Generalized Instantaneous Parameters And Window Matching In ...

1264 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 5, MAY 1997

Generalized Instantaneous Parameters and WindowMatching in the Time-Frequency Plane

Graeme Jones,Member, IEEE, and Boualem Boashash,Senior Member, IEEE

Abstract—In this paper, the concept of instantaneous param-eters, which has previously been associated exclusively with 1-Dmeasures like the instantaneous frequency and the group delay,are extended to the 2-D time-frequency plane. Such general-ized instantaneous parameters are associated with the short-time Fourier transform. They may also be interpreted as localmoments of certain time-frequency distributions. It is shown thatthese measures enable local signal behavior to be characterized inthe time-frequency plane for nonstationary deterministic signals.The usefulness of the generalized instantaneous parameters isdemonstrated in their application to optimal selection of windowsfor spectrograms. This is achieved through window matchingin the time-frequency plane. An algorithm is provided thatillustrates the performance of this window matching. Resultsbased on simulated and real data are presented.

Index Terms—Instantaneous parameters, short-time Fouriertransform, time-frequency analysis, window matching.

I. INTRODUCTION

GLOBAL signal power representations in time or fre-quency (the instantaneous power and energy density

spectrum) provide the analyst with only a limited amount ofsignal information and cannot adequately characterize the be-havior of nonstationary deterministic signals. For example, theenergy density spectrum of a linear FM signal simply revealsa broadband spectral character and provides no information asto the direction (increasing or decreasing) of the modulation.This information is contained in the signal’s phase (in timeor in frequency).

By utilizing both the amplitude and phase of a signal in thetime or frequency domains, all is revealed about the signalof interest. The signal information is, however, presented inan inconvenient and confusing way since phase is a difficultquantity to interpret. Since a nonstationary signal may beconsidered as a signal with a spectrum which varies withrespect to time, this has lead to an alternate signal represen-tation—a time-frequency distribution (TFD). A TFD attemptsto represent the amplitude and phase of a signal together inthe 2-D time-frequency plane—it displays the evolution ofa time signal with respect to frequency (the spectral phase

Manuscript received February 10, 1993; revised October 14, 1996. Thiswork was supported by the Australian Research Council and the DefenceScience and Technology Organization. This work was completed while bothauthors were with Signal Processing Research Centre, Queensland Universityof Technology. The associate editor coordinating the review of this paper andapproving it for publication was Dr. Jelena Kovacevi´c.

G. Jones is with Raytheon Canada Limited, Waterloo, Ont., Canada N2J4K6.

B. Boashash is with the Signal Processing Research Centre, QueenslandUniversity of Technology, Brisbane Q 4001, Australia.

Publisher Item Identifier S 1053-587X(97)03332-1.

function) and, conversely, the evolution of the frequencysignal with respect to time (the time phase function). In otherwords, signal or spectral phase information is combined withamplitude information in the time-frequency plane to create arepresentation displaying all signal information.

Unfortunately, the use of a TFD to represent the non-stationary behavior of deterministic signals leads to severalinterpretive problems. For cases where the signal phase char-acteristic is not monotonic (in timeand frequency), the TFD iscomplicated and confusing, indicating the presence of multiplesignals. Such nonlinear behavior may also be described interms of the so-called cross-terms, which manifest themselvesin TFD’s through interaction of signals in some region of thetime-frequency plane with those in others.

Although there are many types of TFD’s, we shall hereconsider those of Cohen’s class—they fall into (or between)two classes: (energy) density distributions and energy distri-butions. TFD’s that are of the density type are expressible inthe form [21]

(1)

for , where is the Wigner–Ville distribu-tion (WVD, a member of the class under the limit )and is expressed as

(2)

The other limiting member of this class occurs whenand is known as the Rihaczek distribution (RD) [32]. All otherdistributions of this class have thus been generated by allowingthe parameter to vary. It is not necessary to consider anyother values for due to the cyclic nature of in (1).

Density distributions are so named because they satisfy thefollowing properties:

(3)

(4)

(5)

(6)

1053–587X/97$10.00 1997 IEEE

JONES AND BOASHASH: GENERALIZED INSTANTANEOUS PARAMETERS AND WINDOW MATCHING 1265

where is the TFD of , which is any valid1-D signal. These results demonstrate, respectively, that ap-propriate integration of the functions will yield the totalenergy, the instantaneous power, energy density spectrum, orlocal energy in some region [specified by the time-frequencysmoothing function in (6)]. Viewed in such a way,it is seen that these distributions satisfy necessary criteriato be called densities. Additionally, they may even be real(the WVD) and are invertible uniquely to within a phaseconstant [12].

The result of (6) actually specifies the other type of TFDavailable—the (local) energy distribution. It is often the case intime-frequency analysis that one signal is designated a windowand applied to determine the shape and form of the localregion of energy distribution of the other signal. The energydistribution is in actual fact the well known spectrogram,which will be defined here as

(7)

The spectrogram has been used for many years in speech,underwater acoustic, and time-varying spectrum estimationproblems. Many TFD’s bear a dichotomous relation to thesetwo limiting TFD types and actually fall in between thetwo. The Born–Jordan–Cohen distribution (BJC) [6], [12], forexample, satisfies (3)–(5) but not (6). Similar comments maybe made as regards other common distributions—the reducedinterference distribution (RID) [9] and the Zhao–Atlas–Marksdistribution (ZAM) [35] [which only satisfies (4) and (5)].

A. Density Distributions versus Energy Distributions

For representational purposes, an energy distribution hasmany advantages over a density distribution. This is especiallytrue when it is realized that density TFD’s are complex (theWVD may be thought of as a complex function with thephase restricted to 0 or ). A positive energy distributionwould show regions where there was significant local signalpresence and would be useful for time-frequency filtering,pattern recognition, and template generation. Since an energydistribution is a magnitude squared quantity with an arbitraryanalysis window, unique distribution inversion is not possible,and a density function should always be retained if furtherpostprocessing is required. Nevertheless, as a representationaland analysis tool, energy distributions are extremely useful,and their positivity satisfies our intuitive expectations of theform of such a function [14].

Since energy distributions (that is, spectrograms) are de-pendent on the window function, it will be fruitful to lookat ways to generate improved spectrograms that are optimalin some sense. With this and other goals in mind, the paperwill describe the general theory of instantaneous parametersin time-frequency and the notion of window matching forTFD’s. An example application generates an adaptive energydistribution with certain optimal properties. The first relevantconcept to be investigated is that of instantaneous parameters.

II. I NSTANTANEOUS PARAMETERS AND THEIR

APPLICATION TO NON-STATIONARY SIGNAL ANALYSIS

There are many parameters that are inevitably used todescribe the global behavior/nature of a signal. For example,practical signals are often described in terms of their band-width and duration. Such common measures use the algebraicmoments (and cumulants) of the time and frequency powersignal representations (the instantaneous power, energy densityspectrum). The bandwidth, for example, is the (algebraic) vari-ance (second-order cumulant) of the energy density spectrum.

Such global signal parameters are most convenient fordescribing a stationary signal in time and frequency butprovide inadequate characterizations of nonstationary signalssince they cannot reflect the local behavior. To describe thisnonstationary or instantaneous behavior, a set of parame-ters may be defined that are perceived to be the instan-taneous equivalents of such global moment and cumulantmeasures—the instantaneous parameters (IP’s).

The IP’s may be derived through the general forms of the(Fourier) power theorem, which are written as [31]

(8)

(9)

By re-expressing the above equations in the form

(10)

and

(11)

the technique of “extracting” instantaneous parameters fromglobal moment calculations may be illustrated. The instan-taneous contributions to the moment calculations, that is, theIP’s, are simply those quantities contained in the square braceson the right-hand sides of (10) and (11), which correspond tothe time and frequency moment operators on the left handsides. This form ensures that the instantaneous frequencymoments are a function of time and vice versa and that equalityholds under the integrals. The best (and most successful)example of the application of these IP’s are the instantaneous


frequency (the instantaneous first frequency moment withrespect to time ) and the group delay (the instantaneousfirst time moment with respect to frequency, [4], [5],[7], [29].

These IP’s are not the only ones derivable. By rotatinga signal, it is possible to analyze it at different orienta-tions—between the time and frequency domains. A time-frequency rotation operator is expressible in the form [30]

(12)

A. Instantaneous Parameters andTime-Frequency Distributions

One of the significant features of TFD’s is that IP’s maybe derived from them by determination of conditional mo-ments. This is only universally true of TFD’s that are densityfunctions [see (1)]. In other words, instantaneous frequencymoments are expressible as [6], [12]

(13)

with the instantaneous time equivalents of the form

(14)

The point of interest with these results is that, dependenton the TFD employed, the IP’s calculated will be different(and, thus, not in general equal to the IP’s derived classicallyfrom the generalized Power theorem). Since each of theseTFD’s contain the same information and are invertible (asalready mentioned to within a phase constant), the inherentnonuniqueness of IP’s is highlighted. Each IP value generatedthrough the use of density TFD’s is correct under the globalmoment calculation, that is

(15)

and

(16)

In other words, there are many IP’s that satisfy the globalintegral equalities of (10) and (11). Each of these IP’s essen-tially yield the same information since these kernels of thedensity distributions are simply related by a phase change inthe ambiguity plane [32].

B. Use of Instantaneous Parameters to CharacterizeSignal Nonstationary Behavior

The question remains as to how these IP’s may be usedto describe a nonstationary signal. By utilizing IP’s frommany time-frequency directions, the complete nonstationarybehavior of a signal may be documented. IP’s for a given time-frequency orientation characterize the instantaneous behavioralong that 1-D “slice” in the plane (for example, the IFcharacterizes instantaneous behaviorin time).

The IP’s are not always useful for characterising the nonsta-tionary behavior of a signal since they are only instantaneouswith respect to time, frequency, or some other line throughthe time-frequency plane. It is envisaged that a large numberof them would be needed to enable satisfactory analysis ofan arbitrary signal. What is indeed required are IP’s that areinstantaneous in the plane (that is, with respect to specific time-frequency locations), rather than instantaneous with respectto a line. Such a class [the generalized IP’s (GIP’s)] areintroduced and discussed in the next section.

III. GENERALIZED INSTANTANEOUS

PARAMETERS IN TIME-FREQUENCY

In the time-frequency plane, IP’s are required to representthe signal nonstationarities, which cannot be convenientlyachieved (as has been discussed), with 1-D measures. The needfor parameters that are instantaneous in the plane requires thatthe IP’s are functions of both time and frequency—that is, 2-D. The generation of 2-D GIP’s proceeds in exactly the sameway as for the 1-D IP’s, with the notable difference that now,the general Fourier power theorem must be realized in twodimensions, i.e.,

(17)

where

(18)

It should be noted that in the form given above, variableson both sides of the equation may be interpreted as timeand frequency quantities. Since both sides are related byintegration, arbitrary variables may be assigned since they haveno bearing on the equality relation. Such a form allows thederived GIP’s to be functions of time and frequency, and itwill be shown that this is a valid form.

If we are to derive appropriate definitions of GIP’s throughextension of the 1-D instantaneous quantities’ definition,TFD’s would be used in (17). The chosen types of TFD’swould need to be expressible as a product of complexconjugates (as ). A local energy distribution (i.e.,a spectrogram) is expressible in an appropriate form. We will


here redefine the short-time Fourier transform as

(19)

A 2-D relationship may then be expressed as

(20)

The quantity on the left side of (20) is a scaledSTFT with the time-reversed window (similar to across-WVD) and is expressible as

(21)

Equation (20) allows 2-D GIP’s to be derivedin the same manner as the 1-D IP’s of Section II. It shallbe shown that the relations above are not arbitrary but formappropriate definitions of the GIP’s, suitably extending theclass of 1-D IP’s. One important interpretation issue is therelation of these GIP’s to conditional moments of TFD’s,which is addressed in the next subsection.

A. Generalized Instantaneous Parameters and TheirRelation to Time-Frequency Distributions

It has been seen in an earlier section how 1-D IP’s arederivable through the conditional moments of certain TFD’s[4], [6], [11], [12]. It was also noted that they may differfrom those derived through the power theorem but satisfies theintegral relations of (10) and (11). It will now be shown howGIP’s are also derivable through conditional TFD moments,exhibiting the same properties and behavior as their 1-Dcounterparts. Whereas for the 1-D IP’s the moments wereconditional over the time or frequency axes [see (13) and(14)], for the GIP’s, measures are over local areas in the time-frequency plane—ensuring their representation of localizedsignal characteristics.

To determine the relationship of the GIP’s to TFD’s, wewill initially employ Nuttall’s results [26], which are of aform expressed here as

(22)

This result may simply be generalized for any TFD with aphase product kernel [see (1)] to yield

Utilizing the above relation, the GIP’s may then be ex-pressed in terms of density TFD’s as

(23)

(24)

This result is the 2-D analog to the 1-D IP’s derived in(13) and (14). Similar comments apply to their properties. Forexample, it may be deduced that substitution of the GIP’s of(24) into (20) is valid. If we employ the WVD, which is a realenergy density distribution, all GIP’s will be real.

From (23), it is noted that these GIP’s may be rendered analternate interpretation as the local moments of a density TFDdefined over a region specified by the TFD of the window.Such a result once again illustrates that the GIP’s are a naturalextension of the 1-D results, in fact forming a broader class.For example, if a window of the form

(25)

was applied, the GIP’s generated are given by

(26)

which has simply reduced to the frequency IP’s, which arefunctions of time. This illustrates that the (1-D) IP’s areequivalent to the conditional moments of density TFD’s takenover the time or frequency lines (which are the limiting casesof the window being a delta function or a constant in time).This observation reinforces the notion that the GIP’s are a2-D extension of the 1-D IP’s.

The GIP measures that have thus been derived can essen-tially characterize the instantaneous signal behavior atanypoint in the time-frequency plane and will be shown to be ableto be of more practical use than the 1-D IP’s. An applicationof these GIP’s will now be illustrated through the developmentof the general theory of instantaneous window matching.


IV. GENERATION OF AN ENERGY

DISTRIBUTION THROUGH WINDOW MATCHING

One of the major drawbacks of application of the spec-trogram to time-frequency analysis is its nonuniqueness andarbitrary window. In addition, one choice for the analysiswindow cannot usually provide a high-resolution distributionacross all regions of the time-frequency plane. This is wherethe concept of window matching may be employed.

A matched filter [15] is the classic example of windowmatching. The best response (for detection in white noise) oc-curs when the impulse response of the filter is the time reversedincoming signal conjugate—that is, the filter ismatchedto thesignal. We are faced with a slightly more complicated situationhere, although the idea behind the method is fundamentally thesame. A trivial extension of the matched filter solution is tosimply use the signal as the window for matching; this willproduce an energy distribution of the form [18]

(27)

which is a squared and scaled WVD. Such a technique isgenerally unsuitable for most signals due to the nonstationarynature of their WVD’s. For example, if a signal were com-posed of two linear FM components, application of a windowrepresenting their combined form would be inappropriate formatching to each individual component (over their respectivetime-frequency locations).

By effectively performing global matching, as has been donein (27), the local behavior of the signal is not taken intoaccount and thus not reflected in the choice of the window.The example just discussed illustrates the necessity to matchthe window to the signal at all locations in the time-frequencyplane. If we thus employ a window function that matches thesignal locally in the time-frequency plane, the general form ofsuch an energy distribution would be

(28)

where is the WVD of the location-dependentwindow function, where is the equivalent time-varying filter. The window will depend on the signal, butthe criterion through which such windows may be chosenis as yet unknown—in the next section, the concept ofwindow matching in the time-frequency plane will be furtherdeveloped.

V. CONCEPTS OFWINDOW MATCHING

The desire to employ the most appropriate window toanalyze the local energy at every point in the time-frequencyplane requires the selection of windows that are optimum insome sense. It is thus important to choose windows that aretightly bound in time-frequency to ensure that smoothing by

such functions will cause a minimal amount of spread. Theobvious example of such a window is a Gaussian

(29)

which has a WVD of the form

(30)

(The Gaussian function is completely specified by only twoparameters— and —and, in the development of windowmatching that follows, it will be seen how these parametersvary to generate the local signal and window matching in thetime frequency plane.) The Gaussian has the smallest possiblevariance in any time-frequency direction and satisfies, withequality, the uncertainty principles [16]

(31)

and [27]

(32)

where represents the normalized average operator, andand are the time and frequency (magnitude squared) signalaverages. The generalized uncertainty principle of (31) is wellknown to the radar community [13], [31].

It is these considerations that also serve to illustrate whythe technique of matching a signal to itself could not generallyyield a well-concentrated distribution. Unless the signal (andthus the window) is of an “optimum” form (for example,is Gaussian), such a method can be expected to introduceexcess smoothing due to the fact that it will most likely havepoor concentration properties (at least in some parts of theplane). If, however, the signal and window were Gaussian andidentical, the best match is achieved, and the resulting energydistribution is optimally concentrated (in terms of second-order measures). We shall now examine an interesting andnovel approach to the concept of window matching beforedemonstrating how the GIP’s may be used to produce awindow-matched spectrogram.

A. Window Matching in Terms of Signal andWindow Moments and Cumulants

In a similar approach as is used for probability density func-tions, any finite energy signal representation may be expressedthrough its associated characteristic function. (Although weare here referring to algebraic and not probabilistic measures,we shall retain the statistical nomenclature). For example,the characteristic function of the WVD of a signal (undernormalization) would be defined as

(33)

The above characteristic function may be expressed in theform

(34)


or

(35)

Since it is known that the global moments of a WVDare equivalent to those of the actual signal and its spectrum[6], [12], the parameters of (34) are the moments of thesignal/spectrum, which are known to be defined as

(36)

(37)

(38)

The above relations may be equivalently expressed in termsof the cumulants of the signal and its spectrum. The cumulants[ see (35)] are related to the moments but are oftenpreferred as descriptors due to their more convenient repre-sentation of characteristics. In essence, theth cumulant givesan indication of the th-order properties by “removing” fromthe th moment the distorting influences of the lower ordermoments. Variance measurements, for example, are cumulantsof the second order. The general moment-cumulant relationsmay be expressed in forms given in [24].

It is through using such measures that window matchingmay be performed. If a Gaussian window is used, it is entirelydetermined in the time-frequency plane by its second-ordercumulants and no others (indeed, all higher order cumulantsare zero for a Gaussian [24]). Thus, when applying a Gaussianwindow, which has the minimum (second-order) spread intime-frequency, the matching could be based on GIP’s—inthis case, the instantaneous second-order cumulant measures.

Such a window matching approach may now be devisedand theorized. The idea is that the window should matchsome local set of properties associated with the signal in thetime-frequency plane. This approach will thus require that theinstantaneous moments or cumulants of the signal-windowspectrogram at a given time-frequency point be matchedto those instantaneous parameters associated with the thewindow itself (that is, the self-windowed spectrogram). Toformulate this, one needs an “instantaneous characteristicfunction” through which the instantaneous parameters maybe expressed. It should be recalled from earlier sections thatfor both 1-D IP’s and the GIP’s, the actual measures varydepending on the TFD from which they are derived. We shallemploy the WVD-derived measures, which are real and havespecial physical significance (as will be seen later). From (22),the instantaneous characteristic function through which the

(WVD-derived) GIP’s are calculated is

(39)

Using earlier results, the instantaneous characteristic func-tion may be written as

By thinking about window matching in such ways, theproblem will simply reduce to the requirement that the ap-propriate (real, or WVD-derived) GIP’s of the signal-windowspectrogram be matched to those of the window-windowspectrogram. This is equivalent to locally approximating thesignal in time-frequency by the window. The window used willbe Gaussian, which implies that the relevant GIP’s to matchwould be measures of the second order.

The matching task, however, is not altogether straightfor-ward. One possible algorithm will now be discussed, followedby simulations that illustrate the worth of the method.

VI. A N ALGORITHM TO GENERATE AN

ADAPTIVE ENERGY DISTRIBUTION (AED)

It will now be examined as to how the GIP’s can beemployed to generate a window-matched spectrogram, whichwill be referred to as the adaptive energy distribution (AED).

The problem, as has been stated, requires that the relevantinstantaneous moments or cumulants of the signal-windowspectrogram be matched with those of the window-windowspectrogram. For a Gaussian window, only matching of theinstantaneous second-order cumulants is required since theycompletely characterize the Gaussian. Thus, the GIP’s requiredto be matched are the instantaneous time varianceITV

, the instantaneous frequency varianceIFV , andthe instantaneous time-frequency covarianceITFReferring to the expression of (23) and (24), where thechosen distribution is the (real) WVD , the variancemeasures may be calculated from the instantaneous momentsand expressed in the following succinct forms, i.e.,

ITV (40)

IFV (41)

ITF

(42)


where indicates the real part of the expression. The equationsto be solved for the matching then become

ITV ITV (43)

IFV IFV (44)

ITF ITF (45)

Satisfaction of these equations ensures that the final solutionhas locally approximated (in a second-order sense) the signalby the window. Since, in the Gaussian window situation, twoparameters [ and ; see (29)] are required to be solved,the above set of three equations is overdetermined. They are,however, satisfied with equality if the signal is itself Gaussian,with the trivial result that the window parameters becomeidentical to those of the Gaussian signal.

In general, since both sides of the equations depend on thewindow parameters, determination of a general solution wouldnecessarily employ an iterative method. The technique requiresselection of a Gaussian window (through the parametersand ) such that the resulting instantaneous second-ordercumulants approach those that would be generated if the signalwas the same as the window (for the specific time-frequencypoint under analysis).

For signals that are not simple Gaussians, interpretive prob-lems may exist with application of the described method. It ispossible that the calculated values for the instantaneous timeand frequency variances may become zero or even negative.This would seem to render the measurements meaninglesssince it does not appear possible that a negative variance couldever occur. This conceptual stumbling block may be resolved,however. The theorem given below expresses a relationshipbetween the ITV and IFV that will be shown to allow asensible interpretation of these quantities:

Theorem 1: The second-order (real) GIP’s, the ITV, and theIFV, for an arbitrary signal and any Gaussian window ofthe form satisfy

IFV ITF (46)

The proof of this theorem is given in the Appendix.Lemma 1: The above theorem may be generalized such that

(46) is true for any Gaussian window of the formby defining the measures IFV and ITV over a

new coordinate basis

(47)

where

(48)

and

(49)

The theorem above is valid for a Gaussian window thatpossesses rotational symmetry in the time-frequency plane(that is, any TFD appears circular on the scale). Toshow that it may be generalized for any Gaussian, it is simply

a matter of shifting and rotating the time-frequency axes suchthat this Gaussian window has rotational symmetry (whichensures the new time and frequency axes have the samescale). By measuring the spread of the Gaussian in time andin frequency, as well as the time-frequency covariance, astraightforward plane rotation and then axis scaling achievesthe desired affect (that is, ).

The special results given above are true even if one of theITV or IFV is negative. It may also be observed that it isof the form of the addition uncertainty principle of (32)—itthus may be regarded as a type of local uncertainty principle,which is independent of the signal.

This result allows effective use to be made of the GIP’s. Itspecifically illustrates that although one of the ITV or IFV maybe negative, they are never both so. As a result, no problemswill occur with application of the GIP’s. A negative valuefor one of the instantaneous variance measures means thatthe complementary variance is very large and positive. Thisimplies a bad signal-window match and, as will be shown,prompts the appropriate change in the window.

The iterative window matching algorithm may now beformulated. It was noted that the two-parameter problemrequired the solution of an overdetermined set of equations.This is circumvented by using only two of the second-ordermeasures to derive the parameters for each iteration. Thus,the equation involving the ITF is used, and either of the IFVor ITV yield a positive (or the largest) value (they can neverboth be negative). As a result, after measuring the IP’s for the

th iteration, the Gaussian parameters for the next iterationare directly calculated, assuming that the Gaussian

window is matched to itself. This leaves the iterative equations,and the algorithm, in a very simple form. The algorithm maybe succinctly stated:

1) Form the spectrogram [see (28)] at the desired point inthe time-frequency plane (with noa priori information,initial conditions are , and

2) Determine the GIPS—ITV, IFV, and ITF [see(40)–(42)].

3) At the th iteration, update the window parametersand through

ITF (50)

IFV (51)

or, if ITV IFV , then replace (51) by

ITV (52)

4) Repeat the above procedure to the desired level ofmatching.

This iterative method continually reduces the error betweenthe window and the best fit matched condition. If, for example,at one stage the ITV was very large (and the IFV smallor even negative), this would indicate that the window has toogreat a time spread compared with the signal. Equation (52)would then reduce this spread (since ITV This


Fig. 1. Wigner–ville distribution of simulated signal.

occurs since the Gaussian parameters derived assume that thewindow is matched to itself. As a result, the action of thesignal on the window at thenext iteration further adjusts theestimate and selects an improved window.

Through the use of derivatives [see (40)–(42)], rather thanmoment calculations in the time-frequency plane, the full(discrete) algorithm is computations [20], [21],where is the signal length, and is typically Theiterations are stopped when there is less than a 3-bin-widthchange in window (3 dB) spread—it has been found that fouriterations are generally adequate.

VII. AED A PPLIED TO SYNTHETIC AND REAL DATA

The performance and behavior of the AED will now beillustrated through its application to the analysis of simulatedand real signals. Many TFD’s have recently been formulated[2], [3], [19], [33], [34] that employ adaptivity to improveresolution. The AED is also of such a type, although it ispresented here mainly to demonstrate the applicability of theGIP’s. Further investigation of its performance and propertieswill not be undertaken here.

A. Simulated Test Signal Results

The simulated signal to be analyzed is of the form

where ms, Hz, Hz, ,and

The performance of the AED will now be compared withthe WVD and the spectrogram. The spectrogram employs, asits window function, the window used for the first iterationof the AED algorithm. This window is Gaussian, with anassociated WVD that has circular symmetry over the time-frequency scale with ms, and Hz. TheAED is calculated using the algorithm described earlier—fouriterations of the method are used, which have been shownthrough simulation to produce good results. All plots shownhave been normalized.

The TFD’s for this signal are displayed in Figs. 1–3. TheAED is similar to the spectrogram but has improved resolutionsince the window function is locally matched to the signal.

Fig. 2. Spectrogram of simulated signal.

Fig. 3. Adaptive energy distribution of simulated signal.

The AED is unlike the WVD, which displays large cross-terms midway between the signal autoterms. In this case, theAED seems to provide the most intuitive representation. Theeffect of the window matching allows a sharper picture of thelocal energy in the time-frequency plane than is afforded bythe spectrogram.

B. Results for the Analysis of Humpback Whale Sounds

The sounds of the humpback whales are analyzed in thissection. These sounds are nonstationary deterministic (butsubject to various noise interferences) and are generally non-transient and multicomponent—this makes them well suited toa time-frequency analysis. The humpback whale data has beensupplied by the Defence Science and Technology Organization(Australia). The data has been decimated to yield an effectivesampling rate of 5000 Hz. One of the fundamental sounds inhumpback whale songs will be investigated here—this soundhas been described by biologists as a deep throat growl or amoan [28]. The moans are generally low frequency, relativelybroadband, and extremely loud. Two such sounds were isolatedand placed into separate 64-point data files. They will now beanalyzed by four TFD’s—the WVD (fixed window) spectro-gram, ZAM (with Gaussian smoothing parameter[35]) and the AED.

The results for the first test signal are displayed in Figs. 4–7.The WVD is not easily interpretable due to its positive-negative (density) nature and the large number and size of


Fig. 4. Wigner–Ville distribution of humpback whale sound 1.

Fig. 5. Spectrogram of humpback whale sound 1.

Fig. 6. ZAM distribution of humpback whale sound 1.

the cross-terms that are inherently generated. The spectrogramis well smoothed, but it does provide some indication as to thenature of the moan—it is composed of a series of linear FMpulses, which occur at various times and frequency locations.The ZAM provides excellent resolution for signals alignedwith the time or frequency axes but degrades for signals withlarger FM gradients, as is evidenced in Fig. 6. The signalcomponents appear sharp due to the use of a lateral inhibitionwindow function [35]. It is also of significance to note that thebroadband component at about 5.5 ms is well represented bythe ZAM but subject to some frequency smoothing introducedby the distribution. The AED (Fig. 7) in this case has yielded

Fig. 7. Adaptive energy distribution of humpback whale sound 1.

Fig. 8. Wigner–Ville distribution of humpback whale sound 2.

Fig. 9. Spectrogram of humpback whale sound 2.

what may be considered the best result; it is a better resolvedversion of the spectrogram since no assumptions (as regardswindow selection) have been made on the signal. This has thebenefit that no direction of signal evolution is attenuated orfavored over another.

The second signal is a more complicated moan. Theseresults (Figs. 8–11) vividly illustrate the fundamental limita-tions of nonadaptive distributions. The WVD is cluttered withcross-terms to the point of incomprehensibility, whereas thespectrogram shows many characteristics of the signal but is


Fig. 10. ZAM distribution of humpback whale sound 2.

Fig. 11. Adaptive energy distribution of humpback whale sound 2.

smoothed to the point that much fine detail is lost. The ZAMhas failed to characterize the signal—the linear FM’s withsignificant gradients have been poorly resolved. Once again,the AED has well-represented the signal in time-frequency—itappears to be composed of multiple components, with linearFM behavior, occurring at various regions throughout theplane. There is much detail present that has not been resolvedby the other TFD’s. Most of the modulated signals collapseinto one high-energy sinusoid (at about 9 ms), which may bea characteristic of some resonant oscillation associated withthe humpback whale’s vocal tract. The linear FM signals mayreflect some intrinsic modulating behavior of the animal’s vo-cal system, especially considering they are of similar (positiveor negative) gradient.

This section has provided some results that have demon-strated the successful application of the theory of generalizedinstantaneous parameters and window matching to generationof the adaptive energy distribution.

VIII. C ONCLUSIONS

This paper has attempted to explain how signal repre-sentations of 1-D signals in the time-frequency plane maybe interpreted. All such TFD’s are either energy densitydistributions, (local) energy distributions, or lie somewherebetween those two limiting classes.

Since a TFD is expected to represent nonstationary behaviorin the time-frequency plane, there is an inherent notion ofinstantaneous measures. Such quantities, as functions of timeor frequency, may describe the instantaneous or local behaviorof a signal in the same manner that global moment measuresdescribe the overall signal characteristics. Unfortunately, suchinstantaneous measures, as currently defined, are only instan-taneous with respect to timeor frequency but never both. Anew class of instantaneous parameters—the generalized instan-taneous parameters—have been defined in this paper. Theyare functions of both time and frequency and are associatedwith the short-time Fourier transform in an analogous mannerin that the 1-D parameters are associated with the signalor its Fourier transform. Their particular advantage is thatthey provide measures with localization in the time-frequencyplane.

These generalized instantaneous parameters may be em-ployed to characterize local behavior in the time-frequencyplane. An example application is given in this paper—windowmatching. Generation of a time-frequency energy distribution(a spectrogram) requires use of an analysis window. If thewindow does not match the signal well at the given time-frequency point under analysis, the representation is smearedas a result. By applying the ideas of matched filtering on thefine time-frequency scale, better spectrogram representationsmay be achieved, which are sharper than any using an arbitrarywindow. The generalized instantaneous parameters are usedfor this purpose (with a Gaussian window) to produce anadaptive energy distribution. The simulated and real dataresults provided demonstrate both window matching in thetime-frequency plane as well as the usefulness of the newlydefined parameters.

APPENDIX

The theorem of (46) shall be proved in this Appendix. Theproof requires the decomposition of the arbitrary signal intoa complete series of orthonormal Hermite functions [17] ofthe form

(53)

where is a Hermite polynomial of the form [17]

(54)

The signal could then be expressed as

(55)

where are the resulting coefficients of the orthonormalHermite decomposition. To prove the theorem, the ITV andIFV must be calculated. Let us commence with a Gaussianwindow, which has equal “spread” in time and in frequency

and Introducing a normaliz-ing scalar, this window may be expressed as a zeroth-orderHermite function, that is,

(56)


By first rewriting the STFT defined in (19) in the equivalentform

(57)

an STFT using the window of (56), and the form of in(55), may be expressed as

(58)

where

By making the variable substitution the aboveequation may be expressed as

which can be simplified to yield

By realizing that , the STFT of thewindow and one Hermite polynomial of the orthogonal de-composition of can be written as

where is a Laguerre polynomial [17] such thatAs a result, the signal-window IFV calculated for a

WVD, the expression for which is given in (41), is expressible

as

IFV

Similarly, the ITV [using (40)] becomes

ITV

Combining these quantities, one obtains

IFV ITV

which proves the theorem.


REFERENCES

[1] L. E. Atlas, P. J. Loughlin and J. W. Pitton, “Truly non-stationarytechniques for the analysis and display of voiced speech,” inProc.ICASSP91, Toronto, Ont., Canada, 1991, pp. 433–436.

[2] R. G. Baraniuk and D. L. Jones, “A signal-dependent time-frequencyrepresentation: Optimal kernel design,”IEEE Trans. Signal Processing,vol. 41, pp 1589–1602, 1993.

[3] R. N. Czerwinski and D. L. Jones, “Adaptive cone-kernel time-frequency analysis,”IEEE Trans. Signal Processing, vol. 43, pp1715–1719, 1995.

[4] B. Boashash, “Estimating and interpreting the instantaneous frequencyof a signal: Part 1—Fundamentals,”Proc. IEEE, vol. 80, pp. 520–538,1992.

[5] , “Estimating and interpreting the instantaneous frequency of asignal: Part 2—Algorithms and applications,”Proc. IEEE, vol. 80, pp.440–568, 1992.

[6] , “Time-frequency signal analysis,” inAdvances in SpectrumEstimation, S. Haykin Ed. Englewood Cliffs, NJ: Prentice-Hall, 1990,ch. 9.

[7] B. Boashash, G. Jones, and P. J. O’Shea, “Instantaneous frequencyof signals: Concepts, estimation techniques and applications,” inProc.SPIE, Advanced Algorithms Architectures Signal Processing, San Diego,CA, 1989, vol. 1152, pp. 382–400.

[8] R. N. Bracewell,The Fourier Transform and Its Applications. NewYork: McGraw-Hill, 1986.

[9] H. L. Choi and W. J. Williams, “Improved time-frequency representationof multicomponent signals using exponential kernels,”IEEE Trans.Acoust., Speech, Signal Processing, vol. 37, pp. 862–871, 1989.

[10] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wignerdistribution—A tool for time-frequency analysis (Part III),”Philips J.Res., vol. 35, pp. 217–250, 1980.

[11] L. Cohen and C. Lee, “Instantaneous Bandwidth,” inMethods andApplications of Time-Frequency Signal Analysis, B. Boashash Ed. NewYork; Halstead, 1992, ch. 4

[12] L. Cohen, “Time-frequency distributions,”Proc. IEEE, vol. 77, pp.941–981, 1989.

[13] C. Cook and M. Bernfeld,Radar Signals, an Introduction to Theory andApplication. New York: Academic, 1966.

[14] J. L. Flanagan,Speech Analysis and Perception. New York: SpringerVerlag, 1965.

[15] S. Haykin,Communication Systems. New York: Wiley, 1978.[16] C. Helstrom,The Statistical Theory of Signal Detection. New York:

Oxford Univ. Press, 1968.[17] I. S. Gradshteyn and I. M. Ryzhik,Tables of Integrals, Series and

Products. New York: Academic, 1965.[18] D. L. Jones and T. W. Parks, “A resolution comparison of several time-

frequency distributions,”IEEE Trans. Signal Processing, vol. 40, pp.413–420, Feb. 1992.

[19] D. L. Jones and R. G. Baranuik, “A simple scheme for adapting time-frequency representations,”IEEE Trans. Signal Processing, vol. 42, pp.3530–3532, Dec. 1994.

[20] G. Jones and B. Boashash, “The adaptive spectrogram,” inProc. 1992IEEE Digital Signal Processing Workshop, 1992.

[21] G. Jones, “Instantaneous frequency, time-frequency distributions andthe analysis of multicomponent signals,” Ph.D. dissertation, QueenslandUniv. Technol., Brisbane, Australia, 1992.

[22] G. Jones and B. Boashash, “Instantaneous quantities and uncertaintyconcepts for time-frequency distributions,” inProc. SPIE, AdvancedAlgorithms Architectures Signal Processing, San Diego, CA, 1991, vol.1566, pp. 167–178.

[23] , “Instantaneous frequency, time delay, instantaneous bandwidthand the analysis of multicomponent signals,” inProc. ICASSP, 1990,pp. 2467–2470.

[24] M. Kendall and A. Stuart,The Advanced Theory of Statistics—Volume1. London, U.K.: Griffin, 1977.

[25] P. L. Loughlin, J. W. Pitton, and L. E. Atlas, “New properties to alleviateinterference in time-frequency representations,” inProc. ICASSP91,Toronto, Ont., Canada, 1991, pp. 3205–3208.

[26] A. H. Nuttall, “Wigner distribution function; Relation to short-term spec-tral estimation, smoothing and performance in noise,” Naval UnderseaSyst. Cent. Tech. Rep. 8225, 1988.

[27] T. W. Parks and R. G. Shenoy, “Time-frequency concentrated basisfunctions,” Proc. ICASSP90,1990, pp. 2459–2463.

[28] R. Payne and S. McVay, “Songs of the humpback whale,”Sci., vol.173, pp. 583–597, 1971.

[29] M. A. Poletti, “The application of linearly swept frequency measure-ments,”J. Acoust. Soc. Amer., vol. 84, no. 1, p. 238–252, 1988.

[30] F. Reis, “A linear transformation of the ambiguity function plane,”IRETrans. Inform. Theory, vol. 8, p. 59, 1962.

[31] A. W. Rihaczek,Principles of High Resolution Radar. New York:McGraw-Hill, 1969.

[32] , “Signal energy distribution in time and frequency,”IEEE Trans.Inform. Theory, vol. IT-14, pp. 369–374, 1968.

[33] B. Ristic and B. Boashash, “Kernel design for time-frequency signalanalysis using the Radon transform,”IEEE Trans. Signal Processing,vol. 41, pp 1996–2008, 1993.

[34] E. J. Zalubas and M. G. Amin, “Time-frequency kernel design by thetwo-dimensional frequency transformation method,”IEEE Trans. SignalProcessing, vol. 43, pp 2198–2203, 1995.

[35] Y. Zhao, L. E. Atlas and R. J. Marks II, “The use of cone-shapedkernels for generalized time-frequency representations of nonstationarysignals,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp.1084–1091, 1990.

Graeme Jones(M’91) was born on the Gold Coast,Queensland, Australia, on November 29, 1966. Hereceived the B.E. and Ph.D degrees from the Univer-sity of Queensland and the Queensland Universityof Technology (QUT), Brisbane, in 1987 and 1992,respectively.

He remained at QUT as a lecturer in signal pro-cessing in 1992. From 1992 to 1994, he was a post-doctoral fellow at McMaster University, Hamilton,Ont., Canada, where he received an NSERC In-ternational Fellowship. He then returned to QUT,

remaining for a year as a research fellow and lecturer in the Aerospace andAeronautical Division of the School of Electrical and Electronic Systems Engi-neering. Since September 1995, he has been a Senior Systems Engineer in theAdvanced Systems Department of Raytheon Canada Limited, Waterloo, Ont.His research interests include time-frequency and general higher dimensionalsignal analysis and radar signal processing, detection, and tracking.

Boualem Boashash(SM’89) received the Diplomed’ingenieur–Physique–El´ectronique from the ICPIUniversity of Lyon, France, in 1978, the M.S. de-gree from the Institut National Polytechnique deGrenoble, France, in 1979, and Doctorate (Doc-teur–Ingenieur) from the same university in May1982.

In 1979, he joined the Elf-Aquitaine GeophysicalResearch Centre, Pau, France. In May 1982, hejoined the Institut National des Sciences Appliqueesde Lyon, Lyon, France, where he was a Maitre-

Assistant associe. In January 1984, he joined the Electrical EngineeringDepartment of the University of Queensland, Brisbane, Australia, as alecturer, Senior Lecturer (1986), and Reader (1989). In 1990, he joined BondUniversity, Graduate School of Science and Technology, as a Professor ofElectronics. In 1991, he joined the Queensland University of Technologyas the foundation Professor of Signal Processing and Director of the SignalProcessing Research Centre. His research interests are time-frequency signalanalysis, spectral estimation, signal detection and classification, and higherorder spectra. He is also interested in wider issues such as the effect ofengineering developments on society.

Dr. Boashash was technical chairman ofICASSP 1994. He is the editor oftwo books, has written over 200 technical publications, and has supervised 20Ph.D. students and five Masters students. He is fellow of both IE Australiaand of the IREE.

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