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-1

- 1I I. Example of adar opera tional scenario: transmitter, receiver, target, noise,and clutter.r 7

Trihedral

Cavity

I -IA 2. Examples of manm ade targets: surface, edge, corner,

trihedral, and cavity.

where R s the range from the radar to the point targ,etand c is the wave propagation velocity.

When the target moves with a velocity v relative to theradar, called the radial velocity, the radar signal musttravel an extra longer or shorter distanceto reach the tar-get. Therefore, the signal received at time t is reflectedfrom the target at time (t-T t ) /2),nd the received signalbecomes sR(t )pps, [t-~( t)], here the round-trip traveltime is a time-varying delay [2],[7].

In addition to the signal reflected by the target, there 1salso additive noise. The signal-to-noise ratio (SNR)at theradar receiver is determined by th e received power re-flected from the target, the noise figure, and the band-width of the receiver. Improvement in the SNR willincrease the probability of target detection and the accu-racy of parameter estimation.A radar usually transmits aisequenceof pulses or other signal waveforms with a pulserepetition frequency(PRF)usually required by the maxi -mum range of detection. Target information may be exmined directly from the radar-range profile, i.e., the.distribution of target reflectivity along the radar line ofsight to the target, o r from its frequency spectrum by ap-plying the Fourier transform [81- [111.

In general, the radar processes the received signal andextracts information about the target. The range to thetarget, i.e., the distance from the radar to the target mea-sured along the radar line of sight, is estimated by thetime-delay between the transmitted signal and the re-

ceived signal. For a moving target, the mea-surement of th e targets velocity is based on thewell-known Doppler effect. If the ra-dar-transmitted signal is at frequencjrf, the re-flected signal from the moving target issubjectedto a Doppler frequency shift from itstransmitted frequency,f+L. his frequencyshift is induced by relative motion between theradar and the target [21. In the case where a tar-get has a radial velocity v, the Doppler fre-quency shift is determined by the radialvelocity of the target and the wavelength of theradar transmitted waveform

VC

f,=-2 f . (3)Thus, if a target is closing on th e radar at a velocity v =-1000 (ft/s) = -304.8 (m/s) , the Doppler frequency shiftfor X-band radar at 9842 MHz is 20 kHz.

Radar targets, especially man-made targets, usuallycan be considered as a collection ofpoint scatterers. Thesescatterers may have a wide variety of reflecting or back-scattering behaviors 1121, [13]. They can include sur-faces, edges, corners, dihedrals, trihedrals, and cavities(Fig. 2 ). Each type of scatterer has a different backscatter-ing behavior, which can provide a way to recognize thetarget. For example, when cavities or duct-type structuresare present in targets, multiple bounces appear in the ra-dar image as cclo~d xtended in the range domain [43].However, the same mechanism may provide importantfeatures that, if properly interpreted, can be importantfactors in the target recognition process. In the next sec-tion, we will further describe radar backscattering and in-troduce time-frequency analysis applications to radarbackscattering and feature extraction.Range profiles have been examined for target recogni-tion, especially in cases when a radar image cannot begenerated due to various reasons, such as short radardwell time on the target [SI. With high range resolution,the range profile can provide target information abouttarget length and positions of stronger scatterers, such asa radar dish, engine intakes, and other strong scatteringcenters (Fig. 3 ) . The dimension transverse to the radarline of sight is called the cross range. Because the Dopplershift of a scatterer in a target is proportional to the cross

Range (m)3. Example of an aircrafts range profile.

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range of the scatterer by a scale factor, the projection ofthe target reflectivity distribution on the cross-range di-mension can be obtained from the distribution of Dopp-ler shif ts , ca l led the Doppler prof i le . Withhigh-resolution Doppler profrles, the locations of stron-ger scatterers and the target's extents in the Doppler1-mension can be obtained as illustrated in Fig. 4.Bycombining range profiles and Doppler profiles, a2-D-radar image may be generated [13]-[16]. The radarimage is a spatial distribution of the target's reflectivitymapped onto a range and Doppler plane. Therange-Doppler image can be convertedto this 2-D imageif we have accurate knowledge of the scale factor. This isdetermined by the rotation rate, the distance of the scat-terer from its rotation center, and the wavelengthof thetransmitted signal 1151, [16].

An important factor of the image quality is its resolu-tion. Resolution is the ability to separate closely spacedscatterers in range and in cross range. The minimum dis-tance in the range AVcombined with in the cross range orazimuth A

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(a) ScatteringCenter (b) Resonance I.6 6

s?l i LLa, a,U U

Time Time(c) Material Dispersion (d) Structural D ispersion

C 63 Jp! FxCTLL LL

Time TimeA 5.Electromagnetic mechanisms are manifested in the oint

time-frequency mage as distinct features. (a) Scattering center,(b) resonance, (c) material dispersion, and (4 structural disper-

approaches. Others use modern spectral analysis toachieve sharper images with shorter data samples[27]-[30]. Sets ofradar data are available from http://air-borne. nrl. navy.mil/-vchen/tftsa. html. In another sec:-tion, we introduce an image formation based ontime-frequency analysis, which can resolve the in-i-age-blurring problem without resorting to sophisticatedpreprocessing algorithms.

Applications of JTF Analysis toRadar BackscatteringTime-Frequency Analysis of Radar Range ProfilesWith the radar operating at a sufficiently high bandwidlh(i.e., equivalent to a short pulse), the resultinghigh-resolution range profile can be interpreted as a map-ping of the reflectivity of the target along the radar line o fsight. With simple targets, a range profile typically coq-sists of a series of distinct peaks that can be related spa-tially to the scattering centers on the target. Thesefeatures are often examined for signature diagnostic aridtarget recognition applications. For real targets, howevcr,the scattering characteristics are usually more complex.For example, the scattering from some components on atarget can be strongly dispersive as a function of fre-quency, and may give rise to extended returns in the rangedomain. These dspersive scattering phenomena can bedifficult o interpret from the time-domain range profile.

JTF methods described in [11have been used to aria-lyze electromagnetic backscattered data with good suc-cess [31]-1421. The JTF representation of a signal is a2-D-feature space that facilitates the visualization and Ln-terpretation of complex electromagnetic wave phenom-enology. In this feature space, discrete time events such asscattering centers, discrete frequency events such as reso-nances, and dspersive mechanisms due to surface wavesand guided modes can all be displayed simultaneously.This can lead to more insight into the complex electro-

A 6.Joint time-frequency (JTF) image of the backscattering datafrom a coated plate with a gap in the coating. TheJTF image isgenerated by the short-time Fourier transform. Those featureswhich show slanting in the JTFplane are associated with thedispersive surface wave mechanisms in the coating.

magnetic wave propagation and scattering mechanismsthan the information available from either time or fre-quency domain representations alone.

Shown in Fig. 5 are the time-frequency features ofsome commonly encountered scattering mechanisms. Adiscrete event in time, for example, could be due to wavescattering from a spatially localized scattering center on astructure. It shows up as a vertical line [Fig. 5(a) ] in theimage because it occurs at a particular instance, but overallfrequencies.A target resonance, e.g., the return from apartially open cavity, is a scattering event that becomesprominent at a particular frequency.It shows up as a hori-zontal line in the JTF plane [Fig. 5(b)]. Dispersive phe-nomena, on the other hand, are characterized by slantedcurves in the time-frequency image. For instance, surfacewave mechanisms due to material coatings are character-ized by curves with a positive slope [Fig. 5(c) . Anothertype of dispersion arises from waveguide structures.These (structural dispersion mechanisms are character-ized by curves with a negative slope in the time-frequencyimage [Fig. 5 (d)].All of the above phenomena have beenobserved in a wide variety of structures, from simulationdata on canonical structures to measurement data oncomplex platforms [311 [421. Below, two examples arepresented to demonstrate the unique features of electro-magnetic scattering mechanisms in the JTF plane.

In the first example, the backscattered data from a di-electric-coated plate with a gap in the coating is consid-ered [35] (Fig. 6). The radar signal is incident edge-onfrom the left, with the incident electric field polarized inthe vertical direction. The radar frequency response hasbeen generated by computer simulation and verified bylaboratory measurement. The simulation result is shownalong the vertical frequency axis. The time-domain re-sponse, or equivalently, the range profile (where range =cz/ 2), is obtained by Fourier transforming the1.7-to-18-GHzdata. The resulting range profile is shownalong the horizontal time axis. It appears that three dis-tinct pulses are present. However, the second and thirdpulses are spread out in range.

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In order to resolve in finer detail the dispersive scatter-ing mechanisms in this coated plate, the spectrogram ofthe backscattered signal is generated using the short-timeFourier transform. As can be seen, the scattering mecha-nisms are much more apparent in the 2-D JTF plane thanin either the time or the frequency domain. In particular,it is observed that the third broad pulse in the time do-main actually consists of three separate scattering mecha-nisms (labeled as 3a, 3b, and 3c). As the frequencyapproaches zero, the propagation delays of mechanisms3a, 3b, and 3c approach the same value.As the frequencyincreases, the pulses have different propagation delays,and become clearly separated.

Slanted curves in th e time-frequencyplane (like mech-anisms 2, 3b, and 3c) are characteristic of dispersive be-havior. In the case of the coated plate, surface wavesexcited in the coating give rise to the dispersive mecha-nisms. At frequencies well above cutoff, the surface waveis tightly bound to the dielectric and the wave velocity ap-proaches the slow dielectric velocity. Near cutoff, the sur-face wave velocity approaches that of free space andexhibits a shorter propagation delay. Therefore, in thetime-frequencyplane, the surface wave phenomena showup as slanted curves with a positive slope.

Based on the propagation delay considerations and theabove observation, it is possible to pinpoint the five domi-nant scattering mechanisms. They are shown on the left inFig. 6,which clearly indicates that mechanisms 2,3b, and3c include surface wave propagation. For the polarizationunder consideration and the frequency range of the data,the TM, surface wave mode that has zero cutoff is the only

mode that can propagate in the dielectric (the TM, modehas a cutoff frequency of 23.4 GHz). Finally, higher-orderscattering mechanisms can be observed during thelate-time portion of the data, but they are very weak.

A discrete event in time, forexample, could be due to wavescattering from a spatiallylocalized scattering centeron a structure.

In the second example, the scattering from a slottedwaveguide structure is considered [391. The geometry isshown in Fig. 8, where a long rectangular waveguide isflush mounted in a conducting ground plane. Two nar-row slots are opened on each end of the ground plane.The structure is excited by a horizontally polarized radarsignal occurring a t an angleof 30"with respect to thever-tical. The backscattered data are generated by a computa-tional electromagnetic simulator based on the method ofmoments. The data are generated from 0.025 to 10 GHzin 25-MHz increments.

The time-frequency representation of the data, ob-tained using the short-time Fourier transform, is shownon the right in Fig. 7. The two early-time vertical linescorrespond with the exterior scattering centers from theslots. The other curves are related to signals coupled into

SlottedWaveguide Time-Frequency Image

30" Off-NormalEn d HH-polSlots

Waveguide Size: 5.25 cm x 4.5 cm x 96 cmSlot Size: 4.5 cm x 2.5 cm

9.5

-f"926 5.0U$

0.5 0 5 10 15Time (ns)

I7.Joint time-frequency image of the backscattering data from a slotted wavegu ide structure. The da ta we re simulated using amethod of moments solver. The JTF image is generated by the short-time Fourier transform. The JTF image shows both theearly-time discrete-time returns from the slot exterior a nd the late-time dispersive mechanisms due to modal p ropagationinside the waveguide.

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eRange

4

Frequency-/3 R F r e q u e n c y -Independent Depend nFeatures(Large oQA Smalleatures)Cleaned Nonpoint-ScatteringE A R Image Mechanisms

ca,3Q

Range FrequencyA 8. Joint time-frequency processing is applied to the range di-

mension of the conventional range/cross-range ISAR image tcgain an additional frequency dimension.By examining howthe resulting images vary as a function of frequency, the fre-quency-independent eatures can be separated from the fre-quency dependent ones and displayed in an appropriatefeature space.

the waveguide (Fig. 7,again). When the wave reaches thefirst slot, some energy is coupled into the waveguide,propagating to the other end as a sum of waveguidemodes. The energy carried by these modes begins toreradiate through the other slot after a time given byL/Gwhere L is the length between the two slots. However,this is the time delay only for frequencies well above themodal cutoff for which the modal group velocity ap-proaches G . For frequencies approaching the cutoff fre-quency of the respective mode, the group velocity teiidstoward zero and the time delay goes to infinity. Corse-quently, each modal dispersion behavior is manifestedas atime-frequency trajectory with negative slope [as illus-trated earlier in Fig. 5(d)]. This behavior can be clearlyidentified in the spectrogram, where the presence of t w omodal dlspersion curves with cutoffs at 3 and 6 GHz areobserved. They correspond to the TE,, and TE,, modes inthe waveguide. Note that the amplitude variation of lhesignal along these curves is governed by the couplingmechanisms hrough the slot apertures and is considerablymore complex.

Because multiple reflections occur,we also see other clis-persion curves with greater delays during the late-time por-tions of the return.The first one corresponds with the energythat, upon reachmg the other end, reflects back and radiatesthrough the slot on the left. The next is the three-bouncemechanism. Note that energy is also coupled into the wave-p d e hrough the slot on the right, and through s imi l a rmechanisms, generates dispersion curves in tiletime-frequency plot depending on the number of bounces.

From the above two examples, it can be seen that th eJTF representation can aid in the interpretation of corn-

plex electromagnetic phenomena. Furthermore, the JTFfeatures can be well understood in terms of the tar-get-scattering physics. For radar applications, thetime-frequency representation is particularly effective inidentifying scattering mechanisms in targets containingsub-skmline tructures such as inlet ducts, antenna win-dows, and material coatings.

Physics-Based Feature Extraction fromRadar ImageryThe JTF processing of the one-dimensional range profiledescribed above can be further extended to deal with 2-Dradar imaging. ISAR imaging, as discussed earlier, is a ro-bust process for mapping the position and magnitude ofthe point scatterers on a target from multi-frequency andmulti-aspect backscattered data. However, for complextargets containing other scattering phenomena such asresonances and dispersive mechanisms, image artifactsare often encountered in the resulting ISAR image. Oneimportant example is the scattering from the engine in-let/exhaust duct on aircraft.It is a dominant contributorto the overall scattering from the target, yet its wave-guide-like structure and the associated fre-quency-dependent scattering mechanisms make it anonpoint-scattering feature.

When processed and displayed by the conventionalISAR algorithm, the inlet return results in an image fea-ture which is not well-focused, is not related to the spatiallocation of the scatterer, and can often obscure other im-portant point features on the target. Therefore, it wouldbe useful to automatically emove these artifacts from theISAR image, le al ng to a clean ISAR image containingonly physically meaningful point scatterers. Purther-more, the inlet features extracted can be better displayedin a more meaningful feature spaceto identify target reso-nances and cutoff phenomena.

JTF processing can be applied to ISAR image process-ing to accomplish the above objective [43]. The concep-tual idea behind the JTF ISAR algorithm is to apply JTFprocessing to the range (or time) axis of the conventionalrange/cross range ISAR image to gain an additional fre-quency dimension. The result is a 3D range/cross-range/frequency matrix, with eachrange/cross-range slice of this matrix representing anISAR image at a particular frequency (Fig. 8 ) . Conse-quently, by examining how the ISAR image varies withfrequency, we can distinguish the frequency-independentscattering mechanisms from the frequency-dependentones. In the actual implementation of the JTF ISAR, thechoice of the JTF processing engine is critical to preserv-ing range resolution. This is demonstrated below usingthe adaptive Gaussian representation proposed by Qianand Chen [44].

The signal-adaptive Gaussian representation is ahigh-resolution time-frequency representation. (A simi-lar algorithm called matching pursuit was developed in-dependently at around the same time by Mallat and

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Zhang [45]). he objective of this method is to expand asignalf(t) in terms of Gaussian-modulated exponentialbasis functions h,(Q with an adjustable standard deviation0,and a time-frequency center (t,,f,):

(4)where

Note that the basis function has a dual form in its Fou-rier transform representation:

Therefore, these basis functions have a time-frequency ex-tent given by0, nd (1/20,), respectively. The coefficientsBp are found one at a time by an iterative procedure. Onebegins at the stagep = 1and chooses the parameters o,, ,and&, such that bp (t) is most similar tof(t), that is:

wheref,(t) = (t). Forp > l,& (t) is the remainder afterthe orthogonal projection off,.,(t) ontob, (t) has been re-moved from the signal:(b)Conventional SAR Image

ISAR(a) VFY-218 Airplane -30-35

4 -404 5-50

4 24 1-E 3 9

a, dBsm-p 3 8m% 3 7ffl 3 6 -55

3 5 -603 4 -653 3 3 3 3 4 3 5 3 6 3 7 3 8 3.9 4 4 1 4 2 -70

Inlets

Down-Range(m)

(c) Enhanced SAR ImageAfter JTF Proce ssingEnhanced ISAR

4.27.14

E 3.9Ua,mp 3.82 3.7v)5 3.6

3.53.43.3

3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4Down-Range m)

-30-354 0-45-50-55-60-65-70

dBsm

(d) Extracted Inlet Return nthe Frequency Aspect PlaneFrequency-AspectPlane

11 12 13Frequency

-0-5-1 0-1 5-20 dB-25-30-35

14 15 16 4 0

9.Adaptive JTF ISAR processing using the test range data from a 1:30scaled model of the VFY-2 18 airplane. (a) Geometry of theVFY-218 with two deep inlet ducts. (b) Conventional ISAR image for frequency from 8 to 16GHz, aspect from 0 to 403 The dispersivecloud over the right wing is due to the return from the left inlet duct. (c) Enhanced ISAR image obtained by removing non-point scat-tering mechanisms using the AJTFISAR algorithm. The inlet return has been removed from the image. (d) Frequency-aspect displayshowing the extracted features from the engine return. Prominent resonant frequencies are observed in this feature space.

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a pp ying sophisticated focusingalgorithms.

Th s procedure is repeatedto generateasmany coefficientsas needed to accurately represent the original signal.

The adaptive Gaussian representation has two distinctadvantages over conventional time-frequency techniquessuch as the short-time Fourier transform. First, it is aparametric procedure that results in very hightime-frequency resolution. More importantly for our $ 1 ~ -plication, the adaptive spectrogram allows us to dist n-guish the frequency-dependent events from thefrequency-independent ones automatically through theextent of the basis functions. From (5), t can be seen thatscattering centers, i.e., signals with very narrow lengths intime, will be well represented by basis functions with verysmallor.Frequency resonances, on the other hand, wiU bebetter depicted by large op. herefore, if we reconstructthe ISAR image using only those Gaussian bases withsmall variances, a much cleaner image can be obtainedshowing only the scattering centers. The remainingmechanisms, i.e., those related to the large variaxeGaussians, are more meaningful when viewed in a dualfrequency-aspectdisplay,where resonances and other Tre-quency-dependent mechanisms can be better identified.

The algorithm is demonstrated by using the chambermeasurement data of a 1:30 scale model LockheedVFY-218 airplane provided by the ElectromagneticCode Consortium [46].The airplane has two ong engineinlet ducts, which are rectangular at the open ends, butmerge together into one circular section before reaching asingle-compressor face [Fig. 9(a)] . As we can clearly seein the conventional ISAR image for the horizontal polar-ization at 20" near nose-on, the large cloud outside of theairframe structure is the inlet return [Fig. 9(b) ].Fig. 9(c)shows the enhanced ISAR image of Fig. 9(b),obtainedby applying the JTF ISAR algorithm and keeping clnlythe small variance Gaussians.We see that only the scatter-ing center part of the original signal remains in the imageas expected. Notice that the strong return due to engineinlet has been removed, but the scattering from the rightwing tip remains. Fig. 9(d) shows the frequency-aspect

display of the high variance Gaussians. A number ofequispaced vertical lines can be seen between 10.5 and13.5 GHz. Given the dimensions of the rectangular in-let opening, we estimate that these frequencies corre-spond approximately to the second cutoff frequency ofthe waveguide-lilce inlet. This information may beunique to the particular inlet structure under consider-ation and may be useful as an additional feature vectorin ta rget classification [47]. The processed images forobservation angles varying from 0-360" have beenmade into an MPEG movie that can be downloadedfrom http ://ling0.ece.utexas.edu.

In this section we presented a JTF ISAR algorithm toprocess data from complex targets containing not onlyscattering centers, but also other frequency-dependentscattering mechanisms. The adaptive JTF ISAR algo-rithm allows the enhancement of the ISAR image byeliminating nonpoint-scatterer signals, thus leading to amuch cleaner ISAR image. Moreover, the extractednonpoint-scattering mechanisms can be more appropri-ately displayed in an alternative feature space to show tar-get resonances and frequency dispersion. This isaccomplishedwithout any loss in down-range resolution.In the next section, the JTF processing will be applied tothe cross range (or Doppler frequency) instead of therange dmension of the radar image for the purpose of re-focusing the image due to complex target motion.

ApplyingJTF Analysis to RadarImaging of Moving TargetsTime-Frequency-Based Radar Image FormationTo achieve high cross-range resolution, SAR uses syn-thetic array processing that coherently combines signalsobtained from sequences of small apertures at differentaspect angles, with respect to the target to emulate the re-sult from a large aperture [141, [151. In radar images, thecross-range resolution is obtained from the Doppler reso-lution [131, [151.The differential Doppler shifts of adja-cent scatterers of the target can be observed in the radarreceiver; therefore, the distribution of the target's reflec-tivity can be measured by Doppler spectra. The conven-tional method to retrieve Doppler information is theFourier transform.

Based on the radar-received signal from a single pointscatterer, the received signal from the entire target can berepresented as the integration of the returned signalsfrom all scatterers in the target [151

(9)where r is the range of a point scatterer in the target. It canbe expressed by the range of the target's center of rotationR( t ) , he rotation angle of the target O ( t ) , and the scat-terer's location (%,?) in the target inertial coordinate (Fig.10). Because the range R( t ) nd the rotation angle O ( t )

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X

___.- -.___,---Radar

Center of Rotation

10.Geometry of radar imaging.

are functions of time, the range of a point scatterer in thetarget is also a function of time

where the frequency components are defined byf =-2f cosQ(t)

and

From the radar-received sig-nal S(t ; t ) , if the targets rangeas a function of time t isknown exactly over the imag-ing time duration, then the ex-traneo us phase termexp{-j4jR(t)/c) can be re-moved by multiplying its con-j u g a e with the re c e ve dsignal, i.e., G(f, 6) Sgt)exp{j4jR(t)/c). This is re-ferred to as gross focusing ormotion compensation [151,[181.Then, the targets reflec-tivity density function can bereconstructed simply by tak-ing the inverse Fourier trans-form of the mot ioncompensated signal

In order to apply the Fourier transform properly, scat-terers must remain in their range cells during the imagingtime, and their Doppler frequency shifts must be constant[17]-[20]. To obtain a focused Fourier radar image,range tracking must be used to pull scatterers back intotheir range cells and Doppler traclung must be used tocorrect any time-varyingphase change. Thus, a constrainthas been imposed on the Doppler frequency spectrum,requiring it to be constant. Then, a clear radar image canbe captured (Fig. 11).

The range traclung and Doppler tracking form the ba-sis of the standard procedure for focusing radar images[131, [151.If the target is moving smoothly, the standardfocusing is good enough to generate a clear image of thetarget by using the Fourier transform. However, when atarget exhibits complex motion, such as rotat ion and ma-neuvering, the standard procedure is not sufficient togenerate an acceptable image for viewing and analysis.Scatterers can still drift out of their range cells and theirDoppler frequency shifts can still be time-varying. Thus,the Doppler spectrum obtained from the Fourier trans-form will be smeared, and the radar image will be blurred.In this case, more sophisticated procedures for individualscatterers, such as polar reformatting [161, are needed.Thus, each scatterer may remain in its range cell and itsDoppler frequency shift may be constant. Then, the Fou-rier transform can be applied properly to reconstruct aclear image of the target. However, to perform the polarreformatting, the knowledge required of the initial lune-matic parameters of the target may not be always avail-able. Some individual scatterers may still drift throughtheir range cells and their Doppler frequency shifts may

i-adar ReceiverRange Profiles 1

r 1Fourier TransformRange TrackingDoppler Spectrum

toppler Tracking1 1. Conventional radar ima ging of moving targets with the Fourier transform.

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Time-Domain Radar Signal

-5

50-5-1 0 J

50 100 15 0 200 250 300 350 400 45 0 500Time __+Fourier Transform

PsL

(b) Time -----+A 12. Time-varying spectrum of radar data represented by (a)

the Fourier transform and (b) the time-frequency transform.

and the rotation-induced Doppler shift [5 11fd, =

C (16)- [x ( -nsin~,-~2 tcosWO)--Y(RcosWO-~2f tsinv,,)]

where is the initial aspect angle of the target. It is clearthat even if the rotation rate Q and the radial velocity ofthe target v y re constant, the rotation-induced Dopplerfrequency is still time-varying.Other sources oftime-variat ion n the Doppler frequency shift may resultfrom uncompensated phase errors due to irregularities in

Radar Receiver

Range TrackingTime-Frequency AnalysisndI DopplerTracking I ,

Range-Doppler-Time mage Cube

____+

- I Range ProfilesN Pulses

I ____J

L 13.Time-frequency based radar image formation.

still be time-varying. Hence the resulting image cac. stillbe blurred when the Fourier transform is applied.

However, the restrictions of the Fourier transform canbe circumvented if the time-frequency transform de-scribed in [ l ] s used to replace the Fourier transform[48]-[51]. Due to the time-varying behavior or theDoppler frequency shift, an efficient method of solvingthe problem of the smeared Fourier frequency spectrumand, hence, the blurred image, is to apply ahigh-resolution ime-frequency ransformto the Dopplerprocessing. By replacing the Fourier transform with ahigh-resolution time-frequency transform, the imageblurring caused by the time-varying Doppler frequencyshifts can be mitigated without applyingsophisticated fo-cusing algorithms.

The Doppler frequency spectrum of moving tarigets isalways time-varying. The relationship between thetime-varying Doppler spectrum and the targets motioncan be described by the translation-inducedDoppler shift

the targets motion, the fluctuation of the tar-gets rotation rate, fluctuation in localizing therotation center, inaccuracy in tracking thephase history (i.e., a collection of phases of re-turned signals during imaging ime), and othervariations of the system and the environment.Because the residual phase errors may varywith time, the Doppler frequency can also varywith time.

The Fourier transform indicates which fre-quency components are contained in the sig-nal, but it does not tell how frequencies changewith time. By representing the time-varyingDoppler frequency spectrum with the Fouriertransform, the Doppler spectrum becomessmeared as in the example shown in Fig. 12,where the Fourier transform ( a) and atime-frequency ransform (b) are applied to aradar data. We can see that the Fourier trans-form of the data is actually the integral of thetime-frequency transform of the same dataover its duration. This is due to the frequencymarginal condition [511.

From the insight into the image blurring, we can seethat in order to achievea clear image, the time-frequencytransform should be used in place of the Fourier trans-form. The time-frequency transform introduced in [11isan efficient way to resolve the image blurring problem

I I14.Rotation motion represented by roll, pitch and yaw.

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Ib) ISAR Time-Frequency Based Images

14.Rotation motion represented by roll, pitch, and yaw.

caused by the time-varying Doppler's behavior withoutapplying sophisticated focusing algorithms for individualscatterers.

Fig. 13 illustrates the time-frequency-based radar im-age formation [50], [51]. Standard range tracking andDoppler traclung are necessary prior to performing thetime- requency image formation. The difference betweenthe time-frequency-based mage formation and the con-ventional Fourier-based image formation is that the fo-cusing procedure for individual scatterers is no longerneeded in the time-frequency-based mage formation andthe Fourier transform is replaced by the time-frequencytransform followed by time sampling. The Fourier-basedimaging approach generates only one image frame from aradar data set G{rm(n)} , here r v s (n )s the m-th rangeprofile o f M range profiles and n = 12,..N. However,the time-frequency-based image formation takestime-frequency transforms at each range cell and gener-ates an NXN Doppler frequency (or Doppler) anddwell-time (or time) distribution. By combining the MDoppler time distributions at M-range cells, theMxNXN-range-Doppler-time cubeQ(r,,,fn,t,,)can be formed

where TF denotes the time-frequency operation with re-spect to n. At a particular time instant t,,only one rangeDoppler image frameQ(rm,L,,,= ti) an be extractedfrom the cube. There are a total o f N mage frames avail-able, and every one represents a fullrange-Doppler imageat a particular instant. Therefore, by replacing the Fouriertransform with the time-frequency transform, a 2-Drange-Doppler Fourier image frame becomes a 3Drange-Doppler-time image cube. By sampling in time, atime sequence of 2-D range Doppler images can beviewed [49] [ 511. Each individual time-sampled framefrom the cube provides not only a clear image with supe-rior resolution, but also time-varying properties from onetime to another.

Time-Frequency Analysis forMoving TargetsA moving target always has translational androtational motions during the imaging time.Assume the radar is located at the origin ofthe (u,v,w) frame of reference, and a target islocated at a distance R(u,v,w,t) from the ra-dar at time t. The inertial coordinate of thetarget is the (x,y,z) frame. Thus, rotationalmotions of the target can be described byroll, pitch, and yaw [15], [18]. For an air-craft heading along the x-axis, roll corre-sponds to a rotation 8,about thex-axis, pitchcorresponds to a rotation 8, about they-axis,and yaw corresponds to a rotation 8, aboutthe z-axis (Fig. 14) . If the order of rotationsis a roll, followed by a pitch, and finally, a

yaw, then the composite roll, pitch, and yaw motions canbe represented by a rotation matrixRot(8, Op ,e, )=ROLL,(e, )pitch, e p raw,ey . (1s)According to this rotation matrix, any point scatterer onthe target will move from its current location to a new lo-cation. When the target is rotating, the Doppler fre-quency spectrum of the returned signal from the targetbecomes time-varying, and the reconstructed image willbe blurred. For example, if a target has pitch motion only,i.e., 8, = OY = 0 and OP = Qp ,where Qp s the pitch rate,th e rotation matrix becomesr cos8, 0 sine, 01

1 0 0 0 1JFor a point scatterer at x,y,z, the Doppler frequency shiftinduced by the pitch motion becomes

C

where we assume the initial pitch angle is zero. Even if thepitch rate Qp is a constant, the motion-induced Dopplerfrequency shift fdpirihis still time-varying. Assume thepoint scatterer at x = 5 m, z = 0 and the target rotatingwith apitch rate ofQ, = 0.14 rad/s, from (20), he Dopp-ler bandwidth is approximately 6 Hz. Consequently, ifthe Doppler resolution is 1Hz, corresponding to a imag-ing time of 1 , the motion-induced Doppler frequency ofthe scatterer will be smeared over 6 Doppler resolutioncells, and the measurement of its true position will be un-certain.Fig. 15(a) shows the conventional ISAR Fourier im-age of a target taken by a X-band radar operating at 9000MHz [52]. Because the target is maneuvering, the Fou-rier image of the target is still blurred even after applying

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From the insight into th e iblurring, we can see that in ordert o achieve a clear image, thetime-frequenbe used in pltransform.the standard procedure for focusing. By using thetime-frequency-based image formation, because eachscatterer has its own range and Doppler frequency shift ateach time instant, without knowing any targets kine-matic parameters and resampling the data, a blurred Feu-rier image becomes a sequence of clear time-frequencyimages. Fig. 15(b)shows one ofthese time-frequency m-ages in which the nose, wing-tips, fiiselage, and enginesof the aircraft can be seen very clearly. The blurred imagedue to target maneuvering can be refocused withoui: ap-plying sophisticated autofocusing or motion compe nsa-tion algorithms.

SummaryThe Fourier transform has been widely used in radar sig-nal and image processing. When the radar signals exhibittime- or frequency-varyingbehavior,ananalysis that canrepresent the intensity or energy distribution of signals inthe joint time-frequency domain is most desirable.In thisarticle, we showed that JTFanalysis is auseful tool for im-proving radar signal and image processing for time- andfrequency-varyingcases. We applied JTF analysis to radarbackscattering and feature extraction; we also examinedits application to radar imaging of moving targets. Otherapplications of time-frequency analysis to radar can alsobe found in [53]-[56].

Most methods of JTF analysis are non-parametric.However, parametric or model-based methods oftime-frequency analysis, such as adaptive Gaussian andchirplets, are more suitable for radar signals and images[571 [591.VictorChen is an Electronics Engineer and Staff Consul-tant at the Radar Analysis Branch of the Radar Divisionof the Naval Research Laboratory. Ha0 Ling is a Profes-sor in the Department of Electrical and Computer lhgi -neering at the University of Texas at Austin.

AcknowledgmentsThe authors would like to thank Dr. Merrill Skolnili forhis helpful comments. Victor Chens work was supportedby William Miceli of the Offlce of Naval Research. HaoLings work was sponsored by the Joint Services Elec-

tronics Program under contract no. F49620-95-C-0045.The United States Government is authorized to repro-duce and distribute reprints for governmental purposesnotwithstanding any copyright notation hereon.

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