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1206 IEEE TR ANSACTIONS ONNTENNAS AND PROPAGATION,VOL. AP-35, NO. 11, NOVEMBER 1987

Superresolutionof Near or Far Field CoherentTargetsUsing Aperiodic Antenna ArraysHARISH M. SUBBARAM AND BERNARD D. STEINBERG, FELLOW,EEE

Abstruct-A technique for obtaining high resolution imagesof point-likeoherentargets sing aperiodic antennarrays is developed.Whereas existing superresolution techniques can be sed to imagecoherent targets located only in the far field of periodic receiving arrays,this technique can be used to obtain high resolution images of targets inthe near field also. The key to this technique is the use of multipletransmitters. Completely correlated (coherent) targets are decorrelated bytransmitting sequentially from dif ferent locations, thereby enabling theuseof modern superresolution echniquestoestimate the target locations.An expression for estimating the variance of the arget locations isderived, and isverified by empirical methods. For the same amount oftransmittedpower and thesamenumber of receivers, this variance isshown to be an order of magnitude mallerhanhat of existingsuperresolution techniques. The practicality of this technique is demon-strated with experimental microwave data.I. LWRODUCIIONI MOST LONG wavelength microwave imaging scenarios,a transmitter illuminates the field of view, and the complexfield amplitude (CFA) of the echoes from the targets isrecorded along a line array. These CFA values are thenprocessed to estimate the positions and amplitudes of thetargets. If the targets are stationary over a period of severaltransmitted pulses, their echoes will be unchanging relative to

each other, and they may be modeled as coherent targets.Conventional (Fourier or Fresnel) microwave imagingtechniques estimate the amplitudes and locationsof the targetsby focusing the array to various points n the field of view[20]. This procedure is adequate to resolve widely separatedtargets; however, if the targetsarewithin a beamwidthof eachother, one has to resort to superresolution (SR) techniques inorder to resolve them. Several SR techniques havebeendeveloped over the last two decades; most of them model thereceived CFA as a complex valued autoregressive (AR)process, andconsequently obtain a high resolution all-poleimage by estimating the AR parameters. Among he ARsuperresolution techniques, the covariance method (CM) 111-[4], and the modified covariance method (MCM) (also calledthe least squares method) [2], [5], [6] have proved effectivefor resolving coherent targets located in the far field (Fraunho-fer zone) of a periodic array. In many applications, the arraymay be too close to the targets for the far field approximationto be valid. Therefore, it is useful to develop a SR techniquethat canbeused to detect targets in the near field also.

Manuscript received J anuary 24, 1986; revised December 1, 1986.The authors are with the alley Forge Research Center, The Moore Schoolof Electrical Engineering,University of Pennsylvania,Philadelphia, PA19104.IEEE Log Number 8716772.

Both CM and MCM require the antennas to be placed3tequidistant points. In order to avoid aliased images, themaximum interelement spacing should beM2 , whereX is thewavelengthof he transmitted radiation. By removing therestriction of locating the antennas on a X/2 grid, the length ofthe array can be increased for a fixed number of receivers. It isintuitive to expect the increase in length to 1) decrease themean squared error in the target location estimates, and 2)improve the resolving power of the system. Therefore, it isadvantageous to develop aSR technique that does not imposeany restrictions on array geometry. It will beshown later thatby usingNelement aperiodic arrays, it is possible to spreadthe elements over a length much larger than ( N- 1)M2andstill obtainunaliased images. Among the existing SR tech-niques, the MUSIC algorithm171and the maximum likelihoodmethod [8] are independent of array geometry; however bothtechniques fail to resolve coherent targets [9], [101. Atechnique that uses moving aperiodic arrays is described in[ll]. The MUSIC algorithm wasmodified to accomodatecoherent targets in [12]; this modified version is computa-tionally intensive as it requires an M-dimensional search forlocatingA4targets. Therefore, existing SR techniques that arenot computationally ntensive and can resolve coherent targetsrequire the receiving array to be periodic and/or moving.These techniques could be used for detecting near field targetsby focusing the array to a particular point in the near field andsubsequently forming an all-pole estimator for the targetlocations; however, there will be a residual error in the targetlocation estimates even under noiseless conditions if thetargets are away from the focal point in the near field [13].In this paper, we develop an SR technique for resolvingcoherent targets in the near or far fields of arbitrarily shapedarrays. Multiple transmitters are employed to achieve thisgoal. Both CM and MCM change the center of phase positionalonghe array by using identical periodic subarrays toperform spatial smoothing. This subarray smoothing proce-dure decorrelates the coherent targets and estimates theARparameters to generate an all-pole estimator for the targetlocations 111-[4]. We notehat the subarray smoothingprocedures limit the size of the receiving array to (N - 1)M2,where N is the number of receivers. The SR techniquedevelopedn this paper (called RAM for random arraymethod) decorrelates the coherent targets by sequentiallytransmitting from several different locations. During eachtransmission, the di st ance from the transmitter to the targetcorresponds to an equivalent phase delay. The sum of thisphase delay and the phase of the reflection coefficient of the

OO18-926X/87/11OO-1206$01oO 0 1987 IEEE

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A ND STEINBERG: APERIODIC ANTENNA ARRAYS 1207considered to be the overall phase of the targetthat particular transmission. Because this phase depends onstance between the target and the transmitter, the phaseference betweenany two targets can be changed bynging the locationof the transmitter, thereby decorrelatingrent targets. Anall-pole estimator for the targetons can be formed after decorrelating the coherent. Wenote hat his procedure for decorrelating the

s independent of the geometryof theiving array. Therefore the size of the receiving array canincreased for a fixed number of receivers by placing theivers aperiodically over the aperture. (If the array isriodic, the grating lobes destroy the potential for usefulaging.) This increase in the size of the aperture leads to aean squared error; however, the lower sampling ratell introduce spurious peaks in the all-pole estimator for thet locations. The RAM overcomes this problem by usingiver location diversity. Several all-pole estimators for thelocations are formed by splitting the receiving array intorrays (which are not identical to each other) and using theeasured at these subarrays to estimate the AR parame-Each of hese estimators peaks at the target locations, andous peaks at locations that depend on the geometrythe individual subarrays. Therefore, a parallel resistor typeof the all-pole estimators obtained by using subarraysth different receiver geometries willotontainnyicant spurious peaks. Further, because the size of theay canbe ncreased to more than ( N- l)h/2 if thevers are aperiodically ocated over the aperture, the meanared error (MSE) in the target location estimates will bealler than the MSE for the MCM and CM. In Section Vwethat the RAM performs as well as the MCM and isor to the CM when all methods use arrays of the samegth. Further, we show that by increasing the length of theused for the RAM, it is possible to decrease the meanred error in the RAM target location estimates by morean order of magnitudeascompared to the MCM andCM.The paper is structured in the following format. The RAMormulated for noiseless conditions in Section I and is thencomparedwith heMLM. The mean squaredin the target location estimates is derived in Section ILI.a criterion to determine the number of targets isveloped, and the statistics of theRAMestimator near its

ma are derived. Section IV presents somesimulationts and discusses the dependence of the mean squared errorength of the array, the SNR, and heseparationtween the targets, for the two target case. Simulation resultsso presented to compare the RAMwith Fourierssing, MCM, and CM. Section V confirms the theoryng measured microwave data obtained from real targets atValley Forge Research Center, University of Pennsylva-II . SUPERRESOLUTIONNDEROISELESSONDITIONS

.Assumptions andNotationThe SR technique willnowbe formulated for noiselessitions. Consider an array of K transmitlreceive antennasat x = xi, i = 1, - - ,K, and (N - K ) receiving

I-

M TARGETSK TRANSMI T/ RECEI VE

ANTENNASN- K RECEI VERS

I /I /1 : '

iI ,

k-th Transmzt / Recei ve iAnt enna n- t h Recei ver7 1

1 1Y Y Y i Y Y Y Y y y y yY Y Y Y Y Y Y

Fig. 1 . Array and target geometry.

antennas atx =xi, i =K +1, - *, N. We note here thatNreceiving andK transmitting antennas could also be used; thepresent format is used for convenience of notation.) Let Mtargets be located in the same range bin at a distanceR fromthe origin, atu = ui, i = 1, - * * ,M, whereui = sinSi andBiis the angle from broadside. Each target may be parametrizedby a complex amplitudeSi , = 1, * - - ,M, which depends onits cross-section area and reflection coefficient. The geometryis depicted in Fig. 1.Let the CFA at the nth antenna, located atx =x,, while themth antenna is transmitting, bedenoted by em,. If thereciprocal of the bandwidth of the transmitted signal is muchgreater than the transit time of theRF pulse along the array,e, may be written as

Memn=C i exp [- jk(rm(~i)+rn(~i))l (1)i = 1

whererm(ui)=[R2+-x;-2Rxmui] I2 (2 )

is the distance between the mth antenna and the ith target, k= 2n/h where h is the wavelength of the transmittedradiation, and the term inside the exponent in (1) is a phasedelay corresponding to the round trip distance from thetransmitter to the receiver [141.B. Superresolution of Two Coherent Targets

A technique to resolvetwocoherent targets atu = u1andu=u2usingtwo transmit/receiver antennas atx =xI andx =x2, and a receiving antenna atx =x3will nowbedeveloped.The objective is an estimator that tends to infinity at the targetlocations and is smalleverywhere else. This is the familiar all-pole estimator used n various SR techniques [l], [6], [15].These estimators are the reciprocal of thesquaredmagnitude

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1208 IEEE TRANSACTIONSONANTENNASND PROPAGATION,VOL. AP-35,NO. 11. NOVEMBER 1987of the dot product of a weight vector anda beamsteeringvector, the weights being determined from the measured CFAso that they are orthogonal to the beamsteering vector at thetarget locations. With this in mind, consider the followingsystem of equations:

w3(l)e11+ 3(2)e12=-e13 (3a)w3(l)e21k w3(2)e22=-e23 (3b)

wherew3 1) andw3(2)are unknown weights o be determinedby solving3), and the subscript 3 indicates that the CFA at thereceiving element located atx =x3 is assigned a weightequalto unity. Using (l),we may rewrite (3) asW T B ( u l )+- xp [jk(rl(ul) l(uz))]WTB(u2) 0 (4a)2S

where Wis the weight vector equal to [w3(l), w3(2),1 T, B(u)is a beamsteeringectorqualoexp( - jkr~(u)) ,exp (-.J %(u)), exp ( - jk(u) ) l and thesuperscriptT indicates transposition of a matrix.The constants attached to the WTB(u2) term in the twoequations in (4) will be different ifr2(u2) 2(u1) rl(ul) l(u2)#nX, n=k1, -t2, - - - . ( 5 )

Therefore, if (5) is satisfied, the two equations in (4)demand that WTB(u l )= WTB(u2)= 0. Furthermore, if (5)holds, the system of equations (3) may be solved uniquely forw3(1)and w3( 2) so that WTB(ul) = WTB(u2) = 0.Therefore, an all-pole intensity pectrum estimator of the form

I (u)=c/ W%(u) 12 (6)where C isa positive constant, tends to infinty as uapproaches uI and u2. In the presence of additive Gaussiannoise in the measured CFA, E {I (ui)}s proportional to ISi I ,as shown in Appendix I ( eq. (40)). Thisdesirable propertyofI(u) is absent in the least squares estimator [161.C. Superresolutionof M Coherent Targets

The SR technique willnow be extended to resolve Mcoherent targets, usingK transmitlreceive antennas( K2 M )and one receiving antenna. The weightswK+(n),n = 1, - - ,K are obtained by solving the following matrix equation:

orEWK+= - E K + 1 (7)

where thee,, are given by (1).Further insight may be gainedby decomposingE andE K + in(7).After some manipulations,we may rewrite (7) as

STWK+=-SXK+ (8 )

whereS is aK x M matrix whose elements ares = s, exp(-jkr,(u,)), Tis anM x K matrix with elementstmn=exp= [exp(-jkrKcl(u1)), ...,exp ( - j k r ~+~( u ~) ) I .tmaybeshown that if(-jkrn(um)),W;+1 = [ w K +I ( ~) ,.*,~K+~( K) I ,ndXz+l

rm(U k)-r m(u/)+r,(ul )-r n(uk)#ih,i =+1, +2, * . - (9)

then the rank of E , given by (7), isM. Because the likelihoodis small that thesumof the four distances n (9)is exactly equalto an integral number of wavelengths,(9)will be satisfied formost target scenarios. If (9) is satisfied, the rank of E andhence T equalsM, and (8)may be written asTWK+,+XK+,=O, orDK +I(ui)=O,=l , * - - , M

(10)where

K+1&+1(u) = w + ( i ) exp (.hkfu)) (1 1)

i = I

is the dot productof a beamsteering vector and a weight ectorand wK+ (K+ 1 ) = 1. Therefore, anyall-pole ntensityspectrum estimator I(u)of the formI (u)=C/I DK+I (u)l212)

whereC is a positive constant, tends to infinity as u tends toui, = 1 , * - ,M. BecauseTandX,+ are independent of theSi, we observe from (10) that the weight vector is alsoindependent of the target amplitudes.Until now no far field restrictions have been mposed; hencetheSR technique developed above can be usedto detect targetslocated in thenear and far field of the array. However, fromthis point onwards, we shall assume that all the targets arepresent in the far field of the array. Thisassumption simplifiesthe mathematical expressions to be derived and enables thecomparison of thistechnique withtheMCM andCM , both ofwhich yield erroneous estimates of the locations of near fieldtargets even under noiseless conditions.I f the targets are in the far field of the array, the all-poleestimator (u),given by (12),simplifies considerably by virtueof the fact that the denominator reduces to

K+ 1DK +l(u)= wK+l ( i ) exp (jkmi). ( 13)

i = 1D. Artifactsof the All-Pole Ektimator

If the poles of (u)are located only atu =ul, - - - ,uM,thenI(u) may be considered to be a good estimator for thetargetlocations. However, as shown below, I(u)will have poles foru differentfromthe target locations, especially if the array issparsely filled( L% (N - 1)N2whereL is the length of thearray). The locations of the spurious poles will be determined.Armed with this knowledge, a technique will bedeveloped tosuppress these poles of I (u).Because the poles of I(u)occur atthezeros of DK+(u), t is-

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ND STEINBERG: APERIODIC ANTENNA ARRAYS 1209icient to enumerate the zeros of DK + (u) to find the polesZ(u). It will now be shown that DK+u) is zero for thoseues of u where the quantityAui 4 - u;, i = 1, e , M ,multaneously equals theK quantities mk/(Xk - X I ) , k = 2,- e , K + 1, where the m k are integers. In other words, well prove that the zeros of DK+l(u)re enumerated by the

Am2 Am3 hmK+ Aui 4 u - u j = - - --=. . = (14)X2 -X1 X3 -X1 X K +1 -X1= 1, e . . , M , andmk = 0, kl , k2, *..,k = 2,K + 1.There aretwo conditions under which the equalitiesin (14)satisfied. The conditionmk =0, allk, mpliesM =ui all i;r pole locations correspond to mk =. The second condition includesal l sets of nonzero values ofk that satisfy (14). For each such' set all themk./(Xk - xl )ona nonzero value, implyingu # ui; in other words well prove that a false target or spurious pole occurs for each

those sets. If u # ui,we may write (14) asXk =x1+ h n k /mk = k l , k2, * - - , k 2, * - * , K 1. Usingthisinin

=exp (jkxlAui)DK+l(ui)=O(15)m = 0. Therefore, the zeros of DK + (u) occur for al lgiven by (14). The spurious zeros can be ignored if all ofm occur outside the visible range, i.e., for JuI>1.We can

sure this by using a periodic array withan nterelementcing of M2. However, it is shown later that under noisytions, the mean squared error in the estimate of the targetons decreases as the length of the array increases. Weld therefore like to usearrays whose interelement spacingsuch larger than X/2 while at the same time keep all therious zeros of DK+I(u)utside the visible range. Over-cing the elements in a periodic manner leads to grating, which replicate the target poles abouteach grating lobe.the element locations are randomized it will be difficult tond a set of integersmk, k = 2, - ,K + 1, which satisfies) for IuI 1). Let the transmitters radiatesequentiallyand for each ransmission let the N receiversrecord the CFA samples simultaneously. A total of KN CFAsamples are stored. Only K(K +1)

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1210 IEEE TRA NSACT IONS ON ANTE NNASAND PROPAGATION, VOL. AP-35, NO. 11,NOVEMBER 1987RAM. The MLM estimator can also be written as the parallelresistor summation ofseveral all-pole estimators, however theorder of these estimators varies from 1 toK [17].BecauseMtargets can be correctly represented by anall-polemodelwhose order is at eastM,we would expect he RAM to yield abetter estimate of the target locations than the MLM.For future use, we note that (16)may also be written as

STWi=-SX ; , i =K +l , - . e , N (19)whereS is aK X M matrix, Tis anM x K matrix, andX; sa vector of lengthM. These quantities are defined below:

smn=Sn exp (jkunxrn) (20a)tmn=~ X Pjkumxn (20b)

1 (20c)e-jhlx;,... e-jk~l,wxi].m UPERRESOLUTION UhWER NOISY ONDITlONS

A.Mean Squared Error in the Target LocationEstimatesThe behavior of theRAM intensity spectrum under noisyconditions willnow be examined, and an equation for the-meansquared error of the target location estimates will bederived. In the presence of noise, the CFA at the qth antennawhen the pth antenna is transmitting, denoted GP,, may bewritten as

Mgp,= Siexp [ jkui(xp+x,)]+np, (21)i = 1

where the quadrature components of np, are assumed to beindependentnddentically distributed Gaussianandomvariables each with zero meanand variance 0:. The SRtechnique involves solving the ollowing (N - K) matrixequations to obtain ( N- K ) weight vectors W ;EWi=Eil i =K +l , -..,N (22)

where is aK X K matrix whose pqth element s equal togP,, J!?L = [ & i , - - , K ; I T , andR.= [R.(l),- e , Wi (K ) ]The Wj obtained by solving (22)are substituted nto (19)toobtain theRAM intensity spectrumI(u).Clearly, the presenceof noise in the measured CFA will make he weight vectorsW ;deviate from their noiseless valuesW;.Therefore the peaks inI(u) will be shifted from the actual target locations, and theamount of the shift will depend on the noise samplesnp,.Weshall now evaluate the variance of this shift in the peaklocations of T(u).Assume that W; = W; + 6W;, i.e., the noisy weights arethe sum of the noiseless weights plus an error weight vector.Assume further that the peakposition error is a smoothfunction of he noise samples. Then it may be represented by aTaylor's series within a finite region of convergence. Forsufficiently high N R the peak positionerror may be estimatedby truncating this series after the linear term. Once hisapproximation is made, the variance of the peak positionerrorcanbe calculated given the second-order statistics of the noise

process. The validityof uch a linear approximation issupported by computer simulation results in SectionI V.The variance of the peakposition error is derived inAppendix I, and s given by (40).This expression does notlend itself toward making simplifying observations about thenonlinear RAM intensity spectrum estimator. It will now besimplified for the two target case. Assume that thewo targetsto be resolved are at u = u1and u = u2and have complexamplitudesS and S2, respectively. Let the imaging systemconsist of two transmitheceive antennas atx =0andx = L ,and(N - 2)receivers atx =Xk, k = 3, a , N,where thexkare independent uniformly distributed random variables be-tween0andL. Equation(40)will now be simplified for twocases, viz: closely separated and widely separated targets.When Iu l - u21 < 0.15X/L , i.e., the two targets areseparated by less than0.15 beamwidths, (40) canbeapproxi-mated by [18]0 ; P E{( u - u ; ) ~}=( ~u~/ I S~~~) ( ~/ L ) ~( CI

i = l , 2 (23)whereC1 s a dimensionless constant which depends on thex;,i = 3, - - e, N , andsusuallybetweenandWhen the targets are separated by more thana beamwidth,i.e., (ul - z.421 = (n +p)X/L,wheren is a nonzero integerand0

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ANTENNA ARRAYS 12115.0-

-5.0-

-15.0-

-25.0-

- 35. 0-

/INCOHEREST

-45.0-0.6 ' -81.5 ' -81.4 ' -8.3 ' -d.2 ' -dl ' d.0 ' a'l ' 012 ' 0.'3 ' 014 ' n.'5 ' B.'EuFig. 2. Comparison ofRAMwith Fourier beamforming. Intensity spectra obtained by using two transmitkeceive antennas and fourreceivers. Len@ of array = 45 wavelengths. Twounit amplitude targets at u = - .05 and 0.05.Q PE (k - l), k > 1, wemayconclude hat kts are present. If all the PE(m) are of the order of thegnal power, we may conclude that more thanK - 1 targets

Statistics of I@)Near Its MaximaIn Appendix I it is shown that the expected value of I(&),

Eii is the estimate of the location of the ith target, islSi12.Further, it is shown hat f the noisecan be estimated (by estimating the variance of theived signal from empty rangebins), it is possible to obtainlue for the constant in the numeratorof I(u) so that

I (&)} = ISi *. Thisvalueof C isgivenby (42).(42) into (le),we obtain the final form of theAM ntensity spectrum estimateI(u), or N - K > 1, as=(4U3N-K- l ) /K}[ lDi(u)l2] -2,

i = K + IN-K> 1. (26)

Further, from (43)and (44) we have

computer simulations. This variance is also compared to thevariance of the target location estimation error for the MCMand CM. Theoretical expressions for the variance for theMCM and CM are available for very closely separated andwidely separated targets [19];however, we shall be dealingwith target separations where the approximations leading tothe expressions for the variance n [19] maynotbe valid.Therefore, empirical values for the variance for the MCM andCM are obtained from Monte Carlo simulations.Inmostof the simulations, two targets, each of unitmagnitude,were located at u = -0.05 and 0.05: respec-tively. The imaging system consisted of two transmitheceiveelementsat x = 0 and x = L , and four receivers at x =0.93L, 0.69L, 0.54L, and 0.18L. Therefore, the beamwidthof the array is X/L, the separation between the targets inbeamwidth units isO.lL/X, and the thinning ratio2L/X(N -1 ) is 0.4L/h. The SNR is defined as 1/2a$ where ut, is thevariance of each of the quadrature components of the noise.Because two transmitters are needed to estimate the intensityspectrum using the RAM and only one transmitter is requiredfor the MCM and CM, the SNR for the MCM and CM wasalwaysdoubled, thus keeping the total transmitted power

E{I (C i )}~I Si\' ,N-K >l (27) constant. Sixquispaced receiving elementseresed for theMCM and CM. Theseparation between adjacent elements wasE{[I (ai)- Sil']'} Si12/(A'-K-2), N-K>2. M2. Therefore the lengthof the array wasixed at2.5 X for

(28) both the MCM and CM, and was a parameter of the lengthLFrom (27)we observe that the expected value of the peaks Fig. 2 is a plot of the RAM intensity spectrum for variousRAM intensity spectrum is proportional to the target SNR, for a thinning ratio of 18. The separation between thensity; this advantage is not found in the MCM and CM. targets is 4.5 beamwidths. Fig. 2 demonstrates that thinnedarrays canbeused for imaging purposes without any signifi-I V . REsULTs CoMPrUUSoN MCM cant spurious peaksn the intensity spectrum.taye

CMbservedhat the sidelobe characteristics of all the imagesnIn this section, the validity of the variance of the target Fig. 2 are similar. Thismaybeexplained as follows. Thestimation error, given by (40) is first confirmed by sidelobe artifacts occur at the spurious zeros of all theDi(u) n

for the RAM.

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IEEE TRANSACTIONSON AhTENNAS A ND PROPAGATION, VOL. Ap-35, NO. 11, NOVEMBER 1987- TBEORY

1 . ~ ~ 1 1 1 ~ 1 . 1 . I .10 20 10 $0 so 60 !O (SSR dB)

Fig. 3. RMS angular error in target ocationestimates versus SN R forvarious array lengths. Twounit amplitude targets at u = -0.05 and 0.05,Solid lines indicate heoreticalvalues for RAM, CM , and MCM. Thevarious symbols represent empirical values from Monte Carlo runs.

(18), which depend on the element and target locations via(14). Since the array and target geometry were the same for allthe three images in Fig. 2, their sidelobe characteristics aresimilar.Alsoshown in Fig. 2 is a noiseless intensity spectrumobtained by using conventional (Fourier) beamforming. Theintensity spectra obtained from thetwo transmissions wereaveraged to form this image. Although the SNR is infinite, thepeak artifact of the Fourier intensity spectrum, due to the smallnumber of receivers, is -1.3 dB. The RAM clearly outper-forms conventional Fourier imaging.Fig. 3confirms the validity of he theoretical variance of thetarget location estimation error u i = 1, 2, given by (40).The root mean squared angular error (RMSE) in estimating thelocation of the target at 1c = 0.05 is plotted against SNR forvarious array lengths. The solid lines represent the theoreticalRMSE obtained from (40). The various symbols epresentempirical valuesobtained by usingMonte Carlo methods.Fiftytrialswere conducted for each SNR and , and the targetlocation estimates i j , i = 1, 2, were obtained by looking forthetwo highest peaks of I(u) in the visible range. Empiricalvalues for the variance were obtained by averaging the 50valuesof (u i - It can be seen that the theoretical andempirical values of RMSE agree closely. The symbols + and x represent empirical values of the RMSEnestimating the location of the target at u = 0.05 by using theMCM and CM respectively. The RMSE for the RAM with a165 wavelength array is abouttwoorders of magnitude lowerthan the RMSE for the MCM and CM.

When the RMSE is approximately0.05, the widths of thetwo peaks in I (u) near the target locations are of the sameorder as the separation between them. When thishappens, thetwo peaks merge to form a single peak about midway between

the two targets, i.e., theRMSE in estimating each targetlocation is half the distance between them. This phenomenonis evident in thedatafor the three 2.5 wavelength arrays, allofwhich taper off at low SNR to about0.05.The empirical variance also deviates from theory for lowSNR. This behavior may be explainedn the followingmanner: while evaluating the theoretical variance, the peakposition error was approximated by truncating its Taylorsseries after the linear term. This approximation is valid forlarge SNR, however, when theSNR is small, (40) will yieldan inaccurate value for variance sincethehigher order termsof the Taylors series will also contribute to the peak positionerror.As mentionedearlier, the variance of the error in estimatingthe target locations s much higher fortheMCM andCM. Thisindicates that he accuracy in estimating the target location canbe vastly improved by usingheRAMas opposed to the MCMand CM. Further, because the widthsof the peaks in theestimated ntensity spectrum are proportional to the RMSangular error, the RAM can be used to resolve targets whichmaynotbe resolvable byheMCM and CM. This isdemonstrated in the following figure.Fig. 4 is a plot of the estimated intensity spectrum obtainedby using the RAM, MCM, and CM. In order to keep thetransmitted power constant, the noise power in the measuredCFA for the MCM and CM was half the noise power in theCFA for the RAM. The narrow widths of the peaks in theintensity spectrum for largeL clearly indicate that the RAMoutperforms the MCM andCM. The spurious peaks thatoccurin the intensity spectrum when the length of the array is largehave been suppressed to a level of - 19 dB, i.e., the noisepower level. This confirms the observations made previouslyon suppressing the spurious peaks while using thinned arraysto estimate the target locations.

V. EXPERIMENTALESULTSThe theory developed previously will now be verified withreal data. The following experiment was conductedat the fieldsite of theValley Forge Research Center, University ofPennsylvania. A 16 element array with adjacent elementsseparated by 0.4m (length of array = 16 m) was used forimaging purposes. The wavelength of the transmitted radiationwas 3.125 cm. Three one foot comer reflectors were placed inthenear field ofthe array as shown in Fig. 5. The rangeresolution of the radar system was 75 cm. The distances inFig. 5and the element locations were measured by tape, andhence are only approximate. The SNR per element in thereceived signals was estimated to be 20dB.For each range bin, two sets of echoes were recorded ateach of the 16 elements by transmitting sequentially from twodifferent locations. The comer reflector at the62m range binwas usedas a beamformer to phase cohere the array. Fig. 6 isthe ntensity spectrum of the targets in the 65 m range binobtained by 1) adding the Fresnel intensity estimates obtained

from the two sequential transmissions, labeled Fresnel beam-forming plus incoherent addition, 2) the RAM developed nSectionIV-B, and 3) CM. The two targets were separated byeight beamwidths (length of array = 6m for the first image;

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AND STEINBERG: APERIODIC ANTENNA ARRAY S 12135.0-

-5.0-

2 -25. Rz, i35. a

-0ls - a l 5 I 4 4 - 013 1 -0:2 -;. I ;0 3 011 0.12 ' 013 8 01 4 015 I 0;s uFig. 4. Comparison of RA M wth MCM and C M. RA M ntensity spectrum obtained by using two transmit/receive antennas andfour receivers, CM and MCM wth six receivers. SNR =20 dB for RAM. 23 dB for CM and MCM . Two unit amplitude targets atu = -0.05 and 0.05.

x j y . 5 ~ 1 4 65mT 62mx CORNERREFLECTORRECEIVER ELEMENT

RECEIVING ARRAY

I- X6m ------IFig. 5. Arrayand argetgeometry for microwave experiment.

wewouldexpect this image to estimate the targettions accurately. The weights for the RAM and CM wereined by using three and four adjacent elements, respec-two targets were separated by about oneamwidth for both these methods. RAM yields an accuratetimate of the target locations and intensities, and does notn the sidelobe artifacts present in the diversity image.yields a distorted image as the quadratic curvature of thevefront leads to errors in the subarray averaging performedng its impIementation.V I . SUMMARY

An SR technique, called the random array method, forIving coherent targets located in the near or far fieldsof

__---/. . , El /-.----..-.( 4 SEC EI YER S)

- 4 L e-2.5 -$.B -1.5 - ! .a -a 5 e.a x5 1.8 1 5 Z B 215CROSSRABGE (RETERS)Fig. 6. Experimentalverification of RAM and comparisonof RAM wthFourier beamforming and CM . Intensity spectra obtained by using RA Mwith two transmittersand three receivers, CM with one transmitter and ourreceivers, and incoherent addition of hvoFourier images obtained by usingtwo transmitters and 16 receivers.

periodic andfor aperiodic arrays has been developed in thispaper. The ey to its success is the useof sequentialtransmissions from different locations for decorrelating thecoherent targets. ExistingSR techniques that can be used toresolve coherent targets such as CM [11, [4], or the MCM [ 5 ] ,[6]require that 1) the array be periodic, and2) the targets belocated in the far fieldof the array. Other techniques require amoving array [111or a multidimensional search for the targetlocations which could be computationally cumbersome [121.In addition to removing these restrictions, it is shown that forthe same amount of ransmitted power andthe same numberofreceivers, the M SE in the RAM target location estimates ismore than an order of magnitude lower the MSE in the MCMand CM target location estimates.A theoretical expression for

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1214 IEEE TRA NSACT IONS ON AhTENNAS AND PROPAGATION, VOL. AP-35, NO. 11, NOVEMBER 1987the contribution of noise to theMSE is derived, and confirmed by setting the derivative of the truncated series to zero, i.e.,by empirical methods. The practicability of this technique sdemonstrated with measured microwave data. iii- ui= - A '(u;)/A (u ; ) . (34)

APPENDIX UsingAOk(u;)= 0 for k = K + 1, - a , N, ubstituting(32) into (36), andneglecting terms containingA;,(ui) inThe RA M intensity spectrum estimate in the presence of favor of A I ( ui )we obtainnoise is Ok

where the weight vectors W ; are obtained by solving (22).Substituting mi = Wi + 6Wi ?V i is the noiseless weightvector given by (16)), I? = E +N (E s the noiseless CFAmatrix defined by (7)andN is aK x K noise matrix whosepqth element isnp4) ,and ?; =E i +N; hereE,? = [eli, - - ,eKj]ndN = [nl;, ,nKi]nto(29),and usingE =STandE ; = SXi,we obtain

From (29) and (31) wecanshow that E {Al k(ui))= 0,therefore the mean value of the right hand side of (35) is zero(note that Ao~(u)s deterministic). ThereforeE{&}= u;,i.e., I(u) is an unbiased estimator of the target locations. Themean squared error in the target location estimates is

whereS, T, andX;are defined by (20).UsingSTW; = -SX;and ignoring the product of twosmall quantities viz.: the noisematrixnd the error weight vector webtain * ~ & ( u ; ) ] ] / [5 l A i k (U i ) l2 ] 2- (36)k=K+1ST6 W;=-N; - W;. (29) Wehallow evaluate the terms on the numerator of the

The intensity spectrum estimate can be rewritten as right hand sideof (36). Using (20) for the matricesS and T,and (29)we can show thati=K+ 1

whereKAoj(u)=exp(jkux,)+x wi(n)exp (jkux,,) (31a)

n = I

andKA ~~(U >=6wi(n)exp (j kux, ). (31b)where the superscript indicates transposition andomplex

conjugation of a matrix, and the subscript i stands for the ithwewish to determine the maximaof I ( ~)ear the target diagonal element of a matrix. In a similar fashion we obtainn=llocations, i.e., the minima ofA (u)nearu = u,, r n = 1, - ,Mywhere E {(A ~~(u ; ) ) ~)E {A~~(u; )A I I (u; ))=O38)

C Nand

A(u)=-= x IAoi(u)+Al;(u)I2* (32) E { A ~ ~ ( ~ ; ) A ~ , ( U ; ) ) = ( ~ ' T Z ~ / I S ; ~ ~ )'(1 j =K+1ExpandingA (u) in a Taylor's series aroundu = u;, = 1, wk(n)w,*(n) (T*TT),;'. (39). M,and neglecting erms higher than he second order, weobtain (37) to (39) into (36), we obtain the following

)A ( u)=A u; )+ U-u;)A (u;)+0.5 (U-u ~) ~A(u;) (33) expression for theMSE in the target location estimate

whereA'(u;) = dA(u)/duevaluated at u = ui, etc. Let hevar {tii )=(aZ,/I Si12)(T *T 7),1Cominimum ofA (u) in the neighborhood ofu;occur at ti;. If t i i issufficiently close to ui so that the truncated Taylor's seriesreasonably approximatesA (u) at t i ; , then i j cane determined k=K+I

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ND STEINBERG: APERIODIC ANT ENNA ARRA YS

Equation (40) s the mean squared error in the estimate oftarget located at u = ui,and is oneof the main results ofAs T, W,, andAik(ui)do not depend on thet strengths Si . we observe that the MSE is nverselyonal to the SNR of the target.Wenow evaluate the meanand variance of the target(& ) for largeM andN - K , withK 2 M.

(33)and (34)into (32),we obtainI ( a i ) / ( A ( ~ i ) - [ ( A ( u ; ) ) / 2 A ( u i ) ] ) .

For large M and N - K , the seconderm in theinator can be neglected[181, and

akiandbkiare the real and imaginary parts of A Ik(ui).e aki and bk are the weighted sum of Gaussian randomables, therefore they are also Gaussian. It can be shown

a,; and bki are iid zero mean Gaussian andomables whose variance is half of(A-9). Further, for largeKcan be shown that) (T*T7)-l tends to I / K (l is theK X Kntity matrix), and 2) CIwk(k(n)l (the summation stakenn from 1 o K ) tends to unity.Using these in theding equation we obtain E{aii! = E{bi;}= 2ui/2. Finally, the denominatorof I(ui) s the sum of2(N -) id zero mean Gaussian random variables, and has a chi-f with 2(N - K )degrees of freedom[21].Using

l(& )}=03 for N-K= 1, E{I(ti,)}=CKISi12/4u2,(N-K-1),orN-K>l.

If the noise power can be estimated, the value of C can beso that E{Z(&)}= 1S;I. This value of C isC = ~U ~( N - K -)/K,-K>l. (42)

If this value of C is used in (25),we can show thatE{I (ti;)}=03, N- K =1 (43d

- ( S i l 2 , N-K>l (43b)var{(Lii)}=o3,- K 5 2 (44d=S;I, N-K>2. Wb)

.1215 REFERENCES

A. K . Luthra, Maximum entropy method n the space-angle domainand a new technique with superior performance, Ph.D. dissertation,Syst. Eng. Dept., Univ. Pennsylvania, Philadelphia, 1981.J . E. Evans, J . R. J ohnson, and D. F. Sun, Application of advancedsignalprocessing echniques ,to angle of arrival estimation n ATCnavigation and surveil lance systems, L incoln Labs., Lexington, MA,Tech. Rep. 582, une 1982.J . Makhoul, L inear prediction: A tutorial review, Proc. IEEE, vol.63, no. 4, Apr. 1975.T. J .Shan, M.Wax, and T . Kailath, On patial moothing fordirection of arrival estimation of coherentsignals, IEEE Trans.Acoust., Speech and Signal Processing,vol. ASSP-33, no. 4, Aug.1985.A. H . Nutall , Spectral analysis of a univariate process with bad datapoints, Naval U nderwater Syst. Center, NewLondon, CT,Tech.Document 5419, May 1976.T. J . UlrychndR. W.Clayton, Timeseries modeling andmaximum entropy. Phys. Earth Planetary Interiors, vol. 12, Aug.1976.R. 0. Schmidt, Multiple emitterocation and signalparameterestimation, in Proc. RADC SpectrumEstimation Workshop, Oct.J . Capon, High resolution requency-wavenumber spectrum estima-tion, Proc. IEEE, vol. 57, Aug. 1969.F.Haber,R. S. Berkowitz,and 3.S. Meagher,Radio location inmultipathand interferenceenvironments, Valley Forge ResearchCenter, Univ.ennsylvania,hiladelphia, ep.UP-VFRC-18-81,Mar.1981.S. M. Kayand S. L . Marple J r., Spectrum analysis-A modemperspective, Proc. IEEE. vol. 69, A pr. 1981.F. Haber and M . Zoltowski, Spatial spectrum estimation in a coherentsignal environment using an array n motion, IEEE Trans. AntennasPropagat., vol. AP-34, no. 3, Mar. 1986.M. Zoltowskiand F. Haber, A vector spaceapproach to directionfinding in a coherent multipath environment, IEEE Trans. AntennusPropagat., vol. AP-34, no. 11, Sept. 1986.C. in, A daptive superresolution n henear ieldusingperiodicarrays, M asters thesis, Elec. Eng. Dept., Univ. Pennsylvania, 1986.B. D. Steinberg: Principlesof Aperture and Array SystemDesign.New Y ork: Wiley, 1976.J. P. Burg, M aximum entropyspectral analysis, in Proc. 37thMeet. Soc. Exploration Geophys., 1967.S. L . Marple J r . , A newutoregressivepectrumnalysis al-gorithm, IEEE Trans. Acoust., Speech, andSignalProcessing,J. P. Bug: The relationship between maximum entropy spectra andmaximum ikelihood spectra. Ceophys., vol. 37, Apr. 1982.H. M. Subbaram, Superresolution of coherent targets using aperiodicantenna arrays, Ph.D. dissertation. Elec. Eng. Dept., Univ. Pennsyl-vania,Philadelphia, 1984.S. W . Lang and J . H. McClellan,Frequency estimationwithmaximumntropypectral estimators, IEEE Trans. Acousr.,Speech, and Signal Processing,vol. ASSP-28, Apr. 1980.B. D. Steinberg. Microwave Imaging with Large Antenna Arrays.New Y ork: Wi ley, 1983.P. J . Bickel nd K . A. Doksum, Mathematical Statistics. CA :Holden-Day, 1977.

3-5, 1979, pp. 243-258.

V O~.ASSP-28, Aug. 1980.

Harish M . Subbaram was born in Mysore, India,on November 8, 1957. He received the %.Tech.degree in electronics engineering from the IndianInstitute of Technology, Madras, in 1980, and theM.S.E. and Ph.D. degrees in electrical engineeringfrom the University of Pennsylvania, Philadelphia,in 1983 and 1986, respectively.He is currently a Postdoctoral fellow at he ValleyForge Research Center, UniversityofPennsylva-nia. H is research interests include signal and imageprocessing, microwave and ultrasonic imaging, andspectrum analysis.

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1216 JEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL. AP- 35, NO. 11, NOVEMBER 1981Bernard D. Steinberg (S148-A5O-SM&I-F66)was born in Brooklyn, NY , in 1924. He receivedthe B.S. and M.S. degrees in electrical engineeringfrom the Massachusetts Institute of T echnology,Cambridge, in 1949, and he Ph.D. degree from theUniversity of Pennsylvania, Philadelphia, in 1971.He worked in the Research Division of Philcothrough the middle 1950s. He was one of thefounders of General A tronics Corporation n Phila-delphia in 1956 and served as i ts V ice PresidentndTechnical Director or 15 years.Currently, he isChairman of the Board and a founder of Interspec Inc., also in Philadelphia.He has been aProfessor of ElectricalEngineering at the University ofPennsylvania, MooreSchool of E lectrical Engineering, since 1971. He isthe

Director of the University of Pennsylvanias V alley Forge Research Center.The primary work f ths laboratory is the developmentof large, self -adaptivemicrowave imagingsystemsbased on the radiocamera technology developedby the laboratory. His work has been in adar backscatter and in signalprocessing techniques and their appli cations to radar, HF communications,hydroacoustics and seismology. His most recent work is in seIf-adaptivesignal processors, particularly in large antenna arrays.Dr. Steinberg isthe authorof hincipla of Aperture and Array SystemDesign (Wiley,976). i n which the random array is analyzed, andMicrowave Imaging with Large Antenna Arrays (Wiley,1983), whichdescriks radiocamera principles and techniques. For many years, he was aconsultant to the A irborne Radar Branch of the Naval Research Laboratory,Washington, D.C. He is a member of U.S. Commissions B and C of theInternational Union of RadioScience.