PAPERS

Wigner Distribution Representation and Analysis of AudioSignals: An Illustrated Tutorial Review*

DOUGLAS PREIS, AES Fellow

Department of Electrical Engineering and Computer Science, Tufts University, Medford, MA 02155, USA

AND

VOULA CHRIS GEORGOPOULOS**

School of Electrical Engineering and Computer Science, Ohio University, Athens, OH 45701, USA

The Wigner distribution provides a visual display of quantitative information abouthow a signal's energy is distributed in both time and frequency. Through its low-ordermoments the Wigner distribution embodies the fundamentally important concepts of bothFourier analysis and time-domain analysis. Signal energy is distributed in such a waythat specific frequencies are localized in time by the group-delay time (from classicalfilter theory), and at specific instants in time the frequency is given by the instantaneousfrequency (from classical modulation theory). The energy spectrum (energy per fre-quency) and the instantaneous power (energy per time) are specified by the zero-ordermoments of the distribution. The net positive volume of the wigner distribution is numeri-

'_ cally equal to the signal's total energy. While the theoretical underpinnings of the Wignerdistribution are mathematically elegant and do merit in-depth study, a substantial amount

\ of practical insight, understanding, and interpretive skill can be gained by carefullyexamining a wide variety of computed Wigner distributions such as those of the audiosignals presented.

0 RETROSPECT trogram, which was simply a smoothed version of theWigner distribution, and 2) it was efficiently computed

The Wigner distribution was "reintroduced" to the by using the fast Fourier transform (FFT) algorithm.international digital signal processing community in The most controversial issue surrounding the Wigner1980 at the IEEE L'Aquila Workshop by Theo Claasen distribution was the appearance of so-called "beats,"and Wolfgang Mecklenbr/iuker [1], who shortly there- which subsequently became known as interferenceafter published a significant three-part paper on the sub- terms. These terms were oscillatory and appeared be-ject [2]. While joint-domain representations like the cause the distribution was based on the quadratic con-Wigner distribution had been rather common in mathe- tent, or energy, of the signal, but some workers foundmatical physics for some time [3], it was only in the them to be seriously objectionable and arguably non-early development of signal analysis that Ville [4] first physical because the distribution became negative,alluded to time-frequency analysis in connection with 1) implying negative energy when viewed as a distribu-his exposition of the analytic signal, and later Woodward tion of energy in time and frequency and 2) not being[5] introduced the mathematically related complex ambi- able to be viewed as a distribution in the probabilisticguity function, sense, but able to exist where the signal had no spectral

There were two reasons for the broad appeal of the content. Although the Wigner distribution provided aWigner distribution. 1) It was shown to underlie and fascinating link between the time domain and the fre-provide more resolution than the commonly used spec- quency domain in signal analysis, its interference terms

were seen by many as unwanted artifacts, which de-* Presented at the 104th Convention of the Audio Engi- tracted significantly from its usefulness. The authors [1]

neering Society, Amsterdam, The Netherlands, 1998 May correctly explained that while these cross terms must16-19; revised 1999 October 29.** Currently at Technological Educational Institute of Pa- exist, they, on average, contain no signal energy.

tras, Koukouli Patras GR26334, Greece. There were, however, five undeniably attractive prop-

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erties of the Wigner distribution of the analytic signal: is questionable how the very concept of phase manifestsits volume represented total signal energy, its time mo- itself or whether phase is, in fact, lost in these newermerits gave the energy spectral density and group delay, signal transformations. One distinct advantage of theand its frequency moments gave instantaneous power Wigner transform is the fact that it is firmly rooted in(signal envelope squared)and instantaneous frequency. Fourier analysis. One of its moments does yield theBecause these moments are so useful in signal analysis, Fourier spectral magnitude squared, and Fourier analysisthe Wigner distribution, or Wigner-Ville distribution phase is given indirectly in terms of its negative deriva-as it became known, has continued to dominate for the tive with respect to frequency, the group-delay time. Onpast 18 years as the most popular time-frequency dis- the other hand, while the group-delay time and its dualplay (actually in the form of the spectrogram--a smoothed variable, instantaneous frequency, are average quan-version of the Wigner-Ville distribution), and it has titles that are not, in general, the same asFourier analy-gained widespread use in fields as diverse as electrocar- sis time and Fourier analysis frequency, they do playdiogram analysis,, acoustics, psychoacoustics, and an important role in identifying whether more than onespeech research, frequencyis presentat a given instant in time or whether

Currently the Wigner distribution is simply referred a certain frequency appears at different times.to as the Wigner transform. When applied to the complex Time and frequency are normally assumed to be inde-analytic time signal, it is known to be particularly well pendent variables in separate domains. The Wigner dis-suited to the analysis and detection of simply modulated tribution joins these two domains and invites the signalsignals such as amplitude modulation and frequency analyst to inquire, for example, 1) at which instant inmodulation, which are easily identified from envelope time a given frequency appears, or 2) what is the fre,and instantaneous frequency moments. In addition the quency of a signal at a given instant in time. The first-low-order time moments yield both the one-sided energy order moments of the Wigner distribution related tospectrum and group-delay time, the latter Of which can group-delay time and instantaneous frequency serve tobe a good estimate of a signal's average time delay, answer these questions in an average sense. Indeed, it

During the past 15 years an emerging trend in digital is remarkable that our very understanding of speech assignal processing has been to use certain mathematical well as our aesthetic impressions of music depend ontransformations, other than the Fourier transformation, our perceived answers to such questions.in signal analysis. Signals such as, for example, thosein medical diagnostics, seismic exploration, synthetic 1 DEFINITIONS _and natural speech, sonar, and radar require more analy-sis than a simple visual inspection of their time-domain Instantaneous frequency is defined as [4] _ ,/?waveforms. As is often the case, information about thephase of the signal or how the phase is altered by modu- 1 d[0(t)]fi(t) = (1)lation or linear distortion plays an important role in re- 2'n dtvealing important underlying physical mechanisms suchas modulation schemes, dispersion, or reflections. Re- where 0(t) is the phase of the analytic signal,searchers in digital signal processing have proposed, forexample, using the Wigner transform, the ambiguity z(t) -- IZ(t)le_°<'_ (2)function, the Gabor transform, and various wavelet trans-forms for signal analysis. Because the Fourier trans- and Iz(t)lis the envelope. The analytic signal of a realform maps, for each frequency, a complex frequency- signal x(t) consists of the real signal as its real partmodulated version of the signal that is averagedover and the Hilbert transform of the real signal _(t) as thethe whole time interval to yield a single (complex) point imaginary part,in the spectrum and, second, because Fourier synthesis(inversion) used to recover or reconstruct the original z(t) = x(t) + j_(t) . (3)signal often depends strongly on cancellation, differenttransformations have been sought that offer certain ad- Group delay 'rs is the delay of the envelope of thevantages in terms of multiresolution, time localization signal x(t) and is given byof frequency events, and more dependence on superpo-

sition rather than cancellation in signal synthesis. Many 1 d_b(f) (4)of these newer transformations seek to represent signals xg = 2_r dfas sums of shifted (translated) and expanded (dilated)or even modulated versions (modulates) of simpler, ele- where _b(f) is the phase of the Fourier transform of x(t).mentary signals that occupy a short time interval. It is Because the instantaneous frequency can, in practice,not that these newer signal transformations are intrinsi- be difficult to estimate accurately, many different tech-cally better, but they are thought to be more suited to niques have been proposed and developed [6]-[13]. Aspecific applications. Since so much insight can be survey and unified treatment of various computationalgained from a thorough examination and understanding methods for estimating both instantaneous frequencyof the various forms of phase alterations or phase and its dual variable, group-delay time, of discrete-timechanges within the general context of Fourier analysis, it signals is given in [14], [15].

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2 AUDIO SIGNALS to it), 5) a phase-shift-keyed (PSK) modulated signal,and 6) a frequency-shift-keyed (FSK) modulated signal.

The Wigner distribution has been used extensivelyin audio signal analysis for spectral estimation [16], 2.1 Two Gausslan Windowed Tone Bursts (Fig. 1)electrocardiogram analysis [17], characterization of au- Two tone bursts, one at 700 Hz and the other at 1500ditory fibers [18], characterization of loudspeakers [19], Hz, each 10 ms long, were windowed separately usingand perception of phase distortion [20]. a Gaussian window of a standard deviation of 1.2 ms.

The Wigner distribution of a signal contains, in a The resulting signal was 50 ms long with the 700-Hzsimple way, the following four properties useful to the burst centered at 15 ms and the 1500-Hz burst centeredaudio signal analyst: 1) frequency response, 2) group at 25 ms. The width and placement of the Gaussiandelay, 3) instantaneous power (the signal's envelope windows were chosen so that the two tone bursts didsquared), and 4) instantaneous frequency, not overlap in time. Fig. 1 shows the time sequence of

Each of these properties can be estimated visually by the tone bursts (bottom), the energy spectrum (left), andtaking a "slice" of the elevation contours of the Wigner the envelope squared (top), which is the instantaneousdistribution parallel to the horizontal time axis or parallel power of the signal. Because power equals energy perto the vertical frequency axis in the time-frequency unit time, this curve is commonly referred to by audioplane. The area under a horizontal slice at frequency engineers who assess loudspeaker performance as thefl gives the numerical value of the frequency response ETC, or energy-time curve. It shows the arrival times(magnitude squared) at that frequency, whereas the cen- of signal energy.ter of gravity of that slice (the point at which all the At right in Fig. 1 there are curves of group-delay timearea could be concentrated to produce the same moment versus frequency and instantaneous frequency versusabout the vertical axis) gives the group-delay time at time. These two quantities are, mathematically, dualthat frequency. Similarly, the area under a vertical slice variables of one another and numerically equal the centerat time ti gives the instantaneous power of the signal's of gravity of the Wigner distribution about the frequencyenvelope, whereas the center of gravity of that slice axis and the time axis, respectively. In the center Fig. 1equals the instantaneous frequency, shows elevation contour lines of the Wigner distribution

The representative signals examined here are 1) two itself as derived from the analytic signal associated withGaussian windowed tone bursts, 2) two Gaussian win- the time sequence. The analytic signal has zero Fourierdowed chirps, 3) the impulse response of an antialias spectrum for negative frequencies and is used as a mathe-low-pass filter, 4) the impulse response of an all-pass matical means to define and extract the envelope andfilter (a filter that only alters the phase of signals applied instantaneous frequency of the signal. The oscillatory

l0ff7.5 nvelopexlO-4

5.0

2.5

0 0 0'01 0.02 0.03 0.04 0.05time, s

::11_42000 _ _ 2 Wigner distribution :_ 2( group delay

o 1500 ........................energy spectral . -_ _1 0,:_._..O _1-_dens?._ _ 100_/ ___ instantaneou:

1000500_ -__ 500[ _ 5t frequency

02 1.5 1 0.5 0 0 _ 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05xl0-3 time,s time,s

0.0_

0.02 signal

o-0.0_

-0'060 0.01 0.02 0.03 0.04 0.05time, s

Fig. 1. Wigner-Ville distribution of two Gaussian windowed tone bursts along with its zero- and first-order moments: energyspectrum, envelope, group delay, and instantaneous frequency.

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terms between the signal terms in the Wigner-Ville dis- below is the impulse response of the low-pass filter. Ontribution are the so-called interference terms of the Wig- top is the instantaneous power of the envelope of thener-Ville distribution, which are alternatively posi- impulse response. To the right are curves of the group-tively and negatively valued and which contain, on delay time versus frequency and instantaneous ftc-average, zero energy, quency versus time. In this distribution a minor,-but

observable, artifact of the analytic signal used in ob-2.2 Two Gausslan Windowed Chirps (Fig. 2) taining the Wigner-Ville distribution is that the Wig-

Two chirps cos(2=700t + 50 000_ and cos(2_1500t + ner-Ville distribution as well as the energy-time curve50 000t 2) were windowed separately using a Gaussian are not causal, that is, they exist before the impulsewindow of a standard deviation of 2.25 ms. The resulting response starts. The small, alternately positive and nega-signal was 50 ms long with the 700-Hz chirp burst cen- tive C-shaped interference terms to the right of the Wig-tered at 19 ms and the 1500-Hz chirp burst centered at ner-Ville distribution produce the peaks and valleys in29 ms. Fig. 2 shows their time sequences, the FFT-based the energy-time curve but tend to cancel in the hori-estimate of the energy spectral density, the computed zontal direction to yield a fiat frequency response (spec-enevelope, the group delay and instantaneous frequency, tral magnitude). The "ringing" in this filter's impulseand the Wigner-Ville distribution. In this case the sig- response near and at 4 kHz is predicted in the uppernals overlap slightly in time but not significantly spec- part of the Wigner distribution by the large, horizontaltrally. The instantaneous frequency becomes oscillatory positive energy concentration, which implies a damped,when the two signals overlap. Based on a cursory exami- nearly sinusoidal oscillation in the time domain. Thenation of the time sequence, envelope, or spectral den- rising group-delay curve follows the significant valuessity it is not possible to determine whether the signals are of the Wigner distribution. The instantaneous frequencychirps or merely tone bursts. However, the instantaneous rises quickly from zero, then oscillates in the region offrequency, the group delay, and the orientation of the the cutoff frequency of 4 kHz.Wigner-Ville distribution clearly show the variation · In qualitative terms, large concentrations in the Wig-with time. This example illustrates important advantages ncr distribution parallel to the frequency axis imply thatof time-frequency analysis using the wigner-Ville the signal is "impulsivelike," whereas concentrationsdistribution, parallel to the time axis imply that the signal is "sinusoi-

dallike" or oscillatory in time (see Fig. 3) [20]. From2.3, Low-Pass Filter (Fig. 3) the standpoint of modulation theory, a horizontal WignerA Wigner-Ville distribution corresponding to a 4- distribution indicates "perfect" amplitude modulation or

kHz low-pass filter (a speech-bandwidth antialias filter) unmodulated carrier because at every instant in timeis shown in Fig. 3. At left is the frequency response and there is only one frequency present (just the carrier

4

x104321 . _velope00 0.01 0.02 0.03 0.04 0.05

time, s

N'250_ _ N25t N25' E..... ' delayWi ner

'=_5_0__ 150_- energy spectral dens___,_10_15,Distribuuo.._ _rlO-_15_ _i5( nstanta_ex)us, I , frequem0'2 1.5 1 0.5 0 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05

xlff3 time,s time,s0.02 ' '

· _0.01 II;.m

-0.01

-0.02 0.01 0.02 0.03 0.04 0.05time, s

Fig. 2. Wigner-Ville distribution of two Gaussian windowed chirps along with its zero- and first-order moments. Instantaneousfrequency, group delay. Orientation of Wigner-Ville distribution shows variation of frequency with time.

1046 d.AudioEng.Soc.,VoL47,No.12,1999December

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wave). On the other hand, a vertical orientation of the distribution corresponds to the arrival of the "packet" ofWigner distribution indicates "perfect" frequency modu- signal energy near the 6.15-ms group-delay time which,lation because every frequency is present at one instant from the impulse response or instantaneous frequency,in time. A change in the orientation of the Wigner distri- is an oscillation at approximately 3.2 kHz. The Wignerbution as time progresses can be interpreted as a conver- distribution contains rather small contributions that os-sion between frequency and amplitude modulation, for cillate in time near the frequencies of 1.5 kHz and 5.3example, as Fig. 3 shows. In summary, the Wigner distri- kHz, which determine the "fine structure" of the enve-bufion in Fig. 3 is a graphic representation of the statement lope and instantaneous frequency.that the impulse response of a filter is, in general, an This example and the previous one clearly show theamplitude-modulated and frequency-modulated signal interplay between the Wigner distribution and impulse re-[21]. While these effects are always evident in the impulse sponse, frequency response, envelope power or energy-response itseff, they ate dramatically illustrated by the time curve, group delay, and instantaneous frequency [20].orientation of the Wigner distribution.

2.5 PSK Signal (Fig. 5)2.4 Ali-Pass Filter (Fig. 4) A PSK signal cos(2'_200t + 2_ri/5), where i = l, 2,

Fig. 4 (bottom) shows the impulse response of an all- . . . , 5, was rectangularly windowed and sampled atpass filter (energy/frequency = constant) whose peak 10 kHz. There are four phase shifts in the signal, eachgroup delay equals 6.15 ms at 3.2 kHz.' Examination of one lasting 0.01 s. Fig. 5 shows the time sequence, thethe impulse response waveform reveals both amplitude FFT-based estimate of the energy spectral density, theand frequency modulation effects. The energy-time computed envelope, the group delay and instantaneouscurve (envelope power in decibels) shows the former frequency, and the Wigner-Ville distribution. Thewhereas the absolute value of the instantaneous fre- phase shifts are not all evident in the plot of the signalquency indicates the latter. The envelope collapses sev- in the time domain. However, there are indications (im-eral times, and each time this occurs, the absolute value pulsive) of the signal's abrupt phase shifts in the Wignerof the instantaneous frequency simultaneously swings transform itself and also in the envelope and instanta-between 3.2 kHz and 0 kHz. The main, positive portion neous frequency [22]. In addition, both the instanta-of the Wigner distribution has a shape similar to the neous frequency and the spectral magnitude reveal .thegroup-delay curve. Between 0.1 and 2.0 ms alternating frequency of the sinusoidal carrier.positive and negative (black and white) impulsive contri-butions between about 2.0 and 4.0 kHz in the Wigner 2.6 FSK Signal (Fig. 6)distribution strongly modify the energy-time curve (en- An FSK signal cos(2'trfit), where f, takes on the fre-velope). The rightmost central portion of the Wigner quency values 300, 1100,700, and 1800 Hz sequentially

0.012

00l

0.0080.0060.0040.00_

0 ' '0 0.002 0.004 0.006 0.008 0.01

time, s

Magnitude '_ 100013 :_ 1000(31000C . Wigner DistributionFrequency _ groupResponse =

o _ delay .5000 $. 5000 5ooo i

_ _ _instantaneousI . _. frequency.°0.01 0.008 0.006 0.004 0.002 0 0 0 0.002 0.004 0.006 01008 0.01 0 0 0.002 0.004 0.006 0.008 0.01

energyspectraldensity time,s time,s0.1

0.0: Impulse

0 _ __ Resp°nse-0.0_

-0.0 0.002 0.004 0.006 0.008 0.01

time, s

Fig. 3. Wigner-Ville distribution of 4-kHz low-pass filter. Wigner-Ville distribution graphically illustrates that the impulseresponse of a filter is, in general, an amplitude-modulated and frequency-modulated signal.

J.AudioEng.Soc.,Vol.47,No.12,1999December 1047

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for 0.0l s, was rectangularly windowed and sampled at nal itself in the time domain reveals that the frequency

10 kHz. Fig. 6 shows the time sequence, the FFT-based varies with time, but it is unclear exactly where there are

estimate of the energy spectral density, the computed transitions from one frequency to the next. The energy

envelope, the group delay and instantaneous frequency, spectral density shows the four frequencies present in

and the Wigner-Ville distribution. Inspection of the sig- the signal but, of course, does not indicate at which time

0810.6 envelope

0.4

,"0 0.002 0.004 0.0060.0080.01

time, s6000 6000

5 ":::_ 5000 :_ Wignerdistribution "_ 5000

4000[ I magnitude _, _ 4000

_ 3000[ I frequencyresp°nse =o o_3000_' 200oi II Vinstantane°us

l°°°ll , U 1 frequency0 3 2.5 2 1.5 I 0.5 O. 0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01

lox-3 time,s time,s0.6

0.4 '

-0.4

-0.6 0.002 0._)40. 0.'008 0.01time, s

Fig. 4. wigner-Ville distribution of all-pass filter with peak group delay of 6.15 ms at 3.2 kHz along with its zero- and first-order moments. The main, positive portion of the Wigner distribution has a shape similar to the group-delay curve.

4

2

1

O0 0.01 0.02 0.03 0.04 0.05time s

spectral densit_ Wigner distribution 1st order moments250( ' '

N _:'. _! _' ' :_J _":'___"_' _._200_-_2500[- 0!0

_2150_ t_i i: ' .!1 Crl___l [group / .

= ' _'_____ _ 150QL instantaneouq frequency

0 150 100 50 0 0 0.01 0.02 0.03 0.04 0.05 U 0 0.01 0.02 0.03 0.04 . 5time,s time,s

signal, in time-doma, in

!150 0.01 0.02 0.03 0.04 0.05time, s

Fig. 5. Wigner-Ville distribution of phase-shift-keyed (PSK) signal along with its zero- and first-order moments. Phase shifts areindicated by the presence of "impulses" in the Wigner-Ville distribution as well as in the envelope and instantaneous frequency.

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they occur. The envelope of the signal is approximately In the literature various techniques have appeared to1 for all time, which is consistent with the presence of reduce interference terms or result in a nonnegative dis-a sinusoidal signal, but there are "glitches" at the transi- tribution. However, in many instances both zero-andtion points from one frequency to the next. The higher, first-order moments of the distribution are not preservedinner contours of the Wigner distribution (that is, those that and valuable information is lost.appear like the letter I rotated 90°) indicate the frequency of Wigner showed that positive bilinear distributions sat-the signal at given time instants. The importance of group isfying the time and frequency marginals do not existdelay and instantaneous frequency in localizing the signal [3]. The spectrogram, for example, is manifestly posi-in time and frequency is clearly seen [22]. tive but does not satisfy the marginals, whereas the Wig-

ner-Ville distribution satisfies the marginals but is not3 NONNEGATIVE APPROXIMATE WIGNER manifestly positive. Claasen and Mecldenbr_iuker [2]DISTRIBUTIONS WITH VERY ACCURATE LOW- have shown that the Wigner-Ville distribution for aORDER MOMENTS chirp, and for that one signal only, is manifestly posi-

tive. Since our experience in everyday life is with posi-The foregoing examples have illustrated the important tive energy densities [23], it is important to obtain

role that the low-order moments, especially those related time- frequency distributions that both are manifestlyto group-delay time and instantaneous frequency, play positive and satisfy the marginals. Cohen and Posch [24]in interpreting the Wigner-Ville distribution. As a tool have defined "proper" time-frequency distributions thatfor time-frequency analysis, the Wigner-Ville distribu- satisfy positivity and the marginals and all of which cantion has its shortcomings, however. Negative values of be parametrized by the function p(x, y). The lack of athe distribution, suggesting negative-energydensity, can single distribution being best for all applications hasexist. Oscillatory interference terms due to quadratic resulted in a proliferation of distributions, each cortes-interaction can appear in regions of the time-frequency ponding to a different mapping from the signal plane toplane where the signal has no spectral energy. The Wig- the time-frequency plane. Some of these techniques arener-Ville distribution of causal signals has noncausal mentioned here. It is important to emphasize that sincevalues due to the use of the analytic signal in the Wig- these techniques do not in general yield good results forner-Ville distribution calculation. Moment accuracy is all classes of signals, the work described in the Appen-impaired due to data windowing. Smoothing of the dis- dix takes an alternative approach.tribution not only reduces its resolution but also intro- One approach is to formulate the solution as a mini-duces further moment inaccuracies, mum cross-entropy (MCE) optimization problem subject

envelope

00 0.01 0.02 0.03 0.04 0.05 -time. s

spectral density Wi_ner distribution Ist order moments

' ' 250_ ,_.. _.'_ group delay '200( r_ 2 ., _e 20C -

150( 15 ". _.--'. : 15C - '

500 - instantaneoUS]ncyO_050 40 30 20 10 0 0 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05

time, s time, s

signal in time-domain

.o?[VVVlIIIIIIIIIIIIIIIIM1-I 0 0.01 0.02 0.03 0.04 0.05

time, s

Fig. 6. Wigner-Ville distribution of frequency-shift-keyed (FSK) signal along with its zero- and first-order moments. Contoursof the Wigner distribution indicate the frequency of the signal at a given time instant, whereas group delay and instantaneousfrequency show the localization of the signal in time and frequency.

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to linear constraints, in particular, the marginal condi- They are easier to interpret and may, in fact, be moretions [25]. In this method all the functions that satisfy, useful in those applications where accuracy in the higherthe conditions are found, and then the one chosen among order moments can be compromised.them is the one that maximizes the entropy, the reasonbeing that the maximum entropy solution is the one that 4 CONCLUSIONS -is unbiased. This technique was extended to satisfy anadditional constraint that along a particular axis the dis- In its largest context, the problem of time-frequencytribution must be a specified function [26]. Another set analysis is to develop a two-dimensional display of one-of joint constraints has also been proposed in the entropy dimensional data that can be computed efficiently, thatmaximization approach, which are equivalent to the two clearly reveals those essential features to be extractedclassical marginals [27]. from the data, that has no distracting artifacts, and thatReduced interference distributions (RID) have been is readily interpreted physically. This research area is

developed by designing kernels to specifically accom- important because it should provide a means to improveplish that. For certain values of the kernels an RID ap- the human visualization and understanding of compli-proaches the Wigner distribution [28]. cated data, and it could even lead to efficient, automatedAnother approach is to approximate positive time- data or signal analysis by expert computer systems. The

frequency distributions (TFDs) through nonlinear corn- Wigner-Ville distribution is suited to the task because,binations of spectrograms [29], where closed-form solu- within the framework of Fourier analysis, it combinestions for the combinations are obtained via optimization the traditional concepts associated with the time andof entropy functions subject to an energy constraint, frequency domains into a single, so-called joint domainThis method gives results superior to that achieved with that contains and illustrates virtually all of the salientindividual spectrograms, and remarkably close to the information about the signal in a very graphic way. Be-positive TFDs obtained via computationally intensive cause rather prominent interference or cross terms con-methods, tinue to be viewed by some workers as distracting arti-Optimal-kernel time-frequency representations rely on facts of the Wigner-Ville distribution, recent research

the idea that for any given signal some TFRs are "better efforts have focused on eliminating them by either modi-than others" and thus, with optimization-based kernel de- fying the distribution or seeking approximations to it.sign, the goal is to extract the maximum possible Sincethelow-ordermomentsoftheWigner-Villedistri-time-frequency information. The optimal kemel may have bution are so rich in information useful to the signala special binary structure, "1/0 kernel," or it can be an analyst, it is essential that they be preserved as accu-additional smoothness constraint that forces the optimal rately as possible when constructing any "approximate" ) _\keme! to be Gaussian along radial profiles [30]-[32]. Wigner-Ville distribution [36].

Wavelets have been used to soft-threshold the Wig- Perhaps the oldest and most common form ofner-Ville distribution [33]. This technique is a poten- time-frequency analysis of audio signals is the musicaltially useful tool for Wigner-Ville spectrum estimation score whose notes indicate the exact timing, duration,of unknown deterministic signals embedded in noise, and pitch (frequency) of vocal or instrumental sounds.For time-varying spectral analysis of noisy signals, non-linear smoothing of the Wigner-Ville distribution _5000should offer higher performance than linear smoothing.However, this approach does not yield a distribution _ _((_ _(_lr"satisfying the marginals, c

The least mean squared (LMS) adaptive algorithm hasbeen used to simulate an exponentially smoothed pseudo _ 0 -Wigner-Ville distribution. This was achieved using 0 0.002 0.004LMS frequency-domain adaptive structures and the auto time, sterms are separated from the cross terms in the Wig-

Fig. 7. Nonnegative approximate Wigner distribution saris-ner-Ville distribution [34]. fying zero-order moments with very high accuracy for low-

The concluding examples illustrate how the shape Of pass filter example of Fig. 3 (see Appendix).the Wigner-Ville distribution shown for the low-pass

filter of Fig. 3 would change if the hard constraints of _ 5000nonnegativity, causality, and very high low-order mo-ment accuracy are imposed [35]. The results shown in _ _ _'7_0_Figs. 7 and 8 are neither Wigner-Ville distributions o_nor are they unique time-frequency representations, butthey do serve to illustrate the results of a simple and _ 0direct way to eliminate all of the objectionable artifacts 0 0.002 0.004

of the Wigner-Ville distribution while preserving, or time, s

even improving, low-order moment accuracy. "Approxi- Fig. 8. Nonnegative approximate Wigner distribution saris-mate" Wigner-Ville distributions such as those in Figs. fying zero- and first-order moments with very high accuracy7 and 8 can be computed iteratively (see Appendix). for low-pass filter example of Fig. 3 (see Appendix).

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PAPERS WIGNER DISTRIBUTION

To a musician, the score by itself is truly music that is [14] V. C. Georgopoulos and D. Preis, "A Compari-seen but not heard. To the signal analyst, the visual son of Computational Methods for Instantaneous Pre-displays of quantitative information in this paper, espe- quency and Group Delay of Discrete-Time Signals,"cially the various tones represented and analyzed in Figs. Dept. of Electrical Engineering, Tufts University, Med-1, 2, and 6, should, with a modest amount of study and ford, MA, Tech. Rep. TR-EEDSPO1-93 (1993).interpretation, elicit very similar impressions related to [15] V. C. Georgopoulos and D. Preis, "A Compari-the essential time-frequency properties of signals, son of Computational Methods for Instantaneous Fre-

quency and Group Delay of Discrete-Time Signals," J.5 REFERENCES Audio Eng. Soc., vol. 46, pp. 152-163 (1998 Mar.).

[16] E. F. Velez and R. G. Absher, "Spectral Estima-[1] T. A. C. M. Claasen and W. F. G. Mecklen- tion Based ontheWigner-VilleRepresentation,"Signal

briuker, "The Wigner Distribution: A Tool for Time- Process., vol. 20, pp. 325-346 (1990).Frequency Analysis," in Proc. IEEE Acoustics, Speech [17] B. Boashash, "Time-Frequency Signal Analy-and Signal Processing Society L'Aquila Workshop sis," in Advances in Spectrum Analysis and Array Pro-(L'Aquila, Italy, 1980). cessing, S. Haykin, Ed. (Prentice-Hall, Englewood

[2] T. A. C. M. Claasen and W. F. G. Mecklen- Cliffs, NJ, 1991), pp. 418-517.bdiuker, "The Wigner Distribution--A Tool for Time- [18] J. J. Eggermont and G. M. Smith, "Characteriz-Frequency Analysis, Parts I-III," Philips J. Res., vol. ing Auditory Neurons Using the Wigner and Rihaczek35, pp. 217-250, 276-300, 372-389 (1980). Distributions: A Comparison,"J. Acoust. Soc. Am., vol.

[3] E. P. Wigner, "On the Quantum Correction for 87, pp. 246-259 (1990).Thermodynamic Equilibrium,"Phys. Rev., vol. 40, pp. [19] C. P. Janse and A. J. M. Kaizer, "Time-749-759 (1932). Frequency Distributions of Loudspeakers: The Applica-

[4] J. Ville, "Theorie et applications de la notion de tion of the Wigner Distribution," J. Audio Eng. Soc.,signal analytique," Cables et Transmissions, vol. 2A, vol. 31, pp. 198-223 (1983 Apr.).no. 1, pp. 61-74 (1948). [20] D. Preis, F. Hlawatsch, P. J. Bloom, and

[5] P. M. Woodward, Probability and Information J.A. Deer, "Wigner Distribution Analysis of FiltersTheory with Application to Radar (Pergamon, Lon- with Perceptible Phase Distortion," J. Audio Eng. Soc.don, 1953). (Communications), vol. 35, pp. 1004-1012 (1987

[6] P. Y. Ktonas and N. Papp, "Instantaneous Enve- Dec.).lope and Phase Extraction from Real Signals: Theory, [21] A. Papoulis, TheFourierlntegralandltsAppli-Implementation, and an Application to EEG Analysis," cations (McGraw-Hill, New York, 1962).Signal Process., vol. 2, pp. 373-385 (1980). [22] V. C. Georgopoulos and D. Preis, "Analysis of

[7] L. B. White and B. Boashash, "On Estimating PSK- and FSK-Modulated Signals Using Time-Frequen-the Instantaneous Frequency of a Gaussian Random Sig- cy Information", Dept. of Electrical Engineering, Tuftshal by Use of the Wigner-Ville Distribution," IEEE University, Medford, MA, Tech. Rep. TR-EEDSP01-Trans. Acoust., Speech, Signal Process., vol. 36, pp. 94 (1994).417-420 (1988). [23] L. Cohen, Time-Frequency Analysis (prentice-

[8] P. Rao and F. J. Taylor, "Estimation of Instanta- Hall, Englewood Cliffs, NJ, 1995).neous Frequency Using the Discrete Wigner Distribu- [24] L. Cohen and T. Posch, "Positive Time-pre-tion," Electron. Lett., vol. 26, pp. 246-248 (1990). quency Distribution Functions," IEEE Trans. Acoust.,

[9] K. M. Wong and Q. Jin, "Estimation of the Time- Speech, Signal Process., vol. 33, pp. 31-37 (1985).Varying Frequency ofa Signal: The Cramer-Rao Bound [25] P. J. Loughlin, J. W. Pitton, and L. E. Atlas,and the Application of Wigner Distribution," IEEE "Construction of Positive Time-Frequency Distribu-Trans. Acoust., Speech, Signal Process., vol. 38, pp. tions," IEEE Trans. Signal Process., vol. 42, pp.519-536 (1990). 2697-2705 (1994).

[10] B. Boashash and L. B. White, "Instantaneous [26] J. R. Fonollosa and C. L. Nikias, "A New Posi-Frequency Estimation and Automatic Time-Varying Fil- tive Time-Frequency Distribution," in Proc. IEEE IC-tering," in Proc. ICAASP '90 (1990), pp. 1221-1224. ASSP-94 (1994), pp. 301-304.

[11] B. Boashash and P. O'Shea, "Use of the Cross [27] J.R. Fonollosa, "Positive Time-Frequency Dis-Wigner-Ville Distribution for Estimation of Instanta- tributions Based on Joint Marginal Constraints," IEEEneous Frequency;" IEEE Trans. Signal Process., vol. Trans. Signal Process., vol. 44, p. 2086 (1996).41, pp. 1439-1445 (1993). [28] H. I. Choi and W. J. Williams, "Improved

[ 12] K. S. Riedel, "Kernel Estimation of the Instanta- Time-Frequency Representation of Multicomponentneous Frequency," IEEE Trans. Signal Process., vol. Signal Using Exponential Kernels," IEEE Trans. Acoust.,42, pp. 2644-2648 (1994). Speech, Signal Process., vol. 37, pp. 862-871 (1989).

[13] L. J. Stankovic and S. Stankovic, "An Analysis [29] P. J. Loughlin, J. W. Pitton, and B. Hanaford,of Instantaneous Frequency Representation Using Time- "Fast Approximations to Positive Time-Frequency Dis-Frequency Distributions-Generalized Wigner Distribu- tributions with Applications,"in Proc. ICASSP-95 (De-tion," IEEE Trans. Signal Process., vol. 43, pp. 549- troit, MI, 1995), pp. 1009-1012.552(1995). [30] R. G. Baraniukand D. L. Jones, "A Signal-

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PREIS AND GEORGOPOULOS PAPERS

Dependent Time-Frequency Representation: Optimal (LMS) algorithm developed by Widrow and Hoffin 1959Kernel Design," IEEE Trans. Signal Process., vol. 41, [37]. In the present application the input is the vectorpp. 1589-1602 (1993). x, and the desired output is the vector y.

[31] R. G. Baraniuk and D. L. Jones, "Signal- The update equation for the elements a0 of matrix A isDependent Time-Frequency Analysis Using a Radially

c eU) [kix(0 (6)Gaussian Kernel," Signal Process., vol. 32, pp. 263- aa[k + 1] = aa[k] + 21_-_284 (1993).[32] D. L. Jones and R. G. Baraniuk, "An Adaptive

Optimal-Kernel Time-Frequency Representation," IEEE where the error eV)[k] is the difference between the jthTrans. Signal Process., vol. 43, pp. 2361- 2371 (1995). element of the desired output vector and the jth element

[33] R.G. Baraniuk, "Nonlinear Wigner- Ville Spec- of the output vector of the algorithm on the kth iteration,trum Estimation Using Wavelet Soft-Thresholding," in ILxl[2 is the square of the norm of the vector x [x(° is theProc. SPIE Syrup. on OR/Aerospace Sensing and Dual /th element of vector x], and c is a Constant that controlsUse Photonics, Mathematical Imaging: Wavelet Appli- stability and speed of convergence. A practical rangecations for Dual Use (Orlando, FL, 1995). for c is 0.05 < c < 0.5.

[34] M. G. Amin, P. J. Davis, and F. Allen, Jr., "An The LMS algorithm has been extended to allow adap-Application of the LMS Algorithm in Smoothing Pseudo tation of the aa by four vectors of errors, one generatedWigner Distribution," IEEE Trans. Signal Process., ' for each moment. Also, a threshold-type nonlinearity isvol. 41, pp. 930-934 (1993). used to ensurenonnegativity.

[35] V. C. Georgopoulos and D. Preis, "A Nonnega- A.2 Iteration Stepstire Approximate Wigner Distribution with AccurateLow-Order Moments," in Proc. 13th Int. Conf. on Digi- An estimate of the matrix A, whose elements representtal Signal Pocessing, DSP'97 (Santorini, Greece, 1997 time and frequency samples of the approximate WignerJuly 1-4). distribution, can be obtained by using the following

[36] D. Preis and V. C. Georgopoulos, "Moment- procedure.Enhanced Spectrograms for Audio Signal Analysis," 1) Choose A[0], the initial estimate of the matrix A.presented at the 106th Convention of the Audio Engi- For example, an A[0] that satisfies exactly the total en-neering Society, J. Audio Eng. Soc. (Abstracts), vol. ergy of the analytic signal and the zero-order moments47, p. 534 (1999 June), preprint 4972. of the Wigner distribution (square of signal envelope

[37] B. Widrow and J. M. E. Hoff, "Adaptive and energy spectral density), is given by [38]Switching Circuits," in IRE WESCON Convention Rec. y, x y_(1960), pp. 96-104. A[0] - _ (7)

[38] V. C. Georgopoulos, "A Time-Frequency Rep-

resentational Basis for Perceptual Systems with Applica- where Ys is the energy spectral density vector of thetions to Neural Networks," Ph.D. dissertation, Tufts analytic signal and y, is the envelope squared vector.University, Medford, MA (1992). Other initializations [38] or the Wigner-Ville distribu-

tion itself can be used.APPENDIX 2) A[k], the estimate of A on the kth iteration, is then

used to generate four error vectors,The Wigner distribution W(t, f) of the analytic signal

z(t) is [2] es[k] = Ys - A[k]xl (8)

W(t,f) = z t + z* t - e -j2*tf_d,. ee[k] = Ye - AT[k]xl (9)--oo

(5) es[k] = Ys - A[k]x2 (10)

W(t,f) is usually interpreted as the signal's energy den- el[k] = yf - AX[k]x3 (11)sity in time t and frequencyf. The double integral overtime and frequency of the Wigner distribution (that is, where yf is the instantaneous frequency vector multipliedits volume) is the total energy E of the analytic signal, by the envelope squared vector element by element, yg isAlthough E > 0, W(t, f) can assume positive, zero, or the group-delay vector multiplied by the energy spectralnegative values. Here the goal is to generate a time- density vector element by element, and x_, x2, x3 arefrequency distribution that is nonnegative but still yields appropriate weighting vectors used to produce thethe correct total energy and zero- and first-order mo- moments.ments of W(t,f) (envelope squared, energy spectral den- 3) Based on the four error vectors, an error matrixsity, group delay, and instantaneous frequency), is defined as

1_1_12 __ x3 TXl T

+ eg[k] ]Pc2[[' +[k]

A.1 Iterative Algorithm e[k] = es[k] x'T + i[x_ll2e; [k]The iterative algorithm used here is based on a modi-fied versionof the well-knownleast mean squared (12)

1052 d. Audio Eng. Soc., Vol. 47, No. 12, 1999 December

PAPERS WIGNERDISTRIBUTION

and the extended LMS update rule becomes date rule is modified to become

ao[k + 1] = au[k ] + 2ce_[k]. (13) aij[k + 1] = S[aii[k] + 2c_/j[k]] (14)with S(u) = u for u _> 0 and S(u) = 0 otherwise. When

4) An additional step for this particular modification viewed as part of a neural network, the neurons contain aof the extended LMS algorithm is to pass each update threshold-type nonlinearity with the threshold set at zero.through a threshold-type nonlinearity (with the threshold 5) Renormalize the matrix A[k] to ensure that the totalset at zero). This is necessary to ensure that the elements energy of the distribution is exactly E.of matrix A[k] are nonnegative. Therefore the usual up- 6) The algorithm then repeats steps 2-6.

THE AUTHORS

!w

D. Preis V.C. Georgopoulos

Douglas Preis was born in Chicago, IL. He received received the award in 1998.the B.S.E.E. and M.S.E.E degrees from the University ·of Santa Clara, CA, where he was teaching assistant,and the Ph.D. from Utah State University, Logan, as a Voula Georgopoulos was born in Lowell, MA. SheNASA predoctoral fellow. During the following 8 years holds a Diploma in electrical engineering from Democri-he was postdoctoral research fellow in applied physics tus University of Thrace, Greece (1984), an SM in EECSon the faculty of the Division of Engineering and Ap- (1987) and an Engineer's Degree from the Massachu-plied Physics at Harvard University, Cambridge, MA. setts Institute of Technology (1988), and a Ph.D. inIn 1978 he joined the faculty of Tufts University, electrical engineering from Tufts University (1992). She

Medford, MA, in the College of Engineering as assistant received a Vinton Hayes Fellowship in communicationsprofessor of electrical engineering and rose to the rank during the academic year 1984-1985 at MIT and be-of professor in 1989, his current position in the Depart- tween 1985-1988 was a research and teaching assistantment of Electrical Engineering and Computer Science. at MIT. In May 1988 she joined the MITRE Corpora-

In 1982 he was Mellon Grant recipient at Tufts Uni- tion, Bedford, MA, as a member of technical staff,versity. In 1984 he was Deutsche Forschungsgem- where she worked in the Optical Communications andeinschaft Professor at Ruhr Universitfit, Bochum, Ger- Photonics Specialty Group. Between April 1994 andmany. In 1985 he was Fulbright Professor at Technische December 1995, she worked as a research scientist inUniversi_t Wien, Vienna, Austria. In 1991 he was Vis- the Electrical Engineering Department of the Universityiting Scholar in Applied Physics, at Harvard University. of Patras in Greece. In January 1996 she joined the

Professor Preis has published or presented more than faculty of the School of Electrical Engineering and Com-150 original technical papers and reports. He won the puter Science of Ohio University, as an assistant profes-Audio Engineering Society's Publication Award in 1978 sor. As of September 1998 she is professor of informa-and in 1985. He served for 5 years as associate editor of tics and associate department head of the Department ofthe IEEE Transaction on Acoustics, Speech and Signal Speech Therapy at the Technological Educational Insti-Processing, and has been amember of the Review Board tute of Patras in Greece. Her current research interestsof the Journal of the Audio Engineering Society for more are audio and speech signal analysis, perceptual model-than 20 years, ing, multimedia speech therapy systems, neural net-He is an eminent engineer of Tau Beta Pi, a national works, and decision-making systems.

engineering honor society, a member of both Eta Kappa Dr. Georgopoulos has published more than 40 publi-Nu, a national electrical engineering society, and Sigma cations in journals and conference proceedings in addi-Xi, a national research society. In 1997 he was elected tion to 10 technical reports. She is a member of Taufellow of the AES for his research contributions and Beta Pi and Sigma Xi.

J. AudioEng.Soc.,VoL47,No.12,1999December 1053