THE WIGNER DISTRIBUTION - A TOOL FOR TIME … Bound...philips j.res. 35, 372-389, 1980 . r1028 the wigner distribution - a tool for time-frequency signal analysis part ill: relations

Philips J. Res. 35, 372-389, 1980 . R1028

THE WIGNER DISTRIBUTION - A TOOL FORTIME-FREQUENCY SIGNAL ANALYSIS

PART Ill: RELATIONS WITH OTHER TIME-FREQUENCY SIGNALTRANSFORMA TIONS

by T. A. C. M. CLAASEN and W. F. G. MECKLENBRÄUKER

372 Philips Journalof Research Vol.35 No.6 1980

AbstractA comparison is made between the Wigner distribution and several othertime-frequency signal transformations. Amongst these are the ambiguityfunction, well-known from radar and sonar, and the spectrogram used inspeech analysis. It is shown that all these signal transformations are relatedto the Wigner distribution by a two-dimensional transformation with dif-ferent kernel functions.

1. Introduetion

The previous parts of this paper have primarily dealt with the description ofthe Wigner distribution and its properties. The aim of this part is to discussthe connection of the Wigner distribution with other time-frequency signaltransformations. First the relation between the Wigner distribution and theambiguity function is investigated. A comparison of some of the fundamentalproperties of these two transformations will be made.

Next, in section 3, a class of time-frequency signal transformations is con-sidered. This class has been discussed before in the context of quantummechanics by Cohen 1,2). Each member of this class is uniquely determined bya two-dimensional kernel function. The class is rather broad and contains,apart from the Wigner distribution itself, other known representations such asthe one proposed by Rihaczek 3) and the spectrogram 4,5) as will be shown insec. 4. Furthermore a number of properties of such a transformation will beconsidered that are of interest in the context of signal analysis. In this applica-tion these transformations are used to represent the distribution of the signalenergy in time and frequency. All properties that are investigated derive theirimportance from this interpretation. The restrictions these properties imposeon a particular representation will be discussed. Although the discussion willbe restricted to continuous-time signals, the results are easily extendable todiscrete-time signals, using the methods described in part Il. Moreover, onlyauto- Wigner distributions are considered since only these can be interpreted asan energy distribution.

If(ç, cv) = F(cv + ç/2)F*(cv - ç/2), (2.2)

The Wigner distribution

2. Wigner distribution and ambiguity function

As was already remarked in part I of this paper, the definitions of theWigner distribution (WD) and the ambiguity function (AF) look very muchalike at first glance. In this section we will discuss the relations between thesetwo time-frequency signal representations, and point out the similarities andthe differences. The ambiguity function has been widely used in the context ofradar and sonar, and its properties are very well understood 6,7,8). Thediscussion here will therefore be focussed only on those properties that are ofimportance for a better understanding of the WD.As before, we denote the signal by f(t) and its spectrum by F(cv).

Introducing the functions yj(t, r) by

yj(t, r) = f(t + r/2) f*(t - r/2) (2.1)

and If(ç, cv) by

the WD can be written as

00

(2.3a)-00

00

= _1 J eW If(ç, cv) dç.21t

(2.3b)-00

Similarly the AF is given by 8)

00

Aj(ç, r) = J e-jçt yj(t, r) dt (2.4a)-00

00

1 J .= 21t eJwr If(ç, cv) dcv. (2.4b)-00

Relations (2.3) and (2.4) are represented in fig. 1, where the operator 8Jirepresents the Fourier transform with respect to the variable t, and similarlyfor the other variables.It follows immediately that the WD and the AF are related by the two-

dimensional Fourier transform

Phlllps Journalof Research Vol.35 No.6 1980 373

T. A. C. M. Claasen and W. F. G. Mecklenbräuker

1., (to T)

y~

f (t) = ± f (- t) (2.6)

Af{~. T) W.,{t. ui)

r-,{~.w)

Fig. 1. Diagram indicating the relation between the signal, the spectrum, the Wigner distributionand the ambiguity function.

00 00

Af(Ç, r) = 2~ J J e-j(Çt-WT) UJ(t, w) dt dw. (2.5)-00 -co

In general eq. (2.5) indicates that the WD and the AF are different signalrepresentations. A noteworthy difference is for example that the WD of anysignal is real, while the AF, even of a real-valued signal, will in general becomplex. However, for signals that are even or odd functions of time, Le.

the AF and the WD are the same up to scale factors, Le. 9)

UJ(t, cv) = ± 2Af(2cv, 2t). (2.7)

As an example the chirp signal

f(t) = ejat2/2 (2.8)

has WD and AF given by

UJ(t, w) = Af(CV, t) = 2n o(cv - at). (2.9)

The differences between the two signal representations become most pro-nounced if shifts in time or frequency of the signal are considered. While suchshifts lead to corresponding shifts in the WD, the effect on the AF is a phase

374 Phllips Journalor Research Vol.35 No.6 1980

""g;i....==:I!!.S-

S;;=--e~...'"Z?'"~""i!!l

~-..JUl

TABLE I

signal spectrum Wigner distribution ambiguity function

original f(t) F(w) U'l(t, w) Aj(ç, r)

.time shift f(t - to) F(w) e-iw1o U'l(t - to,.w) Af(Ç, r) e-iç/o

frequencyf(t) eiWo1 F( co- wo) U'l(t, co- wo) Af(Ç, r) eiWo"l"

shift

co co co

filtering f f(u)h(t-u)du F(w)H(w) f U'l(u,w) Wh(t-u,w)du f Af(Ç, u)Ah(Ç, r-u)du-co -co -co

co co comodula-

f(t) m(t) _1 J F(rt)M(w-rt)drt _1 J U'l(t, rt) Wm(t,co=n) àn ~ J AArt, r)Am(ç -rt, r) drttion 21t 21t 21t

-co -co -co

~~

oi1::1~S.c",'

~5:l:::.....c:ï::1


factor only, as can be seen from table 1. This illustrates clearly that the inter-pretation of the time and frequency variables of the WD corresponds to thatof the signal, while this is not the case for the AF. In fact the magnitude of theAF is completely insensitive to shifts in time or frequency of the signal. Intable I, the effects of filtering and modulation on the WD and the AF,respectively, have been indicated too. Although the corresponding relationslook very similar, their effect on the two functions will in general be verydifferent, as already shown for the two special cases discussed above.

3. A generalized class of time-frequency signal representations

In part I the WD was introduced in a rather axiomatic way and was shownto have some interesting properties. Such an approach leaves the questionunanswered whether the WD is the only such representation, Le. whetherother representations exist that have similar or even more desirable properties.This question will be addressed in some detail in this section.

The starting point is a generalized class of time-frequency signal represen-tations that was introduced by Cohen 1,2). This class is given by

co co co

c.«.w; </J)= 2~ J J J ej(çt-rw-çu) </J(ç,i)f(u + r/2)f*(u - r/2) du didç,-co -co -co (3.1)

where </Jis at present an arbitrary kernel function, determining the particularrepresentation in the class. Using the definition of the ambiguity function offit is easy to see that (3.1) can be rewritten as

00 co

({J(t, co) = 2~ J J ej(çt-wr) </J(ç,r) dç di.-00 -00

(3.4)

q(t, w; </J)= 2~ J J ej(Çt-WT) ct>(ç,r)Af(ç, r) dç dr. (3.2)-00 -IX)

Alternatively q can be expressed in terms of the WD according to

00 co

c-«, w; </J)= 2~ J J ({J(t - i, W - o Wj(r, o dr dé. (3.3)-00 -co

whereco co

The interpretation of eq. (3.3) is that all members of this class of signalrepresentations can be obtained by a linear transformation of the WD charac-

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The Wignet distribution

terized by the kernel (/J(t, w) that is related to the kernel IP(ç, r) by a two-dimensional Fourier transformation according to (3.4).

Clearly the identity kernel

IPw(ç, r) = I (3.5a)

or(/Jw(r, ç) = 21t ó(r) o(ç) (3.5b)

leads to the WD. Allowing the kernel lP to depend on tand wand setting

IPA(f" r; t, w) = 2n ó(r - t) o(ç - cu) (3.6a)

or(3.6b)

the AF is obtained:

(3.7)

In the remaining part of this section several properties of time-frequencysignal representations are stated that are desirable in signal analysis. Each ofthese properties has associated with it one or more constraints on the kernels.In this way it is possible to analyse the properties of a particular representa-tion in a systematic way once its kernel is given. The various constraints willhenceforth be labelled "Ck" and the corresponding property "Pk" . •

The first two properties to be discussed are that shifts in time or frequencyof the signal result in corresponding shifts of the representation:

PI:

P2:

Cff,of(t, co; lP) = Cj(t - to, w; lP)

C,AtwoÁt, w; lP) = Cj(t, co - wo; lP),

(3.8)

(3.9)

where the shift operator 9/0 and the modulation operator ./tfwo are defined insec. 2.1. in part I. Since the WD satisfies both of these properties, it is easilyseen from (3.3) that these properties hold for any Cj(t, cu; lP) if the kernel lPis independent of t or co, Using the notation of eq. (3.6) we get the constraints

Cl:

C2:

IP(ç, r; t,w) does not depend on t,IP(f" r; t,w) does not depend on to,

(3.10)

(3.11)

Henceforth only representations with kernels that are independent of both tand ca will be considered. This excludes, for example, the ambiguity functionwhose kernel does not satisfy either of the constraints Cl and C2. Both prop-erties are essential if we wish the time and frequency variables of the signalrepresentation to correspond to those of the signal and its spectrum,respectively.


The next two properties of interest have already been considered byCohen 1), namely

co

P3: 1 J c«, w; çp) dw = I/(t)12

2n(3.12)

-coand

co

P4: (3.13)

-co

These properties are obtained by representations with kernels satisfying theconstraints

C3:

C4:

çp(ç,O) = 1

f/J(O,r) = 1 Vr.

(3.14)

(3.15)

Properties P3 and P4 are attractive in view of the desire to interpret thetime-frequency representation as an energy distribution of the signalover timeand frequency. P3 states that the integral over all frequencies at time t is equalto the instantaneous power, while P4 states similarly that the integral over alltime at a certain frequency to is equal to the value of the energy densityspectrum at that frequency. An immediate consequence is that if either of thetwo properties is satisfied then the integral over the whole (t,w)-plane

co co

2~ J J c«. w; çp) dtdw = II/W (3.16)

is equal to the signal energy.Apart from the WD for which P3 and P4 were demonstrated in part I to

hold, the representations obtained by taking

f/Ja(ç, r) = eiaçr (3.17)

have these properties as well. For a = t this kernelleads to the representationproposed by Rihaczek 3):

Cj(t, w; f/J!) = f*(t)F(w) eiwt. (3.18)

This representation has the charm of being simple, but, in general, it is com-plex valued which is not very convenient in view of our interpretation of Cjas an energy distribution. If we wish the representations to be real, Le.

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PhllipsJournal of Research Vol.35 No.6 1980 379


PS: q(t, cv; ifJ) = cr«, cv; ifJ) (3.19)

then we must put the constraint

CS: ifJ(ç, r) = ifJ*( - ç, - r) (3.20)

on the kernel of the representation.It would of course be even more desirable for such a representation to be

positive for all time and frequency, because then it could truly be interpretedas an energy distribution over time and frequency. However, such an inter-pretation would even then be questionable, as Heisenberg's uncertainty rela-tion prohibits an arbitrarily sharp frequency discrimination from being pos-sible in an arbitrarily short period of time.Moreover, as will be shown at the end of this section, the positivity require-

ment is incompatible with other requirements to be introduced subsequently,and which in our view are more preferable than positivity. In any case suitableaverages of the WD, incorporating values over a portion of the (t, cv) planethat has a dimension that is in accordance with Heisenberg's uncertainty rela-tion, can be shown to yield positive values 10). This means that locally negativevalues of such a representation need not necessarily be disturbing.A kernel related to that of Rihaczek, but satisfying all constraints Cl to CS

is given by

ifJa(ç, r) = cos (açr) (3.21)

of which the particular case a = t has been considered by Cohen 2). For thisvalue of a the corresponding representation is equal to the real part of theRihaczek representation of eq. (3.18).In section 6 of part I it was shown that the WD has two more properties that

are of importance in signal analysis. Firstly, the average frequency Df(t) ofthe distribution at a certain time is equal to the instantaneous frequency of thesignal. It should be remembered that this holds for complex-valued signals,and that the instantaneous frequency of a real-valued signal is defined as thatof the corresponding analytic signal. Secondly, the average time Tj(w) at acertain frequency is equal to the group delay. These properties of thegeneralized distributions can be expressed by

P6:.f cv q(t, cv; ifJ) dco-~ d-~------= Df(t) = Im-

dtlnf(t),

f c.«, cv; ifJ) dcv(3.22)

-~


co

P7:

f t c.«. cv; CP)dt-co d-co----- = 1f(cv) = -lm -dcv-InF(cv),f o«. cv; CP)dt

(3.23)

-co

CP(ç,O) = 1

Property P6 holds if the following contraints are met:

C6:()()r CP(ç,r)] T=O = 0

Similarly, to have P7 we must require:

CP(O,r) = 1C7:

(3.24)

\lç.

(3.25)

\Ir.

It should be observed that the first parts in C6 and C7 are identical to the con-straints C3 and C4 respectively.

While the kernel CPa in (3.21) satisfies all these conditions, the kernel CPa in(3.17) does not. This means that the Rihaczek representation does not haveproperties P6 and P7, but its real part does.

Finally we will consider the finite support properties. By this it is meant thatfor a signal which has a finite extent (support) in time or frequency itsrepresentation has the same finite support in the corresponding variable.Hence

P8: if f(t) = 0

Cj(t, cv; CP) = 0

F(cv) = 0

Cj(t, cv; CP) = 0

then

P9: whileif

then

Itl >T

Itl >T,

Icvl>Q

Iwl>Q.

(3.26)

(3.27)

(3.28)

(3.29)

Property P8 leads to a constraint on the kernel cP of the form

This constraint is equivalent to the requirement that CP(ç, r) must be an entirefunction of ç of exponential type 7). More specifically this requires that func-tions A(r) and a(r) exist such that A(r) is bounded,

C8:co

f eWCP(ç, r)dç = 0-co

380

[r] <2Itl. (3.30)

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a(r) ::;;Irl/2 (3.31)and

ItP(ç, r)] <A(r)ea(r)lçl

for all complex values of ç. Similarly P9 leads to

(3.32)

C9:CX)

f e -jWT tP(ç, r) dr = 0 lçl <21wl (3.33)-CX)

which is equivalent to the requirement that tP must be an entire function of rof exponential type. Hence, functions B(ç) and g(ç) must exist such that B(ç)is bounded,

«o ::;;lçl/2 (3.34)and

ItP(ç,r)1 < B(ç) ee(ç)iT1 (3.35)

for all complex values of r,As an example we may consider the kernel tPa from (3.17) which satis-

fles CB and C9 only if lal::;; t. The same conclusion holds for the kerneltPa(ç, r) = cos (açr) as well. With this restrietion on a all representationsbased on this kernel satisfy each of the properties PI to P9 discussed so far.For a = 0 the WD is obtained and for a = t the real part of the Rihaczekrepresentation results. It is interesting to note that such a variety of time-fre-quency signal representations share this large number of desirable properties.To conclude this section it will be indicated that the properties P7 and PB

are incompatible with the requirement that the representation is nonnegativefor all time and frequency values. One of the ways to see this is to considera causal signal, i.e. a signal that vanishes identically for negative times.Property PB guarantees that q too will vanish for negative t. Assuming q.to be nonnegative, it follows from property P7 that for such a signal thegroup delay has to be nonnegative for all frequencies. This contradiets thewell known fact that causal signals can have a negative group delay for certainfrequencies 11).

Similarly it is possible to prove that properties P6 and P9 cannot hold forsignal representations q(t, w) that are nonnegative. This follows from thefact that analytic signals may have negative values for the instantaneousfrequency during certain periods of time. Observations related to the abovehave been made by Friberg in the context of radiance functions in optics 12).

4. Investigation of methods for spectral analysis

In this section various methods for spectral analysis of signals will beinvestigated in relation to the Wigner distribution. First in sec. 4.1 it will be


indicated that the spectrogram of the signal 4,5) is a special case of the time-frequency representations introduced in sec. 3. Secondly in sec. 4.2 this fact isalso shown for the pseudo- Wigner distribution that was introduced in part I.Finally it is shown that the output signalof a spectrum analyser that uses achirp signal to sweep the signal past a fixed tuned narrow-band filter can bedescribed elegantly by means of the WD.

4.1. The spectrogramSpectrograms are used extensively for time-frequency analysis of speech

and other signals 4,5,13). They are two-dimensional density functions that arederived from the signal and are considered to represent the energy distributionof the signal in time and frequency. The aim of this section is to show thatthese spectrograms are special cases of the signal representations discussed insec. 3, and hence are a weighted version in time and frequency of the WD (seeeq. (3.3». We will concentrate on the case that the spectrogram is obtained bya short-time Fourier transform 14). It has been shown in the literature thatdifferent techniques for obtaining the spectrogram, such as by means of afilterbank, yield exactly the same result 15).

The short-time Fourier transform requires the use of a sliding window h(t)and computation of the Fourier transform of the windowed signal. Althoughin the bulk of the applications these computations are performed on discrete-time versions of the signals, so that computationally efficient algorithms forthe discrete Fourier transform can be applied, we will restrict the discussion tothe case of continuous-time signals. The results of part 11of this paper give themeans for the necessary modifications to deal with discrete-time signals.

Letting f(r) denote the signal whose spectrogram is to be determined andh(r) the window function, windowing generates the signal

h(r) = f(r) h(r - t), (4.1)

where t is the time instant indicating the position of the window on the timeaxis.

The short-time Fourier transform (SFT) is given by

coFt(w) = J e-jwTf(r)h(r - t)dr.

-co(4.2)

The spectrogram Sf(t, co) is obtained by taking the magnitude squared ofthis SFT for all possible positions of the window, Le. by considering t as arunning variable:

Sf(t, cv) = IFt(w)l2. (4.3)

382 PhIlIpsJournni of Research Vol.35 No.6 1980

(4.6)

The Wigner distribution .

Using eq. (1.2.37), eq. (4.3) can be written as

coSf(t, w) = f Jf'l,(r, w) dr. (4.4)

-co

The Wigner distribution U'l, of the windowed signal is given by eq. (1.4.6),which yields

00 00

Sf(t, w) = 2~ J J Jf'l(r, r/) Wh(r - t, W - Yf)dr d». (4.5)-co -00

Interpreting this equation in the sense of eq. (3.3) it is seen that Sf(t; w) isa member of the class of representations given by (3.1) and is generated bythe kernels

orifJs(ç, r) = Ah( - ç, r) (4.7)

i.e. by the WD or the AF of the window function.The fact that the spectrogram is a representation generated by a kernel that

is itself a WD excludes certain properties discussed in sec. 3 for this represen-tation. Clearly, independent of the window that is used the spectrogram hasthe properties PI, P2 and PS, but properties P3 and P4 can never be obtainedsince

co co

2~ J Sf(t, w) dw = J 1!(rW Ih(r - tW dr (4.8)-co -coand

co co

J Sf(t, w) dt = _1 J IF(YfW IH(w - YfWdYf, (4.9)21t o.

-00 -co

where H(w) is the spectrum of the window.Instead of obtaining the instantaneous power in (4.8) one obtains the

average power over the duration of the window with the square of the windowas weighting function. A similar conclusion holds for the energy density .spectrum in (4.9). Furthermore, since by definition Sf is nonnegative, it can-not simultaneously have properties P7 and P8 nor P6 and P9 as was shown insec. 3. In fact none of these properties is satisfied by the spectrogram. Theaverage frequency of the spectrogram !JSf(t) is given by

Phlllps Jnnrnal of Research Vol.35 No.6 1980 383


co

f Df(.)IJ(.)12Ih('r - t)12drDsj(t) = --co--co------- __ "

f If(.)12Ih(. - t)12dr(4.10)

-co

Le. it is an average of the instantaneous frequency of the signal with theinstantaneous power of the windowed signal as the weighting function.Similarly the average time of the spectrogram is equal to

co

f 1](17)I!(11)12IH(co - 17)12 ànTSf(co) == --co--co------ _

f IF(I1)12IH(co - 17)12 dl1(4.11)

-co

Hence P6 and P7 cannot be obtained.Additionally, condition CS requires Yh(t,.) to vanish for Itl >1.1/2, which

in particular for. = ° requires h(t) = 0, t f 0. It is obvious that no windowfulfils this condition and hence PS is never obtained. Similarly C9 wouldrequire H(co) = 0, co f 0, which is not realizable.

In conclusion, of the properties stated in sec. 3, only the shift properties andthe realness are satisfied by the spectrogram of a signal. As regards the otherproperties, an average of the desired quantities is obtained taken over theduration of the window. This means that for signals for which these quantitiesdo not vary appreciably over the effective length of the window (short-timestationary signals) the spectrogram provides useful information. If this short-time stationarity cannot be guaranteed, then it may be advantageous to usethe pseudo-Wigner distribution (PWD) which is better able to cope with non-stationarities, as will be shown in the following section.

4.2. The pseudo-Wigner distribution

In part I the pseudo-Wigner distribution was introduced as the concatena-tion of slices ofthe WD ofthe windowed signalft(.). It can be computed fromthe signal according to '

- coU'l(t, co)= f e -jw'f(t + ./2)f*(t - ./2) h(./2) h*( - ./2) dr. (4.12)

-co

Moreover it was shown that this PWD is related to the WD byco

~(t, co) = _!_ J U'l(t,·I1) Wh(O,co - 11) d17,21t (4.13)-co

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Phlllps Journal oî Research Vol.35 No.6 1980 385


where as before Wh is the WD of the window. Clearly therefore the PWD is amember of the class of representations of eq. (3.3) as well. The correspondingkernels are

tpp(" ç) = c5(r) Wh(O, ç) (4.14)

and

~p(ç, r) = h(r/2) h*( - ./2). (4.15)

Thus the constraints Cl, C2, C3 and CS are satisfied for all windows andhence the PWD has properties PI, P2, P3 and PS. C6 is satisfied for anywindow that has an instantaneous frequency that vanishes at • = 0 (Qh(O) = 0)which in particular is the case for a real-valued window. In that case theaverage frequency of the PWD is equal to the instantaneous frequency of thesignal.

Property P7 does not hold, however, because the first part of constraint C7is not satisfied. In fact, the average time of the PWD can be computed to begiven by

00

Tp(w)= -_00 __

00(4.16)

-00

Comparing this result with the corresponding expression for the spectrogram(4.11) we observe that in both cases an average of the group delay is obtainedbut the weighting functions are different.

Finally, it follows from (4.15) that P8 holds, but P9 does not, so that thefinite support property is only fulfilled with respect to the time variable.

4.3. Spectrum analyser using a chirp signalA classical way to obtain a plot of the spectral content of an assumedly

stationary signal is to sweep the signal past a fixed tuned narrow-band filter bymodulating it with a chirp signal. A simplified version of such a configurationis sketched in fig. 2, where for ease of analysis it is assumed that a complexchirp signal and a lowpass filter are used. It will be shown now that thisspectral analysis method can easily be interpreted in terms. of the WD of thesignal, which is the case even if the signal is not stationary.

In the scheme of fig. 2 it is assumed that the inputf(t) modulates a complexcarrier with instantaneous frequency - at:

(4.17)


_f_(t_)_~0 x(t) ~If c(t) = e_jct.l/2'-------I

h(t)

y(t) .1 s(t)

Fig. 2. Schematic diagram of a spectrum analyser using a chirp signal (wobbIer).

The WD of this signal can be obtained from eq. (1.4.4) and (1.3.6) to yield

Wx(t, ca) = Utj(t, ca + at). (4.18)

The lowpass filter has the impulse response h(t) which has a WD Wh(t, ca).Using eq. (1.4.2) we find for the WD of the output signal y(t)

co

Wy(t, ca) = f f!'l(r, co + ar) Wh(t - r, co) dr. (4.19)-co

The square of the magnitude of the output signal can be expressed as anintegral of its WD according to eq. (1.2.29)

co

s(t) = ly(t)12 = _1 J Wy(t, ca) dca21t

-co

co co

= ;1t J J f!'l(r, co + ar) Wh(t - r, co)drdca

= 2~ J J f!'l(r, 11)Wh(t - r, YJ- ar)drdYJ. (4.20)

-co -00

co co

-00 -00

This means that the output signalof this spectral analyser can again be de-scribed by an averaging of the WD of the input signal in time and frequency.

A nice interpretation of the averaging is obtained by rewriting eq. (4.20)according to

co co r-(4.21)s(t) = 2~ J J ff'l(r, 11)Wh[t - r, a(t - r) - (ca - YJ)]drdYJlw=at·

-00 -co

We see that s(t) is obtained by a convolution of the WD U1(t, ca) of the inputsignal with the kernel

rp(t, ca) = Wh(t, at - ca) (4.22)

386 PhlllpsJournalof Research Vol. 35 No. 6 1980

________ -----_-~_~ -T .-_

Phlllps Joumol of Research Vol.35 No.6 1980 387


and taking the values on the line co = at.If the impulse response of the low-pass filter h(t) is real-valued, then the

WD Wh has its main support in the cross-hatched area indicated in fig. 3a.According to (4.22) this means that the kernel in the convolution has its sup-port along the line co = at as indicated in fig. 3b by the cross-hatched region.The convolution in (4.21) for a fixed value of t then means an averaging of theWD of the input signalover the area cross-hatched in fig. 3c. Inspection ofthis figure shows that the frequency resolution of the method is of the order of(2nB + al B) while the time resolution is of the order 1/B, where B is thebandwidth of the low-pass filter. To obtain an acceptable resolution in thefrequency a has to be-taken sufficiently small, but this makes the analyserrather slow. A practical figure is a =:: nB2/10 (ref. 16).

w w=cd

a)

w w=oct

b)

T/=OCT

I" 1/8c)

Fig. 3. Illustration of the moving averaging process of the Wigner distribution that takes place ina spectrum analyser. (a) Region of support of the WD of the filter in the spectrum analyser.(b) Region of support of the kernel of the convolution. (c) Region of the averaging that takesplace in the convolution. With increasing time this region moves along the line n = ar. Theregions labelled 1 correspond to the case of a real-valued filter, while the regions labelled 2 areapplicable in case of a complex-valued filter with impulse response given by eq. (4.23).


It is possible to make the frequency resolution independent of the scanningparameter a by using a filter with the complex-valued impulse response,

h(t) = h' (t) eiat212, (4.23)

where h' (t) is as before the real-valued impulse response of a low-pass filter.The relation between the WD of h(t) and that of h' (t) is given by

Wh(t, w) = Wh' (t, to - at) (4.24)

according to (1.4.4). This means that if the WD of h' (t) has its support in thecross-hatched region labelled CD in fig. 3a then the support of Wh(t, w) is theregion labelled @ in this figure. The support of the kernel ({J is then centredaround the line co = 0 which means that the convolution in eq. (4.22) is anaveraging over the area labelled @ in fig. 3c. Clearly in this case the fre-quency resolution is 21tB. However, the filter has now become complex andmust be adapted to the scan speed a of the analyser.It can be seen that by decreasing B the region of averaging becomes longer

in the t-direction and narrower in the eo-direction, which means that forstationary signals a better estimate of the energy density spectrum is obtained.As already shown by Papoulis 7) in the limiting case B = 0 we get .

s(t) = IF(atW. (4.25)

Figures similar to fig. 3 have often been used to explain symbolically theworking principle of a spectrum analyser. The description given in this sectionshows that this figure can be interpreted in a much more concrete way in termsof the WD in view of eq. (4.20).

5. Conclusion

In this paper an attempt has been made to place the Wigner distribution inthe perspective of time-frequency signal representations and spectral analysismethods. Generally it has been shown that all such representations that aim atgiving an energy distribution over time and frequency can be obtained as aweighted average of the.Wigner distribution. In particular this has been shownfor the spectrogram, and also for the output of commonly used spectrumanalysers that use a chirp signal. It was indicated that a number of usefulproperties of the WD could be secured also for other representations if thecorresponding weighting function (kernel) fulfils certain conditions. It thenappeared that apart from the WD a number of representations exist for whichthe kernel satisfies all constraints and hence have all the properties that wereconsidered.

For the spectrogram it was shown that the corresponding kernel could notfulfil most of these requirements. This results in a smearing in both time and

388 Phllips Journal of Research Vol.35 No.6 1980

PhlllpsJournal of Research Vol.35 No.6 1980 389

The Wigner distributton

frequency of the information contained in the WD if the spectrogram is com-puted for nonstationary signals. A better situation is obtained by using thepseudo-Wigner distribution, which, like the spectrogram, can be computedfrom windowed data, but introduces no smearing in the time direction. Thislatter representation is not always positive, however.

On the other hand it was shown that positivity of such a representationexcludes it from having properties that are desirable for the extraction ofinstantaneous information like the intantaneous power, instantaneousfrequency, and the finite support property. If such instantaneous character-istics should be obtainable from such a time-frequency representation then wehave to accept negative values for it. This seems to be the only way to accom-modate Heisenberg's uncertainty relation.

After completion of this manuscript we became aware of the paper byFlandrin and Escudie 17) who report results which are closely related to thosedescribed in this part of our paper.

Philips Research Laboratories Eindhoven, May 1980

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reality, ch. 4, M. Bunge (ed.), Springer-Verlag, Berlin, 1967, pp. 71-89.3) A. W. Rihaczek, Signal energy distribution in time and frequency, IEEE Trans. on Inf. Th.

IT-14, 369-374, 1968.4) A. V. Oppenheim, Speech spectrograms using the fast Fourier transform, IEEE Spectr.,

August 1970, pp. 57-62.5) L. R. Rabiner and R. W. Schafer, Digital processing of speech signals, Prentice-Hall,

Inc., 1978.6) D. E. Vackman, Sophisticated signals and the uncertainty principle, Springer Publ. Co.,

Inc., New York, 1958.'7) A. PapouIis, Signal analysis, McGraw-HilI, New York, 1977.6) L. E. Franks, Signal theory, Prentice Hall, Englewood Cliffs, 1969.9) J. H. McClellan, private communication.10) N. G. de Bruij n, Uncertainty principles in Fourier analysis, in Inequalities, O. Shisha (ed.),

Academic Press, New York, 1967, pp. 57-71.11) L. Weinberg, Network analysis and synthesis, McGraw-HilI, New York, 1962.12) A. T. Fri berg, On the existence of a radiance function for finite planar sources of arbitrary

states of coherence, J. Opt. Soc. Am. 69, 192-198, 1979.13) R.K. Potter, G. A. KoppandH. C. Green, Visiblespeech, D. van Nostrand Co., Inc., 1947.14) M. R. Portnoff, Time-frequency representation of digital signals and systems based

on short-time Fourier analysis, IEEE Trans. on Acoustics, Speech, and Signal ProcessingASSP-28, 55-69, 1980.

15) J. B. Allen and L. R. Rabiner, A unified approach to short-time Fourier analysis and syn-thesis, Proc. IEEE 65, 1558-1564, 1977.

16) K. Küpfmüller, Die Systemtheorie der elektrischen Nachrichtenübertragung, HirzelVerlag, Stuttgart, 1968.

17) P. Fiandrin and B. Escudie, Time and frequency representation of finite energy signals:A physical property as a result of an Hilbertian condition, Signal Processing 2, 93-100, 1980.

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THE WIGNER DISTRIBUTION - A TOOL FOR TIME … Bound...philips j.res. 35, 372-389, 1980 . r1028 the wigner distribution - a tool for time-frequency signal analysis part ill: relations