Space Time Freq Process Barbarossa Farina

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I. INTRODUCTION

Space-Ti me-FrequencyProcessingof Synthetic ApertureRadar Signals

SERGIOBARBAROSSA, Member, IEEEUniversity of RomeALFONSO FARI NA, Member, IEEEAlenia S.p.A.Italy

The subject of this work is thedetection and high resolutionmicrowave imaging of objects movingon the ground and observedby an airborne radar. The proposed approachis based ona combined space-time and time-frequency processing. Thespace-time processing makes use of a linear array antenna and ,exploits the radar motion for filtering the received echoes in orderto improveas much as possible the signal-to-disturbance atio.The signal is then mapped onto the time-frequency domain, bycomputing ts Wigner-Ville distribution, for a further fi lteringand for estimating its instantaneous frequency, necessary for theformation of a high resolution image of the moving object.

Manuscript received July 25, 1991;revised September3, 1991 andMarch3, 1993.IEEE L og No. T-AESL30Ul5482.Authors addresses:S. Barbarossa, INFO-COM Department,University of RomeLa Sapienza,Via Eudossiana 18, 00184 Rome,Itay; A. Farina, Alenia S.p.A. (Aeritalia& Selenia), Radar Dept.,Va Tibunina, Km 12,400,00131 Rome, Itay.0018-!?251/94/$4.00@ 1994 IEEE

The synthetic aperture radars (SA R) [l] chieve ahigh cross-range resolution by exploiting the relativemotion between the radar, placed on an airborneor a spaceborne platform, and the observed scene.The knowledge of the motion allows the synthesisof a long antenna, along the flight direction, and theconsequent high spatial resolution. The presence ofmoving targets within the observed area, however,might be not easily detectable and, even in case ofdetection, the formation of the corresponding highresolution image is not immediate, because the lack ofknowledge of the relative radadmoving target motiondoes not allow the synthesis of a wide antenna withrespect to the target itself. In fact, the synthesis of along aperture is obtained by the coherent processingof the echoes received during the overall observationinterval. The successive echoes are linearly combinedafter a proper phase shift necessary for compensatingthe phase shift induced by the motion. Therefore, theechoes can be properly combined only if the motionis known. In such a case, the energy corresponding tothe target echo concentrates in the due position withinthe scene and produces a focused image of the target.When the lack of knowledge of the target motion doesnot allow a correct coherent processing, the usefulenergy spreads out over a wider region, thus causingtwo shortcomings: a lower detection capability and alower recognition capability.The detection of moving targets, observedagainst the background of a stationary scene, and theconsequent imaging has to be carried out throughtwo main steps: 1) a filtering of the received signalin order to improve as much aspossible the signal(echo from the target) to disturbance (echo from thestationary scene) ratio, and 2) an estimation of thephase modulation law induced by the target motionto properly rephase the received signals before thecoherent combination necessary to synthesize the longaperture with respect to the target.The problem of filtering the radar echo in orderto cancel out the returns from the ground has beenextensively treated in the l iterature referring toairborne moving target indicators (A MTI) [2-4]. Areliable detection of objects moving on the groundrequires the preliminary cancellation of the echo fromthe background. If one antenna is available, the onlypossibil ity to cancel the undesired contributions isbased on a Doppler filtering. However, this produces aseries of shortcomings, namely: 1) a reduced detectioncapability of targets having a small range velocity (i.e.,target moving parallel to the radar), 2) the need ofusing a high pulse repetition frequency (PRF) formaking available a certain visibility region in thefrequency domain, to achieve a consistent cancellationdegree, while it is desirable to work with a PR F as lowas possible, to avoid coverage losses and an excessive

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increase of the data rate, and 3) the so-called azimuthposition uncertainty problem [4],according to whichthe measure of the Doppler shift alone is not sufficientto evaluate the azimuth position of the target withrespect to the fixed ground.more then one antenna, or a phased array, mountedalong the flight direction, by a technique known asdisplaced phase center antenna (DPC A) [2, 31. A nextension of the DP CA technique has been providedin [5,61 where an adaptive space-time filtering is usedto maximize the signal-to-disturbance ratio (SDR).Thecoherent combination of the echoes receivedby the dif ferent antenna elements at different times(space-time processing) allows, in fact, a considerablegain in the cancellation capabil ity. T he approachis here extended in order to introduce a temporalcorrelation between the echoes from the ground.Special care must be taken when the filteringis used jointly with synthetic aperture processing,especially when a high resolution image of the movingtarget is required. In such a case, the fil tering mustnot affect excessively the successive formation of thesynthetic aperture. Having detected the presence of amoving object, in fact, we have to estimate its phasemodulation law, to form a high resolution image of it.In [7,81a method for providing such an estimate hasbeen proposed. T he method was based on a mappingof the received signal onto the time-frequency domain,obtained by evaluating the Wigner-V ille distribution(WVD) [9] of the received signal. T he main goal of themapping was to emphasize some features of the signal,in particular its instantaneous frequency, which is offundamental importance for the ensuing estimation.The time-frequency approach allows us to exploit thedifference between the frequency modulation lawsof the echo from the moving objects and the echofrom the ground. A drawback related to the useofthe WV D is due to the nonlinearity of the WVD,which is bil inear. However, such a problem can becoped by the method suggested in [lo] and analyzed in[ll],where it is shown that, even in the presence of amulticomponent signal, the linearity can be restored byapplying a Hough transform to the WV D. A peculiaradvantage connected with the WV D, beyond thoseones mentioned above, is its capability of dealingwith undersampled signals, that is signals sampled ata rate lower than their bandwidth, as shown in [12].This possibility is valid for signals characterized bya narrow instantaneous bandwidth. Since the echoesfrom moving targets possess this characteristics,this property can be particular appealing in SA Rprocessing related to the imaging of moving objects,because it allows us not to work with the high values ofPR F dictated by the target motion.In [7,81, the signal was assumed to be received byonly one antenna, therefore the already mentioned

A consistent improvement can be obtained by using

problems concerning the cancellation of the stationarybackground were all present. In this work, thespace-time and the time-frequency analyses arecombined in order to overcome some of the problemsmentioned above. M ore antennas are used in order tomake available successive echoes from the same scene,received from different positions and at different times.These echoes are combined to cancel the disturbanceand to improve the estimation accuracy of the targetmotion parameters. The successive time-frequencyanalysis yields a further fi ltering capability, in thetime-frequency domain, and, most important, providesan estimation of the target instantaneous phase,necessary for the successive formation of the syntheticaperture with respect to the target. In summary, thespecific advantages related to the proposed approachare the following ones.

1) The space-time fil tering provides performance,in terms of disturbance cancellation capabilities,much better than the conventional Doppler filtering.Furthermore, it does not require hg h P R F values,thus avoiding the related coverage losses and data rateincrease.2) Since the frequency modulation law impressedon the received signal is due to the relative radarhargetmotion, the modulation law characterizing the usefulsignal iscompletely different from the modulationlaws of disturbance and thermal noise. T he mappingof the signals in the time-frequency domain thenprovides a means for exploiting such a difference fordiscriminating useful signals from disturbances.3) The time-frequency analysis provides animportant tool for estimating the signal instantaneousfrequency, and then the instantaneous phase.4) The estimation of the useful signal parameterscan be carried out even if the received signal isundersampled.

a model for the echoes backscattered f rom pointlikemoving objects and from extended random scenesand received by a moving linear array antenna isderived. The cancellation of the echo from the ground,by space-time processing, is analyzed in Section 111.In Section IV , after a brief review of the WV D, thecompensation of the target translational motion isperformed, by operating in the time-frequency domain.Section V shows the combination of the space-timeand time-frequency processing. T he merit of theproposed approach is validated by simulation results.

This work is organized as follows. In Section 11,

I I . MODEL OF THE RECEIVED ECHOThe aim of this section is to propose a model forthe echoes backscattered from a pointlike movingobject and from an extended stationary scene and

342 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO. 2 APRIL 1994

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SceneFig. 1. Observation geometry.

received by a moving linear array antenna. Thesemodels provide the basis for deriving the structure ofthe processing scheme, shown in the ensuing sections.The echoes from the pointlike target are modeled ina deterministic way. Conversely, the echoes from theextended scene are modeled as a random process,with assigned space and time correlations. Two kindsof correlation are considered, in order to take intoaccount the homogeneity of the illuminated surface(space correlation) and the time fluctuations of thesurface itself (time correlation).

A. Echo from a Pointli ke Moving TargetWith reference to Fig. 1,assuming a pointliketarget, the distance between the kth antenna elementand the moving object, at time t is

where r k ( t ) is the vector indicating the position of thekth antenna element at time t , and r(t) is the vectorcorresponding to the target. I n practical applications,the distance of the radar from the scene center ismuch higher than the distance of the moving objectfrom the scene center, that is

In this case, the radarharget distance can beapproximated byIrk(t) - (t)l Irk(t)l -&(t). (t)

where fk(t) is the unit vector indicating the radarposition. In this expression, the second term representsthe component of the target motion parallel to theradarharget line of sight (LOS),while the third termcontains the square modulus of the component tangentto the LOS.If s ( t ) is the transmitted waveform, the echo fromthe target is

~s c- lrk(t) - (t)l) (4)where A is a complex value which takes intoaccount the target reflectivity, the antenna gain, theradarharget distance, etc. I t isassumed, as usuallyin SAR processing, that A is constant during theobservation interval.For a sinusoidal signal ( s ( t ) =exp(jwot)), we have

(5)e j w o e- 44 dt)- /AThe relative motion introduces then a phase shiftwhich varies with time according to the following

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expression:(6)47$k( t ) =-+t) - (t)l

wherefirst-order approximation (see (3)), the correspondinginstantaneous frequency isis the transmitted wavelength. Within a

where the dots indicate time derivatives. The firstterm is the Doppler variation of the scene center,which is compensated at the beginning of the signalprocessing, both for the fixed scene and for the movingtarget. The second term is the Doppler variation dueto the change of the LOSbetween radar and target.The third contribution is due to the target motionprojected along the LOS. Since&(t) s perpendicularto i?k(t) (being &(t)a unit vector), the second termdepends only on the component of the target trajectoryvector perpendicular to rk(t),which is referred toin the following as the radar vector, while the thirdterm depends on the component of the target velocityparallel to the radar vector.

Let us now suppose that the radar undergoes arectil inear uniform motion, with velocity vector v andthat the antenna elements are located in positionscharacterized by the vectors (see Fig. 1):rk(t)=ro(t)+kAr =ro(0)+vt+kAr (8)

where ro(t) is the position of the first antenna elementat time t. In particular, if the antenna elements arelocated along the flight direction and spaced of anamount Ar =vT,where v is the modulus of thevelocity vector and T is the pulse repetition period(T =l /PRF), the vector rk(t), at time t=nT, can beexpressed in the following way:rk(nT)=ro(nT)+kAr

=ro(0)+nvT +kAr=ro(0)+(n+k)Ar .

rk+m((n-m)T)=rk(nT).

(9)

(10)This implies that

Therefore, the (k +m)th antenna element, at time(n-m)T,occupies the same position as the kthelement, at time nT.Thisproperty will be exploitedin the cancellation of the stationary scene.B. Echo from Extended Scene

The echoes from the fixed scene, due to specklephenomena present in microwave imaging, cannot

be modeled in a deterministic way. They have tobe modeled as a random process with assignedfeatures, depending on the observation geometryand the characteristics of the observed surface. M orespecifically, the observation of the Earth surface by alinear array distributed along the flight direction givesrise to a particular correlation between samples ofthe signals received from different array elements atdif ferent times. A statistical model of the correlationis derived in Appendix A , under the followingassumptions. The illuminated surface is supposedsubdivided into several adjacent cells, of size dependingon the radar geometrical resolution, where each cell ischaracterized by a complex amplitude backscatteringcoefficientA(r,t) that can be modeled as a complexGaussian random field (the value assumed in a pointof coordinates r, at each time instant t, is modeled asthe outcome of a complex Gaussian random variable).The values corresponding to different positions areassumed to be statistically independent, whereas thevalues corresponding to the same position, but atdifferent times, are supposed to be correlated of anamount depending on the time fluctuations of theobserved surface. Furthermore, the hypothesis ofstationarity is assumed to be verified, so that the timecorrelation coefficient between the values assumedat instants t and t depends only on the differencet - . In the case of a homogeneous scene, if weindicate by s(k,n) the sample received from thekth array element in correspondence with the nthtransmitted pulse and by s(i,m) he sample receivedfrom the ith array element, corresponding to the mthtransmitted pulse (s(k,n)and s(i,m)are referred tothe same distance from the radar), the correlationbetween s(k,n) and s(i,m) derived in Appendix A ,assumes the following expression:

E{s*(k,n)s(i ,m)}=Pps((m- n)vT+ i - )d)pt((m -n)T) (11)where the functions p s ( x ) and p r ( r ) assume themeaning of a space correlation coefficient and atime correlation coeffi cient, respectively. A goodmathematical model for the time correlation functionis the Gaussian, especially for its analytical tractability.Furthermore, by assuming a Gaussian antennaradiation pattern, the function p s ( x ) is itself Gaussian,as shown in Appendix A . We then assume thefollowing expressions for the correlation coefficients:

ps(Ax) =e-Ax2/2u: (12)pT(At)= -Ar2/2u? (13)

where the standard deviations nx and nt take accountfor the space and time correlations.I t is important to notice, from a physical point ofview, that the time correlation comes out directly fromthe backscattering process because it is reasonable to344 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO. 2 APRIL 1994

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Fig. 2. Successive observations from a moving multielement antenna.expect a slow time fluctuation of the backscatteringcoefficient. O n the other hand, the backscatteringcoefficients corresponding to different cells may besupposed uncorrelated. However, the observation ofthe surface by an antenna with a radiation patternwith a finite non-zero beamwidth impresses a spatialcorrelation on the received signals which is responsiblefor the presence of the function p s ( x ) in (11). I t isalso important to notice, about (l l ), that the timevariable appears even within the space correlation,owing to the antenna motion. This property is whatmakes the space-time processing a true bidimensionalprocessing, not separable into the cascade of twomono-dimensional, space and time, processing.A n illustrative example can be useful to understandthe correlation properties of the received signals. Withreference to Fig. 2, and supposing that the antennaelements are spaced at a distance d =vT, f the sceneis perfectly still (e.g. no wind at all), the echo receivedat time to from the antenna 2 is equal to the echoreceived at time to+T from antenna 1. In such a case,the correlation coefficient between these two echoesis equal to one. If the scene is not perfectly stil l (somebreeze from the lake on the tree leaves, for example),the echoes are correlated of an amount dependingupon to the time fluctuations of the scene. We referto this kind of correlation as to a time correlation.Withrespect to (ll),his situation corresponds to the casein which, for example, k =2, i =1,n=0and m =1.Inserting these values in (ll),we obtain

E {s*(k,n )s(i,m)}=Pps(O)pr (T) =PPf(T). (14)In such a case, therefore, the correlation dependsonly onthe time correlation. L et us consider now theecho received at time to from the antennas 1and 2.These echoes are correlated of an amount depending

on the homogeneity of the scene, and not on its timefluctuations, because the looks are taken at the sameinstant. In this case, therefore, we have to consider aspatial correlation. In fact, this situation corresponds tothe case n =m and k = +1,and, by using (ll), eobtainand then there is only a spatial correlation.the SA R operating mode as well. I n a spotlightSA R, for example, the same scene is observed forthe whole observation interval, while in a stripmapSA R , successive echoes are relative to different areas.Therefore, the correlation between echoes collectedin a spotlight mode is expected to be higher than thecorrelation observable in a stripmap.

The correlation between samples depends on

Ill. SPACE-TIME PROCESSING FOR SUPPRE SSINGTHE GROUND ECHOA reliable detection of moving objects requiresthe cancellation of the undesired echoes, i.e., thebackscattering from the ground. A conventionalDoppler filtering can be used at this purpose, but itpresents a certain number of shortcomings, like: 1)blind speeds or detectability losses of objects movingparallel to the radar, 2) the need of using high PRFs,while it is desirable to work with the PR F as lowas possible, and 3) the azimuth position uncertaintyproblem [4]. These inconvenients can be overcome,

at the expense of an increased hardware complexity,by using more antennas, or a phased array antenna,located along the flight direction. T he coherentprocessing of the echoes received from the differentantenna elements at different times, a space-timeBARBAROSSA& FARINA: SPACE-TIME-FREQUENCYPROCESSING OF SYNTHETIC APERTURE RADAR SIGNALS 345

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\.................................

Fig. 3. Space-time processing schemeprocessing, allows in fact a considerable gain over theconventional Doppler processing.convenient to locate the antenna elements at a distanced equal to the space vT travelled by the radar duringan interpulse period T. n fact, this arrangement makespossible to observe the same scene from the sameobservation point at different times. Therefore thecoherent combination of successive echoes allows thecomplete cancellation of the fixed scene. However, thiscondition imposes a severe constraint on the choice ofthe PRE Since the PR F affects a number of systemperformance, like swathwidth, signal-to-noise ratio,data rate, it is desirable to relax the choice of the PR Ffrom the constraint imposed by the clutter cancellation.For this reason, in the following we derive the optimalperformance, in terms of improvement of the SDR,achievable by the filtering, for different operatingconditions, e.g. for d not necessarily equal to vT.

From the cancellation point of view, it is

A. Optimum Space-Time FilteringThe signals received from different antennaelements at different time instants can be linearlycombined to maximize the SD R, defined as the ratiobetween the power of the useful signal and the sumof the powers of the disturbances, e.g. noise, clutter,and possibly jammer, assumed to be statisticallyindependent. The processing scheme is shown in Fig.where the delays T are equal to the pulse repetition 3,period and the input samples correspond to scatterersplaced at the same distance from the radar. Thereceived samples can be arranged in a vector form:

ZT =( ZT, ;, . &) (16)where each element Zk, for k =1,2,.. M , is a vectorcontaining the time samples received by the kthantenna element. T he vector Zk can be expressedas the sum of a useful vector Sk plus a vector ckcorresponding to the clutter, plus a noise vector nk.evaluate the optimal weighting vector w, dependingIn general, two approaches may be fol lowed to

upon the optimization criterion and the modelassumed for the useful signal. If the useful waveformis known, the optimal weighting vector is given by [13]w =RJ 's' (17)

where s is the vector containing the samples of theuseful signal and R, is the space-time correlationmatrix corresponding to the disturbance (clutter plusnoise), defined asRoo Rol ' . ' &,M-l

.. . j (18)% =[ . . . ... ... . . .% R11 " '. . . . . . RM-1,M-1

where the generic (k ,m)element of the matrix R;;,represents the correlation between the echo z,; ( kT )received at time kT from the ith element and the echoz,;(mT) at time mT from the jth element:

I t is useful to introduce the matrix of thecorrelation coefficients, obtained by normalizing thecorrelation matrix by the power:1pc =-&.pc

The computation of the weights according to (17),however, requires the knowledge of the useful signal,which is not available. A second approach consistsin modeling the useful signal as a white noise, whichcorresponds to assume that the frequency componentsand the directions of arrival of the useful echo are allequally likely. This is equivalent to find the weightswhich minimize the output clutter power subject tothe constraint that the norm of the weighting vector isequal to one. I n this case, the optimal weighting vectoris equal to the eigenvector associated to the minimumeigenvalue of the clutter space-time correlation matrixI t is worth pointing out that the fi rst approachprovides better performance since it takes intoaccount both the cancellation of the disturbance andthe coherent integration of the useful signal, whilethe second approach does not provide the coherentintegration gain. D ue to the obvious lack of knowledgeof the signal behaviour, in this work we have followedthe second approach. T he coherent integration isthen performed on the clutter-free data, after havingestimated the signal phase history induced by therelative radarharget motion, as shown in Section IV.

~31.

B. PerformanceA quantitative measure of the clutter cancellationand target detectability is given by the improvement

346 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONICSYSTEMS VOL. 30, NO. 2 APRIL 1994

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60. 00,..4 47-0

45.00

30. 00

15.00-v0LDLnL 0.00

\--_---- - - - _ _ _ - - - - _ _ _ _ - - _\\ 'tl = .9-4-20. 00 -17.88 -15.75 -13.63 -11.50 -9.38 -7.25 -5.13 -3

Fig. 4. Behavior of smallest eigenvaluesof space-time correlation matrix (for N =W =2), as function of space correlation coefficient,for some values of time correation coefficient.- 0loglo(l/X1); - - 10loglo(l/Xz).factor, defined as the improvement of the SDRobtained by filtering the received signal:

where the input SDR is defined with respect to thesingle pulse and the single receiving element. If wis the vector containing the fi lter weights, the I Fis [13]wHpswWHPcwI F = ___

where ps is the matrix of the correlation coefficientsof the useful signal, pc is the matrix of the cluttercorrelation coefficient, andoperator (transpose and conjugate).The maximum I F achievable by the scheme ofFig. 3can be evaluated by computing the minimumeigenvalue Ami, of the space-time correlation matrixand using the relationship [13]:

is the Hermitian

(23)1(IF)" =-XminExample: To understand the behavior ofthe filter as a function of number M of antennaelements, number N of time samples, space and timecorrelation, etc., it is useful to analyze an example insome detail. I n this example, the overall number of

samples N x M to be used for the filtering is keptconstant, while the ratio between N and M is varied.In particular, choosing N x M =4, three cases arepossible: 1) N =1,M =4, 2) N =2, M =2, 3) N =4,M =1.

In the case 2), the matrix of the space-timecorrelation coefficients assumes the form:

where p.1 and p2 are the space correlationcoefficients, at distance vT and 2vT, p t l is the timecorrelation coefficient at lag T. In this example thenoise is not present. It is easy to verify that, if pt l =1(still scene) or pS l =1(scene perfectly homogeneous),the correlation matrix is not a full rank matrix,therefore the minimum eigenvalue is zero and, asa consequence, the IF is infinite. In general, theseconditions are not verif ied and the I F depends on thespace and time correlations.In this example, the minimum eigenvalue mayassume one of the following two.expressions:

The absolute minimum, between X1 and X2,depends on the values of the space and timecorrelations. The behavior of these two eigenvaluesis reported in Fig. 4, as a function of the spacecorrelation coefficient, for some values of the timecorrelation coefficient.

BARBAROSSA & FARINA. SPACE-TIME-FREQUENCY PROCESSING OF SYNTHETIC APERTURE RADAR SIGNALS 341

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10 og,O(-p,,)-20.00 - 1 7 .8 8 -1 5 .7 5 -1 3 .6 3 -1 1 .5 0 -9.38 - 7 .2 5 -5 .1 3 -3

Fig. 5. Behavior of mnimum eigenvalueof space-time correlation matrix (for N =1 and M =4) as function of space correlationcoefficient.

To better understand the behaviors shown in Fig. 4,it is useful to give some approximate expression of XZ,when the time correlation coefficient is close to one. Insuch a case:A2 1- t1 . (27)

This explains the constant behavior of A2 as a functionof psi, in Fig. 4. From this figure, and using (23),wecan state that, for a given time correlation, the IF isnot smaller than:(28)1-(IF)max- -.Amin 1 - p t l

For high values of the space correlation coefficient, asshown in Fig. 4, the I F isgreater.In the cases 1)or 3),the correlation matrix is

where the subfix a indicates space (case 1)or time(case 3). The corresponding minimum eigenvalue isX z = - 2- pa1- pa3- (P.1- pa3)z +4(pa1- a2)2 *1

(30)7

This expression is plotted in Fig. 5, as a functionof the correlation coefficient. For comparing thisbehavior with the analoguous one shown in Fig. 4,let us consider the case 1) (pal =psi). From thecomparison, if the time correlation coefficient issufficiently high (which is usually the case), it is

evident that we can achieve better performance byusing a two-element antenna and two time samples,instead of a four-element antenna.In general, the performance strongly depend on thedistribution of the time and space samples to be usedfor the filtering and on the ratio a between the spacingd between two consecutive antenna elements and thespace vT travelled by the radar during an interpulseperiod. M oreover, the presence of the receiver thermalnoise puts a limit on the maximum IEThe limit imposed by the noise presence can beeasily evaluated by summing a diagonal matrix to theclutter correlation matrix and then considering the newmatrix:

Rt =R +o:I =Pc(pc+o:/PcI) (31)where o,s the noise variance.A; for i =1,2,.. N , the eigenvalues of the matrixin the brakets are equal to A i +o,/Pc. herefore,the minimum eigenvalue becomes Amin +u;/Pc. As aconsequence, the maximum I F cannot be greater thanthe clutter-to-noise ratio.As regards the parameter a, a value of a equalto one corresponds to the case in which it is possibleto collect data from the same observation point atdifferent times. In this case, we can exploit the radarmotion for obtaining a better cancellation. On thecontrary, a high value of a characterizes the situationof a radar fixed. In such a case, it is not possibleto observe the same scene from the same point atdifferent times.aforementioned parameters, e.g. a, the space and time

I t is easy to see that, if the eigenvalues of pc are

The relationship between the I F and the


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Fig. 6. IF (dB)

-1 60.00-550.00

40.00

30. 00

2000

- 10.000

mN)m 0.00

ANTENNA ELEMENTS/7 =16/N=1-- --- - -- --- --- z- \ IME SAMPLES\>1.Case 4 slow time fluctuations and (Y >>1.To make the comparison significant, the overallnumber of samples used for the cancellation N x Mis kept constant, while the ratio M / N is varied. Someexamples, relative to the cases listed above, are nowanalyzed. In all the examples, the product N x M isequal to 16and the I F is shown as a function of thespace correlation ps. A thermal noise, 60dB under the

clutter power, has been introduced. This puts a limiton the maximum I F equal to 60dB.Case 1

pf (T)=0.5 and Q =1.The I F obtainable in this case is shown in Fig. 6.From the behaviors shown in this figure, it turns outthat, for a given number of available samples (N .M

=16), it is better to have more antenna elements(M =16, N =1) than time pulses. Due to the lowcorrelation between successive time echoes, in fact,it is idle to compare several successive echoes. I t isbetter to compare echoes taken at the same instant,coming from contiguous areas, as far as the spacecorrelation ishigh.

Case 2pf(T)=0.995 and Q =1.

In this case, with reference to Fig. 7, we obtain thebest performance when we can collect the maximumnumber of echoes from the same scene, observedat different times from the same observation point.This happens when different antenna elements occupythe same position at successive time instants. I n ourexample, this condition is verified for N =M =4.With reference to Fig. 8, in fact, only in that case, wecan collect four echoes from the same observed area,at dif ferent time instants. I n the same fi gure, there areshown the situations corresponding to M / N =2/8and M / N =8/2. In both cases, no more than twoechoes from the same scene can be collected. T his alsoexplains why the performance in these two cases are soclose to each other. The same considerations apply tothe cases M / N =1/16 and M/N =16. In these cases,no more than one time echo per observation pointcan be stored, therefore the performance degradewith respect to the previous cases. W hen the spacecorrelation is high, we have a good cancellation,whatever is the ratio M I N . But, when the spacecorrelation is low, it is better to have M =N .

Case 3pt(T)=0.5 and Q =10.

When Q is much greater than one, it means thatthe radar motion is soslow that it is not possibleto observe the same scene from the same pointat different times. A value of Q going to infinitycorresponds to a radar fixed. Therefore, to understandthe behavior of the I F in such cases it is useful toextremize the situation and think of a radar fixed ina certain position in space. In this case, if the scenefluctuates rapidly (low time correlation), a high IFrequires the use of more antennas. As far as the space

BARBAROSSA & FARINA: SPACE-TIME-FREQUENCY PROCESSING OF SYNTHETIC APERTURE RADAR SIGNALS 349

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70.00

60. 00

50.00

40. 00

3000

20.00

io.oa

Fig. 7. IF (dB) versus spatial correlation coefficient (p l =0.995, (Y =1,clutter-to-noise ratio =60dB), for different values of ratio M I N .

----------

I I

t i m e

t o + Tt o I-9-D -DD -DD D

D DD DD D-D -D-D

( 4

t i m e

t o , t o + T , t o + 2 T , t o + 3 TLDXD --DD D DD D D DD D D-D -D

t i m e

t o , 0 + T , . . - , 0 + 7 T-D-D-D

D DD-D-D

Fig. 8. Position of antenna elements at different time instants. (a) M =8, N =2. (b) M =4, N =4. (c) M =2, N =8.correlation is high, we can obtain the maximum I Fvalue by employing the maximum number of antennaelements (M =16and N =1).Fig. 9exhibits, infact, an example of such a behavior. Due to the lowtime correlation, the results are very close to thoserelative to the case 2. T he only difference is that, forlow values of the space correlation, it is better, for theradar, to remain fixed in a point (a>>1) instead ofmoving (a=1).Therefore, for low space correlationvalues, the results shown in Fig. 9are better than theresults of Fig. 6.

Case4p l (T ) =0.995 and a=10.

If the radar can be considered fixed, and the timefluctuations of the scene are very limited (high timecorrelation), it is better to transmit as many pulsesas possible from a single point. Since successiveobservations of the same scene are highly correlated,it is possible to achieve high IF values for high ratios

N / M . This statement is confirmed by the resultsshown in Fig. 10.C. Degrees o Freedom

I t is important to make some considerations aboutthe choice of the number of antenna elements M andthe number of time samples N . In particular, for afixed value of the overall number of available samplesN .M , t is useful to analyze the criteria for choosingN and M . The cri terion must result f rom a tradeoffamong different requirements and constraints, such asperformance, cost, data rate, and coverage.practical interest, i.e., a strong time correlationbetween successive echoes from the fixed scene, itis better to take N =A4 and a value of a as closeto one as possible. This last condition, however,imposes a strong constraint on the choice of thePRF(PRF =v/d).

As regards the cancellation needs, in cases of


l---m~



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I0.00- 10.00vu

40.00

30.00 ---20.00

1/16

10 'og,J'-P,,)-20 00 -1500 -1000 -5 00Fig. 9. IF (dB) versus spatial correlation coefficient (p , =0.5, (Y =10, clutter-to-noise ratio=60 dB), for different values of ratio M I N .

20. 00

-vR 10.000 'IR 0.00 I 1 10 IoQIJ1-Ppr)-20.00 -15.00 -10.00 -5.00Fig. 10. I F (dB) versus spatial correlation coefficient (p t =0.995, (Y =10, clutter-to-noise ratio =60 dB), for different values of ratioM I N .

-On the other hand, from the point of view of thecoverage needs and the data rate constraint, it isbetter to work with a PRF as low as possible. Thiswould entail a value of N as small as possible. Thelast practical constraint (the last but not the least),of course, is the cost associated to have more thanone antenna and receiving channel. Considerationsabout costs and hardware complexity would makeus tend to use one antenna. The choice of N andM, therefore, must result as a tradeoff among all theprevious considerations.

IV TIME-FRE QUENC Y ANALYSIS FOR THEESTIMATION OF THE SIGNAL PARAMETERSHaving once detected a moving object, a properimaging process requires the knowledge of the phaseshift induced by the relative radadtarget motion.

The aim of this section is to provide a tool forthe estimation of the signal instantaneous phase,based on ? mapping of the received signal onto thetime-frequency domain. T his operation is performedby evaluating the WVD of the signal at the outputBARBAROSSA & FARINA: SPACE-TIME-FREQUENCY PROCESSING OF SYNTHETIC APERTURE RADAR SIGNALS 351



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of the clutter cancell ation filter. T he aim of thetransformation is to emphasize some features ofthe signal which make easier the extraction of thedesired information. In particular, among otherdistributions, the WVD provides the maximumconcentration of the energy around the curve ofthe instantaneous frequency, in the time-frequencyplane, and since the knowledge of the instantaneousfrequency is fundamental for the estimation of thedesired instantaneous phase, this explains the reasonfor choosing the WV D. The advantages obtainable byusing the WVD are paid by the loss of linearity relatedto it, since the WVD is a bilinear transformation.However, this shortcoming can be overcome by theapproach proposed in [l l ], where the detection andparameter estimation of multicomponent signalsembedded in noise by time-frequency analysis is solvedby combining the WV D with the Hough transform.A. Representation of the Signal in the

Time-F requency DomainWhat distinguishes a moving target echo from

the stationary background echo is the differentlaw of variation of the instantaneous phase or,correspondingly, the instantaneous frequency. Ifit were possible to observe the behavior of theinstantaneous frequency of each return, separately,one would be able to detect the presence of movingtargets. T he time-frequency distributions provide,at this purpose, a powerful tool. These distributionsare signal representations consisting in the mappingof the signal onto a plane whose coordinates aretime and frequency (t- plane). The WV D, inparticular, produces a mapping such that the energyis concentrated along the curve of the instantaneousfrequency in the t- plane. TheWVD is defined as[91

where s(t) s intended to be the analytical signal.The following two properties of the WVD areparticularly important in our application.1) Conservation of the Inner Products:

(33)This property, known as the Moyal's formula

[9], allows us to translate any filtering and detectionproblem, defined in the time domain, in thetime-frequency domain.an analytic signal, expressed in terms of envelope and2) Estimation of the Instantaneous Frequency: Given

phases ( t ) =a(t)e'+(') (34)

its instantaneous frequency can be estimated as thecenter of gravity of its WV D [9]:

This property allows the estimation of theinstantaneous phase through two successive integrationsteps, namely, 1)computation of the WVD, and 2)computation of centers of gravity.The representation provided by the WV D offersalso a number of advantages, with respect to theconventional spectral analysis, as shown in the ensuingsections.

B. Time-V arying FilteringIn SAR applications, the echoes correspondingto successive transmitted pulses exhibit a phaseshift varying with time as a function of the relativemotion between the radar and each backscatteringelement. Due to the radar motion, each point onthe ground reflects an echo having a Doppler shiftproportional to the projection of the radar velocityvector along the LOS (the line passing throughthe radar antenna center and the backscatteringelement). Points at dif ferent distances are separatedin time (e.g. by transmitting a chirp signal and thenapplying a range compression). Points at the samedistance, but observed from different angles, exhibitdifferent Doppler shifts. T herefore, the echo from eachrange cell, for each transmitted pulse, has a certainDoppler bandwidth, proportional to the size of theilluminated spot. A moving object has dimensions

much smaller than the illuminated surface, thereforethe instantaneous bandwidth of the correspondingecho is much narrower than the bandwidth relativeto the ground. O n the other hand, however, thetarget motion induces a Doppler shift which varieswith time from pulse to pulse. As a consequence,the overall Doppler bandwidth covered by the targetecho during the whole observation time (i.e., overseveral transmitted pulses) may be much wider than itsinstantaneous bandwidth (the bandwidth correspondingto each transmitted pulse). As an extreme example,the echo from a pointlike target (see ( 5 ) ) , after rangecompression, can be thought of as a sinusoid whosefrequency varies with time according to the law ofvariation of the radarharget distance. Therefore, theinstantaneous bandwidth is practically zero, while theoverall bandwidth can be much greater than zero.Conversely, the echo from the ground, which is initiallycompensated with respect to the radar platformmotion, does not vary with time [14].

352 IEEE TRANSACTIONSON AEROSPACE AND ELECTRONICSYSTEMS VOL. 30, NO. 2 APRIL 1994

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Summarizing, the echo from the ground exhibitsa certain bandwidth proportional to the antennabeamwidth, for each pulse, that does not changefrom pulse to pulse, while the echo from a target hasa narrow bandwidth, for each pulse, but i ts meanDoppler frequency varies from pulse to pulse, sothat the overall bandwidth, covered during the wholeobservation interval, can be much wider.This difference in Doppler frequency allocationwith time between echoes from the ground or frommoving objects can be advantageously exploited if thetime information about the Doppler frequency contentisretained. This information can be maintained onlybe resorting to time-frequency representations of thereceived signals.The signal can be fi ltered in the t - domain bya filter whose bandwidth is proportional to the signalinstantaneous Doppler bandwidth, tuned around afrequency which varies with time according to theestimated instantaneous frequency. In this way, it ispossible to achieve noise rejection capabilities, notobtainable with a conventional Doppler f iltering.However, due to the nonlinearity of the WV D, itis necessary to remove the clutter contribution,before computing the WVD, or otherwise the signalwould be obscured by the clutter. The detectionand parameter estimation of the signals embeddedin clutter, based on the WVD has been alreadyanalyzed in [SI.Some results are here reported forcompleteness sake. As shown in [SI, the optimalfiltering of chirp signals embedded in clutter, basedon the time-frequency analysis, consists in multiplyingthe vector of the received samples by the inverse ofthe clutter correlation matrix and then computing theWV D of the resulting vector. A n example is shown inFig. 11and 12. Fig. 11shows the WVD (representedby contour plots) of signal plus clutter. T he signal isbarely recognizable because is masked by the clutterand by the cross terms introduced by the WV D.But, after multiplication of the input vector by theinverse of the clutter correlation matrix, the resultingvector exhibits the WVD shown in Fig. 12, where thesignal contribution is clearly discernible, whereas theclutter components have been filtered out. Of course,the cancellation degree depends on the frequencyallocation of the signal spectrum with respect to theclutter spectrum. In the case examined in Fig. 12, partof the signal spectrum has been attenuated by the filter,but nevertheless it is still possible to estimate the slopeof the chirp, which is the information we need forfocusing the target image.

V. COMBINED SPACE-TIME AND TIME-FREQUENCYPROCESSlNGThe space-time and the time-frequency processing,described in the previous two sections are now

limeFig. 11. WVD of clutter plus signal at input of clutter cancellationfilter (signal-toclutter ratio=0 dB).

I Itime

Fig. 12. WVD at output of clutter cancellation filter.

combined to form a processor that allows boththe cancellation of the clutter echoes and thecompensation of the target motion, necessary for theformation of a high cross-range resolution image.A. Processing Scheme

The overall space-time-frequency processingis shown in Fig. 13. Theecho from the ground iscanceled at the beginning by means of the schemealready shown in Section 111 Having once canceledthe background echo, the ampli tude of the resultingsignal iscompared with a threshold to decide aboutthe presence of a moving target. A detection enablesthe channel for the estimation of the moving targetfrequency history. This first detection allows us toavoid idle computations of the WVD when no targetdetection occurs. Before proceeding to the estimation,however, some care must be devoted to the rungemigration problem. In fact, the relative radarhargetmotion causes not only a phase shift, but also a rangemigration which cannot be neglected if it overcomesthe range resolution.The radar echoes are sampled and arranged ina matrix, whose columns are relative to successivetransmitted pulses and the entries of each columncontain the echoes coming from different distances.Given a certain point, if its distance from the radar

BARBAROSSA & FARINA SPACE-TIME-FREQUENCY PROCESSING OF SYNTHETIC APERTURE RADAR SIGNALS 353

1 -T~-Authorzed censeduse mtedto: UNIVERSITAT POLITCNICA DECATALUNYA.Downoaded onSe tember 6 2009at 03:02 fromIEEEX ore. Restrctonsa .


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b fu ll resolutioncompression

fine rangemig ra t ioncompensationb space- t imeprocesso rLh t

I+ lo w rangereso lu t ion

?-ocalize range re fe renceeferencedistancew +does not vary by an amount bigger than the sizeof a range resolution cell, the echoes from thatpoint are all stored on one column. If all the pointscomposing the observed scene satisfy this condition,no range migration occurs and the samples can befirst processed in range, column by column, andthen in cross-range, row by row. If, however, theechoes from some point occupy more than one row,the cross-range processing cannot be performeddirectly on the collected data, row by row. Somealgorithm for compensating the range migration mustbe applied in order to realign the data, before thecross-range processing. R ange migration compensationtechniques have already been examined in [15], in theimaging of stationary scenes. The problem is harderin the imaging of moving targets, however, sincethe radarharget distance is not known. Two mainproblems arise when dealing with moving targets, inpresenceof range migration: 1) the migration causesa spreading of the target energy through many rangecells, therefore the signal-to-noise ratio, for each rangecell, decreases and this causes a loss of detectability,and 2) due to the lack of knowledge of the targetrange migration law, it is not known a priori whichsamples of the matrix of the collected data are to betaken for carrying out the estimation of the phasehis tory.for compensating the migration problem: 1) a coarserange resolution, and 2) a fine range resolutionanalysis. The rationale of the proposed approach isbased on the fact that the migration problem canbe neglected if the amount of migration does notexceed the range resolution. Therefore, we can initiallywork in a low resolution mode, where the rangeresolution has been degraded by an amount such asto make the migration negligible. A first detection isperformed on these data: if a detection occurs, the

In this work, we have followed a two-step approach

processing chain for the estimation of the target echophase history is enabled. T he estimation is carri edout in the time-frequency domain, according to thecriterion described in the previous section. Havingonce estimated the phase @(t),we can evaluate thelaw of variation of the distance by exploiting therelationship between phase shift and distance (see (6)):

x4Td(t)= -@(t).

By using d(t),we can compensate for the rangemigration by properly rearranging the data in thematrix containing the received samples. These datacan now be phase shifted to compensate for the phaseshift induced by the target motion. The data undergothen a full range and cross-range compression forobtaining the high resolution image. The price paidby adopting the two-resolution approach is the SNRloss inevitably related to the degradation of the rangeresolution related to the coarse resolution mode.translational motion of a dominant point on thetarget. Since the motion of any rigid body can bedecomposed as the sum of the translation of a pointplus the rotation of the other points around that point,in some cases, depending on the target dimensionand kind of motion, the compensation of the onlytranslational motion is not sufficient and we stillhave to compensate for the rotational motion. Thissecond compensation can be efficiently performed inthe frequency domain, by applying, for example, themethod proposed in [16].I t is important to point out that, even if thecause of both phase shift and migration problems isthe same, namely the variation of the radarhargetdistance, the phase shift is much more sensitive torange variation than the range migration. I n fact, forexample, in an X-band radar (A =3 cm), having a

The proposed method corrects only the



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rangeFig. 14. Imageof moving target, after clutter cancellation.

rangeFig. 15. Imageof moving target, after range mgration and phase compensation.

range resolution of 1m, an error on the distance of7.5mm causes a phase shift of T radians, while thecorresponding range migration is absolutely irrelevant.This means that, having once estimated the phasehistory by the described procedure, the resulting valuefor the distance variation d( t ) is estimated with a goodaccuracy.B. Simulation

The performance of the processing schemehas been evaluated by a simulation program whichgenerates the echoes from an extended surface andfrom a pointlike target and then applies the proposedalgorithm. T he target issupposed moving on theterrain (shadowing effects have been neglected) at aconstant velocity, in a direction oblique with respectto the radar motion. The velocity parameters havebeen chosen in order to make evident the presenceof range migration and of cross-range smearing ofthe target image. T he ground reflectivity has beenassumed equal to the target reflectivity (this is quitea pessimistic assumption, because in many cases ofpractical interest, the target reflectivi ty is higher). Areceiver thermal noise, 40dB below the target return,has also been summed to the received signal.

The ground echo is first canceled, by using atwo-element antenna and two time samples (N =2and M =2 in the scheme of Fig. 3). The two antennasare spaced by d=vT. A SAR image is then formedby conventional techniques. T he result is shown inFig. 14.The smearing of the moving pointlike targetis evident. Given the motion parameters, the target hasmigrated over six range cells. This is the cause for thebroadening of the target image even in range, as wellas in cross range. Anyway, the target echo causes adetection and initializes the motion estimation channel.The high resolution data are initially smoothed inrange to decrease the range resolution. Then theprocessor looks for the range cell with the maximumenergy content and computes the WVD of that cellonly. The frequency history, and then the phasehistory, is evaluated, according to the procedureoutlined in the previous section. T he phase history isthen used for compensating the range migration andthe phase shift on the high resolution range data. Thefinal image is shown in Fig. 15.The sharpening of thetarget image is quite evident. T he appearance of thesmall constant contribution on a range cell close tothe target range cell is due to the not-perfect rangerealignment, caused by the initial filtering used for theclutter cancellation.

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VI. SUMMARYIn this paper a method for detecting and imagingobjects moving on the ground and observed by aSA R has been proposed. The method is based onthe combination of two kinds of processing: 1)aspace-time processing which exploits the motion ofan antenna array for canceling the echo from thebackground, and 2) a time-frequency processing

which exploits the difference in time allocation of theinstantaneous spectrum corresponding to echoes fromthe ground or from moving objects, for an adaptivetime-varying filtering and for the estimation of thetarget echo instantaneous frequency, necessary forproducing a focused image of it. The performanceofthe space-time filter isevaluated and some criteria forchoosing the system parameters are given. As regardsthe time-frequency representation, the advantagesobtained in using it are paid by an increased algorithmcomplexity and an increased memory occupation.Furthermore, being the Wigner-V ille bilineartransformation, one has to deal with the appearance ofundesired cross-product terms. A t this regard, however,it is still possible to estimate the signal parameters,even for multicomponent chirp signals embeddedin noise, by combining the WV D with the Houghtransform, as described in [ll].In the paper, the method has been tested onpointlike targets for the compensation of the relativeradarharget translational motion. If the amount ofrotation is not negligible, a further processing isrequired, after the compensation of the translationalmotion, for example, like the one proposed in (161.

APP ENDIX A. S TATISTICAL MODE L OF THEGROUND ECHO

In the following, we derive a statistical modelfor the echo backscattered by the ground surfaceilluminated by the radar, under the assumptions statedin Section 11Accordingly, if we denote by A(r,t)the value of the complex amplitude backscatteringcoefficient of a cell placed in a point indicated by thevector r ,at time t,we assume thatE{A(r ,t)A*(r ,t)}gO(r)uo(r )pI (t )

(37)where gO(c) is the surface backscattering coefficient,uo(r) s the Dirac pulse, and pt(r) s the timecorrelation coeff icient between A(r,t) nd A(r,t+r).The time fluctuations of A(r,t) an be supposed veryslow, with respect to the duration of the transmittedpulse. For this reason, in the following, the coefficientare assumed to be variable only pulse-by-pulse.L et us indicate by s(t) the transmitted pulse andassume the illumination geometry reported in Fig. 16,

I IFig. 16. Projection of observation geometry onto slant range

plane.which shows a projection of the observation geometryof Fig. 1onto the slant range plane. rnk is the vectorindicating the position of the kth antenna elementcorresponding to the instant of transmission of the nthpulse and r is the vector indicating the generic pointon the illuminated surface. The echo backscattered byan elementary reflector located in r ,characterized bya complex amplitude reflection coefficient A(r, ) andan area dr, s

(38)where K is the constant depending on transmit power,wavelength, etc., GT(.,) is the amplitude antennaradiation pattern, in transmission, depending on theangle between the normal to the antenna and the LOS,GR(.,) is the amplitude antenna radiation pattern, inreception, depending on the angle between the normalto the antenna and the LOS, and c is the speed oflight.corresponding to the nth transmitted pulse, is thenThe overall echo received by the kth antenna,

x A(r,n)s t- - rnk- dr (39)cwhere the integral is extended over the illuminatedsurface, indicated by D. he surface is assumed to beflat, so that the integral is a double integral.A fter range compression, that can be modeled asthe correlation of the received waveform with a replicaof the transmitted pulse, delayed by an amount T, thesignal becomesU( T; n,k) =~ ( t ;,k)*s(t- )


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having indicated by the symbol * the correlationbetween signals and by 9,(T) the autocorrelationof s ( t ) . To achieve a good range resolution, thetransmitted waveform is usually chosen in orderto have a narrow autocorrelation function ( s ( t ) isusually a chi rp signal). T he function as(r) an bethen approximated, by neglecting the sidelobes, by afunction different from zero only within a window,centered around T =0,whose width is proportionalto the range resolution. If we now consider the returnsfrom a certain distance from the radar, the previousintegral can be approximated as follows. First ofall, the antenna gains in transmission and receptioncan be assumed to be proportional. The distance,at the denominator of (40), can be assumed to beconstant, without an appreciable error. M oreover,given the property of 9,(r)described above, thedouble integral can be well approximated by a lineintegral. In particular, with reference to Fig. 16, wedenote by x the line coordinate corresponding to theradar position and by x the coordinate correspondingto r, and we take as the origin of the x axis theposition of the first antenna element correspondingat the instant of transmission of the pulse number zero.The coordinate of the generic kth antenna element atthe transmission of the generic nth pulse is then:

x =kd +nvT (41)where d is the distance between adjacent antennaelements, supposed uniformly spaced, v is the radarvelocity along the x-axis and T is the pulse repetitionperiod. As a consequence, (40) can be rewritten as

u(n,k) =K G 2 ( ~ , ~ ) A ( ~ , n ) d ~ 42)s,where K takes into account all the constant terms.The antenna radiation pattern G(x,x ) , n the azimuthplane, depends on the variables x and x throughthe azimuth angle p,which, by referring to Fig. 16, isrelated to x and x by the relationship:

p=arctg(Ra) - x . (43)Therefore the function G(x ,x )only depends on thedifference x - . Equation (42) can be then rewrittenas

u(n,k) =K G2(x- ) A ( x , n ) d ~ . (44)s,L et us now evaluate the correlation betweensamples corresponding to the same distance, butdifferent antenna elements or transmitted pulses. If weindicate by x1 and x2 the coordinates corresponding to

the positions of the antenna elements k and i , at timesnT and mT , respectively, that is if we setx1 =k d +nvT (45)x2 =i d +mvT (4.6)

=E { K ~ G2(x1- )G2(x2- )

IA(x ,n)A* x ,m)dx dx=Kt2/ / 2(x1- )G2(x2- )d dx E {A(x ,n)A* x,m)}d x d x and, by using (37):= / / 2(x1- )G2(x2- )

d dx CY~(X )UO(X)pf((m- )T)dxdx

=KI2p , (m- )T )/ 2(x1- )G2(x2- )Q(x) dx.d

(47)For a homogeneous surface, that is a surface withconstant statistical parameters, we have I T~( X )CO. nsuch a case, the correlation becomes

E {u(n,k)u (m,ll=Kr2~op,( (mn)T) G2(x1- ) G 2 ( x 2 - ) d ~ J,=K%opt((m - )T)R,z(xz - ) (4)

having indicated by RGz(x) the autocorrelation of thefunction G2(x).By assuming, for analytical simplicity,that the antenna radiation pattern isGaussian, G2(x)is Gaussian and, as a consequence, RGz(x) s Gaussian.Introducing the space correlation coefficient functionp,(x), defined as the function RG2(x) normalized toits maximum value, and using (45) and (&), we canexpress the correlation between the echo receivedby the kth antenna element, corresponding to thenth transmitted pulse, and the echo received bythe ith antenna element, corresponding to the mthtransmitted pulse, as

E{u(n,k)u*(m,i))=Ppr((m- n)T)p,((k - )d +(n-m)T) .

(49)ACKNOWLEDGMENTS

The authors wish to thank Dr. E M enichetti andDr. G. M erenda for their help in running the computersimulation program and for valuable comments on theprocessing method.BARBAROSSA & FARINA SPACE-TIME-FREQUENCY PROCESSING OF SYNTHETIC APERTURE RADAR SIGNALS 357

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REFERENCES[l] Cutrona, L. J . (1990)Syntheticaperture radar.In M. I. Skolnik (Ed.),Radar Handbook (2nd ed.).New York McGraw-Hill,1990,ch. 22.

Airborne MTI.In M. Skolnik (Ed.),Radar Handbook (2nd ed.).New York McGraw-Hill, 1990.

Andrews, G.A., Gerlach, K. (1989)SBRclutter and interference.I n L. J. Cantafio (Ed.),Space-Based Radar Handbook.Dedham,M A Artech House, 1989, ch. 11.Detection and imaging of moving objects with syntheticaperture radar-Part 1:Optimal detection and parameterestimation theory.IEE Proceedings, 139,Pt. F, 1(Feb. 1992), 79-88.

Klemm, R., and Ender, J. (1990)New aspects of airborne MTI.I n Proceedings of I EEE International Radar Conference,Arlington, VA, May 7-10, 1990, 335-340.Adaptive airbome MTI: an auxiliary channel approach.IEE Proceedings, 134,Pt. F, 3(J une 1987), 269-276.A nove procedure for detecting and focusing movingobjects with SAR based on the Wigner-Ville distribution.I n Proceedings of IEE E International Radar Conference,Arlington, VA, May 7-10, 1990,44-50.Detection and imaging of moving objects with syntheticaperture radar - Part 2 Joint time-frequency anaysis byWigner-Ville distribution.IEE Proceedings, 139,Part F, 1(Feb. 1992), 89-97.

[2] Staudaher,E M. (1990)

[3]

[4] Barbarossa,S. (1992)

[5]

[6] Klemm, R. (1987)

[7] Barbarossa,S.,and Farina, A. (1990)

[8] Barbarossa,S., and Farina, A. (1992)

[9] Cohen, L. (1989)Time-frequency distributions-A review.Proceedings of IE EE, 77 , 7 (July 1989).MMV SAR on RPV for battlefield surveillance.InAGARD Conference Proceedings,459, The Hague,Netherlands, May 8-12, 1989, 18-21.A combined Wigner-Ville and Hough transform for crossterms suppression and optimal detection and parameterestimation.I nProceedings of the IEEE I nternational ConferenceonAcoustics, Speech and Signal Processing, ICASSP 92, Mar.1992,San Francisco.Parameter estimation of undersampled signalsbyWigner-Ville anaysis.In Proceedings of the IEEE International ConferenceonAcoustics, Speech and Signal Processing, ICASSP 91, May14-17 1991, Toronto, 3253-3256.Optimised Radar Processing.London: Peregrinus, Ltd, 1987.On the useof joint time-frequency distributions n theanaysis of SAR signas.European P ansactionson Telecommunications,2, 6(Nov.-Dec. 1991), 665474.Doppler parameter estimation for spacebornesynthe ic-aperture radar.IEEE Pansactionson Geoscience and Remore Sensing,GE-23,1(J an. 1985).

Wemess, S.,Carrara, W., Joyce,L.,and Franczak, D. (1990)Moving target imaging algorithm for SAR data.IEEE Transactionson Aerospace and Electronic Systems,26, 1(J an. 1990), 57-67.

[lo] Farina, A., et al. (1989)

[ll] Barbarossa,S., Zanalda, A. (1992)

[12] Barbarossa,S. (1991)

[13] Farina, A. (1987)

[14] Barbarossa,S., Gregori,C., and Nicastro, G. (1991)

[15] Li, E K., Held, D., Curlander, . C., and Wu, C. (1985)

[16]

Sergio Barbarossa (S82-M86) graduated in electronics engineering from the Universityof Rome, Italy and received his Dottorato di Ricerca degree in 1989.In 1984 he joined the Radar System Analysis Group of Selenia, where he worked until1986, when he went to the Information and Communication Department of the Universityof RomeLaSapienza. In 1988,on leave from the University of Rome, he worked asResearch Engineer at the Environmental Research Institute of M ichigan, Ann Arbor. From1989-1991 he taught Signal Theory at the University of Perugia, Italy. Since November1991, he has been an Associate Professor at the University of Rome, where he teachesSignal Theory and Radio Commnication Systems.Hismain research interests are insignal processing algorithms, time-frequency representations and array processing, withapplication to radar.Alfonso Farina (M85) received his doctorate in electronic engineering from the Universityof Rome, Italy, in 1973.In 1974 he joined Selenia S.p.A (now part of A LENIA S.p.A) where he is now aManager in the Radar & C2 Division and Head Deputy of the System Analysis Group.He was Professore Incaricato of Radar Techniques at the University of Naples since1979, and in 1985 he was appointed Associate Professor at the same university. Hismainwork interests are in signal, radar data processing techniques for ground-based and AEWradars, ECCM and synthetic aperture radars. He is the author of more than 100 publishedpapers, the co-author with Dr. E A. Studer of a two-volume book, Radar Data Processing,the editor of Optimised Radar Processors, and the author of the recent book by ArtechHouse, Antenna Based Signal Processing Techniquesfor Radar Systems.Dr. Farina has been Session Chairman at several International Radar Conferences.He received the 1987 Radar System Panel Award of IEEE for development of radar dataprocessing techniques. A lso, he is a member of the International Radar System Panel ofAES, IEEE.


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