Analysis and classification of time-varying signals with multiple time-frequency structures

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Analysis and classification of time-varying signals with multiple time-frequency structures
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  92 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 3, MARCH 2002 Analysis and Classification of Time-Varying SignalsWith Multiple Time–Frequency Structures Antonia Papandreou-Suppappola  , Member, IEEE,  and Seth B. Suppappola  Abstract— We propose a time–frequency (TF) technique de-signed to match signals with multiple and different characteristicsfor successful analysis and classification. The method uses amodified matching pursuit signal decomposition incorporatingsignal-matched dictionaries. For analysis, it uses a combination of TF representations chosen adaptively to provide a concentratedrepresentation for each selected signal component. Thus, itexhibits maximum concentration while reducing cross terms forthe difficult analysis case of multicomponent signals of dissim-ilar linear and nonlinear TF structures. For classification, thistechnique may provide the instantaneous frequency of signalcomponents as well as estimates of their relevant parameters.  Index Terms— Classification, matching pursuit, time–frequency. I. I NTRODUCTION Q UADRATIC time–frequency representations (QTFRs)are well-suited for analyzing signals with time-varyingstructures [1], [2] including signals in radar, sonar, wire- less communications, and biomedical signal processing. MostQTFRs are ideally matched to one or two specific structuresbased on the properties they satisfy. For example, Cohen’sclass of time-frequency (TF) shift covariant QTFRs [1], [2] are matched to signals with linear TF characteristics, whereashyperbolic or power QTFRs [3], [4] are matched to nonlinear TF structures. For successful analysis, it is advantageous tomatch a QTFR with the signal’s TF structure. However, TFanalysis is more complicated when distinctively  different  nonlinear structures are present in a multicomponent signal,especially since the presence of QTFR cross terms [2] mayimpede interpretation. Thus, it is advantageous to design anadaptive, concentrated, free of cross terms QTFR to matchsignal components with different instantaneous frequency (IF).In this paper, we propose a  new  TF method that adapts todifferent TF signal structures based on a modified matchingpursuit algorithm [5].II. B ACKGROUND AND  O BJECTIVES Thematchingpursuititerativealgorithmdecomposesasignalinto a linear expansion of element functions selected from a Manuscript received December 26, 2000; revised January 10, 2002. Thiswork was supported in part by the National Science Foundation under GrantNSF-EIA0074663. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Akbar Sayeed.A. Papandreou-Suppappola is with the Department of Electrical En-gineering, Arizona State University, Tempe, AZ 85287 USA ( B. Suppappola is with Acoustic Technologies, Inc., Mesa, AZ 85201USA(e-mail: Item Identifier S 1070-9908(02)04542-X. complete and redundant dictionary using successive approxi-mations of the signal with orthogonal projections on dictionaryelements [5]. A dictionary of Gaussian atoms with all possibleTF shifts and scale changes was used in [5], and a QTFR wasobtained by summing the Wigner distribution (WD) [1] of eachselected element in the expansion. This modified WD is free of cross terms, and preserves signal energy, TF shifts, and scalechanges [5], [6]. When a signal consists of multiple nonlinear components, the QTFR uses many Gaussian atoms to approxi-matetheIFcomponentofeachterm.Also,thealgorithmisoftenincapable of parsimoniously accounting for the time-dependentcharacteristics of nonlinear frequency-modulated (FM) chirps.To analyze linear FM chirps more efficiently and with feweratoms, rotated Gaussian atoms were added to the dictionary in[7], and wave-based dictionaries were used in [8] to process scattering data.We propose to use an adaptive modified matching pursuit(AMMP)algorithmwithdictionaryelementsmatchedtosignalswith constant, linear or nonlinear IF (e.g., sinusoids, and linear,hyperbolic or power FM chirps). Our objective is to parsimo-niously analyze and classify signals with multiple IF structuressuchasdifferentsignaturewhistlesfromagroupofdolphins[9],or biomedical signals measured simultaneously. The advantageof using dictionaries similar in structure to the analysis signalsisthat the algorithm yields fast andparsimonious results as onlya small number of elements are used in the decomposition. Ateach iteration, we adaptively choose a dictionary element andclassify it based on its IF. If necessary for analysis, we com-pute its matched QTFR using a linear combination of QTFRsmatched to each selected element. Such an algorithm not onlydesigns a QTFR adapted to multiple signal structures, but canalso be used in the classification and detection of signals withspecific nonlinear IF.III. M ULTIPLE  TF S TRUCTURE  A NALYSIS Although the AMMP approach uses the basic concepts of thealgorithm in [5], we form its dictionary using a large class of different nonlinear FM chirps(1)where is a reference time point. The signal has a mono-tonic phase function , an FM rate , and an IF. For example, it simplifies to a sinusoid whenin (1) , a linear FM chirp when[where provides the sign of ], a hyperbolic FM chirpwhen , a power FM chirp when ,and an exponential FM chirp when . The dictionary 1070-9908/02$17.00 © 2002 IEEE  94 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 3, MARCH 2002 Fig. 2. Analysis of real data whistles from a long-finned pilot whale: (a) spectrogram, (b) sum of the IFs of selected AMMP waveforms, and (c) an overlay of thefirst two plots. After AMMP iterations, the signal decompositionis obtained. As the dictio-nary is complete [11], any signal can be represented like this if [5]. In actuality, when the signal components matchthe TF structure of the dictionary elements, the algorithm con-vergesquickly.Amaximumnumberofiterations,andanaccept-ablesmallresidueenergycomparedtothedataenergyisusedasstoppingcriteriatothealgorithm[5].TheQTFR,, at the th iteration is the weighted sum of thematched GWD of each selected dictionary element(3)The algorithm’s success depends highly on the choice of dic-tionary elements. Thus, pre-processing the data may be nec-essary to avoid poor performance due to a mismatch betweenanalysis data and dictionary elements. Although this new tech-nique is a modified matching pursuit [5], once the dictionarymatches the data, it is very powerful not only for analysis butalso for classification. An important AMMP property is its co-variance to certain signal changes. Consider the decomposedsignal with , . If the FM rate of a nonlinear chirp is shifted by a constantamount ,thenitsAMMPissimply .Theexpansioncoefficientsare not affected by this signal change. The parameter vectorchanges to indicating that the time shiftsand the scale changes remain the same, whereas the dic-tionary elements undergo a constant shift in their FM rate fromto . If is a power or a logarithmicfunction,wecanshowthatthecorrespondingAMMPisalsoco-variant to scale changes [11].The ARMUS in (3) satisfies various properties desirable inmany applications. By simply combining the GWDs of each se-lected dictionary element, no cross terms are introduced. Also,the QTFR preserves the underlying TF structure of each signalcomponent, and it provides a highly concentrated representa-tion of each component as it does not apply any smoothing.Specifically, the GWD with the parameter of a nonlinearFM chirp with the phase results in[10] with . If a par-ticular application uses signal components with only one typeof TF structure, the ARMUS satisfies other desirable propertiessuch as the preservation of signal energy, and changes in thesignal’s FM rate [11].Another major advantage of the AMMP technique is that itcan be used for the classification of different nonlinear FM sig-nals as well as for adaptively estimating their parameters. Assuch, the QTFR computation algorithm step in (3) is not re-quired. At the th iteration, the signal componentis chosen with IF and FM rate . Note that the matchingpursuit in [5] cannot directly provide this information withoutfurther processing of the large number of Gaussian atoms it re-sults in.  A. Analysis and Classification Application Examples We demonstrate the performance of our new TF signal pro-cessing method using both synthetic and real data. Synthetic Data:  We form a signal consisting of 13 win-dowed components: four hyperbolic FM chirps, five linearFM chirps, and four Gaussian waveforms, all with varyingparameters. Their “ideal” TF representation in Fig. 1(a) is thesum of the FM chirp IFs (hyperbolae and lines) and the TFcurve of the Gaussians (vertical and horizontal lines). The WDin Fig. 1(b) suffers from cross terms whereas the spectrogramin Fig. 1(c) suffers from a loss of resolution. This complicatesthe classification and parameter estimation of each component.To obtain the AMMP dictionary, we used hyperbolic and linearFM chirps, and Gaussian functions. We could have used moreelement types without affecting the results but the computa-tional time would increase. The decomposition approximatesthe data very well after only 20 iterations. The ARMUS inFig. 1(d) provides a highly concentrated representation for all13componentswithoutoutercrosstermsorlossofresolutionasit adaptively computes the Altes Q-distribution [2] for selectedhyperbolic elements, and the WD for selected linear FM andGaussian elements. The mild spreading of the components andsome inner interference follow from the windowing of the datafor processing. Signal parameters such as FM rates are success-fully estimated from the algorithm. For further comparison, thematching pursuit with only Gaussian dictionary elements wasused to decompose the signal in 300 iterations, and analyze itusing the modified WD [5] in Fig. 1(e). This representation ismatched to the Gaussian signals, but it is not concentrated for  PAPANDREOU-SUPPAPPOLA AND SUPPAPPOLA: ANALYSIS AND CLASSIFICATION OF TIME-VARYING SIGNALS 95 the nonlinear terms. Thus, it cannot easily identify their IF eventhough no cross terms are present. Also, it does not provide FMrate estimates of the signal components for classification.  Real Data:  We use the AMMP algorithm to obtain a closedform IF estimate of real data for classification. The data con-sists of noisy whistles (obtained from [13]) from a long-finnedpilot whale. In Fig. 2(a), the spectrogram shows three whis-tles with dispersive TF characteristics but it does not providethe exact IF or FM rate of each component. The ARMUS inFig.2(b)ishighlyconcentratedalonghyperbolicTFcurves,andFig.2(c)isanoverlayofthepreviousplotsforafaircomparison.Based on the spectrogram analysis, we set the iteration limit tothree. However, the algorithm did not extract the third compo-nentsincei) itislowinamplitude,ii) thehigherfrequencycom-ponent is not exactly hyperbolic, so the algorithm keeps tryingto remove that component first, and iii) the signal-to-noise ratiois poor, as evidenced by the low contrast in the spectrogram. Asdesired, the AMMP provided a  closed form estimate  of the IFand FM rate of the two strongest whistles.IV. C ONCLUSION We have developed a QTFR based on the matching pursuitalgorithm to analyze time-varying signals with multiple com-ponents, each of which has a different TF structure (possiblydispersive). We have demonstrated that this new QTFR tackleswellthisdifficultprobleminTFsignalprocessingwithoutintro-ducing cross terms, or altering the underlying structure of eachsignal component, or suffering from a loss of resolution due tosmoothing. In addition, the method provides good estimates of the signal IF and FM rate provided the dictionary is carefullyformed by preprocessing the data.A CKNOWLEDGMENT The authors would like to thank Prof. G. F. Boudreaux-Bar-tels for valuable discussions on this topic.R EFERENCES[1] L. Cohen,  Time-Frequency Analysis . Englewood Cliffs, NJ: Prentice-Hall, 1995.[2] P. Flandrin,  Time-Frequency/Time-Scale Analysis . New York: Aca-demic, 1999.[3] A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, “The hyper-bolic class of quadratic TF representations, Part I,”  IEEE Trans. SignalProcessing , vol. 41, pp. 3425–3444, Dec. 1993.[4] A.Papandreou-Suppappola,“Generalizedtime-shiftcovariantquadraticTF representations with arbitrary group delays,” in  Proc. 29th Asilomar Conf. Sig., Syst., Comp. , Oct. 1995, pp. 553–557.[5] S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dic-tionaries,”  IEEE Trans. Signal Processing , vol. 41, pp. 3397–3415, Dec.1993.[6] S. Qian and D. Chen, “Decomposition of the Wigner distribution andtime-frequency distribution series,”  IEEE Trans. Signal Processing , vol.42, pp. 2836–2842, Oct. 1994.[7] A. Bultan, “A four-parameter atomic decomposition of chirplets,”  IEEE Trans. Signal Processing , vol. 47, pp. 731–745, Mar. 1999.[8] M. R. McClure and L. Carin, “Matching pursuitswith a wave-based dic-tionary,”  IEEE Trans. Signal Processing , vol. 45, pp. 2912–2927, Dec.1997.[9] W.W.L.Au, TheSonarofDolphins . NewYork:Springer-Verlag,1993.[10] A. Papandreou-Suppappola, F. Hlawatsch, and G. F. Boudreaux-Bar-tels, “Quadratic time-frequency representations with scale covarianceand generalized time-shift covariance,”  Digital Signal Process.: Rev. J. ,vol. 8, pp. 3–48, Jan. 1998.[11] A. Papandreou-Suppappola and S. Suppappola, “Adaptive TF represen-tations for multiple structures,” in  Proc. 10th IEEE Workshop Statist.Signal Array Processing , Aug. 2000, pp. 579–583.[12] R. G. Baraniuk and D. L. Jones, “Unitary equivalence: A new twiston signal processing,”  IEEE Trans. Signal Processing , vol. 43, pp.2269–2282, Oct. 1995.[13] W.A.Watkins,K.Fristrup,M.A.Daher,andT.Howald,“SOUNDdata-base of marine animal vocalizations structure and operations,” WoodsHole Oceanographic Inst., Woods Hole, MA, Tech. Rep. WHOI-92-31,Aug. 1992.
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