Positivity of time-frequency distribution functions

Signal Processing 14 (1988) 243-252 North Holland

243

POSITIVITY OF T I M E - F R E Q U E N C Y

DISTRIBUTION FUNCTIONS

A.J.E.M. JANSSEN Philips Research Laboratories, P.O. Box 80.000, 5600 JA Eindhoven, The Netherlands Received 19 November 1986 Revised 18 May 1987

Abstract. This paper deals with the question how various 'natural' conditions posed on time-frequency distribution functions prevent them to be nonnegative everywhere for all signals. The attention is restricted mainly to distribution functions that involve the signal bilinearly. This paper summarizes and extends results recently obtained; the emphasis is on a coherent presentation of these results, rather than on rigorous proofs of them (which can be found in or obtained from existing literature). Distribution functions of the Wigner-Rihaczek type are considered in some more detail.

Zusammenfassung. Nachgegangen wird der Frage, auf welche Weise verschiedene 'natiirliche' Bedingungen, auf ZeitFrequenz-Verteilungs funktionen angewendet, bewirken k/Snnen, dab diese Verteilungsfunktionen fiir bestimmte Signale, Zeiten oder Frequenzen negativ werden. Die Untersuchung beschriinkt sich im wesentlichen auf Verteilungsfunktionen, die bilinear auf das Signal zugreifen. Im vorliegenden Beitrag werden Resultate jiingeren Datums zusammengefal3t und erweitert; hierbei wird besonderer Weft auf die zusammenhiingende Darstellung dieser Ergebnisse gelegt, weniger auf einer genauen Beweisfiihrung (diese liiSt sich der bestehenden Literatur entnehmen). Verteilungsfunktionen vom Wigner-Rihaczek-Typ werden in etwas gr/SBerem Detail diskutiert. R6sum& Cet article s'intrresse au fait selon lequel ditirrentes conditions 'naturelles' imposres fi des distributions tempsfrrquence les emp~cbent d'etre partout non nrgatives pour tout signal. L'article, qui se restreint essentiellement aux distributions bilinraires, rrsume et &end des rrsultats obtenus rrcemment, l'accent 6tant mis sur une prrsentation cohrbente de ces rrsultats plut6t que sur leur preuve rigoureuse (que l'on peut trouver par ailleurs dans la littrrature). Une attention toute particulirre est portre aux distributions de type Wigner-Rihaczek. Keywords. Time-frequency distribution, uncertainty principle, positivity.

1. Introduction In this paper we consider mappings f ~ L 2 ( [ ~ ) ~ C f E L2(~2), mapping signals f onto time-frequency distribution functions Cs. Since we like to interpret CT as an energy distribution function of f over the time-frequency ((4 to))-plane, we consider the following list of conditions. (a) Nonnegativity, i.e., Cy(t, to) >10 for all (4 to) e R 2 and f ~ L2(R). (b) Correct rnarginals, i.e., for all f c L 2 ( ~ ) ,

I C~(t,to)dto=lf(t),2, t ~ ,

I Cf(t, to)dt=lF(to),2, tocR,

(1)

where F is the Fourier transform of f, given by F(to)= I

e-2~i'°~f(t)dt, toeR.

0165-1684/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)

(2)

A.J.E.M. Janssen/ Positivity of time-frequency distribution functions

244

(c) Bilinearity, i.e., for every (t, to)~ R2 there is a linear operator M(t, to) such that, for all f ~ LE(R), Cf(t, to) = (M(t, (d)

where

to)f,f).

(3)

Shift properties, i.e., for all a c R, b c R and all f E L2(II~), (t, ~) e R 2, Crj(t, to)=Cf(t+a, to), CRj(t, to)=Cf(t, to+b), T~, Rb are time, frequency shift operators, given by (Tof)(t)=f(t+a), (Rbf)(t) = e 2~ib'f(t)

(4)

for all a c ~, b c ~, and all f c L2(R).

(e) Correct support properties, i.e., for all a > 0, b > 0 and all f ~ L2(R) f ( t ) = 0, F(to) = 0,

Itl>a~ff(t, to)=O, Itl>a, toeR, /tol > b ~ Q ( t , to) =0,

Itol>b, tcR.

(5)

(f) Validity of Moyal' s formula, i.e., for all f c L2(R), g ~ L2(R),

f I C f ( t , to)Cg(t, to)dtdto= .f f(t)g(t) dtl 2. (g)

Dilation property, i.e., for all Cz,j-(t, to)= Cf(yt, y-lto),

(6)

3' > 0 and all f c L2(R), (t,

to)ER 2,

(7)

where Zv is the dilation operator given by (Zvf)(t)= 3)/2f(yt) for f 6 L2(R) and 3' > 0. Since the distribution functions are to be used in signal analysis, there is a requirement that cannot be phrased easily in terms of mathematical formulas: the distribution functions should yield pictures of the energy distribution that agree with the signal analyst's intuition for signals on which he can check. It may thus happen that a distribution function, failing to satisfy many of the conditions of the above list, is preferred over one that satisfies almost all conditions. For instance, the distribution functions introduced by Cohen and Zaparovanny [7] and recommended by Cohen and Posch [5,6] satisfy conditions (a), (b), (d), (e), and (g), but they give unsatisfactory results for signals of FM type (see [22]). Since signal analysts have a quite clear idea how the signal energy of the latter type of signals should be distributed, they will hesitate to use distribution functions that do not perform well for these signals. It is further noted that the Cohen-Zaparovanny-Posch distribution functions involve the signal in a nonbilinear way. The condition of bilinearity is a logical one in the context of signal analysis: one likes to see the global property

E.s+~ = [c~12Es +1/312Eg + adEi, g +/3aG.i, with Ef either the instantaneous power i3q2 or the spectral density function IFI 2 and to be reflected locally by

c~f÷~=l,~l=Cs + l/31=G+ ,~fic~,~+ /3~G, s

(8)

Ef,g, Eg,ycross terms, (9)

for all a ~ C,/3 c C and all f ~ L2(R), g ~ L2(R). Since, furthermore, distribution functions involving the signal bilinearly allow for a smoothly running mathematical theory, we shall consider in this paper distribution functions that satisfy condition (c) of the list given above, although it is very well conceivable that, for certain applications, nonlinear enhancement techniques are necessary to obtain easily interpretable results (see, e.g., [26]). Signal Processing

A.J.E.M. Janssen/Positivity of time:frequenO, distribution#unctions

245

A second condition which is highly desired is condition (d). In this paper we shall therefore consider bilinear distribution functions satisfying the shift properties. As prime examples of such distributions we have the Wigner distribution and the Rihaczek distribution (which both satisfy conditions (b), (c), (d), (e), (f), and (g)) and distributions of the spectrogram type: Ci.(4~o;w)=

I;

f(t-s)e

2~'i~°'w(s)

,

( 4 w ) c N 2,

(10)

associated with the window w, which satisfy conditions (a), (c), and (d). Of the remaining conditions only condition (f) needs some further explanation since it is not intuitively clear why this is a desirable property. It turns out (see Theorems 3.5, 3.6, and 3.7) that distributions satisfying condition (f) exhibit negative values to an extent which is in accordance with the Heisenberg uncertainty principle. A further point which makes these distributions interesting is that the right-hand side of equation (10) can be written as the two-dimensional convolution of C c and C,,. This property is useful, for example, for signal detection purposes as in [23]. We note that condition (f) is 'affine' in the sense that if the mapping f ~ CI satisfies condition (f), then so does the mapping f ~ C~R~f for all a c R, b ~ R. This 'loss of the origin' can usually be compensated for by imposing a further, mostly very weak condition, such as condition (28).

2. Bilinear distribution functions satisfying the shift properties The set of distribution functions satisfying conditions (c) and (d) of Section 1 can be parameterized by means of a function qb of two real variables according to the formula

=

e

=II

4(0, u)f(u +½r)f(u-½r) dO du

~ ( t - s , oJ-A)Wy.(s,A)dsdA,

( , , w ) E R 2,

(11)

where ¢ is the double Fourier transform of 4, given by ~0(t,w)=fle

z=i'°'+~'°)@(O,r)dOdr,

( t , w ) 6 ~ 2,

(12)

and Ws is the Wigner distribution of f, given by ~(t,,o)=f

e 2~i ..... Jt t + ' ~sutt-~s) "~" ' " ds,

(t,w)e[~ 2.

(13)

One usually considers smooth, bounded functions qb, while the ~o's must be treated as generalized functions. In order for C~~) to be real-valued for all f, we require that q~ is real. For a rigorous approach, a suitable theory of generalized function is indispensable (see, e.g., [19]). Since this paper aims at giving an easily readable account of known results, such an approach is not adopted here. As a consequence, the results listed in this paper are not always as general as possible, while sometimes not all technical conditions under which the results hold are specified. The distributions given by equation (11) have been studied extensively, in particular with respect to the question how further conditions, such as those given in Section 1, are reflected by restrictions on q> Vol. 14, No. 3, April 1988

A.J.E.M. Janssen/ Positivity of time-frequency distribution functions

246

or ~o (see, e.g., [4], [25], [24], [9], [11], [3], [27], [18], [16], [28], [12], [19]). In this section we are interested in results that show to what extent the nonnegativity condition (a) is violated when one or more conditions of the list in Section 1 are added to conditions (c) and (d). In Section 3 we give results that show how much smoothing over the time-frequency plane is at least required to turn the distribution functions (11) that satisfy either condition (b) or (f) into nonnegative ones, and in Section 4 we give results that show how nonnegativity of Wigner-Rihaczek type distributions of an f in (large subsets of) R 2 is reflected by restrictions on f 2.1. Theorem (Wigner [29]). There is no function q~ such that the mapping f ~ C~f~) satisfies conditions (a) and (b). In fact, Wigner has proved this result for mappings f-~ Cs that are bilinear but that do not need to satisfy the shift properties. From the proof of Wigner's theorem one can easily distill the following stronger version of Theorem 2.1 ( f . g denotes the function f ( t ) g ( t ) , t • ~). 2.2. Theorem (Wigner [29]). Assume that the mapping f-> C~0, ~(7(~)~>0, and f . g=--O, then there are a e C and b • C such that r~(~ g ~--"af+bg takes negative values, unless F . G-~ O. As a particular case we see that C~f+)bg takes negative values for some a • C, b • C when C~~) >~0, ¢7~ ) 1>0 and f ~ 0, g ~ 0 vanish outside two disjoint bounded intervals. It is noted that there are nontrivial, smooth f • L2(R), g • L2(R) with smooth F, G such that f . g---0, F . G---0. The next two theorems give results that are in some sense complementary to Theorem 2.2. 2.3. Theorem (Janssen [21]). Assume that the mapping f-> C~~) satisfies condition (b), while the set S:={( 0, ~')l ~(0, T)#0} is dense in R 2, and that f,,g are smooth signals with C ~ ) > 0, v~¢'~)>0 (strict inequality) everywhere. Then there are a • C and b • C such that ~r<~) af+bg takes negative values, unless f and g are proportional. 2.4. Theorem (Janssen [21]). Let cp be as in Theorem 2.3, and assume that f, g are smooth signals with ) . takes negative values, unless C~~) ~>-0, v g¢~( ) ~>- 0 everywhere. Then there are a e C and b c C such that ~" ~ay+Dg either f and g are proportional o f f . g=-O and F . G=-O. To our knowledge, the next two theorems are new. We therefore give a sketch of their proofs. -" (¢') 2.5. Theorem. Assume that the mapping f _> t~ y satisfies the correct support properties (e). Then there is an f such that C~f~) takes negative values, unless qb ~-O.

Proof. Assume that C}~)~>0 everywhere for all f. When f is smooth we have, according to (11), that

C
~p(t, to)dto,

teR

(14)

A.J.E.M. Janssen/ Positivity of time-frequency distribution functions

247

(in general, this r must be treated as a distribution). Then we have for t c R and all smooth f,

This shows that r is a nonnegative distribution, so that it can be identified with a nonnegative measure d/z. This measure must be concentrated at 0 for otherwise one could find functions f, vanishing outside interval [ - 1 , 1], such that the right-hand side of (15) does not vanish for all t~ [ - 1 , 1], contradicting the correct support property. We conclude that r(t)--CSo(t) for some c ~>0. Similarly,

f

~(t, to) dt = dSo(to),

to~R,

(16)

for some d/-- 0. Hence, qb(0, T) = c, ~(0, 0) = d, so that by smoothness of • we have q0(0, 7) = qb(0, 0) = c. r~a,~ has (within a factor c) correct marginals, so that Theorem 2.1 applies; a This means, however, that "--r contradiction, unless c = 0 , i.e. (by (15)), C ~ - - - 0 for allf. [] 2.6. Theorem. Assume that the mapping f -~ C~f~ satisfies the dilation property (g). Then there is an f such

that C~~ takes negative values, unless c19=_O. Proof. It is not hard to show that there is a smooth function ~: such that q~(0, r ) = ~(0T). Hence, qb(0, 0) = ~(0, 7) = ~(0), and as in the proof of Theorem 2.5 we arrive at a contradiction, unless • --- 0. [] At this point we note that the conditions on correct marginals and on correct support properties are independent in the sense that neither one implies the other.

3. Smoothing time-frequency distribution functions The theorems of Section 2 have shown that the nonnegativity condition is incompatible with most of the other conditions listed in Section 2 for bilinear distribution functions satisfying the shift properties. We consider now averages of the distribution functions over the time-frequency plane. One could hope these averages to be nonnegative when the Heisenberg uncertainty principle is taken into account properly. The actual state-of-affairs seems to be rather subtle as one can see already from the first three theorems of this section for the Wigner distribution. 3.1. Theorem. When f ~ L2(R), y8 <~ 1, a e R, b ~ R, we have (3,8) 1/2 f f exp{-2~r(3,s2+&o2)} Wy(s+ a, to+h) ds dto>~O.

(17)

This theorem seems to have been noted for the first time by Husimi [15]; it has been rediscovered by many authors (see, e.g., [1, 2, 8]). Formula (17) has an interpretation in the context of positivity properties of spectra of nonstationary stochastic processes. We shall amply show that IVy takes negative values, unless f is a (possibly degenerate) Gaussian. Now, consider the stochastic function f - - TsR,of, with Ts and R~ as under condition (d), where (s, to) is jointly Gaussian distributed with probability density function (3,8)1/2 exp{-2"rr(3,s2+ &o E)}. Then, the expected Wigner distribution E W s equals the left-hand side of inequality (17) and is hence nonnegative Vol. 14, NO. 3, April 1988

248

A.J.E.M. Janssen/ Positivity of time-frequency distribution functions

when yr<~ 1. It has been noted by Flandrin [10] that expected Wigner distributions are more often nonnegative everywhere than Wigner distributions of deterministic functions. The following theorems detail Theorem 3.1 further. They are meant to indicate how Heisenberg's uncertainty principle should be taken into account to get nonnegative results. For instance, it follows from theorem 3.3, that it is not sufficient to average Wigner distributions over sets of planar measure/> 1, as is often believed. 3.2. Theorem (Hudson [14], Janssen [20]). When f c L2(R), y~ > 1, and (17) is nonnegativefor all a ~ ~,

b E R, then f is a Gaussian. In this theorem, certain generalized functions f can be allowed; then, also certain degenerate Gaussians must be allowed as functions f for which (17) is nonnegative everywhere with y6 > 1. What Theorem 3.2 says is that the Wigner distribution of a non-Gaussian f exhibits negative values to an extent that smoothing as in (17) only yields nonnegative results when 3,6 is as large as 1. 3.3. Theorem (Janssen [17]). Let y > 0, and assume that K is a measurable function such that exp{2~r(yt2+ y-lto2)}K(t, to)-> 0 as t2+t02->oo. Then, for any a, b ~ R there is an f ~ L2(R) with

f Y K(t, to)Wf(t+a, to+b) dtdoo
(18)

This theorem shows that the strong conclusion in Theorem 3.2 must be replaced by a much weaker one when the smoothing kernel in (18) is non-Gaussian. In the next four theorems we shall give similar results for the averages

f f K(t, to)W~t')(t,w)dt dto, where the mapping f

_> .", ( ~ )

t~s

(19)

satisfies either condition (b) or (f).

3.4. Theorem (Janssen [19]). Assume that the mapping f-> C(y~ ) satisfies condition (b). Let 6 > 0, y > 0 and assume that K ( t, to)= 0 (exp{-2xr(yt2+ &o"2)}). (i) I f y r > 1, then there is an f 6 L2(R) such that (19) is negative. (ii) I f y6 = 1 and (19) is nonnegative for all f c L2(R), then ~=--1 (Wigner distribution case) and K(t, to) = a exp{-2~r(yt2 + y-lto2)} for some a >I O. 3.5. Theorem. Assume that the mapping f-> C~y~') satisfies condition (f), and let K ~ L~(R 2) n L2(R2). In order for (19) to be nonnegative for aH f c L2(R) it is necessary that/3 <~ct 2. Here,

ct=Sf

K(t, to)dtdto,

/3=ffK2(t,

to)dtdto.

Proof. The proof consists of a slight adaptation of the first part of the proof of [19, Theorem 4.2].

(20) []

The condition/3 ~t O. We have for any function G(t, to) >>-0 Signal Processing

249

A.J.E.M. Janssen/ Positivity of time-frequency distribution functions

by the Cauchy-Schwarz inequality

I .I K ( t ,

to)G(t, to) dt dto <~

(fl

G2(t, to) dt dto

)1/2 fl

K ( t , to) dt dto.

(21)

Hence, when for instance G(t, to) = 0 or 1 according to whether (t, to)¢ V or (t, to) E V, we get that

[f fvK(t,to)dtdto]/[ff K(t,,o)dtdto]<~m'/2(V),

(22)

where re(V) is the planar measure of V. One can generate more examples like this one, for example, by taking G( t, to) = (1 + a ( t 2 + to 2))-1 with a > 0, etc. Theorem 3.5 has the following two important special cases and refinements, which can be considered as manifestations of the Heisenberg uncertainty principle. 3.6. Theorem (Janssen [19]). Let K ( t, to) = exp{-2w(yt2+ (5to2)}, and let crp be as in Theorem 3.5. (i) I f y 3 > 1, then there is an f E L2(R) such that (19) is negative. (ii) I f 76 = 1 and (19) is nonnegative for all f E L2(R), then there are a E R and b E R such that C~¢)(t, to) = Wy( t + a, co + b) f o r all f E L2(R).

It follows from the remark to condition (f) at the end of Section 1 that the amount of nonuniqueness of the distributions in Theorem 3.6(ii) is precisely given by time-frequency shifts. This is so since (19) is nonnegative for a l l f if and only if it is nonnegative for a l l f with C(f*)(t, to) replaced by C} a') ( t + a, to + b), with arbitrary a, b E R. 3.7. Theorem. Let ~ be as in Theorem 3.5, and let V C R 2 be a measurable set. In order that

ffv

C~'~)(t' to) dt dto i>0

(23)

f o r all f E L2(R), it is necessary that the measure o f V is greater than or equal to 1.

4. Some further results for Wigner-Rihaczek type distributions

In this section we consider distribution functions of the type (a E R) C ~ ) ( t , to) = C(fO°)(t, to) = f e-2~'is'° f ( t + (½- c t ) s ) f ( t - (½+ a ) s ) ds,

(24)

which are obtained by taking ~ ( 0 , ¢)=exp{2"tria0~'}. These distribution functions satisfy conditions (b), (c), (d), (f), (g), and in addition condition (e) when [a[<~½. We shall be interested in consequences for the smooth signal f of nonnegativity of Re(C} ~)) in certain strips of the time-frequency plane.

4.1. Theorem (Janssen [21]) (i) Let - ~ < a < b < o0, and assume that f E L2(R) with f ( t ) # O, Re(C~")(t, to))/> 0 f o r t ~ (a, b), to E R.

Vol.14,No.3,April1988

A.J.E.M. Janssen/ Positivity of time-frequency distribution functions

250

Write If(0r = exp{~b(t)} with ~l, smooth• Then (l+a2)~b"(t)+2ct2(~b'(t))2<~O,

t~(a, b).

(25)

(ii) Let -oo < c < d < oo, and assume that f is such that F is smooth and F( to ) # O, Re(C~)(t, to))/> 0 for t c R , to ~ (c, d). Write IF(to)l = exp{~(to)} with ~ smooth. Then, (~+ct2)~"(to)+2a2(~'(to))2<~0,

toe(c, d).

(26)

This theorem can be used to locate points where Re(C~")(t, to)) < 0. When, for example, f is as in part (i) of Theorem 4.1 and O"(t)>0, we see that Re(C~)(t, to))<0 for some touR. We also observe that conditions (25) and (26) are weakest for a = 0---the Wigner distribution case. 4.2. Theorem (Janssen [21]). Let a ~ R , a # O , there is no smooth f e L 2 ( R ) everywhere. In case Ic~l = ½, the smoothness condition on f can be dropped.

with Re(C)~)(t, to)~>0,

When we restrict ourselves to functions • such that condition (b) or condition (f) is satisfied, we were not able to find any other function f ¢ L2(R) than the Gaussians (with t/, =-1) such that C~*)~ >0 everywhere. It is remarkable, though, that there do exist generalized functions f and t~ # 0 such that Re(C~ ")) i> 0 everywhere (in the generalized sense) (see [21, Section Ill.D]).

5. Conclusion

We have presented a number of theorems which show to what extent time-frequency distribution functions take negative values when certain conditions, such as correct marginal condition, correct support properties, validity of Moyal's formula or the dilation property, are imposed. Our attention has been restricted to bilinear distribution functions satisfying correct shift properties. As a general conclusion we found that either of the four conditions just mentioned is incompatible with the requirement that the distribution function should be nonnegative for all signals. It has furthermore been shown that the Wigner distribution is the distribution that is closest to being positive of all distributions satisfying either the correct marginal condition or Moyal's formula--in the sense that the Wigner distribution requires the least amount of Gaussian smoothing to become always nonnegative. Hence, if one is to sacrifice the nonnegativity condition in favour of the correct marginal condition or Moyal's formula, the 'best' distribution, from the point of view of nonnegativity, is the Wigner distribution. In this respect, the validity of Moyal's formula is a very interesting condition since imposing it yields distribution functions in which Heisenberg's uncertainty principle is reflected quite nicely (see Theorems 3.5, 3.6 and 3.7). At this point we note that among the distribution functions satisfying Moyal's formula, the Wigner distribution is also 'best' for a different reason viz. it has the least amount of spread. For, it • ~..> ,,-', (,/~) can be shown that among the functions • as in (11) such that the mapping J t~y satisfies Moyal's formula, the expression

f .[ ((t-a)2+(to-b)2)lC(f~)(t,

to)l 2 dt dto

(27)

is minimal for all a, b ~ R and all f when • = 1--the Wigner distribution case. Here, • must be such that, Signal Processing

A.J.E.M. Janssen / Positivity of time-frequency distribution .functions

with f ( t )

251

=f(-t),

c'?"(t, ,,,)= c~'~(-,, -o.,), (t, o,)~R 2,

(28)

a very n a t u r a l a n d n o n r e s t r i c t i v e c o n d i t i o n (recall the d i s c u s s i o n at the end o f Section 1). This t h e o r e m , w h i c h generalizes [19, Section 5], expresses that c o n c e n t r a t e d n e s s o f a signal f, such as a G a u s s i a n , in the t i m e - f r e q u e n c y p l a n e is best reflected by the W i g n e r d i s t r i b u t i o n when e q u a t i o n (27) is t a k e n as a m e a s u r e for the s p r e a d a r o u n d (a, b). L o o s e l y s p e a k i n g one c o u l d say that the W i g n e r d i s t r i b u t i o n has the best resolution. In Section 4, we d e m o n s t r a t e d h o w severe the c o n d i t i o n on d i s t r i b u t i o n s o f the W i g n e r - R i h a c z e k t y p e is o f b e i n g n o n n e g a t i v e in strips o f the t i m e - f r e q u e n c y plane. I n d e e d , c o n d i t i o n s (25) a n d (26) are very restrictive a n d can be u s e d to exhibit p o i n t s in the t i m e - f r e q u e n c y p l a n e where the d i s t r i b u t i o n gets negative. In p a r t i c u l a r , there are no ( s m o o t h ) functions f ~ 0 w h o s e W i g n e r - R i h a c z e k d i s t r i b u t i o n is n o n n e g a t i v e everywhere. A l t h o u g h the W i g n e r d i s t r i b u t i o n is ' b e s t ' in various respects, it severely suffers f r o m the o c c u r r e n c e o f negative values. In the W i g n e r d i s t r i b u t i o n , b u t also in the o t h e r d i s t r i b u t i o n s o f the W i g n e r - R i h a c z e k type, this is a p p a r e n t from w h a t m a y be c a l l e d interference f o r m u l a s (see [18, f o r m u l a (75)], [13, f o r m u l a (5)], a n d [21, f o r m u l a (73)]) which show where the cross terms in the d i s t r i b u t i o n o f a m u l t i c o m p o n e n t signal a p p e a r in the t i m e - f r e q u e n c y plane. This restricts the a p p l i c a b i l i t y o f this t y p e o f d i s t r i b u t i o n s severely, viz. to signals that consist o f only a few c o m p o n e n t s or to m u l t i c o m p o n e n t signals where one has a m e a n s , j u s t as in [26], to s u p p r e s s the d i s t u r b i n g cross terms w i t h o u t too m u c h affecting the signal terms in the d i s t r i b u t i o n (see [13]). It m a y even m e a n that one s h o u l d not restrict o n e ' s attention to b i l i n e a r d i s t r i b u t i o n s (see [5, 6, 7]), but these d i s t r i b u t i o n s have their w e a k points as well (see [22]).

References [1] G.A. Baker, I.E. McCarthy and C.E. Porter, "Applications of the phase space quasi-probability distribution to the nuclear shell model", Phys. Rev., Vol. 120, 1960, pp. 254264. [2] N.D. Cartwright, "A non-negative Wigner-type distribution", Physica, Vol. 83A, 1976, pp. 210-212. [3] T.A.C.M. Claasen and W.F.G. Mecklenbr~iuker, "The Wigner distribution--A tool for time-frequency signal analysis llI", Philips J. Res., Vol. 35, 1980, pp. 372-389. [4] L. Cohen, "Generalized phase-space distribution functions", J. Math. Phys., Vol. 7, 1966, pp. 781-786. [5] L. Cohen "Distributions in signal theory", Proc. I C A S S P 84, San Diego, CA, March 19-21, 1984, pp. 41.B.I.1-4. [6] L. Cohen and T. Posch, "Positive time-frequency distribution functions", IEEE Trans. Acoust., Speech, Signal Process., Vol. ASSP-33, 1985, pp. 31-38. [7] L. Cohen and Y.I. Zaparovanny, "Positive quantum joint distributions", J. Math. Phys., Vol. 21, 1980, pp. 794-796. [8] N.G. De Bruijn, "Uncertainty principles in Fourier analysis", in: O. Shisha, ed., Inequalities, Academic Press, New York, 1967, pp. 57-71. [9] B. Escudi6 and J. Grea, "Sur une formulation g6n6rale de la repr6sentation en temps et en fr6quence dans l'analyse des signaux d'6nergie finie", C.R. Acad. Sci., Vol. 283A, 1976, pp. 1049-1051.

[10] P. Flandrin, "On the positivity of the Wigner-Ville spectrum", Signal Processing, Vol. 11, 1986, pp. 187-189. [11] P. Flandrin and B. Escudi6, "Time and frequency representation of finite energy signals: A physical property as a result of a Hilbertian condition", Signal Processing, Vol. 2, 1980, pp. 93-100. [12] P. Flandrin and W. Martin, "Sur les conditions physiques assurant l'unicit6 de la repr6sentation de Wigner-Ville comme representation temps-fr~quence", Proc. 9~me CoIL sur le Traitemenet du Signal et ses Applications, Nice, 1983, pp. 43-49. [13] F. Hlawatsch, "Interference terms in the Wigner distribution", presented at: lnternat. Conf. on Digital Signal Processing, Florence, Italy, September 5-8, 1984. [14] R. L. Hudson, "When is the Wigner quasi-probability density non-negative?", Rept. Math. Phys., Vol. 6, 1974, pp. 249-252. [15] K. Husimi, "Some formal properties of the density matrix", Proc. Phys. Math. Soc. Japan, Vol. 22, 1940, pp. 264-314. [16] C.P. Janse and A.J.M. Kaizer, "Time-frequency distributions of loudspeakers: The application of the Wigner distribution", J. Audio Engrg. Soc., Vol. 31, 1983, pp. 198-223. [17] A.J.E.M. Janssen, "Positivity of weighted Wigner distributions", S I A M J. Math. AnaL, Vol. 12, 1981, pp. 752-758. Vol. 14, No. 3, April 1988

252

A.J.E.M. Janssen/ Positivity of time-frequency distribution functions

[18] A.J.E.M. Janssen, "On the locus and spread of pseudodensity functions in the time-frequency plane", Philips J. Res., Vol. 37, 1982, pp. 79-110. [19] A.J.E.M. Janssen, "Positivity properties of phase-plane distribution functions", J. Math. Phys., Vol. 25, 1984, pp. 2240-2252. [20] A.J.E.M. Janssen, "A note on Hudson's theorem about functions with nonnegative Wigner distributions", SIAM J. Math. Anal., Vol. 15, 1984, pp. 170-176. [21] A.J.E.M. Janssen, "Bilinear phase-plane distribution functions and positivity", J. Math. Phys., Vol. 26, 1985, pp. 1986-1994. [22] A.J.E.M. Janssen, "Note on a paper by L. Cohen and T. Posch concerning positive time-frequency distribution functions", IEEE Trans. Acoust., Speech, Signal Process., Vol. 35, 1987, pp. 701-703, [23] S. Kay and G.F. Boudreaux-Bartels, "On the optimality of the Wigner distribution for detection", Proc. ICASSP 85, Tampa, FL, March 26-29, 1985, pp. 27.2.1-4.

Signal Processing

[24] J.G. Kriiger and A. Poffyn, "Quantum mechanics in phase s p a c e - - l ' , Physica, Vol. 85A, 1976, pp. 84-100. [25] H. Margenau and L. Cohen, "Probabilities in quantum Mechanics", in: M. Bunge, ed., Quantum Theory and Reality, Springer, Berlin, 1967, pp. 71-89. [26] N.M. Marinovich and G. Eichmann, "An expansion of Wigner distribution and its applications", Proc. ICASSP 85, Florida, March 1985, pp. 27.3.1-4. [27] R.F. O'Connell and E.P. Wigner, "Quantum-mechanical distribution functions: conditions for uniqueness", Physics Lett., Vol. 83A, 1981, pp. 145-148. [28] M. Springborg, "Phase space functions and correspondence rules", J. Phys. A, Vol. 16, 1983, pp. 535-542. [29] E.P. Wigner, "Quantum-mechanical distribution functions revisited", in: W. Yourgrau and A. van der Merwe, eds., Perspectives in Quantum Theory, New York, 1971, pp. 25-36.