Barbarossa Optimal Detection Parameter Estimation

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Detection and imaging of moving objects withsynthetic aperture radarPart 1:Optimal detection and parameter estimation theoryS . Barbarossa, PhD

I ndexing terms: Ai rborne radars, Synthetic aperture radar, Wigner-V il le distribution, Moving target detection, Parameter estimation

Abstract: The aim of the paper is to provideoptimal schemes, according to the maximum like-lihood criterion, for the detection of movingobjects observed by airborne radars, and for thesynthesis of a long aperture with respect to themoving objects, necessary to produce high-resolution images. The theoretical limits of detect-ability and of the accuracies achievable in theestimation of the motion parameters necessary forthe synthesis of the long aperture are provided.Some simple suboptimal schemes, with limitedperformance losses, are then proposed.

1 IntroductionIn many applications of microwave imaging by syntheticaperture radar (SAR) [l, 21, it is desirable to be able todetect and possibly to produce focused images of movingobjects.A moving object is not easily detectable and, ingeneral, its resulting image is smeared and ill-positionedwith respect to the stationary background. These short-comings are a direct consequence of the SAR image for-mation process. The crossrange high resolution in anSAR is obtained by taking advantage of the relativemotion, supposed known, between the sensor and thescene. If, however, there is an object moving in an unpre-dictable manner, the image formation process does notfunction properly.Basically, the main degradations due to the targetmotion are the following:(i) The range migration through adjacent resolutioncells causes a reduction of the signal-to-clutter ratio(SCR), which can seriously impair the detection capabil-ities. Furthermore, range migration causes a decrease inthe integration time and a consequent loss of resolution.(ii) Even in the absence of range migration, or after itscorrection, the phase shift induced by the motion causes:an ill-positioning of the target image with respect tothe ground, mainly owing to the range componentof the relative radar-target velocity;

a smearing of the image due to the uncompensatedcrossrange velocity and/or range acceleration.In the open literature there are several contributions[3-61 concerning the detection of moving targets withPaper 8215F (E15), first received 10th J uly 1990 and in revised form21st March 1991The author is with the Universita di Roma L a Sapienza, INFO-COMDepartment, Via Eudossiana 18,00184, Roma, I talyIEE PROCEEDINGS-F, Vol.139, No. 1, FEBRUARY 1992

SAR. These works are based on the possibility of dis-criminating the moving target signals from the fixedscene returns on the basis of their different Dopplerspectra; in all these cases, the target spectrum is supposedto be located out of the clutter spectrum. These methodssuffer, however, from two main shortcomings: (i) theyrequire the use of a high pulse repetition frequency(PRF); (ii) they do not succeed with targets whose motionhas a small range velocity component, so that their spec-trum is superimposed on the clutter spectrum. The PRFmust be high enough to make available a region in theDoppler frequency domain not occupied by returns fromthe fixed scene. However, the choice of the PRF directlyaffects the size of the monitorable swath. In particular,the maximum unambiguous swath is inversely pro-portional to the PRF, therefore an increase in the PRFcauses a corresponding reduction of the swath. Further-more, an increase in the PRF causes a correspondingincrease in the data throughput. More advanced tech-niques are then necessary for overcoming these draw-backs.If focused images of the moving objects are alsorequired, for recognition purposes, the motion param-eters must be estimated to compensate correctly the rela-tive signals. In References 7-10, algorithms have beenproposed for producing fine resolution images of movingtargets having any translational and rotational motion.In particular, the algorithm proposed in Reference 7requires the presence of multiple prominent points in thetarget image. The echo from a first point is initiallyanalysed. Its phase is computed and subtracted pulse-by-pulse from the phase of the incoming signal. This oper-ation removes the effect of the target translational motionand makes this first point effectively the new centre of thescene. At this point, if the rotational motion is negligible,the polar format processing, described in Reference 9,yields the focused image. If, conversely, the rotationcannot be neglected, two other prominent points arerequired to estimate the rotation parameters. Theseparameters are used to compensate for the rotationalmotion in the frequency domain and help in applying thepolar format processing correctly. The algorithm requiresthat the prominent points be separable and that theirphases might be estimated without interfering with eachother. In some cases, however, this assumption might notbe met: at high resolution, separability is more likely tooccur, but the range migration could complicate thephase estimation problem; at low resolution, the rangemigration could be negligible, but it is more likely thatsome prominent points might occupy the same rangeresolution cell. In this work, we have assumed the rota-tional motion to be negligible and we have concentrated

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our analysis on the estimation of and compensation forthe translational motion in presence of disturbances, suchas noise or clutter residues. In particular, in Part I , theoptimum estimation scheme is analysed according to themaximum likelihood criterion; in Part I1 [17] an algo-rithm based on the time-frequency analysis of thereceived signal is proposed for estimation and com-pensation for the translational motion.Before describing the organisation of this paper, it isuseful to state the hypotheses and the operating condi-tions we have assumed. In general, the range migrationmay be not negligible. In this work we have faced thisproblem according to the following strategy. Eventhough the variation of the radar-target distance causesboth range migration and phase shift on the receivedecho, it is convenient to consider the two effects incascade. Since the amount of migration depends on therange resolution, we can envisage working with twochannels, as shown in Fig. 1: a high- and a low-

It is important to point out that the two successivephases, detection and focusing, cannot be completelyseparated. I t would seem obvious that the signal param-eters should be estimated only after having detected asignal. However, a reliable detection in the presence ofdisturbances (such as background clutter or noise)requires us first to process the received signal through amatched filter, in order to improve as much as possiblethe signal-to-disturbance ratio. The filter impulseresponse, on the other hand, is a replica of the usefulsignal; herefore, before filtering, wehave to estimate thesignal modulation parameters. I t turns out that these twooperations are strictly related to each other and that theymust be carried out together.The problem of contemporaneously detecting andestimating the signal modulation parameters is addressedin the second part of the paper. The approach proposedthere is based on a time-frequency analysis of thereceived signal, carried out by means of the Wigner-Ville

fine-range full resolutionalignment.t.

low resolution channelFig. 1 Double-range resolution pr ocessing scheme or estimation and compensationo target motion

resolution channel (the low-resolution data can beobtained by smoothing the high-resolution data or justdegrading the resolution by using, for example, less integ-ration time). Resolution in the low-resolution channel isdegraded until the migration is negligible for the specificapplications of interest. We can estimate the phasehistory on the low-resolution data and, consequently, thelaw of variation of the distance. Then we can use thisinformation to correct for the range migration on thehigh-range-resolution channel. The compensation can beperformed in the frequency domain or after full rangeresolution compression. The price paid by using thisapproach is the degradation of the signal-to-disturbanceratio in the low-resolution channel. In the following, weshall refer to the low-resolution channel and, as a conse-quence, weshall neglect the range migration.As regards the ill-positioning of the target image withrespect to the ground, the only estimation of the Dopplerfrequency does not provide us with the informationnecessary to position the target correctly. The Dopplerfrequency depends on the projection of the relative veloc-ity along the radar-target line of sight. Therefore, theDoppler frequency depends both on the target positionand on its motion. Therefore, knowing the Doppler fre-quency does not allow us to decouple the contributiondue to the motion from that due to the position. This isthe so-called azimuth position uncertainty problem. Theonly way to solve this problem resorts to the use of morethan one antenna. In this work, we shall limit ouranalysis to the case of one antenna only. The extension tomore antennas has been considered in Reference 11.Within the framework established by the aforemen-tioned operating conditions, we shall now outline thetechnical approach followed in this paper.80

distribution. Such an analysis provides us with thedesired information about the energy and the instantan-eous frequency of the received signal. This information isall weneed for carrying out detection (by comparing theenergy with a suitable threshold) and focusing (by phase-compensating the signal by the estimated instantaneousphase). Owing to the presence of disturbances, the time-frequency analysis must be carried out after havingimproved as much as possible the signal-to-disturbanceratio, otherwise the estimation could be seriouslydegraded. The corresponding filtering, however, alters thesignal behavior and then affects the successive time-frequency analysis. It is then necessary to evaluate thefiltering performance, in terms of improvement of thesignal-to-clutter ratio and of the frequency response, inorder to be able to interpret correctly the time-frequencyanalysis.The aim of this paper is to provide the performance, interms of signal-to-disturbance ratio, filter frequencyresponse and parameter estimation accuracy, of theoptimal scheme, according to the maximum likelihoodcriterion, for detecting and focusing moving objects witha synthetic aperture radar. These performance param-eters provide the theoretical limits of detectability andestimation accuracy, for a given operating condition(PRF, antenna beamwidth, number of integrated pulses,motion parameters, etc.). These limits are useful forevaluating the possibility of detecting moving targets, inassigned operating conditions, and as a comparison withrespect to sub-optimal but simpler processing schemes. Inaddition, the analysis of the optimal filter frequencyresponse provides the key for interpreting the time-frequency analysis, performed by the Wigner-Ville dis-tribution.

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The optimal detection and parameter estimationschemes are based on a model of a useful signal (echofrom the moving object) and disturbance (echoes fromthe background or clutter plus receiver thermal noise). Amodel of the received signal is first provided in Section2,to emphasise the main differences between the echoesfrom the moving object and the fixed scene, namely: (i)the phase modulation law and (ii) the time and spatialcorrelation. On the basis of these differences, the optimalsolution, in the sense of maximum likelihood, is providedboth for the detection (Section 3) and signal parametersestimation (Section4).The structure of the optimal detection scheme is basedon a bank of filters, each one matched to a set of signalparameters, namely the Doppler frequency and theDoppler frequency rate. The knowledge of which filtergives the maximum output provides a first coarse estim-ate of the signal parameters. If a finer estimate is requiredfor focusing the moving object, a closed loop techniquecan be used. The performance of the closed-loop estima-tor, given in terms of lock-in band and noise sensitivity,are provided in Section4.The main drawback of the optimal scheme is relatedto the need for implementing a bank of N (in Dopplerfrequency) by M (in Doppler frequency rate) filters. Sub-optimal approaches are then considered in Section 5, forreducing the number of filters in the bank.

The bottleneck of the optimal detection scheme basedon the maximum likelihood principle, beyond the com-putational and hardware complexity, is that it requiresthe assumption of a parametric model of the usefulsignal. The method works only if the model matches thesignal. This assumption greatly affects the applicationpotential. To cover a wider range of cases, we couldassume more complex models, with a higher number ofparameters, but this would directly affect the processingcomplexity. In this paper, a simple model for the usefulsignal is assumed: a linear frequency modulated signal.This model has only two parameters the mean Dopplerfrequency and the Doppler-rate, or rate of variation ofDoppler with time. A general approach, which does notrequire any modelling of the useful signal modulationlaw, is given in the second part of the paper. The limits ofapplicability of the approach proposed there, in terms ofsignal-to-disturbance ratio and signal parameter estima-tion, can be understood on the basis of the performancegiven here. Simulations of the overall detection andfocusing scheme are also given in the second part.2The point in the radar chain we are interested in, asregards the modelling of the received echo, is after therange compression and before the formation of the syn-thetic aperture or, equivalently, before the crossrangecompression. It is useful to make the range compressionfirst, from the detection point of view, because, in thisway, the range resolution cell decreases, therefore themoving object echo has to compete with a smaller cluttercontribution. The same principle applies to the cross-range compression, therefore it would seem obvious toanalyse the signal after both compressions. However,after the crossrange compression it becomes more diffi-cult, if not impossible, to distinguish between echoes frommoving objects and echoes from the fixed scene. There-fore, in the following, we shall start our analysis beforethe crossrange compression. All the signal variationshave to be intended on a pulse-to-pulse basis.

Model of the received echo

IEE P ROCEEDINGS-F, Vol.139, NO. , FE BRUARY 1992

The relative distance r(t) between the radar and thetarget, at the time instant corresponding to the nth trans-mitted pulse, is approximately [3] :

whereRo=radar-point distance in t =0U, =range component of the relative velocityU, =crossrange component of the relative velocity,a, =range component of the relative acceleration.The approximated form of the distance has been used formaking explicit the relationship between the motionparameters and the main components of the phasemodulation induced on the received echo. All the abovecomponents of the motion are relative to the instantt =0. Eqn. 1 applies to moving and fixed targets. Forfixed targets, the parameters in the equation coincidewith the radar motion parameters, whereas for movingtargets they are given by the difference between thecorresponding parameters of radar and target motions.The signal received from the radar presents a phaseshift proportional to the range variation, through thefactor 47r/I, where I is the transmission wavelength. Con-sequently, the echoes from the fixed scene and themoving objects present a quadratic phase, or linear fre-quency, modulation.The signal corresponding to a pointlike moving objectcan then be modelled as a chirp:

and

s(t)=exp (j27tfDt)exp (j.rrpDt2) (2)where the parametersf, and p , depend on the objectmotion:f~=(2/A)UrPD =(&'ARo)(uf +Roar)

A common approach for obtaining a high crossrangeresolution is the dechirping technique, which consists inremoving the quadratic phase term present in eqn.2, andthen applying a fast Fourier transform (FFT) to theresulting signal. Each FFT output corresponds to a fixedcrossrange co-ordinate. After mixing the received signalwith a reference signal to remove the quadratic phaseterm, the signals from fixed points are sinusoidal, whilethe signal corresponding to a moving object is still achirp because its quadratic phase term is not matched tothat of the reference. The situation is shown in Fig. 2.

fixedf~t scenemovingobject

fD

I movingobject

Dopplerfilterbank filter 1

filter N

I referencesignalFig. 2 Dechirping technique

On the left side of the Figure, is represented thereceived signal, in the frequency-time domain. The usefulecho is that with a different frequency modulation(different slope), due to the object motion. This signal is81



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mixed with a reference signal whose frequency modula-tion is matched to the fixed ground returns. Therefore,after the mixer, the signals from the fixed scene are allsinusoids, whereas the signal from a moving target stillhas a linear frequency modulation, whose slope is the dif-ference between its original slope and the slope of thereference signal. After the mixer, the computation of theFFT corresponds to processing the signal through thebank of filters shown in the same Figure. The contribu-tions from the fixed ground are separated, within thelimits of the Doppler resolution, whereas the signal fromthe moving object gives contributions over several filters,thus causing the smearing of the relative image.After the mixer, the echo from the pointlike object isstill a chirp, but with parameters equal to the differencebetween the received signal and the reference signal. Bycombining eqns. 1 and 2, the bandwidth of this chirpdepends on the relative radar-target crossrange velocityand range acceleration.The signal corresponding to the fixed scene can bemodelled as a random process, whose power spectraldensity follows a behaviour proportional to the antennaradiation pattern (considered in transmission andreception). For a uniformly illuminated antenna, theazimuthal antenna power radiation pattern is:

F(8)=G sinc[n(D/A) sin 81 (3)whereD is the antenna dimension in azimuth. Since theDoppler frequency is related to the angle8by the law:f =(244 sin8 (4)

G( f )=Go sinc2[n(D/2u)f] ( 5 )the Doppler power spectrum is:

This pattern has to be considered twice, in transmissionand in reception, therefore the clutter power spectrum is:G,(f) =G$ sinc4[n(D/2u)f] (6)

B, =(2u/D) (7)The clutter bandwidth is:

For obtaining the optimal weights, it is necessary toknow the clutter correlation coefficient. This can beobtained by inverse Fourier transforming the power spec-tral density:

AT)= 1I (B,T - 2)( 0 elsewhere

whereB, is the clutter bandwidth.Actually, the clutter spectrum depends also on theSAR operating mode. With reference to Fig. 3, in thespotlight mode (Fig. 34, the Doppler histories all start atapproximately the same time instant. Each history has aconstant amplitude since, owing to the antenna steering,the echo from each point has a constant antenna direc-tivity gain during the overall observation time. However,the antenna scanning causes a broadening of the clutterspectrum.Conversely, in the stripmap mode (Fig.3b), the ampli-tude of each return varies during the observation intervalaccording to the azimuth antenna radiation pattern. Ifthe integration interval is chosen in correspondence with

the 3 dB antenna gain, this amplitude weighting has aneffect practically negligible.The value of the clutter bandwidth is found as a resultof all the above considerations. In the following the can-

radaUUUl -

Fig. 3historiesD Spotlightmodeb Stripmapmode

SA R operating modes and corresponding Doppler frequency

cellation performance will be analysed by considering thebandwidth as a variable parameter.The clutter spectrum has been modelled with referenceto a homogeneous scene where each resolution cell iscomposed of a certain number of independent scatterers.Therefore, according to the central limit theorem, theprobability density function is assumed to be Gaussian.Because of the homogeneity hypothesis, the standarddeviation of the Gaussian is constant. The model couldbe improved, trying to match more realistic situations, byconsidering possible nonhomogeneous scenes or the pre-sence of dominant scatterers by assuming other distribu-tions, such as the K-distribution [l2], for example.3 OptimumdetectionI n this Section, the optimal detection scheme, accordingto the maximum likelihood principle, will be derived forthe model of the received signal given in the previousSection.3.1 Appl ica t ion of classical detection theor y to SA RTwo hypotheses can be formulated about the receivedsignal: it contains the echo from a moving target(hypothesisH,) or not (hypothesisH,). In both cases, thereceived signal contains the echo from the background(clutter), plus the thermal noise. If we indicate by s(8), eand n the vectors of time samples, collected at intervalsT = l/PRF, corresponding to moving target, clutter andnoise, respectively (the dependence of s and 8 indicatesthat weknow the useful signal only in a parametric formand the vector 8 contains the unknown parameters f DandpD), he two hypotheses are:

H ,: r =c + nH , : r =s(8) +c +n (9)

wherer is the vector containing the received samples. Thevector 8 contains the parametersf, andpD, ntroduced ineqn. 2.As is known from the classical theory of detection[13,141, the optimal detection scheme, in the maximum likeli-hood sense, is based on the comparison of the likelihoodratio with a threshold. In the case in which the dis-turbance, clutter and noise, has a Gaussian probabilitydensity function (PDF), the optimal scheme reduces to82 I E E PROCEEDINGS-F, Vol.139,NO . , FEBRUARY 1992



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the expression:max IrTR- s* @)I 2 Y8

whereR is the disturbance correlation matrix and y is asuitable threshold, ( indicates transposed, ' inversematrix and * conjugated). Since the vector 8 is notknown, the previous equation must be evaluated for allpossible values of 0. The maximum of all the obtainedresults has then to be compared with the threshold y .Expression 10means that the received signal is linearlyprocessed by multiplying it by the weighting vector

ratio (CNR) has been assumed equal to 20 dB. The targetmean Doppler frequency fD is equal to PRF/4 and itsbandwidth is equal to one fifth of the PRF. The filterfrequency response, as expected, tends to reproduce thetarget spectrum in the region outside the clutter band,whereas it shows a deep null within the clutter band.The noise power affects the filter responses, as shownin Figs. 5 and 6. These Figures refer to the same case as

The computation of the optimal filter weights requiresthe knowledge of the parameter vector 8. Of course, theparameters are not known a priori, therefore they mustbe estimated in some way. Two kinds of approaches canbe followed: (1)an open-loop technique in which a bankof filters, each one matched to a set of parameters, is usedto process the received signal: the filter with themaximum output corresponds to the best matchedparameters; (2) a closed-loop technique which estimatesthe signal parameters in order to minimise some errorfunction.In the following subsections, we shall provide the per-formance of the optimal filters, assuming the parametersvector 8 known. The techniques for estimating theparameters will be shown in Section 4.3. 2 Optimum filter responsesIn this Section we give the performance that can beachieved in the ideal conditions, in which the signalparameters are supposed known. They provide the the-oretical limits of detectability.To understand the behaviour of the optimal filter, it isuseful first to analyse its frequency response. The receivedsignal is supposed to be composed of the sum of a usefulsignal, modelled as a chirp, plus a correlated disturbance(clutter), whose power spectral density (PSD) has beengiven in the previous Section, and thermal noise.An example is reported in Fig. 4 where the clutterPSD, the target spectrum and the relative matched filter

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0 0.2 0.4 0.6 0.8 1.0normalised frequencyFig. 4 Optimalfilter frequency responseTarget mean Doppler frequency=PRF/4; target bandwidth=PRFj5;CN R=20dBU Filter responseb Clutter P SDc Target spectrumresponse are shown. This example refers to the case of aPRF equal to 8 times the clutter bandwidth B, and anumber of samples N equal to 64. The clutter-to-noiseIEE PROCEEDINGS-F, Vol . 139, N o. , FE BRUARY 1992

0 0.2 0.4 0.6 0.8 1onormalised frequencyFig. 5 Optimalf il ter requency responseTarget mean Doppler frequency=PRF /4; target bandwidth=PRF/S;CN R=0dBa Filter responseb Clutter P SDc Target spectrum

normalised frequencyFig. 6 Optimalfi lterfrequency responseTarget mean Doppler frequency=PRF/4; target bandwidth=PRF/S;CN R =40dBU Filter responseb Clutter P SDc Target spectrumFig. 4, except for the value of the CNR, equal to OdB(Fig. 5 ) and to 40dB (Fig. 6). In the first case, the noisePSD covers the sidelobes of the clutter PSD, thereforethe filter response is not affected by the sidelobes. In thesecond case, the clutter sidelobes overcome the noisePSD and then considerably affect the filter response.3.3 Improvement factorA parameter which allows us to assess the filters per-formance is the improvement factor, defined as the gainin signal-to-disturbance power ratio (SDR) achieved byfiltering the received signal. The disturbance is given bythe sum of noise and clutter. If we indicate by S, C andNthe powers of signal, clutter and noise, respectively, andtake into account the statistical independence betweennoise and clutter, the improvement factor ( I F ) is defined

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as

where G and CA indicate the signal gain and the dis-turbance attenuation.The improvement factor is plotted in Fig. 7as a func-tion of the target relative mean Doppler frequencyf, , or

mu 30

0 0 0.1 0.2 0.3 0.4 0.5f D

Fig. 7different values of target bandwidth, both normalised to the PRFClutter bandwidth=PRF /8 ;number of integrated pulses=64different values of i ts relative bandwidth B, (both meanfrequency and bandwidth are normalised with respect tothe PRF). The PRF is equal to 8 times the clutter band-width and the number of samples is 64. The clutter PSDis the sinc4(x). The ripple on this curves is due to theclutter spectrum sidelobes. In this example, the maximumCA is equal to the input clutter-to-noise ratio, equal to20dB, while the maximum G,, due to the coherentintegration of 64 samples, is about 18dB. Therefore, themaximum achievableIF is38dB.Starting from the behaviour shown in these curves,some considerations of general validity can be made.First of all, IF increases with the mean Doppler fre-quencyf, of the moving object signal, as could be easilyguessed, since, as the frequency increases, until the valueof PRF/2, it is easier to separate the target spectrum fromthe clutter spectrum. Secondly, for low values of f,, theIF is bigger for high values of the target bandwidth. Atlow frequencies the signal spectrum is superimposed onthe clutter spectrum,so that it is better to have the signalspectrum as large as possible because, in such a case,there will be some frequency component outside theclutter spectrum which can be properly amplified,without amplifying the clutter. Conversely, at high fre-quencies, it is better to have the signal bandwidth asnarrow as possible, so that it is easier to separate theclutter spectrum from the signal spectrum.It is important to note that there exists an improve-ment in the signal-to-clutter ratio also when the signalspectrum falls exactly inside the clutter spectrum, that isfor f D equal to zero, as shown in Fig. 7.This point hasmotivated the search for ad hoc filtering instead of limit-ing the analysis to the conventional Doppler frequencyprocessing.The IF greater than 0dB whenf, andB, are equal tozero,which corresponds to the case of a stationary point-like target, can look like quite strange. The value of 8dB

Improvement factor against target mean Doppler frequency for

obtained in that case, is entirely due to the coherentintegration gain. In fact, being the number of samplesintegrated coherently equal to 64, the filter has the capa-bility of dividing the unambiguous Doppler bandwidth(PRF) in 64 intervals. Since the clutter bandwidth is oneeighth of the PRF, the signal passes through the filterwhile about 7/8 of the clutter spectrum mainlobe arestopped, making possible a coherent integration gain ofabout 8 dB.The PRF greatly affects the filtering performance. Thehigher the PRF, the higher theIF.This property is easilyunderstood since, as the PRF increases, it increases theregion in the frequency domain where there are no sig-nificant clutter contributions, therefore making the fre-quency fi ltering easier. However, the increaseof the PRFproduces a corresponding decrease of the maximumunambiguous swath. The choice of the PRF must there-fore result from a trade-off between the performanceshown in the previous curves and the coverage needs.The performance is affected by the number of integ-rated pulses as well, in the sense that the higher thenumber of pulses, the better the performance; but at theexpense of increased complexity.The detection capabilities can be directly derived fromknowledge of the IF and the input signal-to-clutter ratio.In fact, in the case of a process with Gaussian distribu-tion, the probability of detection P ,, the probability offalse alarm P,, and the SNR are related in a knownform. Once having specified the P,., , which will deter-mine the threshold value, theP , as a function of theP,,and the SNR, can be evaluated by Marcum's curves [lS].In practical applications, having specified the radarcross-sections of the targets of interest, together with allthe radar parameters, the radar equation allows us toexpress the signal-to-disturbance ratio at the receiverfront-end. The final SNR can then be evaluated by takinginto account the SNR gain shown above as a functionof the mean Doppler frequency and the Doppler rate.Hence, from Marcum's curves, we are able to express thedetection probability as a function of the final SNR, andthen as a function of the mean Doppler frequency andDoppler rate. This relationship allows us to evaluate theminimum detectable target velocity, defined as the veloc-ity under which theP , is less than a given value.

C

4After having detected a target, the formation of a properimage requires knowledge of the signal modulationparameters. We shall now discuss the performance of theclassical maximum likelihood (ML) approach, for thecase in which the useful signal can be modelled as achirp. A general criterion for estimating the signal modu-lation law, or the variation of the instantaneous fre-quency with time, without making any assumption aboutthe kind of modulation, will be given in the second partof the paper.Two general approaches can be followed: an open- ora closed-loop approach. The first technique is moregeneral than the second; it does not require to be initial-ised, but it involves a higher computational cost andhardware complexity, depending upon how many filtersare used. The higher the number of filters, the better isthe estimation, but the computational burden increasescorrespondingly. The second technique is simpler since itis based on a single circuit and yields better estimations,but it needs to be initialised by some initial guess: theclosed-loop estimator is able to lock-in the desired

Estimation of th e signal parameters

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parameters within a certain initial error only. The initialguess, therefore, must be accurate enough to give an errorless than that initial error, otherwise the loop convergestowards a meaningless value.These considerations together lead to the use of bothtechniques: the open-loop technique provides the initialcoarse estimation and initialises the closed-loop circuit,which gives the finer estimation.

4.1 Open-loop estimationThe optimum detection scheme is based on the imple-mentation of a bank of filters whose weights are given byeqn. 11 . Each filter is matched to a pair of signal param-eters. Since the useful signal has been modelled as achirp:

s(t)=A exp (j2nfD ) exp ( npD 2) (13)the vector 0 is constituted by the two parameters D andP D .A possible way for choosing the number of filters inthe bank consists in evaluating the range of possiblevalues forf D and pD and dividing them for the respectiveresolutions inf D andpD .The frequency resolution 6fD isinversely proportional to the time observation interval T ,whereas the resolution dpD n frequency slope is inverselyproportional to the square of T [16]. If the ranges ofpossible values for f D and pD are AfD and ApD, respec-tively, the overall number of filters is:

where the3 dB resolutions are, respectively [16] :dfD =0.88/T (15)

dp D =1.8/T 2 (16)The set of filters in the bank will be then characterised bycoefficients:

sk, ={1, exp(j2nvk + nym ,exp( j h v k +4nym), ,eXp[ j 2~(N )vk +jx(M - )2ym]}

for k = 1, 2, ...,N andm= 1, 2, ..., M (17)with vk and y m chosen uniformly spaced and integermultiples of the resolutions given above.The number of filters could be excessively high insome cases of practical interest, therefore some approx-imations will be discussed later on in order to reduce thatnumber within acceptable performance degradations.4.2 Closed-loop estimation: the ML approachIf the signal is supposed known in a parametric forms ( t ; 8)and the disturbance is a white Gaussian noise, theML equations are[ 31:

where r(t) is the received signal, T is the duration of theobservation interval, and B i are the parameters of thevector 8 In the case of a coloured noise, a whitening ofthe disturbance must be performed first. In such a case,the expression of the M L estimator assumes the sameform as before, except for the expression of the usefulsignal, which must be modified according to the pro-cessing operated by the whitening filter.IEE PROCEEDINGS-F, Vol. 139, No. 1, FEBRUARY 1992

In our case, the model of the useful signal i st:s ( t ; D ,pD )=A exp (j2nfD ) exp( j w D 2) (19)

Therefore the following two M L equations, for the con-tinuous case and for a white noise, are obtainedT i2j- i 2Ti 2s-T I2

tr(t)s*(t;Mr. > P M L ) dt =0t2r ( t )s*( t ; fML,M L ) t =K

(20)(21)

where K = T 3/12. A closed-loop system for the simul-taneous estimation of f M LandpM L s shown in Fig. 8.

update PMLPML I *

I h ' I

Fig. 8Doppler frequency rateMaximum likelihood estimator of mean Doppler frequency and

It is interesting to interpret the previous equations in aphysical sense. The first equation has, on the firstmember, an estimation of the centroid of the signalresulting from mixing the received signal with a referencesignal. Since the interval is symmetrical with respect tothe origin, it is clear that, when the reference signal isperfectly matched to the one received, the centroid mustbe zero. The second equation, on the other hand, rep-resents the measure of spreading caused by error in theDoppler rate. The M L estimator then tries to minimisethat spreading and to make it equal to its minimumvalue, given by K .The performance of these estimators in the presence ofnoise can be evaluated by means of the Cramer-Raobounds. Fisher's information matrix, in this case, is com-posed of the four elements[ 31:dt ] i = 1, 2 (22)

where8, = D , 8, =pD and No is the noise power spec-tral density.It is easy to see that:2n2J , , =- 3A 23N0

. I l2= 2 , =0J 2, = T5A240N,

Therefore the variances of our estimates are:

t T he amplitude is assumed constant over the observation time. Thissimplification isoften implicitely assumed in SAR signal pr ocessing.85



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It is important to notice that, being Fisher's informationmatrix diagonal, the two estimators are decoupled.The standard deviations (SD) can be expressed interms of the signal-to-noise ratio (SNR),defined as:

By substituting eqn. 25 in eqns.24, weobtain:1 1

Of D =f J(sA) *3897 J (SNR)As might be expected, the SDs of the errors are inverselyproportional to the integration time T and to the SNR.It is important to point out that the two SDs are directlyproportional to the corresponding Doppler and Dopplerrate resolutions. The accuracy improvement due to theclosed-loop estimation over the open-loop estimation isthen directly proportional to the square root of thesignal-to-noise ratio.To analyse the lock-in capabilities of the previous twoloops, it is useful to consider them separately, byassuming that one of them has converged to the rightvalue. If the Doppler rate ,! is supposed perfectlymatched, we have only the first equation. The analysis ofthe loop can be done by examining the effect of an errorin the estimated frequency on the amount measured fromthe loop, represented by the first member of eqn. 20. Thatbehaviour is sketched in Fig. 9,where the modulus of the

0.70[I0.50 i \ I iI I IAf

Fig. 9 Modulus of error in the loop or the estimation of mean Dopplerfrequency against the Doppler frequency misalignment normalised to thenumber of integrated pulses Nerror is reported. From that Figure, it is possible to findthe lock-in band of the frequency loop. As could beexpected, that band is inversely proportional to theobservation interval.As regards the loop for the estimation of the Dopplerrate, by following a similar approach, weobtain an errorfunction, given by the difference between the first and thesecond member of eqn. 21, sketched in Fig. 10.From thiscurve, the lock-in band turns out to be inversely pro-portional to the square of the observation interval T, bya factor of about six. The relationship with the intervalduration is coherent with the resolution in Doppler rate,which is inversely proportional to the square of T [161.The two loops must then be initialised by a valuemore accurate than the previous lock-in bands. On the

other hand, these lock-in bands are of the same order asthe resolutions provided by the filters described in theprevious paragraph, therefore the two estimation tech-niques can be used in cascade. To widen the lock-inbands, we could work initially with a reduced integrationtime T. The closed-loop circuit could then track thedesired parameters, with the accuracy given by eqns. 26.

APFig. 10 Error in the loop for the estimation of the Dopplerfrequencyrate against the Doppler-rate misalignment, normalised to the square ofthe number of integrated pulses NThis estimate can be used as an initial guess for the samecircuit, but using the overall available integration time inorder to get the maximum accuracy. This criterion func-tions only if theSNR is high enough to compensate forthe degradation of the estimation accuracy and for SNRlosses owing to the decrease in the integration time. Theinitial integration time can be chosen as a compromisebetween the desired widening of the lock-in band and theSNR losses.

5 Suboptimum detection schemesThe straightforward application of the optimal theory istroublesome from the point of view of computationalcost and hardware complexity: the number of filtersgiven in eqn. 14could be excessively high. A first strongsimplification is then necessary for reducing this numberwithin acceptable performance degradations.The optimal processing canberearrangedsoas to dis-cover some possible simplifications. Essentially, theoptimal filtering can be split into a clutter cancellationfollowed by a coherent integration. The integration isperformed through a set of N +M filters, instead of theN . M filters of the optimal approach. The simplificationsof the hardware produce some performance degradations,which will be evaluated.According to eqn. 11, the quantity to be comparedwith the decision threshold is:While, from the mathematical point of view, the order ofthe products is unimportant, from the implementationpoint of view, a different ordering can lead to more orless simple filtering structures. If we perform the multipli-cation of r by R-' first, a different perspective can begiven to the overall filtering process.First of all, if the clutter power spectrum density(PSD)can be modelled by a first order autoregressive process (a1-pole model), the expression of R- ' can be given in a

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closed form:1R-1 =-1 -p2

0 \- p O i ...- p l+p Z -P 0 . . .iI n -,, 1 -wherep is the clutter correlation coefficient at lag T = 1/PRF. Hence it is not necessary to use any matrix inver-sion algorithm and the only quantity to be estimated isthe one lag correlation coefficient. Owing to the particu-lar structure of R- ' , the vector y resulting from theproduct of R-' by r has elements given by:

Y(1) =41) - r(2)y(i) =(1 +p2)r(i)- (r(i - ) +r(i +1))

y(N) =r (N)- r(N - )i =2, 3, ..., N - 1 (29)

Therefore, except for the extreme samples, the multiplica-tion of the received samples by the inverse of the cluttercorrelation matrix can be obtained by simply filtering thereceived signal through a three-tap transversal filter, withcoefficients- , 1 +p2 and -p. This filter reduces to theclassical three-pulse moving target indicator (MTI) [ 51whenp is equal to 1.In this way, the multiplication of r by R-' has beeninterpreted as a clutter cancellation. The remainingmultiplication for s*, present in eqn. 27, can be thoughtof as a coherent integration.The generic vector sk,,, s expressed by eqn. 17. Themultiplication of a vector y by s k , , , s equivalent to thefiltering of y through a bandpass filter whose central fre-quency is vk and whose bandwidth is proportional to y, .If the target and the clutter spectra are not superimposed,the use of a bank of filters centred over a set of fre-quencies uniformly spaced, with bandwidths so as tocover all the unambiguous Doppler band, allows us toseparate the useful signal contribution from the clutter.We can use a set of filters whose weights are

sk,O =(1, exp(j2zVk),exp( j4zvk)9 . ,exp( j 2 W - l v k ) ) (30)

If the vk are uniformly spaced, the filtering through thisbank of N filters is equivalent to the computation of theDFT of the input, which can be efficiently performed byan FFT algorithm. The number of filtersN can be chosenas the ratio between the PRF and the clutter bandwidth.An efficient way for synthesising the set of filters is givenin [SI .These filters do not allow discrimination betweenuseful and disturbing signals if these have spectra super-imposed. In this case a set of M Doppler-rate filters canbe used for improving the signal-to-clutter ratio beforethe detection. The weights of these filters are chosen as aparticular case of eqn. 17, for vk =0. Therefore, theweights are:

SO, m =1, ~ X PjzY,,,h~ X Pj 4 ~ m ) , .,~ X P+jn(M - 12ym)) (31)

IEE PROCEEDINGS-F, Vol . 139,NO. 1, F EB RUA RY 1992

The performance achievable by this bank of filters isshown in Fig. l l a . For each signal Doppler frequency vand Doppler rate y , the curves represent the maximumimprovement factor achievable over all the filters. Thecase considered refers to a sinc4(x) clutter PSD, and to anumber of integrated pulses equal to the ratio betweenthe PRF and the clutter bandwidth (N=16). The

0:% 0.3 0.4a J 0.1

I I I I I0.520 b '0 0.1 0.2 0.3 0.4fDa

4 0 r

-2 0 1 I I I I J0 0.1 0.2 0.3 0.4 0.5bfD

Fig. 11for different target Doppler rates(I Suboptimal filter bankb Optimal filter bank

Improvement factor against target mean Doppler frequency,

optimal improvement factor is shown in Fig. l lb , as acomparison term. It can be observed that there are sens-ible performance losses when the useful signal spectrumis superimposed onto the clutter spectrum and its band-width is zero. On the other hand, this case is not of inter-est, because the useful signal cannot have both meanDoppler frequency and Doppler rate equal to zero,because such a case corresponds to a fixed object.When the signal and clutter spectra are not superim-posed, there are performance losses increasing with thesignal Doppler rate, because the suboptimal filters arenot matched to signals having a Doppler rate differentfrom zero. To improve performance in these cases, wecan get a coarse estimate of the Doppler rate by analys-ing the outputs of the FFT bank of filters. In fact, we canestimate the signal bandwidth by evaluating the numberof filters centred on the signal spectrum (counting thefilters with the highest outputs). Since the bandwidth isdirectly proportional to the Doppler rate, we have anestimate of the Doppler rate. This parameter can be usedfor rephasing the signal, before filtering it with the FFT.The corresponding scheme is sketched in Fig. 12. Thesignal bandwidth is estimated by counting the number of87



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filters with the highest moduli, and dividing that numberby the total number of filters. The Doppler rate y is equalto the bandwidth divided by the number of integratedpulses. If a finer estimate of the Doppler rate is desired,the coarse estimate obtained by this simple method canbe used to start up the closed-loop technique shown inFig. 8.

6thresholdFig. 12Doppler rateSuboptimal filter bank, with a coarse estimation of the target

6 ConclusionsIn this paper, the performance of the optimal scheme fordetecting and focusing moving targets with SAR has beenderived. This provides the basis for assessing the detec-tion and focusing possibilities in prescribed operatingconditions, and as a comparison term for analysing sub-optimal schemes.As regards the detection problem, it has been shownthat a considerable gain in the signal-to-disturbance ratiocan be obtained if the processing takes into account notonly the Doppler frequency but also the Doppler rate,especially when the target spectrum is superimposed ontothe clutter spectrum. This entails the use of a bank offilters in Doppler and Doppler rate. The filter bank alsoprovides a coarse estimate of the target Doppler param-eters. The method for obtaining a finer estimate, based ona closed-loop technique, is also provided. While theopen-loop approach provides an accuracy in the estima-tion of the Doppler and Doppler rate parameters in theorder of the inverse, or the square of the inverse, of theobservation time, respectively, the closed-loop techniqueyields an accuracy improvement proportional to thesignal-to-noise ratio.The range migration problem has been faced byresorting to a double-range resolution approach: an esti-mation of the phase history was carried out on the low-resolution data (where the resolution was low enough tomake migration negligible); the estimated parameterswere used for compensating the range migration on thehigh resolution data. An advantage related to the use of acoarse range resolution is the possibility of monitoring alarger area. The disadvantage is that the useful signal hasto compete with a stronger disturbance.In the second part of the paper the optimal structure istransferred onto the time-frequency domain, where theextraction of information about the instantaneous fre-quency of the received signal is easier. Mapping from the

received time waveform onto the corresponding time-frequency representation is carried out by the Wigner-Ville distribution.

7 AcknowledgmentsThe author gratefully thanks Dr. A. Farina of Alenia SpAfor having followed this work from the beginning and forcontinuous suggestions and help up to the final form ofthis paper.Additionally, he wishes to thank the anonymousreferees for their valuable comments and suggestions thathave undoubtedly improved the paper.The work reported in this paper was funded by AleniaSPA.

8 References1 CU TRONA , L.J.: Synthetic aperture radar, in SKOLNIK, M.I.(Ed.): Radar handbook (McGraw-Hill , New York, 1970),chap. 232 AUSHERMAN, D.A., et al.: Developments in radar imaging,IE EE Trans., 1984, AE SU ), (4),pp. 3634333 RA NEY , R.K.: Synthetic aperture imaging radar and movingtarget, IEEE Trans., 1971, A E SI , pp. 499-5054 FR EE M AN, A .: Simple M TI using synthetic aperture radar. Pro-ceedings of IGA RSS 84Symposium, ESA SP-215, 19845 FRE EM AN, A.: A digital prefilter for M TI with SAR. P r o d i ngsof International Conference on Digital Signal Processing, Firenze,Sept. 1984.6 FREEM AN, A., and CURR IE, A .: Synthetic aperture radar (SAR)images of moving targets, GE C J . Res., 1987,5, (2),pp. 1061157 WERNESS, S., CARRA RA, W., J OY CE, L., and FRA NCZA K , D.:M oving target imaging algorithm for SA R data, IEEE Trans.,1990, A E S26, (I), pp. 57478 WA RD, K.D., TOUG H, R.J.A., and HAY WOOD, B.: HybridSAR-ISAR imaging of ships. IEEE International Radar Con-ference, Arlington, May 7th-loth, 1990,pp. 64-699 WA LK ER, J .L .: Range-Doppler imaging of rotating objects, I E E ETrans., 1980, AE S16, (l ),pp. 23-5210 CH EN , C.C., and ANDREWS, H.C.: Target-motion-indud radarimaging, I EE E Trans., 1980, AE S16, (l ),pp. 2-1411 BARBAROSSA, S., and FARINA, A.: Space-time-frequency pro-cessing for detecting and imaging moving objects with SA R, sub-mitted for publication to the IEEE Trans., AE S12 PUSEY , E., J AK EM AN, P.N .: A model for non-Rayleigh sea echo,

IE EE Trans., 1976, AP 24, (6),pp. 806-81413 VAN TRE ES, H.L .: Detection, estimation, and modulation theory(J ohn Wiley, New Y ork, 1968)14 FA RINA, A.: Optimised radar signal processing (Peter Peregrinus,UK , 1987)15 SKOLNIK, M.I. : R adar handbook (McGraw-Hill , New Y ork,1970)16 BARBAROSSA, S.: Doppler-rate filtering for detecting movingtargets with synthetic aperture radars, i n HUDDLESTON, G.K.,TA NENHAUS, M ., and WIL L IA MS, B .P. (Eds.): M illimiter waveand synthetic aperture radar, Proc. SPI E, 1989,1101, pp. 140-14717 BARBAROSSA, S., and FARINA, A.: Detection and imaging ofmoving objects with synthetic aperture radar. Part 2; J oint time-frequency analysis by Wigner-Ville distribution, IEE Proc.-F, 1992,139, (l),pp. 89-97

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