592 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 4, APRIL 1997

Wave-Oriented Signal Processing of DispersiveTime-Domain Scattering Data

Lawrence Carin,Senior Member, IEEE, Leopold B. Felsen,Life Fellow, IEEE, David R. Kralj,H. S. Oh, W. C. Lee, and S. Unnikrishna Pillai,Senior Member, IEEE

Abstract—Phase-space data processing is receiving increasedattention because of its potential for furnishing new discrimi-nants relating to classification and identification of targets andother scattering environments. Primary emphasis has been ontime-frequency processing because of its impact on transient,especially wideband, short-pulse excitations. Here, we investigatethe windowed Fourier transform, the wavelet transform, andmodel-based superresolution algorithms within the context ofa fully quantified and calibrated test problem investigated byus previously: two-dimensional (2-D) short-pulse plane wavescattering by a finite periodic array of perfectly conductingcoplanar flat strips. Because the forward problem has been fullycalibrated and parametrized, some quantitative measures can beassigned with respect to the tradeoffs of these time-frequencyalgorithms, yielding tentative performance assessments of thetested processing algorithms.

Index Terms—Scattering, signal processing, time-domain anal-ysis.

I. INTRODUCTION

I NCREASED attention within the electromagnetics commu-nity has been given recently to phase-space signal pro-

cessing techniques [1]–[11] for dealing with propagation andscattering in complex environments. Explorations have usu-ally been carried out in the time-frequency subdomain ofthe full configuration (space time)—spectrum (wavenumber-frequency) phase space, although some work has also beendone in the space wavenumber phase space [3], [5], [11].Demonstrations of utility have primarily (and understandably)been supported by results that reflect favorably on the partic-ular technique used for a particular problem, but there haveyet to emerge more general criteria as to how the phase spaceshould be systematically parametrized and calibrated. In thispaper, we make an attempt to move in this direction.

Two useful classifiers of propagation and scattering arewhether the event is local or global. The wave objects as-

Manuscript received May 4, 1994; revised March 14, 1996. This workwas supported in part by the Air Force Office of Scientific Research underGrant F49620-93-1-0093, the Army Research Office under Grant DAAH04-93-02-0010, the Office of Naval Research under Grant N66001-94-C-0013, theNational Science Foundation under Grant ECS-9211353, and by the RaytheonCompany.

L. Carin is with the Department of Electrical and Computer Engineering,Duke University, Durham, NC 27708 USA.

L. B. Felsen is with the Department of Aerospace and Mechanical Engi-neering, Boston, MA 02215 USA.

D. R. Kralj is with the Missile System Division, Raytheon Company,Tewksbury, MA 01876 USA.

S. U. Pillai, H. S. Oh, and W. C. Lee are with the Department of ElectricalEngineering, Polytechnic University, Brooklyn, NY 11201 USA.

Publisher Item Identifier S 0018-926X(97)02492-7.

sociated with local and global events in the time domain arewavefronts and space-time resonances, respectively. Havingmade this basic classification, the question arises as to howthe corresponding phase-space footprints are parametrized andhow these footprints can “best” be extracted from propagationand scattering data. Because the phase space is accessed mosteffectively from the data by windowed transforms, the windowshape and size is one of the essential parameters. Becausephase-space distributions are subject to the configuration-spectrum tradeoff imposed by the uncertainty relation [12],the relative emphasis on configuration or spectrum is a secondparameter. Quantitative assessment of the influence of theseparameters requires analytical and numerical experimentation,with the goal of arriving at criteria that may eventuallyserve as standards (benchmarks) for testing wave-oriented dataprocessing algorithms. The analytical-numerical experimentsare best carried out for problems which can be solved underfully controlled conditions.

Our controlled problem has been a finite array of perfectlyconducting coplanar flat strips arranged periodically alongin the plane (Fig. 1), with subsequent generalization toinclude perturbations around this periodic prototype [13]–[15].This test configuration involves several scales (strip width,strip separation, and array size), collective effects due toperiodicity, and edge scattering due to the individual stripedges, as well as the truncation edges of the array. Under short-pulse plane wave excitation, the corresponding time-domainscattered fields can be parametrized phenomenologically eitheras a sum of time-gated primary and multiple scatterings dueto individual strips or as collective scattering from the entire“aperture” of the periodic array. The latter yields a sum ofdispersive sustained wavetrains (time-domain Floquet modes)each of which accounts via its characteristic frequency profilefor the global effects of infinite periodicity, complementedby Floquet-mode modulated edge diffractions due to thetruncations. For the phenomenology-based (wave-oriented)data processing of this fully calibrated scattering model, wehave utilized the short-time Fourier transform, the wavelettransform, and a windowed superresolution algorithm, withoutand with the addition of white Gaussian noise.

For the present discussion, we return to this canonicalproblem to address more critically the phase-space calibrationissues detailed above. Our earlier studies, as those of others,have shown that phase-space processing can work, but thesestudies have not dwelled on the usually painstaking trial anderror that has to be pursued to make them work. A fully

0018–926X/97$10.00 1997 IEEE

CARIN et al.: WAVE-ORIENTED SIGNAL PROCESSING OF TIME-DOMAIN SCATTERING DATA 593

Fig. 1. Finite array of coplanar perfectly conducting infinitesimally thinstrips in free space with strip widthw = 0:64d. In all computations, aTE-polarized pulsed plane wave (pulse shape and spectrum in Fig. 2) isincident normally and the scattered fields are observed at a distance 29.58ddirectly above the left-most edge of the array.

parametrized and calibrated forward scattering problem is thebasis for quantitative parametrization of the wavefront andresonance alternatives that characterize, respectively, the early-and late-time scattered fields at the observer. We show in thepaper how critically the processing window size influences thephase-space footprints, and how conclusions from these trialscould not have been stated with reasonable confidence withoutthe backup of a forward calibrated model.

The remainder of the paper is organized as follows. Theforward model for the target under study is summarizedin Section II, highlighting information that is relevant forthe subsequent phase-space processing. The fixed resolution,multiresolution, and superresolution algorithms used for time-frequency processing are summarized in Section III, withresults presented in Section IV. Conclusions are summarizedin Section V.

II. STATEMENT OF THE PROBLEM

The test problem involves a pulsed plane wave incidentnormally upon a finite periodic array of perfectly conductingflat, infinitesimally thin, coplanar strips (Fig. 1). For the spe-cific problem parameters here, we have taken strips arrangedwith period and width . The fields are observedat a distance of 29.58 directly above the left-most edgeof the array for TE polarization (electric field parallel tothe strip edges). The Rayleigh pulse in Fig. 2 describes theincident pulse shape with time and frequency normalized to

; note that the pulse duration is shorter thanso that it is possible, in principle, to resolve strip edges.Scattered field reference data have been computed using thetime-domain physical-optics approximation for the inducedcurrents; the accuracy of this approximation for the presentproblem conditions has been validated [4]. For the collec-tive phenomenology associated with truncated periodicity,the calibrated geometric theory of diffraction (GTD)-typeasymptotics for the scattered field summarized in Section Iinvolve -indexed truncated time-domain Floquet modes andFloquet-mode modulated diffractions from the array edges;the latter propagate essentially as weakly dispersed widebandwavefronts with short time support [13]–[15]. The lowest order

time-domain Floquet mode (which represents thespecular return) is also nondispersive, and its pulse shaperesembles that of the incident wave. The higher order

Fig. 2. Pulse shape and spectrum of the Rayleigh wavelet used as theexcitation pulse in all computations. Time and frequency are normalized toT = d=c, whered is the array period (see Fig. 1) andc is the speed of lightin vacuum.

modes are dispersive and have time-dependent instantaneousfrequencies [13]

(1)

where is the speed of light in vacuum andis the height ofthe observer above the array.

From (1) it is seen that as the modal resonant fre-quency , the cutoff frequency of the th Floquetmode under conditions of normal incidence; similar behaviorhas been observed in other scattering scenarios [1], [6], [7],[10]. The expression in (1) applies to the infinite periodic arraywhich generates infinitely extended nontruncated plane wavetrains. Nevertheless, as the calibrated forward model shows[13]–[15], the truncations for the finite array do not noticeablyaffect the Floquet-mode frequencies in (1) over those portionsof the array where these modes can be separately resolved(i.e., away from ).

III. PROCESSINGOPTIONS

As stated earlier, we consider three processing options: theshort-time Fourier transform, the wavelet transform, and awindowed superresolution algorithm. The particular superres-olution algorithm involves an eigenvector-based matrix-pencilmethod [16], which is similar in spirit to the matrix-pencilmethod developed by Hua and Sarkar [17], [18], and is oneexample of several parametric techniques [19], [20] that canbe applied to time-frequency processing. These processingoptions are reviewed briefly below and, in Section IV, theyare applied to the problem discussed in Section II for noiselessas well as noisy data. Concerning the choice of the processingwindow, narrow windows emphasize temporally localized,wideband phenomena (wavefronts) while wide windows em-phasize temporally prolonged narrowband phenomena (modesor resonances), with corresponding alternative parametriza-tions of the scattering process. The definition of “wide” or

594 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 4, APRIL 1997

“narrow” is tied to characteristic scales of the target. Forour prototype strip array, there are two relevant characteristicscales: the strip width and the array period; the third scale,the overall size of the array, is not relevant for the near-zone observations in this problem. To isolate edge diffractions,the window size must be small relative to the travel time

between the edges of each strip. To resolve individualstrips but not individual edges, the window size should liebetween and , where is the array period. Toaccommodate near-zone collective effects as expressed by thetime-domain Floquet modes, we have previously [4] foundthat the window size should be large enough to include thescattered fields (at the observer) from at least three strips; thislast condition is consistent with the minimum number thatplaces the center strip in a locally periodic environment, andit suffices (though possibly with poor resolution) to exhibitthe Floquet-mode footprints of an infinite periodic array withthat interstrip spacing. Tuning the window size to extract aparticular parametrization from data presents difficulties evenfor fully controlled model problems witha priori knowledge.Nevertheless, it is hoped that some insights and guidelinescan emerge from this study for dealing with data derived fromunknown scattering configurations.

A. Short-Time Fourier Transform

The Gaussian-windowed short-time Fourier transform(STFT) of a function is expressed as

(2)

where is the center position of the sliding Gaussian windowwith standard deviation. The well-parametrized phenomenol-ogy pertaining to the diffraction problem in Section II allowsus to investigate the accuracy of the STFT footprints through-out the phase space, for early and late times, varying windowsize, and additive noise.

B. Wavelet Transform

The continuous wavelet transform [21], [22] of a functionis given as

(3)

where is the “scale” and is the “mother wavelet.” Thewavelet transform behaves as a bank of bandpass filters (onefor each scale), and if the filter response is narrow enoughthe scale can be related to frequency, generating a time-frequency distribution. Under such conditions, small scales

correspond to high frequencies (narrow time resolutionand wide frequency resolution) and large scales correspond tolow frequencies (wide time resolution and narrow frequencyresolution). To extract time-dependent spectra, the motherwavelet should have good filtering properties in the spectraldomain; we have utilized the Morlet-type mother wavelet [21]

(4)

For scale , the frequency identifies the centerfrequency of the equivalent bandpass filter [21], [22], and theparameter determines the time-frequency resolution at thatscale (frequency); the center frequencies of wavelet filters for

are . Although (4) is expressed as a two-parametertransform and , in practice it is described by a singleparameter: the number of oscillations desired for the Gaussianmodulation (this can be realized by varyingfor a fixed ,or vice versa).

A demonstration of the wavelet transform has been providedby Kim and Ling [6], [7]. They applied the transform inthe frequency domain for which case it corresponds in thetime domain to a variable-window-size STFT; the windowscaling was chosen so as to satisfy the “wavelet condition”and, at early times, the window size was made small to em-phasize temporally narrow, spectrally wide wavefront returns,while at late times, the window size was made progressivelywider to emphasize temporally wide, spectrally narrow targetresonances. The manner in which their scale-based schemeaccommodates both extremes simultaneously seems to workbest when the early-time and late-time portions of the scatteredfields are “clean”; however, this is frequently not the case,especially with noisy data. The Morlet wavelet transform in(3) and (4), which we have used, is implemented directly inthe time domain, but parametrized in terms of the samplingfrequency , which determines its filtering properties. Wehave found this multiresolution scheme to be robust even inthe presence of moderate noise levels (see Section IV-B).

C. Windowed Superresolution Algorithms

The windowed Fourier transform and the wavelet transformare well suited to extracting wave phenomenology from data,but the achievable time-frequency resolution is subject to theuncertainty relation [12]. The resolution achieved with thesefirst-pass algorithms can be improved by fitting properly cho-sen model-based algorithms locally to the data. We concentratehere on models that relate to oscillatory wave phenomena. Thetime-domain Floquet modes are characterized by dispersivewavetrains with time-dependent frequency [see (1)] and alge-braic damping [13], [14]. To extract the time-varying modalstrength and time-dependent oscillation frequency, we applymodel-based superresolution to locally windowed portions

of the data , using a constant amplitude windowwith finite support

(5)

We have also considered several other types of windowfunctions. An obvious choice, consistent with our STFT, isa Gaussian window; however, it was found that the Gaussian-modulated data could no longer be well represented as a sumof damped sinusoids as assumed in the model and, therefore,the phase-space results were inaccurate. Phase-space resultswere found to be most reliable when the model-based schemesare applied to the unmodulated data [i.e., with a rectangularwindow, as in (5)].

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Fig. 3. Short-time Fourier transform (STFT) of the time-domain fields scattered from the array in Fig. 1; the bottom plot is the time-domain scatteredfield, the left plot is the global Fourier transform of the scattered field, and the center plot is the STFT of the scattered field using a Gaussian windowwith standard deviation� = 1:14T (the real part of the modulated Gaussian window is shown inset for frequencyf = 2:14=T ). The instantaneousFloquet mode frequenciesfm(t) from (1) are shown by the solid curves.

Proceeding with the superresolution processing above forsampled time-domain data , we have

(6)

where is the number of modes contributing inside eachwindow (usually unknown), is the rectangular windowfunction in (5) centered at, the sampling rate is set bythe Nyquist criterion, and is the complexfrequency of the th mode inside theth window (correspond-ing to a modal pole in the complex frequency plane). Manyparametric algorithms [19], [20] can deal with data of the formin (6). We have found the matrix-pencil method developed byHua and Sarkar [17], [18] to be particularly useful for noisydata and have utilized a modified form of this algorithm [16].It should be noted that in addition to being useful for theextraction of modal information, parametric algorithms havealso found utility for the estimation of wavefront arrivals [24],[25].

IV. RESULTS

A. Noise-Free Data

Short-Time Fourier Transform (STFT):We first considerthe scattering scenario discussed in Section II under noise-free conditions. Fig. 3 displays the time-frequency distributionresulting from application of the STFT in (2), for a Gaussianwindow-size , which ensures that scattered fieldsdue to at least three strips reach the observer for nearly allwindow positions. The time-domain scattered field is shownin the bottom plot, the global Fourier transform of this entire

wavefield is shown on the left, and results from the STFTprocessing of the bottom waveform are shown in gray scale inthe center, with the solid curves representing the anticipatedtime-frequency profiles from (1) for the time-domain Floquet modes. The modulated Gaussian window (i.e.,

isshown inset for frequency ; at otherfrequencies the modulation changes while the window sizeremains constant. The initial temporally narrow, spectrallywide response is due to the weakly dispersivespecularly reflected mode and, also, the overlapping left-edge diffraction [13], [14]. Thereafter, one observes sustainedoscillations, generated by successive emergence of the time-domain Floquet modes whose phase-space distribution bandssurround the solid dispersion curves in (1). The truncation ofthese waveforms at the right edge of the array gives rise againto a localized wideband signature.

Although the forward scattering for our test case has beenfully parametrized, the STFT results in Fig. 3 were obtainedonly after several tunings with regard to window size, de-spite thea priori knowledge of the target environment. Todemonstrate the effect of different window sizes on the phase-space results, in Fig. 4 we apply the STFT to the same dataconsidered in Fig. 3, but using a narrower Gaussian windowwith whose modulated form for a particularfrequency is shown inset as in Fig. 3. Comparingthe phase-space results in Figs. 3 and 4, we see, as expected,that the temporally localized wavefront response from theinitial specular return is better resolved by the smaller windowin Fig. 4 than by the larger window in Fig. 3. However, alsoas expected, the corresponding loss in frequency resolutionleads to more widely spread Floquet mode bands than thoseextracted in Fig. 3. We also considered still smaller STFT

596 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 4, APRIL 1997

Fig. 4. Short-time Fourier transform of scattered field (bottom of Fig. 3)using a Gaussian window with standard deviation� = 0:47T (shown insetfor f = 2:14=T ).

window sizes than that in Fig. 4 up to the regime, wherethe Floquet mode dispersion bands begin to overlap to suchan extent that they are no longer distinguishable. Instead,the STFT now tends to isolate individual strip scatteringswith short time support and wide frequency spread. For anunknown scattering environment, where one does not knowthe appropriate scalesa priori, we suggest sampling the datafor wave phenomenology content by sliding a range of fixedwindow sizes across it. If a particular window size appears toextract a recognizable local- or global-phase space signaturefrom the sampled portion of the scattered field, processingalgorithms with better resolution can be applied locally thereto hone in on the suspected wave phenomenology.

Wavelet Transform:In the STFT, for each fixed-windowpass the spectral content at different frequencies is extractedby the corresponding variable modulation of the Gaussian win-dow. On the other hand, the Morlet wavelet transform keepsthe number of oscillations in the modulated Gaussian fixed,but shrinks the scale to extract high-frequency informationand dilates the scale to extract low-frequency information. InFigs. 5 and 6, the Morlet mother wavelets have been tunedso that at frequency , they have exactly the sameshape as the real part of the modulated Gaussian used in theSTFT results of Figs. 3 and 4, respectively (i.e., the Morlet-wavelet basis at (showninset) is identical to the real part of the modulated Gaussianin Figs. 3 and 4 at the same frequency). Thus, at frequency

the results of the STFT and Morlet-waveletprocessing are identical (strictly speaking, these transformsare not identical since the STFT is complex, and the Morletwavelet, as defined, is real; however, the real part of theSTFT is the same as the Morlet-wavelet transform and, ifdesired, the Morlet-wavelet transform could be redefined interms of a complex exponential). In other regions of thetime-frequency phase space, the results of STFT and waveletprocessing are fundamentally different because of the differentparametrizations in these respective algorithms. The Morlet-wavelet transform results in Fig. 5 reveal time-domain Floquetmode bands as in Fig. 3, but now the time-frequency resolutionchanges as one moves through the phase space; in particular,the variable resolution of the wavelet transform causes a

Fig. 5. Time-domain Morlet-wavelet transform of the scattered field data atthe bottom of Fig. 3. The wavelet basis is shown inset forf = 2:14=T .Instantaneous Floquet-mode frequenciesfm(t), as in Fig. 3.

Fig. 6. As in Fig. 5, but utilizing a wavelet with better temporal res-olution and commensurate poorer frequency resolution (wavelet inset forf = 2:14=T). Instantaneous Floquet-mode frequenciesfm(t) as in Fig. 3.

distortion of the specular response. As one improves thefrequency resolution by increasing the number of oscillationsin the mother wavelet, there is a commensurate reduction intemporal resolution and vice versa. This is demonstrated inFig. 6, where the Morlet wavelet at correspondsto the modulated Gaussian in the STFT results in Fig. 4. In thiscase, because of the reduced number of oscillations and thecorresponding reduced frequency resolution, the time-domainFloquet modes are not as easily distinguished whereas, dueto the increased temporal resolution, the temporally localized

specular response is more sharply defined.Along the lines discussed above, one may establish other

interesting relations between the STFT and Morlet-wavelettransform responses. When the large Morlet wavelet in Fig. 5is shrunk for the extraction of high-frequency components,there is a frequency for which the results of the waveletprocessing correspond to STFT processing with the smallwindow in Fig. 4; this frequency is (we recallthat at scale the Gaussian in the Morlet wavelet has standarddeviation and the modulation frequency is , whereand are the standard deviation and frequency at scale ;by adjusting the scale, the Gaussian in the Morlet waveletcan be compressed to correlate to the size of the narrowSTFT window, and the scale required for this translation

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determines the frequency at which the wavelettransform and STFT are similar). Similarly, when the smallMorlet wavelet considered in Fig. 6 is dilated to extract low-frequency information, there is a frequency forwhich the wavelet processing is equivalent to STFT processingwith a large window (Fig. 3). These properties underscore theinterrelationship between the Morlet wavelet and the STFT,which may be exploited when deemed useful.

Windowed Superresolution Processing:The time-depend-ent dispersion bands extracted in Figs. 3–5 straddle the time-dependent dispersion curves predicted by (1). In fact, for oursimple test example, the resolution of the STFT or wavelettransform suffice for discrimination of the different well-separated time-domain Floquet modes. As a comparison, wechoose here an alternative route with broader implications,namely, model-based windowed superresolution processingthat is applied to the data in Figs. 3–6. In particular, we usethe windowed eigenvector pencil method [16].

The superresolution results are shown in Fig. 7. The modalpoles extracted from the windowed data in (6) are plottedas dots in the phase space at the time corresponding to thecenter of the sliding window, and they are weighted (seeshading) by the modal excitation strength (spectral residue).The window size in (5) was set to , which iscomparable to the size of the Gaussian window used in Fig. 3for the STFT computations; concerning the window size, if thewindow is made too small (in an effort to improve temporalresolution), the amount of data available for accurate modelperformance may be insufficient, whereas if the window ismade too large, one will sacrifice temporal resolution andthereby compromise the benefits of superresolution processing.The model order was determined by performing a singular-value decomposition (SVD) at each window position andsetting the model order for that window position equal tothe number of nonzero singular values. As shown by theresults in Fig. 7, this technique works very well for the noise-free data considered here; however, highly noisy data requiremore sophisticated techniques [23]. We see that over mostof the range, the windowed superresolution processing (dots)is in close agreement with the expected Floquet-mode time-frequency dispersion curves in (1). Whereas theSTFT and wavelet transforms extract bands that are centeredabout the predicted curves, the windowed superresolutionscheme homes in on these curves as such. Additionally, themodel-based algorithm extracts the weakly excitedmode, which was not seen in either the STFT or waveletresults. Note, however, that the specularly reflectedFloquet mode isnot extracted in Fig. 7 while the nonmodel-based STFT and wavelet transforms extract this mode easily(Figs. 3–6). The failure occurs because the superresolutionmodel is based on spectra of the form in (6), which does notmatch the mode physics. This illustrates the limitationsof model-based algorithms: their performance is contingentupon matching the model to the phase-space parameters thatcharacterize the data. If the underlying phenomenology isnot known a priori, we have sought to establish it via theSTFT and/or the wavelet transform. When this sorting outindicates the possible presence of previously explored model-

Fig. 7. Windowed eigenvector pencil method time-frequency processing ofthe noiseless scattered field at the bottom of Fig. 3 using a rectangularwindow of duration 1.2T ; the model order was set to the number of nonzerosingular values found via singular-value decomposition (SVD) at each windowposition. The extracted frequencies (dots), plotted at the center of each slidingwindow, are weighted by their respective residues (shading). Instantaneousfrequenciesfm(t) as in Fig. 3.

based phenomenologies, the appropriate algorithms can beinvoked to improve resolution. This strategy is pursued inSection IVB for noisy data.

B. Noisy Data

STFT and Wavelet Transform:Most previous investiga-tions of the STFT and wavelet transform have dealt withnoise-free synthetic data [1]–[10] or data measured in a very-low-noise anechoic chamber [1]. In practice, the data will becontaminated with noise. To account for this, we reconsiderthe test problem investigated in Section IV-A, adding whiteGaussian noise. Up to 10-dB SNR additive noise, we havefound that the STFT and Morlet-wavelet transform continue toresolve the underlying phenomenology provided that one usesa “proper” STFT window size and Morlet wavelet as discussedearlier. However, for 5-dB SNR, the performance of STFT andwavelet processing degrades, even for the “proper” choice.

These observations are substantiated in Figs. 8 and 9 whichshow results of STFT and Morlet-wavelet processing on the5-dB SNR data using the same parameters as in Figs. 3 and5, respectively; the time-dependent modal dispersion curvesfrom (1) are plotted as previously. Evidently, due to blurringof the phase-space footprints, it is now much more difficult toextract the predicted time-dependent dispersion. Thedispersive mode is excited most strongly and is least affectedby the noise; the dispersion of the and modes isoutlined much less clearly but there is at least a suggestionof bands with confined frequency and extended temporalsupport. This suffices to hypothesize that the underlying datais characterized by time-dependent modes and that the datashould, therefore, be subjected to an appropriate model-basedalgorithm for improved resolution, as done below. It should benoted that the noise-free phase-space results in Figs. 3–7 wereplotted down to 90 dBm, while the noisy results were plotteddown to 60 dBm. If the results in Figs. 8 and 9 are plottedto 90 dBm, the modal bands are entirely obscured in thepresent gray-scale format. This points out another difficulty (inaddition to window-size selection) in portraying the outcome

598 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 4, APRIL 1997

Fig. 8. Short-time Fourier transform of the time-domain fields in Fig. 3–7, with 5-dB additive white Gaussian noise. The Gaussian window size is thesame as in Fig. 3, as are plots of the instantaneous frequenciesfm(t).

Fig. 9. Morlet-wavelet transform of the time-domain fields in Fig. 8 usingthe same Morlet wavelet as in Fig. 5. Instantaneous frequenciesfm(t) as inFig. 3.

of phase-space processing: the plots are often sensitive to thescaling used in the graphics. While the results in Figs. 8 and9 are for a specific 5-dB noise realization (and therefore couldbe termed qualitative), their quality is typical of results foundon an ensemble of other test examples.

Windowed Superresolution Processing:Applying the win-dowed eigenvector pencil method to the data in Figs. 8 and9, we note that for these high-noise conditions, it is difficultto rely on a SVD of the data matrix to determine the modelorder. We have, therefore, followed the procedure discussed byothers when model-based algorithms are applied to noisy data[26], [27]: the model order has been set to a value larger thanappropriate for the scattered signal in the absence of noise. Ithas been found that the higher order model mitigates againstthe effects of the noise and that the additional spurious polesoften lie outside the band of the true system poles [27]. Forthe results in Fig. 10, the model order was set to 12 for allwindow positions and the window size was 1.2; we alsoconsidered model orders up to 20, and found little variation

Fig. 10. Windowed eigenvector pencil method processing of the data inFig. 8 using the same window size as in Fig. 7 and a constant model order of12 for all window positions. The results are plotted as in Fig. 7.

of the phase-space signatures in the vicinity of the dispersioncurves of modes , , and . However, the signatureswere relatively strongly perturbed when considering variousmodel orders less than ten.

The results in Fig. 10 show that our superresolution pro-cessing accurately extracts the mode dispersion overmuch of the time-frequency range; the andmodes are also extracted, but somewhat less accurately dueto their relatively weak excitation strength compared to the

mode. We note that the SNR has been defined withrespect to the entire scattered waveform; however, the localSNR degrades with increasing time because the scattered fielddecays with time (see the bottom of Fig. 3): at early times,the SNR is greater than 5 dB, while at late times, the SNR issignificantly smaller (after , the SNR is less than 0dB). Thus, in Fig. 10, the poles extracted by the pencil methodare in relatively good agreement with the dispersion curves ofthe , , and modes for while, for later times,the agreement diminishes considerably.

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To quantitatively assess the properties of a superresolu-tion algorithm for noisy data, one must consider algorithmperformance for an ensemble of realizations, relative to theCramer–Rao lower bound [28]. In the context of the windowedsuperresolution algorithm considered in this paper, we haveperformed such an analysis for a similar problem [29].

V. CONCLUSION

In this paper, we have taken some first steps toward asystematic assessment of the performance of time-frequencyphase space processing schemes for extraction of footprintsassociated with time-domain target scattering. Such informa-tion is useful for classification of forward target scatteringphenomenology, and for extraction of this phenomenologyfrom scattering data for the purpose of target identification.Our demonstrations have been for the particular canonicalexample of a truncated periodic array of flat coplanar perfectlyconducting strips illuminated by a normally incident short-pulse plane wave, but the methodology and conclusions, sofar, should be applicable as well to other scattering scenarios.We have emphasized the importance of fully validated andcalibrated forward scattering algorithms, parametrized in termsof robust wave physics. Assessment of the performance of thevarious schemes for our particular example has been givenat appropriate places in the text. We have tried to distinguishprocedures based on detailed or at least partial knowledge ofthe connection between the target geometry and correspondingforward scattering data from procedures advocated when thereis no a priori information. Resolution is the principal issuethroughout, and is strongly dependent on the sampling windowsize. Our test conditions have been purposely selected so asto highlight signatures with short-time wideband phase-spacefootprints versus those with long-time narrowband footprints,which are associated with weakly dispersive and strongly dis-persive wave phenomena, respectively. Even these favorableconditions—where the various wave types do not overlapand, thus, can be individually identified—are accompaniedby resolution tradeoff and other difficulties inherent in thetime-frequency phase space. It is here that fully calibratedforward scattering models can help substantially toward theunderstanding of features and resolution ambiguities arisingin the inverse treatment of the data, thereby suggesting thatconsideration might be given to eventual benchmarking ofphase-space methodologies.

In this paper, we have taken what we regard as somefirst steps toward this goal. Trials similar to those reportedby us here have undoubtedly been carried out also by otherphase-space practitioners. However, the detailed execution andcalibration, if any, of these trials has generally not beenreported. We hope that such reporting will become part ofthe phase-space literature so that experiences and quantitativeassessments can be shared and critically compared. Calibrationof phase-space methods becomes ever more important if thesemethods are to be applied to real-target scattering scenariosof substantial complexity. While we have concentrated hereon time-frequency processing, similar considerations apply tospace-wavenumber processing.

ACKNOWLEDGMENT

The authors would like to thank D. Youla of PolytechnicUniversity, Brooklyn, NY, for several helpful comments con-cerning superresolution processing. They would also like tothank the two anonymous reviewers for their careful readingof our original manuscript and for their specific constructivecomments which led to substantial revision and improvement.

REFERENCES

[1] A. Moghaddar and E. K. Walton, “Time-frequency distribution analysisof scattering from waveguide cavities,”IEEE Trans. Antennas Propagat.,vol. 41, pp. 677–679, May 1993.

[2] L. Carin, L. B. Felsen, S. U. Pillai, D. Kralj, and W. C. Lee, “Dispersivemodes in the time domain: Analysis and time-frequency representation,”IEEE Microwave Guided Wave Lett., vol. 4, pp. 23–25, Jan. 1994.

[3] M. McClure, D. R. Kralj, T.-T. Hsu, L. Carin, and L. B. Felsen,“Frequency domain scattering by nonuniform truncated arrays: Wave-oriented data processing for inversion and imaging,”J. Opt. Soc. Amer.A, vol. 11, pp. 2675–2684, Oct. 1994.

[4] D. R. Kralj, M. McClure, L. Carin, and L. B. Felsen, “Time domainwave-oriented data processing for scattering by nonuniform truncatedgratings,”J. Opt. Soc. Amer. A, vol. 11, pp. 2685–2694, Oct. 1994.

[5] L. B. Felsen and L. Carin, “Wave-oriented processing of scattering data,”Electron. Lett., vol. 29, pp. 1930–1932, Oct. 1993.

[6] H. Kim and H. Ling, “Wavelet analysis of backscattering data froman open-ended waveguide cavity,”IEEE Microwave Guided Wave Lett.,vol. 2, pp. 140–142, Apr. 1992.

[7] , “Wavelet analysis of radar echo from finite size targets,”IEEETrans. Antennas Propagat., vol. 41, pp. 200–207, Feb. 1993.

[8] S. R. Cloude, P. D. Smith, A. Milne, D. M. Parkes, and K. Trafford,“Analysis of time domain ultrawideband radar signals,” inUltra-Wideband Short-Pulse Electromagnetics, H. L. Bertoni, L. Carin, andL. B. Felsen, Eds. New York: Plenum, 1993, pp. 445–456.

[9] G. C. Gaunaurd, H. C. Strifors, A. Abrahamsson, and B. Brusmark,“Scattering of short EM-pulses by simple and complex targets usingimpulse radar,” inUltra-Wideband Short-Pulse Electromagnetics, H. L.Bertoni, L. Carin, and L. B. Felsen, Eds. New York: Plenum, 1993,pp. 437–444.

[10] E. K. Walton and A. Moghaddar, “Time-frequency-distribution analysisof frequency dispersive targets,” inUltra-Wideband Short-Pulse Elec-tromagnetics, H. L. Bertoni, L. Carin, and L. B. Felsen, Eds. NewYork: Plenum, 1993, pp. 423–436.

[11] L. Carin and L. B. Felsen, “Wave-oriented data processing forfrequency- and time-domain scattering by nonuniform arrays,”IEEEAntennas Propagat. Mag., vol. 36, pp. 29–42, June 1994.

[12] L. Cohen, “Time-frequency distributions—A review,” inProc. IEEE,vol. 77, pp. 941–981, July 1989.

[13] L. Carin and L. B. Felsen, “Time-harmonic and transient scattering byfinite periodic flat strip arrays: Hybrid (Ray)-(Floquet Mode)-(MOM)algorithm and its GTD interpretation,”IEEE Trans. Antennas Propagat.,vol. 41, pp. 412–421, Apr. 1993.

[14] L. B. Felsen and L. Carin, “Diffraction theory of frequency- and time-domain scattering by weakly aperiodic truncated thin-wire gratings,”J.Opt. Soc. Amer. A, vol. 11, pp. 1291–1306, Apr. 1994.

[15] , “Frequency and time domain Bragg-modulated ray acoustics fortruncated periodic arrays,”J. Acoust. Soc. Am., vol. 95, pp. 638–649,Feb. 1994.

[16] S. U. Pillai and H. S. Oh, “Extraction of resonant modes from scatterednoisy data,”Ultra-Wideband Short-Pulse Electromagnetics II, L. Carinand L. B. Felsen, Eds. New York: Plenum, 1995, pp. 585–594.

[17] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method forextracting poles of an EM system from its transient response,”IEEETrans. Antennas Propagat., vol. 37, pp. 229–234, Feb. 1989.

[18] , “Matrix pencil method for estimating parameters of exponen-tially damped/undampled sinusoids in noise,”IEEE Trans. Acoust.,Speech, Signal Processing, vol. 38, pp. 814–824, May 1990.

[19] S. L. Marple Jr.,Digital Spectral Analysis with Applications. Engle-wood Cliffs, NJ: Prentice-Hall, 1987.

[20] W. A. Gardner,Statistical Spectral Analysis. Englewoods Cliffs, NJ:Prentice Hall, 1988.

[21] I. Daubechies, “The wavelet transform, time-frequency localization andsignal analysis,”IEEE Trans. Inform. Theory, vol. 36, pp. 961–1005,Sept. 1990.

[22] S. G. Mallat, “Multifrequency channel decompositions of images andwavelet models,”IEEE Trans. Acoust., Speech, Signal Processing, vol.37, pp. 2091–2110, Dec. 1989.

600 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 4, APRIL 1997

[23] S. U. Pillai and T. I. Shim,Spectrum Estimation and System Identifica-tion. New York: Springer-Verlag, 1993.

[24] M. P. Hurst and R. Mittra, “Scattering center analysis via Prony’smethod,” IEEE Trans. Antennas Propagat., vol. 35, pp. 986–988, Aug.1987.

[25] M. McClure, R. C. Qiu, and L. Carin, “On the superresolution iden-tification of observables from swept-frequency scattering data,”IEEETrans. Antennas Propagat, pp. 631–641, this issue.

[26] M. Van Blaricum and R. Mittra, “A technique for extracting the polesand residues of a system directly from its transient response,”IEEETrans. Antennas Propagat., vol. AP-23, pp. 777–781, Nov. 1975.

[27] A. J. Poggio, M. L. Van Blaricum, E. K. Miller, and R. Mittra,“Evaluation of a processing technique for transient data,”IEEE Trans.Antennas Propagat., vol. AP-26, pp. 165–173, Jan. 1978.

[28] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[29] T.-T. Hsu, M. McClure, L. B. Felsen, and L. Carin, “Wave-orientedprocessing of scattered field data from a plane-wave-excited finitearray of filaments on an infinite dielectric slab,”IEEE Trans. AntennasPropagat., vol. 44, pp. 352–360, Mar. 1996.

Lawrence Carin (S’85–M’89–SM’96) was born March 25, 1963 in Wash-ington, DC. He received the the B.S., M.S., and Ph.D. degrees in electricalengineering from the University of Maryland, College Park, in 1985, 1986,and 1989, respectively.

In 1989 he joined the Electrical Engineering Department at PolytechnicUniversity, Brooklyn, NY, as an Assistant Professor and became an AssociateProfessor there in 1994. In September 1995 he joined the Electrical Engineer-ing Department at Duke University, Durham, NC, where he is an AssociateProfessor. His current research interests include quasi-planar transmissionlines, short-pulse scattering and propagation, and ultrafast optoelectronics.

Dr. Carin is a member of the Tau Beta Pi and Eta Kappa Nu honor societies.

Leopold B. Felsen (S’47–A’53–M’54–SM’55–F’62–LF’89) was born inMunich, Germany, on May 7, 1924. He received the B.E.E, M.E.E, and D.E.E.degrees from the Polytechnic Institute of Brooklyn, Brooklyn, NY, in 1948,1950, and 1952, respectively.

During World War II he was concerned with work on electronic ballistics-calibration devices in the U.S. Army. Starting in 1948 he was with thePolytechnic Institute of Brooklyn (Polytechnic University), NY, and from1978 he held the position of Institute Professor, later renamed UniversityProfessor. From 1974 to 1978, he was Dean of Engineering. On a leave ofabsence from 1960 to 1961, he served as a Liaison Scientist with the LondonBranch of the Office of Naval Research. His research has dealt with a varietyof areas in electromagnetic radiation and diffraction theory, and his recentinterest is centered primarily on general techniques for wave propagationin various disciplines, including optics, acoustics, mechanics of submergedstructures, and seismology, in addition to electromagnetics. He is author orco-author of nearly 300 papers, and author or editor of several books. Hewas an Associate Editor ofRadio Scienceand ofWave Motion, and an Editorof the Wave Phenomena Series(New York: Springer-Verlag). He has heldVisiting Professorships at universities in the United States and abroad. In1967, 1971, and 1988, he was in the Soviet Union as an Invited Guestof the Soviet Academy of Sciences, and in 1981 he was invited for a six-week stay in the Peoples Republic of China. In 1994 he became an EmeritusProfessor at Polytechnic University and joined the Department of Aerospaceand Mechanical Engineering at Boston University, MA.

Dr. Felsen is a member of Eta Kappa Nu, Tau Beta Pi, Sigma Xi, and aFellow of the Optical Society of America as well as the Acoustical Societyof America. He is listed in numerous biographical volumes. In 1974 he wasa Distinguished Lecturer for the IEEE Antennas and Propagation Society. Hewas awarded a Guggenheim Fellowship for 1973, the Balthasar van der PolGold Medal from URSI in 1975, an honorary doctorate from the TechnicalUniversity of Denmark in 1979, a Humboldt Foundation Senior ScientistAward in 1981, an IEEE Centennial Medal in 1984, a Sackler Fellowship fromTel Aviv University in 1985, an IBM Visiting Fellowship from NortheasternUniversity in 1990, and the IEEE Heinrich Hertz Medal for 1991. Also, awardshave been bestowed on several papers authored or co-authored by him. In1977, he was elected to the National Academy of Engineering. He has servedas Vice-Chairman and Chairman of both the U.S. and the International URSICommission B.

David R. Kralj was born May 14, 1968 in Mar del Plata, Argentina. Hereceived the B.S.E.E, M.S.E.E., and Ph.D. degrees in electrical engineeringfrom Polytechnic University, Brooklyn, NY, in 1993, 1995, and 1996,respectively.

He is now with the Missile System Division of Raytheon Company,Tewksbury, MA.

H. S. Oh, photograph and biography not available at the time of publication.

W. C. Lee, photograph and biography not available at the time of publication.

S. Unnikrishna Pillai (S’83–M’85–SM’93) received the B.Tech. degree inelectronics engineering from the Institute of Technology (BHU), India, in1977, the M.Tech. degree in electrical engineering from I.I.T. Kanpur, India,in 1982, and the Ph.D. degree in systems engineering from the Moore Schoolof Electrical Engineering, University of Pennsylvania, Philadelphia, in 1985.

From 1978 to 1980 he worked as a Radar Engineer with the Research Di-vision of Bharat Electronics Limited, Bangalore, India, and since 1985 he hasbeen with the Department of Electrical Engineering, Polytechnic University,Brooklyn, NY, where he currently holds the rank of Professor of ElectricalEngineering. His present research activities include blind identification anddeconvolution, spectrum estimation, system identification, radar imaging, andnetwork theory.