THE WIGNER DISTRIBUTION - A TOOL FOR … Bound...philips j.res. 35,276-300, 1980 r 1021 the wigner...

Philips J. Res. 35, 276-300, 1980 R 1021

THE WIGNER DISTRIBUTION - A TOOL FORTIME-FREQUENCY SIGNAL ANALYSIS

PART II: DISCRETE-TIME SIGNALS

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

AbstractIn this second part of the paper the Wigner distribution is adapted to thecase of discrete-time signals. It is shown that most of the properties of thistime-frequency signal representation carry over directly to the discrete-timecase, but some.others cause problems. These problems are associated withthe fact that in general the Wigner distribution of a discrete-time signalcontains aliasing contributions. It is indicated that these aliasing com-ponents will not be present if the signal is either oversampled by a factor ofat least two, or is analytic.

1. Introduetion

In part I of this paper 1) the Wigner distribution (WD) of continuous-timesignals was discussed, and it was shown that this function has some veryinteresting properties. The determination of this distribution functionrequires, like the spectrum, an integral of the Fourier type to be evaluated.Ideally this requires the signal to be known for all. time, but in practicewindowing techniques can be used to relax this requirement. The effects ofwindowing on the WD were discussed in part 1.

In general two different approaches can be distinguished to compute theseFourier-type integrals. The first is by means of analogue signal processing,and recently optical signal processing methods have been proposed for deter-mining suitable approximations to the WD 2). The second approach is basedon digital signal processing. This opens the way to apply computationallyefficient methods for evaluating the discrete Fourier transform, but requiresthe concept of the Wigner distribution to be transferred to the case of discrete-time signals. This is the aim of this part of the paper.

As can be expected, the WD for discrete-time signals shows much similaritywith that for continuous-time signals, but in some respects it has characteristicdifferences.

To emphasize the similarities and point out the differences we will try tofollow as closely as possible the same lines as in part I, and give commentsonly on those results that differ from that of the continuous-time counterpart.

276 Phillps Journalof Research Vol.35 Nos.4/5 1980

Philips Journalof Research Vol.35 Nos.4/5 1980 277

The Wigner distribution

Also the numbering of the equations is made such that correspondingequations have the same number. This has the consequence that sometimesequation numbers are not successive if equations have been deleted, and thatequations that do not occur in part I have a special numbering. If reference ismade to an equation in part I the equation is given the prefix I.All sections, except sec. 7, have the same topic and heading as in part I.

Section 7, which in part I deals with the WD of band-limited signals, nowdeals with the WD of finite duration sequences. Equations in this section donot correspond in general with an equation of part I.

2. The Wigner distribution for discrete-time signals

2.1 Preliminaries

In this paper Weconsider in general complex valued, discrete-time signalsf(n), feC, n e Z for which the (Fourier) spectrum is defined by 3)

00

F(e) = (fJdf) (e) = L f(n) e -jnlJ. (2.l.a)n=-c:o

The inverse transform is given by11

f(n) = (fJd-l F) (n) = _1_ J F(e) ejnlJ de,21t (2.l.b)-11

Inner products are defined for the signals and spectra by

00

(I, g) = L f(n) g*(n) (2.2.a)n=-oo

and 11

(F, G) = _1 J F(e) G*(e) de21t (2.2.b)-11

respectively. Norms and Parseval's relation are then the same as in eqs (1.2.3)and (1.2.4) respectively.The following operators will be used.

The shift operator for the signals

(9{f) (n) = f(n - k), kEZ (2.5.a)

and for the spectrum

(.9'çF) (e) = F(e - Ç), ÇE R,

(complex) modulation in the time domain

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

( .At!!,!) (n) = f (n) ein!!,

and in the frequency domain .c;«; F) (0) = F(O) einD

çeR (2.6.a)

neZ, (2.6.b)

differentiation of the spectrum

1(fi))F) (0) = -;-F 1(0),

J(2.7)

multiplication by the running variable

(Rlf) (n) = nf(n), (2.8)

time reversal(f!JlJ)(n) = f( -nl· (2.9)

There are several different ways to link analogue and. digital signals andsystems, and hence a variety of ways to define a discrete-time version of theWigner distribution. What one would like with such a definition is(1) to obtain a simple concept;(2) to retain as many as possible of the properties of the WD of continuous-

time signals;(3) to find a simple relation between the discrete-time and continuous-time

WD's for discrete-time signals that are obtained by sampling of analoguesignals.

The definition which, in our opinion, best matches these requirements is theone suggested by eq. (1.7.10).

2.2. Definition of the Wigner distribution

The cross-Wigner distribution of two discrete-time signals f(n) and g(n) isdefined by

00

u-j,g(n, 0) = 2 L e-i2kDf(n + k)g*(n - kl. (2.10)k=-oo

The auto- Wigner distribution of a signal is then given by

00

u-j(n, 0) = Wj,f(n, 0) = 2 L e-i2kDf(n+k)f*(n - kl. (2.11)k=-oo

Both functions will be called a Wigner distribution (WD). Aiming at obtaininga relation similar to (1.2.13) the WD for the spectra must be defined by

278 Philips Journalor Research Vol.35 Nos.4/5 1980


so that

11

wF,G(e, n)= ~ J ej2nt; F(e + Ç) G*(e - Ç) dç (2.12)-11

wF,G(e, n) = w"g(n, e). (2.13)

2.3. Properties of the Wigner distributionFor the WD as defined in (2.10) the following properties hold.

2.3.0. Periodicity

The WD is a function of the discrete variable n and the continuous variablee. With respect to the latter variable the function is periodic with period 1t

w"g (n, e) = w"g (n, e + n), V n, e. (b.1)

The period 1t is distinct from the period 21t that all spectra of discrete-timesignals have. This discrepancy could have been avoided by deleting the factor2 in the exponent in eq. (2.10), but this has the disadvantage that frequencycomponents in f at e will occur at 2e in the WD.

2.3.1. Symmetry

W,(n, e) = w,' (n, B)

W, (n, 8) = wt· (n, - e).is real,

(2.14)

(2.15)(2.16)

2.3.2. Time shiftWBk!, 9/.:g (n, e) = w"g(n - k, B). (2.17)

2.3.3. Modulation

(2.18)

2.3.4. Inner productFrom the definition of the WD and that of the inner product it follows that

w"g(O,O) = 2(j, f!llg)

which, combined with (2.17) and (2.18), yields

w"g(n, e) = 2(~n utt_()j, f!ll ~n.AL() g).

(2.21)

(2.22)

2.3.5. Sum formula

U'it+g.,h+g2(n, e) = U'it,gt (n, e) + U'i.,g2(n,B) + W,2,gt (n, e) + W,2,g2(n,B)(2.23)

Philips Journalof Research Vol.35 Nos.4/5 1980 279

Wj+g(n, B) = Wj (n, B) + Wg(n, B) + 2 Re *I,g (n, B). (2.24)


and in particular

2.3.6. Multiplication by n or ei26

2n *I,g(n, B) = J#l/;g (n, B) + Wj,92g (n, B). (2.25)

The relation corresponding ' with (1.2.26), i.e. the multiplication by thefrequency variable, cannot simply be copied. Because the WD is a periodicfunction with period n, the multiplication of the WD by a function of B canonly be expressed in terms of WD's if this function is periodic with this periodas well. The most elementary such function is ei26, yielding

(2.26)

2.3.7. Inverse transform in the time

The WD evaluated at frequency B/2 can be considered as the Fouriertransform of the sequence 2f(n + k) g*(n - k) with fixed n. Hence, from theinverse transform relation (2.l.b) we find

rt

_1_ J eik6 *I,g(n, B/2) dB = 2f(n + k) g*(n - k).21t

-n

This can be written in the form

n/2

;1t J ei(nl-n2)9 U'i,g( nt : n2 , e) de = f(nt) g*(n2)'

-n/2

(2.27)

where (nt + n2)12 must be an integer.

Three special cases are:

(i) n, = n« = n, yielding

n/2

_1_ J U'i,g(n, e) dB = f(n) g*(n)21t

(2.28)

-n12

and in particular

280. PhilipsJournal of Research Vol.35 Nos.4/5 1980

rr/2_1-./ U'l(n,O)dO= lJ(n)12.2n

-rr/2

(2.29)


This shows that the integral over one period of the WD in its frequencyvariable is equal to the instantaneous signal power.

(ii) ni = 2n, n2 = 0 givesrrl2

_1_ f ej2nO U'l,g(n, 0) dO = f(2n) g*(O).2n

-rr/2

(2.30a)

(iii) n1 = 2n - 1, ns = 1 givesrrl2

_1_ / ej2(n-l)O U'l,g(n, 0) dO = f(2n - 1) g*(l).2n

-rr/2

Recovery of the signals from the WD up to constant factors is thus possible,but requires a different procedure for the even and odd numbered samples.

(2.30b)

2.3.8.

Summation of eq. (2.28) over n yieldsco rr/2

2~ I /U'l,g(n, 0) dO = (f, g) (2.31)n=-co -rr/2

which has the same special case as (1.2.33).

2.3.9. Recovery of the spectrum

Using the fact that the WD of the signals is equal to that of the cor-responding spectra (eq. (2.13» it follows from (2.12) that the WD can be con-sidered as the inverse Fourier transform of F(O + Ç) 0*(0 - Ç) considered as afunction of ç for fixed 0, and evaluated at 2n. This means that only the evennumbered samples of this function are available, from which the functionF(O + Ç) 0*(0 - Ç) cannot simply be recovered in general. Nevertheless itholds that

co

L e-j2nç U'l,g(n, 0) = F(ë + Ç) 0*(0 - Ç) + F(O + ç + n) 0*(0 - ç + n)n=-c:o

(2.34)

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which means that an aliased version can be reobtained from the WD. Equa-tion (2.34) can be rewritten as

co

L ejn(81 -82) U'i,g [n, (Ol + (2)/2] ,; F(OI) G*(02) + F(OI +. n) G*(02 + n).

n=-oo

(2.35)

Again three special cases are of importance.

(i) Ol = O2 = 0, yielding

co

L U'i,g(n, 0) = F(O) G*(O) + F(O + 1t) G*(e + n)n=-oo

(2.36)

and in particularco

L U'i(n, 0) = IF(O)12 + IF(O + n)12•

n=-oo(2.37)

The aliasing term in (2.37) is necessary to make both sides of the equationperiodic with period rr. This term will cause some problems later on in variousequations. There are two important situations, however, where the aliasingdoes not cause any problems, because in these situations the spectrum F(O)occupies only an interval of length n and is zero in the remaining part of oneperiod of the spectrum. The first case is that of an "oversampled" signal, i.e.a signal with a band-limitation to less than n/2. Such a signal can be obtainedeither from an analogue signal by sampling it with a sampling frequency thatis larger than twice the Nyquist rate or by interpolation of a discrete signal bya factor 2 (ref. 4). It should be remarked that the definition of the WD wasinspired by a formula that was derived for this case (eq. (1.7.10» and it there-fore need not surprise us that for such a signal the WD behaves in the sameway.as that of an analogue signal.A second situation where only one of the terms on the right hand side in eq.

(2.37) differs from zero arises if we consider analytic signals, the spectrum ofwhich vanishes for negative values of 0 over the period of the spectrum. TheWD of these signals will be discussed in sec. 5.

(ii) Ol = 0, O2 = 0, which gives

co

L ejn8 U'i,g(n, 0/2) = F(O) G*(O) + F(O + n) G*(n).n=-oo

(2.38a)

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Phillps Journalof Research Vol.3S Nos.4/5 1980 283


(iii) el = e + n, e2 = 0 which gives00

• (J (e + 1t)eJn (-It U'i,g n, -2- = F(e + n) 0*(0) + F(e) 0*(1t).(2.38b)n=-c:o

If 10(0) I f= I 0(1t) I these two equations permit the recovery of the spectrumof f. For interpolated or analytic signals only one of the two equations sufficesfor recovering the spectrum.

2.3.10.

Integration of (2.36) with respect to e yields

lt12 00 rt

_1_ J '"' 'U'i,g(n, e) de = _1_ J F(e) o*(e) de = (F, 0) (2.39)21t ~ 21t

-lt12 n=-oo -lt

which is the same result as (2.31) ..

2.3.11. Moyal 's formula

The discrete-time analogon of Moyal's formula is somewhat more com-plicated than the original version (eq. (1.2.40». It reads

lt12 00

2~ J I ~l,gl(n, e) U'i;,g2(n, e) de-lt12 n=-oo

= U.,f2) (g., g2)* + (f .. .Atltf2) (g.. ..Aln g2)*

= (P., F2)(G., O2)* + (F.. 91cF2)(01' 91c O2)*,

(2.40a)

(2.40b)

The operation ..Aln in (2.40a) is defined by (2.6) and changes the sign of theodd numbered samples, which is equivalent to a shift of the spectrum over n,expressed by the operation 91c in (2.40b).

2.4. Effects of time- and band-limitations on the WD

2.4.1. Time-limited signals

If both f and g are time-limited (finite duration) signals, Le.

f(n) = g(n) = 0, n < na or n > nb (2.43)

then


U'l,g(n, 0) = 0, (2.44)

2.4.2. Band-limited signals

In the definition of the WD no restrietion on the spectra of the signals wasassumed, but discrete-time signals have a spectrum that is periodic with period21t. On the other hand, the WD was found to be periodic with period 1t, and asdiscussed in the previous subsection this causes aliasing to occur, Le.frequency components that are 1t apart have the same influence on the WD. Infact we have

U'l,g(n, 0) = W.A4tt,.A4tg (n, 0). (b.2)

If, however, the signal spectrum is nonzero only over an interval of length lessthan 1t, then aliasing will not occur. Thus if there exist Oa and Ob such that

(b.3)

F(O) = G(O) = 0, (2.45)

and

then

U'l,g(n, 0) = 0, (2.46)

This is illustrated in fig. 1, where the shaded areas indicate the frequencyregions where the corresponding functions differ from zero.The two cases that we discussed before, i.e. oversampled signals' and

analytic signals, have Oa = 1t/2, Ob = 31t/2 and Oa = 1t, Ob = 21t, respectively,and hence have Ob - Oa = 1t so that generally the WD of these signals has nogap in the frequency domain, but no aliasing either.

FrB), GrB)

~~~o Ba Bb -'1C 1C Ba + 1C Bb 21C

Fig. 1. Illustration of the areas where the WD has zero contributions in the frequency direction, ifthe spectra have no contributions in the range Oa < 0 < Ob.

284 PhlllpsJournal of Research Vol.,35 Nos.4/S 1980


2.5. The WD of sampled analogue signals

Iff(n) and g(n) are obtained by sampling the analogue signalsfa(t) and ga(t)respectively with a sampling period T, then the WD off and g can be related tothat of fa and ga according to

00

(b.4)k=-oo

. This formula resembles very much the relation between the input and outputspectra of a uniform sampler (eq. (1.28) in ref. 3), but differs in one importantrespect, namely that the folding (aliasing) in the frequency occurs around TtITrather than 2TtIT. This means that to avoid aliasing in the WD the signalsmust be sampled at twice the Nyquist rate at least. For signals that aresampled at this rate eq. (b.4) reduces to

1Jif'l,g (n, wT) = T Jif'l",g.(nT,co),Tt

Iwl<-· .2T

(b.5)

3. Examples

Most of the examples given in part I carry over rather straightforwardly tothe discrete-time case. Therefore we will discuss only two simple exampleshere

3.1.

f(n) = {1

o1nl < N1nl ~ N,

(3.1)

for which the WD is given by2 sin [2e (N - In I + f)]

sin eWi(n, e) ~{o

1nl <N

1nl ~ N

(3.2)

3.2.f(n) = A eian212,

Le. the discrete-time version of a chirp signal. This signal has the WD

Jif'l(n,e) = IA 12 2TtL a(e - an - kTt)

. k(3.6)

which has been plotted in fig. 2 for the case that a is non-rational. In thisfigure each dot represents. a a-function. This chirp signal has only onefrequency component lying at namod Ttin a period of its WD.

Phlllps Journal of Research Vol.35 Nos.4/5 i980 285


/ e ,_., t /',/ _,..- 7r/2 /'

/' ,,!I'.11"/'

'.- /' ,,!I' /'/'/' / /'_,

,//'

,/ -6 -5 -4 -3 -2 -1/ 0 1 2 3 4 5 _., 7... / _n/' .#'... /'

/' ..-,/ -7r/2 /'/' ,,!I'

/..-,/

fc(n) = (f*hl)(n) = L!(n - k)hj(k)k

gc(n) = (g *hg) (n) = L g(n - k) hg(k)k

(4.1.a)

(4.1.b)

Fig. 2. Wigner distribution of a discrete-time chirp signal.

4. Effects of linear operations on the WD

4.1. Linear filtering

In this section the effect of filtering of the signals f and g on their WD willbe described. In general, Le, if neither f and g nor the filters are band-limitedto n/2, a relation similar to (1.4.2) cannot be obtained. If we consider both theoutputs of the original filters, Le.

and the outputs of two complementary filters, obtained by changing the signof the odd numbered samples of the impulse responses hl and hg, Le.

!CO(n) = (f *hJ)(n) = Lf(n - k)( -ll hj(k)k

gg(n) = (g * h~)(n) = L g(n - k)( _l)k hg(k)k

(d.1.a)

(d.1.b)

we can derive that

w,c,gc(n, 0) + w,~,g~(n, 0) .= L W"g(k, 0) Whj,hg(n - k, 0). (4.2)k

From (d.1.a) and (d.1.b) it follows that the spectra of the output of thecornplementary filters are given by

F'c°(e) = F(O) HI(e + n)

Gg(O) = G(O) Hg(e + n).

(d.2.a)

(d.2.b)

286 PhllIpsJournal of Research Vol,35 Nos,4/5 1980


Hence if f and g and also hf and hg are band-limited to nl2 or less, the outputsof these complementary filters are zero, in which case eq. (4.2) gets the simplerform

(4.2)

4.2. Multiplication in the time domain

If f and g modulate the carriers mj and mg respectively we get

fm(n) = fen) mf(n)

gm(n) = gen) mg(n).

(4.3.a)

(4.3.b)

The WD of the modulated signals is given by

1t/2

U'lm,gm(n, e) = _1_ J U'l,g(n, o Wmf,mg(n, e - o dç. (4.4)2n

-1t/2

4.3. Windowing in the time domain; the pseudo-Wigner distribution

For computational purposes windowing has to be applied in the discrete-time case as well in order to evaluate the WD. Therefore sliding windows areapplied to both signals f and g, yielding

fn(V) = f(v) 9:. wf(v) = f(v) wf(v - n)

gn(V) = g(v) 9:. wg(v) = g(v) wg(v - n).

(4.5.a)

(4.5.b)

The corresponding WD according to (4.4) is equal to

1t12

U'ln,gn(V' e) = _1_ J U'l,g(V, ç) WWj,Wg(V- n, e - o dç. (4.6)2n

-1t12

Again considering this WD only at the instant V = n the pseudo-Wignerdistribution for discrete-time signals is obtained

û1,g(n, e) = .U'ln,gn(V, e) I v=n1t12

= _1_ J U'l,g(n, o WWf,Wg(O,e - o dç. (4.7)2n

-1t12

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

ja(n) = f(n) + jJ(n), (5.1)

5. The Wigner dlstribution for analytic signals

A discrete-time analytic signal is of the form 3)

wherej(n) is real and j'(n) is the discrete Hilbert transform of j, defined by

J(n) = (.Yf.tj) (n) = " j(m) sin21t(m -:-n)/2

~ 1t (m ~ n)/2 (5.2)

This means that the spectrum of the analytic signal is given by

{

2F(O)Fa(O) = F(~)

O<O<1t0=0

-1t < 0 < O.(5.3)

Equation (5.2) describes the ideal case, but the corresponding filter is not. realizable. Realizable approximations are described in the literature, however,and approximations of analytic signals are therefore easily obtainable bymeans of digital signal processing 5,6). Here we will restrict the discussion tothe ideal case in which the analytic signal is related to the original signal by alinear filtering with the transmission function

21o

O<O<1t0=0

-1t < 0 <0.(e.l)

Apart from the usefulness of the analytic signal for describing single-side-band modulation systems and in other system and signal analysis applications,the analytic signal plays an important role.in the framework of the WD. Aswas demonstrated already in sec. 2, the spectral occupancy of the analyticsignal as given by (5.3), taking nonzero values only over an interval of length1t, assures that the WD of the analytic signal does not contain aliasing com-ponents.

We can now make a comparison between the two types of signals that havea WD without aliasirig, i.e. the interpolated signal and the analytic signal.Both can be obtained from the original signal by a linear filtering operation.This is shown in fig. 3, where the boxes with an arrow indicate sampling rateincrease and decrease by a factor of 2 (ref. 7). The interpolated signal is real ifthe original signal is real, but has the double rate, while the analytic signal iscomplex but has the same rate as the original signal. The number of realvalued samples per second is therefore the same for both signals. Also, ingeneral, the complexity of the low-pass filtering to obtain the interpolated

288 Philip, Journal of Research Vol.35 Nos.4/5 1980


realrate 21r

realrate 11r

complexrate 11r

interpolated signal. t.In)

x

Re

analytic signaltarn)

Fig. 3. Relations between a discrete-time signal, the corresponding interpolated version and thecorresponding analytic signal.

signal is quite comparable to that of the Hilbert transformer if the sameattenuation in the stopband is required.

6. Global and local moments of the WD

6.1. General remarks

In this section local and global central moments of the WD for discrete-timesignals are discussed. These moments allow a coarse characterization of theWD without the need to evaluate it. For discrete-time signals, however, we areconfronted with the fact that the WD has a different behaviour with respect toits time and frequency variable, i.e. its time variable is discrete, while itis aperiodic function of the continuous frequency variable. This will require somespecial care in defining the different moments.

Phlllps Journal of Reseorch Vol.3S Nos.4IS 1980 289

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

6.2. Central moments

6.2.1. Central moments of a periodic function

Let K(O) be a periodic function with period 1t and a positive mean given by7t12~I K(O) dO = Mo > O.

-7t12

(6.1.a)

7t/2

OK= targ [1 ei2!1K(O) dO]-7t12

(6.3a)

In part I the moments were obtained by the minimization of a weightedintegral of the function K. Pursuing the same approach, we have to search foran expression that satisfies the following conditions.(1) Weighting functions that it contains should be periodic with period 1t.

(2)" It should have a unique minimum.(3) For positive valued functions K the value of the minimum should be

smaller if K(O) is more concentrated.From a number of candidates for such an expression that satisfy these criteriawe will choose the following one

7t12f sin" (0 - (0) K(O) dO

-7t121(00) = --------

7t12f cos" (0 - (0) K(O) dO

-7t12

This choice is rather arbitrary and a justification for it can only be found inthe results that are obtained and that satisfy our expectations reasonably well.However, replacing in the denominator cos" (0 - (0) by 1 does not have adramatic influence on the results. In fact the position of the minimum will notbe changed at all by this replacement.

The expression (6.2a) has a unique minimum in the interval under the con-dition (6.1.a) at 00 = OK,where

(6.2.a)

and the value of the minimum is given by7t/2

Mo -I ~1ei2!1 K(O) dol.-7t12

I(OK) = -----7t-/~2 -----

Mo + I ~ 1ei20 K(O) dol-11/2

(6.4a)

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From (6.1a) and (6.4a) it follows that

-1 < I(OK) :s 1 (f.1)

and if, moreover, K(O) is positive then

rc12

I ~ J ej26

K(8) dol :S Mo-rc/2

(f.2)

so that(f.3)

The lower bound is only attained if K(O) is fully concentrated in the point OK,i.e. K(O) = 0(0 - OK), 101 :S rr/2.

6.2.2. Central moments for discrete-time functions

Let k(n) be a function of the discrete variable n e Z with

L k(n) = mo >0. (6.1b)n

In this case the central moments are found by minimization of the expression

i(no) = L (n - no)2k(n) / moon

(6.2b)

In principle it is possible to restrict no to Z , but this is not necessary and it iseasily seen that the location of the minimum is then not unique. Minimizing(6.2b) for noER yields a minimum at n« = ne, 'With

n« = L n k(n) / mo (6.3b)n

and the minimum equals

i(nk) = L (n - nk)2k(n) / mo = L n2k(n) / mo - nE.n n

(6.4b)

6.3. Moments of the WD in the frequency variable

6.3.1. Local moments

For a fixed time n the WD is a periodic function of 0 with period rr, forwhich we know from (2.29) that

rc12

_1_ J U'i(n, 0) dO = If(n) 12 = p,(n) ~ O.2rr

-rc12

(6.5)

Phllips Journalof Research Vol.35 Nos.4/5 1980 291


(Llf (/J) (n) = [(/J(n + 1) - qj(n)] mod 27t (f.4)

Hence, for values of n for whichf(n) f:. °we can define the average frequencyOf(n) of the WD at time n as the first-order moment with respect to thefrequency of the WD. From (6.3a) we find

Of(n) = targ [ 72

ej28 u:j(n, 0) dO].-ItI2

From (A2) in the appendix it follows that

Ot(n) = targ (f(n + 1) f*(n - 1».

(6.6)

(6.7)

For real valued signals this value is zero or 7t/2 and contains therefore no in-formation. Considering complex valued signalsf(n) can be written in the form

f(n) = v(n) ejq>(n), (6.8)

where v(n) and (/J(n) are real functions. Using this representation of fwe findfrom (6.7)

. 0 [(/J(n + 1) - (/J(n - I)J'f(n) = 2 mod 7t (6.9)

where [.] mod 7t means that the expression must be evaluated modulo 7t.(Arbitrarily it has been assumed that Of takes values in the interval (0, n), butany other interval of length 7t could have been taken).

In the continuous-time case the counterpart of eq. (6.9) has the interpreta-tion that the average frequency of the WD is equal to the instantaneousfrequency of the signal. For discrete-time signals we are faced with theproblem that a notion like the instantaneous frequency is not defined. Boththe forward difference

and the backward difference

(Llb (/J) (n) = [(/J(n) - (/J(n - 1)] mod 21t . (f.5)

are candidates for such a definition, but suffer from the fact that they areasymmetrical. (If for example (/J(n) = cos nOm then one would like the instan-taneous frequency to be zero in a maximum of (/J(n) like n = 0).

Equation (6.9) yields the result that the average frequency of the WD is thearithmetic mean of the forward and backward differences, taken modulo 1tbecause the period of the WD is equal to 7t.Thus

(f.6)

292 . Philip, Journol of Research Vol.35 Nos.4/5 1980


For arbitrary complex valued signals the "instantaneous frequency" canassume values over an interval of length 21t. Taking the average modulo 1t in(f.6) reflects the inability of the WD to discriminate between frequencies in theinterval (0,1t) and those in the interval (1t,21t) because of the aliasing thatoccurs (see sec. 2). If, however, the signal f is either band-limited to re/2 byinterpolation or is an analytic signal, then the instantaneous frequency is alsoconstrained to an interval of length re; see table I. In both cases the sym-metrical difference

[ffJ(n + 1) - (/J(n - I)J

(LIs (/J)(n) = 2 mod 2re (f.7)

is a suitable candidate for the instantaneous frequency. Since this frequencymust lie in the intervals indicated in table I, a restrietion modulo re instead ofmodulo Zrt is possible. From equations (f.6) and (f.7) it then follows that theaverage frequency of the WD is again equal to the instantaneous frequency ofthe signal.

TABLE I

Intervals of the instantaneous frequency for various complex valued typesof signals

f(n)interval of instantaneous

frequency

arbitrary (-re, re)

interpolated ( - 1t/2, re/2)

analytic (0, re)

The local second-order moment with respect to e is

71/2

pj(n) -12~ J ej211

Wj(n, e) de I-71/2

mj(n) = ------71-/2------

pj(n) + 12~ J ej211

Wj(n, e) del-71/2

(6.11)

Phlllps Journal of Research Vol.35 Nos.4/5 1980 293

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

This can be expressed in terms of the signal according to

I f(n)12 - I fen + l)f*(n - 1)1mj(n) = If(n)12 + If(n + l)f*(n - 1)1 (6.12)

6.3.2. Global moments

The global average of the WD is equal to

11/2

P, = 2~ J I Jf1(n, e) de = 11f 112 = 11F112.-11/2 n

(6.18)

The global average frequency is given by

[11/2 ~Öj = t arg f L ej21J U1(n, e) de

-11/2 n

= targ [~( ej21J IF(e)12 de],

(6.19)

(6.20)

where use is made of (2.37) and the periodicity of ej21J to derive the lastequation. Using Parseval's relation yields

~ = hrg [Lf(n + l)f*(n - 1)]n

= targ [L ej2lJj(n) Ifen + l)f*(n - 1)1],n

(6.21)

= targ [L ejarg(f(n+I)!*(n - I» If(n + l)f*(n - 1)1]n

where the latter expression follows from (6.7). Therefore the global averagefrequency of the WD is on the one hand equal to the average frequency ofthe spectrum of f (eq. (6.20)) and on the other is the weighted average ofthe instantaneous frequency, where If(n + l)f*(n - 1)1 is the weightingfunction.The global second-order moment is given by

11/2

P, -12~ J I ej21J

Jf1(n, e) del-11/2 n

fly = -------------11/2

P, + I ;n J I ej21J

Jf1(n, e) del-11/2 n

(6.22)

294 PhllipsJournal of Research . Vol. 35 Nos.4/5 1980


which can be written in the form1t _

f sin" «() - (}f) IF«(})12 d(}-1t

;rIJ = ----------1t .

f cos" «() - Ot) IF«(})12 d(}-1t

(6.23)

6.4. Moments of the WD in the time variable

6.4.1. Local moments

For a fixed frequency () the average of the WD over the time is

Pj«(}) = L U'l(n, (}) = I F«(})12 + I F«(} + 1t)12;:: O. (6.25)

n

The average time is given by the first-order moment

1/«(}) = L n U'l(n, (})/ Pj«(}).n

(6.26)

It is rather straightforward to derive an expression in which 1/«(}) is expressedin terms of the spectrum of the signal. For the general case this expression issomewhat complicated and not easy to interpret. This is due to the presence ofaliasing in the WD of arbitrary signals. If the discussion is restricted to signalswith a spectrum that occupies an interval of length 1t only, then the aliasingterms vanish. Therefore considering interpolated or analytic signals theexpression simplifies to

F'«(}) d1/«(}) = - Im F«(}) =_- Im d(} In F«(}), (6.27)

which holds for ()E ( -1t/2, 1t/2) in the case of interpolated signals and() E (0, 1t) for analytic signals.Writing F«(}) in terms of amplitude and phase

F«(}) = A«(}) ejlf/({!) (6.28)

this yields

1/«(}) = -IfI' «(}) (6.29)

with the same restrietion for ()as in (6.27).For these signals we obtain again the result that the average time of the WD

is equal to the group delay of the signal.The second-order moment is given by

\

Mf«(}) = L [n - 1/«(}»)2 U'l(n, (})/ Pj«(}).n

(6.30)

Philips Journal of Research Vol.35 Nos.4/5 1980 295

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

For interpolated or analytic signals this can be brought into the form

'.MJ(O) = - 1.Re ~ F '(0)2 dO F(O)

d2- } d(J2 In IA(O)I . (6.31)

6.4.2. Global moments

The global average time of the WD is given by

1t/2

1; =I :n J n WJ(n, 0) dO / p!n -1t/2

(6.32)

= L n If(n)12 / Ilf112.n

(6.33)

For interpolated or analytic signals this can be written as

1t

1; = _1 J 1j(O) I F(O)12 dO/11 F W21t

(6.34)

-1t

which means that 1;is equal to the weighted average of the group delay of thesignal.

The global second-order moment with respect to the time is equal to

1t/2

M! =I 2~ J (n - TW Uj(n, 8) dO/ p! (6.35)n -1t12

=I (n - 1;)2If(n)12 / IIfl12 ~ O. (6.36)

n

6.5. Inequality for the global second-order moments

The global second-order moments of the WD of a continuous-time signalsatisfy a Heisenberg type of uncertainty relation, given by (1.6.39). Such aninequality generally does not hold for the moments of the WD of discrete-timesignals. This can simply be demonstrated by the example f(n) = c5(n) forwhich frIJ = 1, M.t = O. For functions that are barid-limited to n/2 an equalityof this type can be derived, however, using the inequality

296 PhilipsJournal of Research Vol.35 Nos,4/5 1980

Philips Journal of Research Vol.35 Nos.4/5 1980 297


7t12f IF(0)12dO

-7t12

(f.7)

Hence, using the same arguments that led to (1.6.39) we obtain

mf + ~;::: 2 (mf~)!;::: 2/1t.

7. Wigner distribution of finite duration signals

If the signals f and g have a finite duration, Le. if

f(n) = g(n) = 0 1nl;::: L (7.1)

then it follows from (2.44) that

Wj,g(n, 0) = 0 1nl ;:::L. (7.2)

As remarked in sec. 2, the WD T-fj,g(n, 0/2) can be interpreted as the Fouriertransform of.the sequencef(n + k) g* (n - k) considered as a function of k forfixed n. It is known that the Fourier transform of a finite duration sequencecan be recovered from its samples taken at a sufficient number of equidistantpoints on the interval ( -1t, n), by means of interpolation. The interpolationformula is 3)

s-«

T-fj,g(n, 012) = I (7.3)

m=-N+l

where

(7.4)

and

M=2N-l (7.5)

with M taken larger than or equal to the duration of the sequence. For thesequencef(n + k)g*(n - k) this means that

N;::: L - Inl. (7.6)

To compute the WD of two finite duration sequences it therefore suffices todetermine the samples occurring in (7.3). These samples are given by

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

N-1

U'i,g (n, m ~) = 2I e-jkm(2rt/M)j(n + k)g*(n - k). (7.7)

k=-N+1

The right-hand side of eq. (7.7) can be interpreted as the M-point discreteFourier transform (DFT) of the sequence j(n + k) g*(n - k), k E ( - N + 1,...,N - 1) for fixed n (see ref. 3). This means that the evaluation of the WD ofa finite sequence is equivalent to determining one DFT for each value of n,which can be done efficiently using fast transform algorithms like the FIT 3)or Winograd algorithm 8).

The above reasoning can also be applied to find a computationally efficientway to determine the pseudo- Wigner distribution that was introduced insection 4.3. From its definition it follows that it can be computed according to

w(k) = 0 I kl === L (7.9)

00

û"l,g(n, 0) = 2 I e -j2k() w(k) f (n + k) w*( - k) g*(n - k). (7.8)k=-oo

If the windows have a duration 2L - 1, Le. if

then the same reasoning as before can be applied to show that the PWD can be.determined from the samples

L-1

=2I e-jkm(2rt/Mlw(k)w*(-k)f(n + k)g*(n - k)

k=-L+1 (7.10)

with M= 2L -1, (7.11)

and using the interpolation eq. (7.3). This means that to compute the PWD ata certain instant n, a DFT of length at least M = 2L - 1 must be computed.If w(k) w*( - k) is stored, then for each n such a computation requires 2Mmultiplications to determine w(k) w*( -k)f(n + k)g*(n - k) and tMlog2Mto perform the DFT by means of an FIT, where it is assumed that its length istaken a power of 2. Exploiting the fact that the PWD of any function is real itis possible to compute simultaneously the values of the PWD for two distinctvalues of n using only one FFT. This results in a further reduction of the com-putational complexity by a factor of 2.

8. Conclusions

The concept of the Wigner distribution has been introduced for the case ofdiscrete-time signals, and has been shown to share most of the properties ofthe WD for continuous-time signals. Deviations from this behaviour were

298 Phlllps Jnurnnl of Research Vol.35 Nos.4/5 1980

----------~-~-----~-- ~-- -----_ --,-- ----


observed at some points, however, due to the fact that the WD cannot dis-criminate between high and low frequencies because of aliasing componentsthat are present in the general case. It was shown that these aliasing termsvanish in two important cases, namely in the case of oversampling by a factorof at least 2, and in the case of analytic signals. Oversampling can be realised ifthe discrete-time signal is obtained by sampling an analogue signal at a rate ofat least twice the Nyquist rate. It can also be accomplished by means of aninterpolation process applied to a digital signal. For such signals the averagefrequency and average time that were defined for the WD could be given thesame interpretation as for the continuous-time WD. This led to a useful defini-tion of the instantaneous frequency of a discrete-time signal, which was thenagain equal to the average frequency of the WD. The average time of the WDwas shown to be equal to the group delay of the signal. Finally it was shownthat the WD of a finite duration sequence at a fixed time n could be determinedfrom a finite set of samples taken in the frequency interval (-n, n), Thesesamples of the WD can be determined from the signals by means of a discreteFourier transform, so that efficient algorithms like the FFT can be used. Aparticular example of such a WD is the pseudo-Wigner distribution (PWD)which was defined for arbitrary (infinite duration) sequences by means ofwindowing. This PWD was shown to be a filtered version of the WD of thesignals, where' filtering takes place in the frequency variable. If finite lengthwindows are used this PWD is not only computable, but can be computed witha complexity of the order of W(lOg2M + 4), where M is the length of thewindow. This complexity is of the same order of magnitude as that of the short-time Fourier transform, for which the PWD forms a viable alternative.

The relation between these two notions will be clarified in part III of thispaper, where different time-frequency signal representations will be con-sidered and will be related to the WD.

Philips Research Laboratories Eindhoven, May 1980

REFERENCES1) T. A. C. M. Claasen and W. F. G. Mecklenbräuker, The Wigner distribution - a tool

for time-frequency signal analysis, Part I: continuous-time signals, Philips J. Res. 35,217-250, 1980..

2) H. O. Bartelt, K. H. Brenner and A. W. Lohmann, The Wignerdistribution function andits optical production, Optics Commun. 32, 32-38, 1980. •

3) A. V. Oppenheim and R. W. Schafer, Digital signal processing, Prentice Hall, EnglewoodCliffs, 1975. .

4) R. W. Schafer and L. R. Rabiner, A digital signal processing approach to interpolation,Proc. IEEE, 61, 692-702, 1973.

G) O. Herrmann, Transversalfilter zur Hilbert-Transformation, AEU 23, 581-587, 1969.

PhllipsJournal of Research Vol.35 Nos.4/5 1980 299

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

6) L. R. Rabiner and B. Gold, Theory and application of digital signal processing, PrenticeHall, Englewood Cliffs, 1975, Chapter 3.37.

7) T. A. C. M. Claasen and w. F. G. Meckien b r äu ker, On the transposition of linear time-varying discrete-time networks and its application to multirate digital systems, Philips J. Res.33, 78-102, 1978. .

8) J. H. McClellan and C. M. Rader, Number Theory in digital signal processing, Prentice-Hall, Inc. Englewood Cliffs, N.J., 1979. .

Appendix

In this appendix several relations for the moments of the WD will be statedwithout proof. The proofs follow easily by using the properties of the WDgiven in sec. 2.3.

1t12

_1 J ei28Uj,g(n, e) de = f(n + 1) g*(n - 1)2n

-1t12

(A2)

Tt/2. IJ- Uj,g(n, e) dO = f(n) g*(n)2n

-Tt12

(Al)

1

L Uj,g(n, 0) = L F(O + kn) o-te + kn)n k=o

1

L n Ji'l,g(n, e) = -t L [(9lJF) (0 + kn) G*(O + kn)n k=O

(A4)

(A5)

+ F(e + kn) (9lJG)* (e + kn)]1

L n2 Ji'l,g(n, e) = t L [(9lJ2F) (e + kn) o*(e + kn)n k=O

(A6)

+ 2(9lJF) (e + kn) (9lJG)* (0 + he) + F(e + kn) (9lJ2G)* (e + kn)]

Tt12

_2_ J ~ fVt,g(n, 0) dO = (J, g)2n L.,;

-Tt12 n

(A7)

Tt12

2~ JIn fVt,g(n, fJ) de = ~J, g)-Tt12 n

Tt12

2~ J I ei28~,g(n, fJ)dO = (g;_1J, !lig)-1t12 n

(A8)

(A9)

Tt/2

2~ J I n2 fVt,g(n, 0) de = (~J, £il.g)-Tt12 n .

(AIO)

.-300 Phlllps Jeurnnl of Research Vol.35 Nos.4/5 1980

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