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Detection and imaging of moving objects withsynthetic aperture radarPart 2 : Joint time-frequency analysis by Wigner-VilledistributionS . Ba rba rossa , PhDProf. A . Farina

Indexing terms: Wigner-Vi lle distribution, Synthetic aperture radar, Moving objects, Detection

I 1Abstract: The aim of the work is to show howtime-frequency representation by Wigner-Villedistribution of the echoes received by a syntheticaperture radar provides a useful tool for detectionof moving objects and the estimation of theinstantaneous phase shift induced by relativeradar-object motion. The phase history is thenused to compensate the received signal and toform a synthetic aperture with respect to themoving object, necessary to produce a highresolution image.

1 int roduct ionThe main problems related to the detection and imagingof moving targets with synthetic aperture radars (SAR)are analysed in the first part of the paper. This secondpart is entirely focused on the use of the Wigner-Villedistribution (WVD) in the SAR signal processing formoving target detection and imaging.

The possibility of focusing a moving target observedby radar requires knowledge of the phase modulationinduced on the received echo by the radar-target relativemotion. The time behaviour and/or the spectrum of thetarget echoes alone are not sufficient to provide thisinformation. The knowledge of the phase history of theechoes received during the whole observation intervalallows us to form the synthetic aperture with respect tothe moving object, necessary to produce a high cross-range resolution image.A method for extracting the instantaneous phase canbe based on analysis of the time-frequency (TF) distribu-tion of the received signal. Several distributions are avail-able and an excellent review is given in a recent paper byCohen [ 3. The Wigner-Ville distribution has beenchosen in this work because it presents some importantfeatures concerning detection and estimation issues, asalready pointed out [2-51. There are simpler methods foranalysing signals in the TF domain, such as the short-Paper 8216F (ElS), first received 10th J uly 1990 and in revised form 8thApril 1991Dr . Barbarossa is with the Univerita di Roma L a Sapienza, INFO-COM Department, V ia E udossiana 18, 00184, Roma, I talyProf. Farina is with Alenia SpA., Radar Factory, Via Tiburtina, Km12400,00131 R oma, Italy

time Fourier transform (STFT) [l], but they do notexhibit the same resolution capabilities in the TF domainas does the WVD. In particular, since the STFT is basedon a Fourier transform (FT) applied to a time windowedversion of the signal, with the window central instantvarying with time, the frequency resolution is inverselyproportional to the window duration. The narrower thewindow, the better is the time resolution, but the worse isthe frequency resolution and vice versa. Conversely, theWVD does not suffer from this shortcoming. On theother hand, the WVD poses other problems since it isnota linear transformation. This causes the appearance ofundesired cross-products when more than one signal ispresent.With respect to other T F distributions, such as Rihac-zeks (e.g. Reference l) , the WVD provides a higher con-centration of the signal energy in the TF plane, aroundthe curve of the signal instantaneous frequency (IF). Thisallows a better estimation of the IF in the presence ofnoise and this information is fundamental for the syn-thesis of the long aperture with respect to the movingobject.Mapping of the received signal in the TF plane pro-vides a tool for the synthesis of the optimal receiver filterwithout a priori knowledge of the useful signal, providedthat the signal-to-noise ratio be sufficiently large. The TFrepresentation provides a unique tool for exploiting oneof the most relevant differences between useful signalsand disturbances in the imaging of small moving objects,namely the instantaneous frequency and the bandwidth.In fact, it can be shown that, while the bandwidthoccupied by a target echo during the observation intervalnecessary to form the synthetic aperture mainly dependson radar-object motion, the instantaneous bandwidth isproportional to object size. Therefore the echo corres-ponding to a small target can occupy a large band duringthe overall observation time, but its instantaneous band-width is considerably narrower (i.e. the echo back-scattered by a point-like target has a zero instantaneousbandwidth but it may exhibit a large overall bandwidth).Conversely, the echo from the background and thereceiver noise have a large instantaneous bandwidth.Therefore, even if the useful signal and the disturbancemay have a large total band, the possibility of trackingthe instantaneous bandwidth, made available by the TFrepresentation allows a discrimination of the useful signalfrom the disturbance not possible by conventional pro-cessing.

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Another important and unique advantage related touse of the WVD is that it allows the recovery of the echophase history even in the case of undersampling, asshown in Reference 6. This is particularly important inSAR applications since it allows us to work with pulserepetition frequencies (PRF) lower than the limitsimposed by the signal bandwidth occupied during theobservation interval. Owing to the target motion, thisbandwidth may be considerably larger than the band-width occupied by the background echo. According toconventional processing, we should then use a corre-spondingly higher PRF. Conversely, if the useful signalhas a large total bandwidth, but a narrow instantaneousbandwidth, the TF representation prevents superpositionof spectrum replicas created by undersampling because,even if the replicas occupy the same bandwidth, theyoccur at different times. This property allows us torecover the desired information even from undersampledsignals. Since the PRF value imposes a limit on the sizeof the monitorable area, due to time, and then range,ambiguities, the possibility of using a low PRF preventsthe reduction of the region to be imaged, as well as theincrease of the data rate.The WVD is first recalled in Section 2. In particular,its main properties in the presence of noise are derived.Section 3 is then concerned with recasting the optimaldetection and parameter estimation in presence of noisein the TF domain, based on the WVD. Use of the WVDfor the detection of chirp signals embedded in white noisehas already been considered [2-51. The approach is hereextended to different signal modulations and to the caseof correlated noise (the echo from the background). Onthe basis of the matched filter theory, transferred into theTF domain, it is shown that, in the presence of whitenoise, the received signal can be mapped directly onto theTF domain. In the presence of a correlated disturbance(the echo from the background), a cancellation must beperformed before evaluating the WVD. A possible algo-rithm for filtering the received signal in the TF plane isalso proposed. Finally, the application to the SAR case isconsidered in Section 4. Simulation results are shown forevaluating the capabilities of the proposed approach.2 Timefr equen cy analysis by means of theWigner-Vil le distr i but ion2.1 Definit ion and main properties of the W VDIn a recent paper [l], a thorough overview of TF dis-tributions is provided. An extensive analysis of theWigner-Ville distribution is also given in References 7-9.The WVD of a signal is defined as:

W(t,f)=I tms(t +;)* -;) exp( - j2nf~) z (1)- m

wheres ( t ) represents the analytic signal.Two properties, particularly important in detectionand estimation problems are: (a) conservation of thescalar or inner products; (b) estimation of the instantan-eous frequency. These properties are now briefly recalled.(a) Conseroation o the scalar products: Given twosignals x i ( t ) , i =1, 2, the square modulus of their scalarproduct is equal to the scalar product between their dis-tributions:I J - r x l ( t ) x m dtJ 2=~ ~ k ( t 9 f ) w 2 2 ( t 9 f )t df (2)

having indicated with WJ t, o) he WVD of the signalsx i ( t ) , i =1, 2.90

This property will provide the basis for the formula-(b) Estimation o the instantaneous frequency: If wetion of the detection scheme in the TF domain.express the signal in terms of its envelope and phase:s(t) = exp {jdt) j (3)

it can be shown that the local or mean conditional fre-quency of the WVD distribution, defined as[I ]

is equal to the signal IF:

The estimate of the mean conditional frequency of theWVD then provides the information about the signal IF.This is exactly the information we need for rephasing thereceived signal in order to produce a focused image.

2.2 W VD of signal plus noiseA problem which arises when using the WVD is that,since it is not a linear transformation, the distribution ofthe sum of more than one signal is not equal to the sumof the distributions of each signal, but contains all thecross-products.In general, if a signal s& is given by the sum of Ncontributions

Ns J t ) =2 &

i = 1its WVD is

K is,(t,f)= - m@s i ( t+ t-I)xp(-j2nf7) d7In detection problems, it is important to analyse themain statistical properties of the WVD of deterministicsignals superimposed on stochastic processes (thedisturbance). Therefore it is useful to review the mainproperties of the WVD of a random process.In the case of a stationary random process c(t), theexpected value of the WVD is constant over the time axisand is equal to the power spectral density of the processover the frequency axis. In fact:

E{W,(t,f)}= +$*(t -mx exp(-j2nf7) dT

=I - yRc(7)exp( - j2nf~)7 =S,(f) (8)whereR,(z) and S,(f) are the correlation and the powerspectral density of the random process, respectively. Inthe case of white Gaussian noise, the WVD is constant.In the case of a deterministic signal superimposed on azero mean additive noise, the cross-product term has azero mean value. The corresponding average WVD isthen equal to the sum of the WVD of the signal plus the

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(9)

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timeb

WV D of a linearfrequency modulation signalig . 1a Contour levelsb Perspective viewwhereS,(f) is the noise power spectral density. This doesnot mean that there are no cross-products between signaland noise of course, it only means that these terms have azero mean value.2.3 Estimation of the instantaneous frequency in thepresence of noise or mo re than one signalGiven two deterministic signals sl(t) and s 2 ( t ) with con-stant envelope:

sd t ) =Ai exp {j4i( t)} i =1 2 (10)the IF of their sum, estimated by using eqn. 4, is (seeAppendix):

A?fl (t )+A:f2(t) +AIA,(f l ( t)

In general, this expression is not easily interpretable. Twoextreme cases can be considered:(a)A , =A , =A :

(first-order approximation) (13)

If the two signals have the same amplitude, the estimatedfrequency lies exactly in the middle between the twoinstantaneous frequencies, otherwise it lies between them,in a position depending on the relative amplitudes andon their phase modulation. If the signal phases are inde-pendent random variables evenly distributed in [O 2x3,the average value of f12(t) is approximately equal to thecentre of gravity of the various I F components, weightedby the corresponding power:

The presenceof noise inevitably affects the estimation ofthe IF. In fact, in such a case, we have:J '-rfEi W,+n ( t , f ) } df

=j':f~t./, df + J ' - z f s n ( / ) df

sinceS,(f) is an even function. FurthermoreS f l E ( K + n ( t , f ) )df

=J-z ( t > f )f +J - z s n ( f ) df=d ( t )+P , (16)

where P n represents the noise power. Consequently, as afirst approximation, we have:

For a signal with constant amplitude A, the expectedvalueof the instantaneous frequency isthen:

having indicated by SNR the signal-to-noise ratio. Fromthis expression it turns out that, in presence of noise, theaverage value is always smaller than the true value.

2.4 ExamplesSome examples of distribution can be helpful to under-stand the representation by WVD. An importantexample in SAR applications is a signal with a constantenvelope and a phase modulation law composed by thesuperposition of a polynomial term (slow timefluctuations) plus sinusoidal contributions (fastfluctuations)~(t)A exp[ - j { & +2nfDt +zpDt2+nq, t3+k sin (271fm )}] I t I


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(a) Linear frequency modulation (fD #O, pD #0,qD=0 k =0): In this case, the WVD is known in closedform:w(t,f)=2AZ(T- l t l )sinc{27r(T- I t l )

x ( f - f D - ~D t ) ) ItI


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As regards the effect of disturbances in the receivedsignal, it is useful to recall that matched filtering canalways be performed by cascading two kinds of oper-ations: a clutter cancellation followed by a coherentintegration (see the first part of the paper). The first oper-ation does not require knowledge of the useful signalshape, whereas the second operation does. I n particular,a correct coherent integration requires knowledge of thesignal phase history. The time-frequency analysis aims tofacilitate extraction of the signal phase history. Therefore,the sequence of operations to be performed on thereceived signals is the following: a clutter cancellation isperformed first and the output of the cancellation filter isanalysed in the TF domain. If the clutter has beenreduced to a power level sufficiently smaller than theuseful signal, the parameters estimated by the WVD arecorrect, within an error depending on the achieved SCR.These parameters provide the information necessary toset up a correct coherent integration.This rather intuitive reasoning for using the WVD indetection and estimation problems is now formalised intothe framework of the matched filter theory.3.2 Match ed f i l tering based on the W VDThe received signal can be modelled as the sum of auseful signal s ( t ; 0) (supposed known in a parametricform, but this restriction will be removed later on), plus aGaussian process with zero mean value and autocorrela-tion matrixR , (see Part 1). The optimal detection schemeconsists of taking the inner product between the vector r,containing samples of the received signal, and theoptimal weighting vector $e), given by:

$0) =RC-'s*(0) (21)where s(0) is the vector formed by samples of the usefulsignal, supposed known in parametric form, and thevector 0 contains the unknown signal parameters. Themaximum modulus of all the scalar products

y(e)=P - w(e) (22)as a function of the parameters is then compared with athreshold to determine the presence of a useful signal.By using the property of conservation of the inner pro-ducts (eqn. 2) for Wigner-Ville distributions, the squaremodulus of y can be evaluated alternatively as the innerproduct between the WVD of r and the WVD of w . Thisallows us to recast the matched filter theory in the TFdomain.The case of an uncorrelated disturbance is now exam-ined, and then the theory will be extended to correlateddisturbances.3.2.1 White noise c ase: In the case of a white noise, thecorrelation matrix R, is diagonal, therefore the weightingvector is directly proportional to the useful signal. Detec-tion is carried out by thresholding the modulus of thescalar product between the received vector and the con-jugated value of the useful signal. According to eqn.2, thesquare modulus of that scalar product is

(23)According to the TF formulation, we evaluate the WVDof the received signal W(t , f ) ,he WVD of the usefulsignal W,(t,f), take their inner product and compare theresult with a threshold. Since the energy distribution of

IjHe)12= l-ydt)s*(t; ) dt=[-: [-rW(tf)wI(LI.L 0)dt df

the signals of interest is concentrated around the curve ofthe signal IF, the double integral can be reduced to a lineintegral.In the case of a chirp signal, the energy is concentratedalong a straight line in the TF plane (see Fig. la). Thevector 0 in this case is composed by two parameters: themean Doppler frequencyf, and the Doppler rate ,U .Theparameters of the line of maximum energy, constant termand slope, coincide exactly with the chirp parameters.These parameters can be estimated by analysing the TFdistribution of the received signal. As suggested in Refer-ence 3, the detection and estimation would consist, in thiscase, in the evaluation of all the line integrals over the TFplane, (by varying the line parameters fD and ,U thesearch for a maximum, and the comparison with athreshold. The parameters of the line which gives themaximum energy are exactly the parameters of the targetIF. The WVD then offers a means for simultaneouslydetecting and estimating the modulation parameters ofthe chirp.When the SNR is not too small, it is possible tosimplify the aforementioned approach as follows:(i) compute the WVD of the received signal (ii) estimatethe signal IFf(t) by computing the centre of gravity ofthe WVD, for each time instant, by eqn. 4 (5)valuatethe instantaneous phase, apart from an unimportant con-stant term, by simply integrating the estimated frequency:

M)=27c f(4 t +4 ( t o ) (24)1Finally, (iv) generate a reference signal :

sR@ =exp{- j W ) (25)and phase-compensate the received signal by mixing itwith this reference signal.The great advantage of this approach is that it doesnot require any assumption about the kind of phasemodulation. In particular, we can analyse signals whosefrequency modulations do not necessarily follow astraight line.We would point out that the simplification in theestimation procedure is achievable only if the signal-to-noise ratio is sufficiently large, otherwise the frequencyestimation could be seriously degraded. However, asmoothing of the WVD samples can be used for improv-ing the estimation accuracy in the presence of noise, aswill be shown in Section 3.3.3.2.2 Correlated noise case: The strategy followed inSection 3.2.1 can be applied to this case by simplyrearranging the terms in eqn. 22. In fact, we can writeeqn. 22as

y(0) =rT {RC-ls*(0)}=(Rc-Tr)T~(0)xTs*(0) (26)where the superscript - T indicates inverse and trans-pose. Therefore, the square modulus of y(0)can be evalu-ated as the scalar product between the WVD of x and theWVD of s. At this point, the procedure followed in thewhite noise case can be applied to this case as well, pro-vided that the vector r is substituted by the vector x.Since the multiplication of the vector r by the inverse ofthe clutter correlation matrix corresponds to a cluttercancellation (see the first part of the paper) this meansthat, as might be expected, the estimation procedure mustbe applied not directly on the received data, as in thewhite noise case, but on the data at the output of aclutter cancellation filter.

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3.3 Instantaneous frequency estimation in presenceof noiseAccording to classical estimation theory, based on themaximum likelihood principle, the signal parameters canbe estimated as that set of parameters which maximisethe likelihood ratio or, equivalently for Gaussian-distributed processes, eqn.23. The bottleneck of this tech-nique is that the likelihood ratio is a multimodenonlinear function of the parameters to be estimated.Therefore, as in any nonlinear optimisation problem, thesolution is hard to obtain if the number of parameters islarge, and the search can lead to a local maximum.The TF representation makes the estimation problemeasier. In fact, since the WVD provides a distributionwhose energy is concentrated along the curve of the IF, itmakes the extraction of the frequency modulation param-eters easier. Of course, the accuracy of the estimationdepends on the SNR.In the presence of noise, some smoothing or filteringhas to be applied to reduce the effect of the noise on theestimation accuracy. Different kinds of filtering can beenvisaged, in the time domain or in the TF domain, thatis directly on the received time samples or on their TFdistribution. Furthermore, different strategies have to befollowed depending on the noise correlation properties.As shown in the previous Section, the parameterestimation is applied on the received time samplesdirectly, in the presence of white noise, while a cluttercancellation filter has to be used first in the case of acorrelated disturbance. However the filtering modifies thesignal modulation behaviour and the consequent estima-tion. In the following, a strategy for improving the esti-mation accuracy in the presence of white noise isproposed. Then, the correlated noise case is analysed, andin particular the effect of the clutter filtering on the suc-cessive I F estimation.3.3.1 Wh ite noise case: The WVD provides a powerfultool for the estimation of the signal IF in the presence ofwhite noise since it allows us to exploit one of the mostrelevant differences between signal and noise, that is theirIF behaviours. While the bandwidth occupied by thetarget echo during the overall observation interval maybe large, the instantaneous bandwidth is usually muchnarrower than the total bandwidth. Conversely, a whitenoise exhibits both total and instantaneous large band-widths. Therefore an energy concentration in the TFplane reveals the presence of a useful signal.On the basis of this difference, an iterative procedurecan be envisaged for improving the estimation accuracy,based on the following steps:(i) the WVD of the received signal is evaluated(ii) the instantaneous frequencyf(t)and the instantan-eous bandwidth B(t) are estimated by evaluating themean and the standard deviation of the distributionalong the frequency axis, for each time instant(iii) the WVD samples are multiplied by a window inthe TF plane equal to one within an interval centredaroundf(t) and having a width proportional to B(t),andzero outside(iv) steps (ii) and (iii) are repeated until some con-vergence on the estimatedIF is observed.The rationale of the procedure is that, if the window iscorrectly positioned, some noise contributions areremoved in step (iii), while the signal is retained, thusimproving the SNR. Therefore, at the second iteration ofthe procedure, the estimation of f(t) and B(t) improves.

This allows the synthesis of a narrower window which, inturn, produces a further noise rejection. Therefore, aslong as the windows are correctly positioned (whichmeans that no useful signal contributions are discarded),the estimation accuracy improves at each step. Finalaccuracy is dictated, at least in principle, only by thenoise falling within the signal instantaneous bandwidth,instead of by the noise falling within the band occupiedduring the whole observation interval. The methodworks properly if the initial estimate is accurate enoughto position the window correctly.3.3 .2 Correlated noise case: In the presence of a corre-lated noise, the received signal must be filtered to cancelthe disturbance, before mapping the signal onto the TFdomain. Of course, the filtering modifies the signalbehaviour. In particular, the filtering attenuates all thesignal components falling within the clutter band. There-fore, it is possible to recover the desired information onlyif the signal spectrum is not entirely overlapped with theclutter spectrum. Once having cancelled the clutter, thesame procedure outlined for the white noise case can beapplied to this case.

4

4.1 Echo modelThe signal received by a SAR is given by the sum of theechoes from the ground, the echoes from a possiblemoving object and receiver thermal noise. The echo fromthe ground can be modelled as a correlated randomprocess, whose power spectral density is proportional tothe antenna power radiation pattern (see Part 1). Theecho from a moving object is characterised by anunknown modulation, induced by the relative motionbetween the radar and the object. In SAR imaging, weare interested in the phase modulation induced by therelative radar-target motion. If the transmitted signal is

Appl icat ion t o detect ion and focusing ofmoving objects wi th SAR

dt)=4) xp (97!f, ) (27)where a(t) is the analytic signal andf, is the carrier fre-quency, the echo received by a pointlike scatterer at adistancer(t)from the radar is proportional to

(28)During the observation interval, the amplitude of eachbackscattering coefficient can be assumed constant sincethe aspect angle does not vary by an amount such as tojustify a change in the reflectivity characteristics (thisassumption underlies all SAR signal processing). Theonly modulation of interest then is phase modulation.In the formation of the synthetic aperture, we areinterested in the last term of this expression. From Kirkswork on motion compensation for SAR [lo], it turns outthat the motion fluctuations which need to be compens-ated have periods comparable with the observation time(the analysis developed in Reference10mentions an SARobserving a fixed scene, but the same considerations canbe applied to the imaging of moving objects, providedthat the relative radar-target motion is taken intoaccount). The conventional techniques for autofocusingextended scenes compensate only linear and quadraticphase shifts. This means that only slow fluctuations (with

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respect to the integration time) are compensated. Con-versely, the TF approach allows estimation and com-pensation of any kind of phase history [l 13.The distance r(t) can be modelled as the sum of slowand rapid fluctuations, with respect to the observationinterval. Slow variations can be expressed as a low-orderpolynomial, whereas fast variations follow a sinusoidalbehaviour, whose period is shorter than the time observa-tion interval.The change in the distance causes a phase modulation,which must be compensated to obtain a correct image.The variation of distance can cause migration of thereceived echo over different range cells, depending on theratio between the amount of variation and the rangeresolution. The range migration problem can be facedaccording to the double-resolution strategy outlined inthe Introduction to Part 1 of this paper. According tothat strategy, all the phase estimations are carried out ondata whose range resolution is such as to consider themigration negligible. Examples of WVDs of signals withphase modulations containing slow and rapid fluctua-tions, with respect to the duration of the observationinterval, have already been shown in Section2.4.4.2 Processing scheme and simu lation resultsAccording to the formulation of matched filtering interms of the WVD, the processing scheme for detectingand estimating the parameters of the echoes from movingobjects is sketched inFig. 5.The received signal is first range-compressed. Then itis processed by a moving target indicator (MTI) filter toreduce the clutter contribution. A preliminary detectionis performed by comparing the canceller output with asuitable threshold. A detection enables the processingchain based on the WVD. The aim of the first detection isto avoid idle computations in cases in which the presenceof a moving object is unlikely. The analysis of the signalin the TF domain allows estimation of the signal instan-taneous frequency. As mentioned in Section 3.3, somesmoothing is applied on the WVD to improve the estima-tion accuracy. The instantaneous frequency is then inte-grated to obtain the phase modulation to be used for

target

phase compensation of the received signal. At this pointan FFT is sufficient to provide the high crossrangeresolution image. A second thresholding is then appliedto confirm the first detection.An example is helpful to understand the TF analysis ofa signal embedded in a correlated noise. To this purpose,the WVDs of clutter plus signal at the input and at theoutput of the clutter canceller are reported in Figs. 6and7, respectively (the clutter has been generated as a

.--- -+-

rn -a ,------I

compresseddata

I

in eg rat on

generatereferencesigna1e-l+(t)

timeFig. 6lation ilterWV D o clutter plus si gnal at the input of the clutter cancel-SCR =0dB

$(t)

random process with a given power spectral density). Theexample is relative to the case of a pulse repetition fre-quency (PRF) equal to 16 times the clutter bandwidth.The number of samples is 64 and the signal-to-clutterratio (SCR) at the input is 0dB. The signal has a linearfrequency modulation with a bandwidth, due to theDoppler rate, equal to PRF/8. The mean Doppler fre-quency has been chosen equal to PRF/8 (in the analysisof these ditributions, it is worth recalling that the discreteWVD is periodic of PRF/2 [8]).

preliminarydetection

firstestimate I

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From these Figures, it is evident that the filtering hasimproved the signal-to-clutter ratio. Since the processingprocedure is based on the theory of optimal filtering, the

I Jtime

Fig. 7lation ilterW V D of clutter plus signal at the output of the clutter cancel-

Fig. 8moving target superimposed on a stationary backgroundI mage of a simulated extended scene containing a point-like

Fig. 9 I mageof the same scene as in Fig.8,after clutter cancellation

0 crossrangeFig. 10like target, after clutter cancellation and autofocusing96

Imagecorresponding to the range cell containing the point-

achievable improvement in the signal-to-clutter ratio(SCR) can be deduced from the optimal performanceshown in Part 1.The filter has also modified the shape of the signal dis-tribution. In particular, the signal spectrum falling withinthe clutter spectrum is strongly attenuated. The resultingdistribution exhibits a mean Doppler frequency higherthan the true value and a Doppler slope very close tothat of the undistorted signal. Since the focusing dependsonly on the Doppler slope, but is not affected by themean Doppler frequency (which affects only the relativeposition of the target image with respect to thebackground) the successive estimation of the modulationparameters still produces a focused image.An example of an image relative to a moving point-like target superimposed on a fixed extended scene isshown in Fig. 8. The image obtained after clutter cancel-lation is shown in Fig. 9.The target signal has barelycome out, but it is still smeared in crossrange. By com-paring the modulus of each output sample with a thresh-old obtained by averaging the moduli of the contiguouscells, as in the cell-averaging CFAR (constant false alarmrate) thresholding scheme, wecan detect the presence of amoving target on the corresponding range cell. Thatrange cell is analysed by the proposed approach and theresult is shown in Fig. 10(relative to the only range cellof interest). The sharpening of the target image is quiteevident.

5 ConclusionsIn this paper a scheme has been proposed for detectingand focusing moving targets with synthetic apertureradars based on a joint T F analysis of the receivedsignals. The TF formulation has provided us with apowerful tool for getting the desired information aboutthe useful signal instantaneous frequency, which is whatweneed for producing a focused image of the target. Themethod works for point-l ike targets (targets whosedimensions are smaller than the radar resolution) or forextended targets characterised by a dominant scatterer orby independent scatterers having similar reflectivity char-acteristics. The method provides a compensation of therelative radar-target translational motion, whatever itwould be. Once having compensated the translationalmotion, the remaining rotational motion can be com-pensated by, for example, the method proposed in Refer-ence12.The range migration problem has been overcomeusing the double-resolution strategy outlined in the firstpart of the paper. The WVD is used to estimate theinstantaneous frequency, and then the instantaneousphase, on the low-range resolution data. The estimationis then used to recover the law of variation of the radar-target distance, to be used in the compensation of therange migration on the full range resolution data.Further analyses are necessary in the imaging ofextended targets when more dominant scatterers occupythe same resolution cell. In such a case, in fact, the bilin-earity of the WVD creates undesired crossproduct termswhich can seriously impair the estimation of the instan-taneous frequency. A reduction of these undesired termscould be achieved by resorting to other time-frequencyrepresentations, such as the Choi-Williams distribution[l], for example, or by applying a Radon or a Houghtransform on the WVD. Since the received signal isintrinsically two-dimensional (range and crossrange), it is

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possible to envisage the use of a Wigner-Ville distribu-tion defined for two dimensional signals. The new dis-tribution could provideus with a powerful tool for facingstrong range cell migrations.

6 AcknowledgmentThis work was funded by Alenia SPA.

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ReferencesCOHEN, L.: Time-frequency distributions - review, Proc.KUMAR, B.V.K.V., and CARROL L, C.W.: Performance of Wignerdistribution based detection methods, Optical Engineering, 1984, 23,(6),pp. 732-738KAY, S., and BOUDREAUX-BARTELS, G.F.: On the optimalityof the Wigner distribution for detection. ICASSP 85, March 1985,pp. 27.2.14FLANDRI N, P.: A time-frequency formulation of optimum detec-tion, IE EE Trans.,1988, AS P -36, (9),pp. 1377-1384BARBAROSSA, S., and FARINA, A.: A novel procedure fordetecting and focusing moving objects with SAR based on theWigner-Ville distribution. IEEE 90 International Radar Con-ference, Arlington, VA, 7th-10th May 1990, pp. 44-50BARBAROSSA,S.: Parameter estimation of undersampled signalsby Wigner-Ville anaysis. IEEE International Conference onAcoustics, Speech and Signal Proc., ICASSP 91, Toronto, May,1991,pp. 3253-3256CLAASEN, T.A.C.M., and MECKLENBRAUKER, W.F.G.: TheWigner distribution- tool for timefrequency analysis; Part I -

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con&nuous time signals,Philips J .Res., 1980,35, pp. 217-2508 CLAASEN, T.A.C.M., and MECKLENBRAUKER, W.F.G.: TheWigner distribution- tool for time-frequency analysis; Part I1-discrete time signals,Philips J .Res., 1980,35, pp. 276300CLAASEN. T.A.C.M.. and MECKLENBRAUKER. W.F.G.: TheWigner distribution- tool for timefrequency analysis; Part I11- elations with other time-frequency signal transformations,PhilipsJ .Res., 1980,35,pp. 372-389KIRK, J .C.: Motion compensation for synthetic aperture radar,IE EE Trans., 1975, AE Sl 1, (3),pp. 338-348BARBAROSSA,S.: New autofocusing technique for SAR imagesbased on the Wigner-Ville distribution, Electron. Lett., 1990, 26,(18),pp. 1533-1534WERNESS, S., CARRARA, W., JOYCE, L., and FRANCZAK , D.:Moving target imaging algorithm for SAR data., IEEE Trans.,1990,AES26, (I),pp. 57-67BARBAROSSA,S.: Detection and imaging of moving objects withsynthetic aperture radar. Part 1: Optimal detection and parameterestimation theory, IE E Proc. F , 1992, 139, (1).pp. 79-88

Appendix : Estimation o f t he instantaneousfrequency of th e sum of several signalsGiven a signal s ( t ) equal to the sum of two signals sl(t)and s 2 ( t ) , expressed in terms of their envelopes and phase:

(29)si(t)=ai(t)exp{4i(t)} i = I , 2its WVD is (see eqn.7):

W,+z(t,f)=Wl(t,f)+Wz(t,f)+2 Re {Wl2(t7f)) (30)where we have indicated by Wl+z(t,f)he WVD of thesum, by w.(t, ) the WVD of the signal si(t),and byWl2(t , f ) he cross-WVD:Wlz(t,f)= l+msl (ta +i)s : ( t -i)exp(-j2nfz)dz

(31)IEE PROCEEDINGS-F, Vol. 139, NO. , F EB RUA RY 1992

The instantaneous frequency of the sum signal is evalu-ated by using eqn. 4. In particular, the numerator of eqn.4 in this case is equal to:1-rf1+z ( t J 1df

wherefl(t) andfz(t)are the instantaneous frequencies ofthe two separated signalssl(t)and s2( t ) . The contributiondue to the crossproduct terms is:+m +mj-:fwl2(t>f) df =1 fs,(t +z/2)sf(t- /2)- m

x exp (-j2nfz) dz dBy exploiting the derivation property of the Fouriertransform, we can write this expression in the followingform:

r +m 1

where the prime indicates a derivative.In the case of constant envelopes (al(t)=A ,a2(t)=A2),and taking the real part, according to eqnwe have:and.32,

=AlAZ{Cfl(t) +fi(t)l/2) cosC4l(t)- 4zWl (34)The numerator of the estimated instantaneous frequencyis then:

J - m=A f f N +4 Z W+A,AZCfl(t) +f20 )1 cosC4l(t)- 42(t)l

By proceeding in a similar way for the denominator, we

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