y: IR ~ 1R2 and a continuous function g defined on C= (y(s) ISE IR] suchthat f f H),~ (t,w) oe; co)dt dw = f g(y(s) G(y(s)) Iy '(s) I ds for all smoothfunctions G of compact support. Let So E IR be such that g(y(so)) > 0 and suchthat the tangent line at C through y(so) = (to,wo) is perpendicular to the t-axis.This is no serious restrietion since we can apply a rotation of the plane if neces-sary. Also assume there is a c)> 0 such that the vertical Ct = {(t, w) IWEIR]intersects C in at most 2 points if I t - toI < C). (This restrietion simplifies thepresentation of the proof; however, the arguments can be modified so as toapply as well if the verticals interseet C in finitely or, if proper assumptions onconvergence are made, in infinitely many points.) Finally assume that C has iny(so) a finite radius of curvature. We get the following picture.

w

c

Fig.9.

A. J. E. M. Janssen

We shall derive a contradiction now. We get by Fourier inversion (integra-tion along Ct; see picture)

f(t + !v)f(t - !v) = f exp(2n i to v) H),~(t, w) dw= d1(t) g(Cl(t» exp(2n i Cl(t) v) + d2(t) g(C2(t» exp(2n i C2(t) v) (69)

for all t with to - ö < t < to and all v, and

f(t + !v)f(t -lv) = 0 (70)

for all t with to< t < to + ö and all v. Here d;(t) is the modulus of the deriv-ative of the tangent at C through c;(t). Since g is continuous and non-zero at(to,wo) and since c;(t)-+wo(t1' to), d;(t)-+ co(tj to) we see that

f(t + lv)f(t - !v) 4= 0

for (t, v) in a set (to - Öl, to) X (-Ö2,Ö2). This implies thatf(t) 4= 0 in an inter-val (to - ös, to + ös). But this contradiets (70) 0).

We now consider approximations of the Wigner distribution of f(t) =exp(2n icp(t». It is not unreasonable, in view of the statement just proved, toexpect a large value for the spread of the Wigner distribution around the curve(t, cp'(t» at those points to where cp"'(to) is large. Let to E IR, and assume thatcp"'(tO) 4= O. Replace cp in the integral (63) by its third order Taylor approx-imation around to. We get as an approximation for H),~ (to,w) the integral 00)

f exp(2n i (cp'(to) - w) s + ~; cp1I'(tO) SS) ds. (71)

We can express (71) in terms of Airy's function

Ai(z) = 2~ f exp(j i t3 + iz t) dt

as Ai«cp'(to) - w)fJ-l)fJ-l where, fJ = /3(to) = cp'"(to)/32n2. A similar approx-imation was used by M. V. Berry in ref. 5 to obtain a semi-classicallimitingform of the Wigner distribution of the bound energy eigenstates for integrablesystems.

Although we have to keep in mind that (71) only provides an approximationof H),~ we make the following observations 000). Also see fig. 10.

(72)

0) A similar argument is used in ref. 3. The conclusion in ref. 3 that this implies that a Wignerdistribution cannot be concentrated on curves other than straight lines is wrong 18).

00) Formula (71) is useful as an approximation for H),j(lo,w) if (to, w) lies not too far away fromthe curve (t, cp '(I» and if the integral (63) (with to instead of t) has at most two stationarypoints s close to zero. When (to, w) is not close to the critical curve, or if there are more thantwo stationary points, the stationary phase method is more appropriate.

000) Similar observations were made by M. V. Berry in ref. 5.

106 Philip. Journalof Research Vol.37 No.3 1982

Philip, Journal of Research Vol. 37 No.3 1982 107


Ai{tJ

Î 10

4 6 8 10-t

-1.0

Fig. 10. Sketch of Airy's function (72) for te [-10,10].

(1) Assume that (/J'''(to) varies only slowly with to. Then (71) depends to alarge extent only on (/J'(to) - co, Hence the levellines of (71) tend to bereplicas of the curve (z, (/J'(t)) shifted in vertical direction. The larger thevalue of I (/J" '(to) I the larger the distance between the levellines when thedifference between the various levels is kept fixed.

(2) There is a dramatic difference between the behaviour of Ai(z) for z <0and z> O. We have Ai(z) > 0 for z > 0, Ai(z) oscillates for z < 0, Aitakes its maximum for a z close to -1, Ai has an asymptotic expansion

Ai(z) - !n-! z-i exp( -1zl) :f (_I)k ck(i zitk (73)o

for z > 0, z-+ 00, and Ai has an asymptotic expansion

Ai(z) - n-l( -zti {sin (i( -z)i + v:f (- It C2k(i( -Z)Jt2k, 0

-cos (i(-z)i + ~):f (-ItC2k+1(i(-z)it2k-1 (74)o

for z < 0, z -+ - 00. Here Ck = r(3k + i)/54k k! Ftk +~)(these formulas aretaken from ref. 1, sec. 10.4). Hence, Ai(z) decays fast for z > 0 and ratherslowly for z <O.Since (71) involves «(/J" '(to)ti we have fast decay of (71)at the convex side and oscillatory behaviour of (71) at the concave side ofthe curve (t, (/J'(t)) (assumed that this curve is indeed convex or concave).

(3) When (/J"'(to) is close to zero, the set of all W for which (71) is significantlyunequal to zero in a large lopsided interval around (/J'(to). In practice, how-ever, we always deal with windowed signals. Ifthe window-length is ratherlarge, but still small compared to 11/ (/J" '(to) I , it is better to approximate (/Jby a second order Taylor expansion. Then we get oo(w-(/J'(to)) instead of(71).

A. J. E. M. Janssen

The slow decay of H),~ at the concave side of the curve (t, rp'(t)) must beconsidered as a kind of interference phenomenon. A relevant formula in thisconnection is

4 IIH)}> (2a - t, 2b - w) H),~>(t, w) dt dco = (H),~(a,b))2, (75)

which holds for generaljand which can be derived from Moyal's formula in astraightforward manner. What the formula shows is this. If we have anjforwhich H),~ is large and positive around two points (thWI) and (t2,W2), then wecan expect IH),~I to be large around the point (!{tl + tz), HWI + (2)) = (a,b)since the neighbourhood of the point (t,w) = (tl,WI) gives a large contribu-tion to the double integral in (75). Note that (75) does not give informationabout the sign of H),~ at (a,b). Hence if j(t) = exp(2nirp(t)) and (a,b) is apoint of the form (Ht+ s), Hrp'(t) + rp'(s))) then the value of H),~ at (a,b) willnot be negligible in general by "interference" of the "mass" at the points(t, rp'(t)) and (s, rp'(s)).

In sec. 4 we claimed that integrals of H),~ over regions far away from thecurve (t, rp'(t)) are negligible. This claim will now be substantiated. To that endwe consider 'P, defined by

'P(t, w) = 21 1 H),~ (t+s, W + À) exp( -2n (S2+ À2)) ds d).. (76)

It is well-known that (see ref. 8, sec. 27.12.1)

'P(t, w) = 12t1 f(s) exp( -2n iW s - n (s - t)2) ds12; (77)

in particular, 'P(t, w) ~ 0 for all it,w) E fR2. Hence the oscillations of H),~around the curve (t, rp'(t)) can be appeased by convoluting with the Gaussian2 exp( -2n(s2 + À.2)). Consider the case t = 0 and let the integral I(w) bedefined by

I(w) = 2t 1 exp(2n i rp(s) - 2n i cv s - n S2)ds, (78)

and assume that rp(O)= rp'(O)= 0 (note that If'(O,w) = II(w)12). Now I(w)can be approximated by inserting the yd order Taylor polynomial !as2 + lbss

, of rp around 0 in the exponential. Some elementary manipulations show thatthe resulting approximation J(cv) can be expressed as

J(w) = 2t(nb)-l exp(in i b gS + 2n i g co)Ai( -2n (nb)-l (w + !b (2)), (79)

where {]= (a+i)b-l• The approximation J(w) for I(w) is far more accuratethan the one given by (71) is for H),~ since we are now dealing with rapidlyconvergent integrals.Let us assume b>O. Using the asymptotic expansions (74) and (73) (valid

in Iarg z - nl < in and I arg z I <n respectively) one can show that

108 PhllIps Journalof Research Vol.37 No,3 '1982


J(w) = O(w; exp( -27t b-1 co + 7t I al (2/b)l w!»

for co ~ 00, and that

Ha2 - 1) 7t(2/b)llwli»(81)

for to ~ - 00. Note that J(w) decays fast at both sides of the critical curve(t, (/J '(t». The decay is faster at the convex side of the curve than at the concaveside, but the difference is far less dramatic than for H),~. See also fig. 2 inref. 9, Part I where an FM signal is considered.

Acknowledgement

The author wishes to thank his colleagues of Philips Research Laboratories,Eindhoven, especially T. A. C. M. Claasen, C. P. Janse and A. J. M. Kaizerfor stimulating discussions and G. F. M. Beenker and B. P. A. Boonstra forproducing the pictures.

Philips Research Laboratories Eindhoven, March 1982

REFERENCES1) M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publica-

tions, New York, 1970.2) G. S. Agarwal and E. Wolf, Calculus for functions of noncommuting operators and general

phase-space methods in quantum mechanics I, Phys. Rev. D 2, pp. 2161-2186, 1970.3) N. L. Balazs, Weyl's association, Wigner's function and affine geometry, Physica 102A, pp.

236-254, 1980.4) R. Beals and C. Fefferman, Spatially inhomogeneous pseudo-differential operators I,

Commun, Pure Appl. Math. 27, pp. 1-24, 1974.5) M. V. Berry, Semi-classical mechanics in phase space: a study of Wigner's function, Phil.

Trans. Royal Soc. London A 287, pp. 237-271, 1977.6) R. P. Boas, Entire functions, Academic Press, New York, 1954.7) N. G. de Br u ij n, Uncertainty principles in Fourier analysis, in Inequalities, ed. by O.

Shisha, Academic Press, New York, pp. 57-71, 1967.8) N. G. de Br uij n, A theory of generalized functions, with applications to Wigner distribution

and Weyl correspondence, Nieuw Archief voor Wiskunde 21, pp. 205-280, 1973.9) T. A. C. M. Claasen and W. F. G. Mecklenbräuker, The Wigner distribution - A tool

for time-frequency signal analysis, Part I, 1I, lIl, Philips J. Res. 35, pp. 217-250, 276-300,372-389, 1980.

10) T. A. C. M. Claasen and W. F. G. Mecklenbräuker, On stationary linear time-varyingsystems, IEEE, Transactions Vol. CAS-29(3), pp. 169-184, 1982.

") L. Cohen, Generalized phase-space distribution functions, J. Math. Phys, 7, pp. 781-786,1966.

12) I. Daubechies, On the distributions corresponding to bounded operators in the Weyl quan-tization, Commun. Math. Phys. 75, pp. 229-238, 1980.

13) H. J. Groenewold, On the principle of elementary quantum mechanics, Physica 21, pp.405-460, 1946.

Phllips Journalof Research Vol. 37 No.3 1982

(80)

109

A. J. E. M. Janssen

14) A. Grossman, G. Loupias and E. M. Stein, An algebra of pseudo-differential operatorsand quantum mechanics in phase space, Ann. Inst. Fourier 18, pp. 343-368, 1969.

15) L. Hörmander, The Weyl calculus of pseudo-differential operators, Commun. Pure Appl.Math. 32, pp. 359-443, 1979. •

16) A. J. E. M. Janssen, Positivity of weighted Wigner distributions, SIAM J. Math. Anal. 12,pp. 752-758, 1981.

17) A. J. E. M. Janssen, Application of the Wigner distribution to harmonic analysis of general-ized stochastic processes, MC-tract 114, Mathematisch Centrum, Amsterdam, 1979.

18) A. J. E. M. Janssen, A note on Hudson's theorem about functions with non-negativeWigner distributions, submitted to SIAM J. Math. Anal.

19) P. Johannesma, A. Aertsen, B. Cr an en and L. van Erning, The phonochrome: Acoherent spectro-temporal representation of sound, Hearing Research 5, pp. 123-145, 1981.

20) M. J. Levin, Instantaneous spectra and ambiguity functions, IEEE Trans. on Inf. Th. IT-10pp. 95-97, 1964.

21) H. Margenau and L. Cohen, Probabilities in quantum mechanics, in Quantum theory andreality, ed. by M. Bunge, eh. 4, Springer-Verlag, Berlin, pp. 71-89, 1967.

22) J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45,pp. 99-132, 1949.

23) A. W. Rihaczek, Signal energy distribution in time and frequency, IEEE Trans. on Inf. Th .. IT-14, pp. 369-374, 1968.

24) G. Szegö, Orthogonal polynomials, 4th ed. Colloquium Publication, Vol. 23, Am. Math.Soc., Providence, Rhode Island, 1975.

25) J. Ville, Théorie et applications de la notion de signal analytique, Cables et Transmission 2,pp. 61-74, 1948.

26) E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys, Rev. 40, pp.749-759, 1932.

110 Phlllps Journalof Research Vol.37 No.3 1982

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