MIMO Radar a Idea Whose Time Has Come

8/2/2019 MIMO Radar a Idea Whose Time Has Come

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MIMO RADAR: AN IDEA W HOSE TIME HA S COMEEran Fishlert, Alex Haim ovicht, Rick Blumt, Dmitry C hizhik*, Len Cimini', Reinaldo Valeneuela*

t New Jersey Institute of Technology, Newark, NJ 07 102, e-mail: [email protected], [email protected]$ Lehigh U niversity, Bethlehem, PA 18015-3084, e-mail: [email protected] University of Delaware, New ark, D E 197 16, e-mail: [email protected]* Bell Labs - Lucent Technologies, e-mail: chizhik,[email protected]

It has been recently shown that multiple-input multiple-output(MIMO ) antenna systems have the potential to dramaticallyimprove the performance of comniunication systems oversingle antenna systems. Unlike beamforming , which pre-sumes a high correlation between signals either transmit-ted or received by an array, the MIMO concept exploitsthe independence between signals at the array elements. Inconventional radar, target scintillations are regarded as anuisance parameter that degrad es radar performance. Thenovelty of MIMO radar is that it takes the opposite view,namely, it capitalizes on target scintillations to improve theradar's performance. In this paper, we introduce the MIMOconcept for radar. The MIM O radar system under consid-eration consists of a transmit array with widely-spaced ele-ments such that each views a different aspect of the target.The array at the receiver is a conventional m a y used fordirection finding (DF). T he system p erformance analysis iscarried out in terms of the Cram er-Rao bound of the mean-square error in estimating the target direction. It is shownthat MIM O radar leads t o significant perfomiance improve-ment in DF accuracy.

I. IntroductionThe idea of active direction finding for radar or active sonaris not new (see, for example [1,2]).In radar or active sonar,a known wa veform is transmitted by an omnidirectional an-tenna, and a target reflects some of the transmitted energytoward an array of sensors that is used to estimate some un-known parameters, e.g., bearing, range, or speed. There aretwo common approaches for estimating the unknown pa-rameters. In the first appro ach, high resolution techniques,e.g., MUSIC o r maximum likelihood (M L) [3], are used toestimate parameters of the target of interest. In the secondapproach, the array of sensors is used to steer a beam to-ward a certain direction in space and look for some energy,

WORK BY RICK BLUM AND ALEX HAlMOVlCH WAS SUP-PORTED IN PARTBY THE AIR FORCE OFFICE OF SCIENTIFICRE-SEARCH.

0-7803-8234-x/04/$17.00 0 2004 IEEE

essentially the same way a s a conventional radar with a di-rectional antenna. It i s well known that an array of receiver scan steer a beam toward any direction in space by using aprocess known as beantfor-nzirzg [4]. Unlike high resolutiontechniques, beamforming is based on a fixed transforma-tion.The advantages of using an array of closely spaced sen-sors at the receiver are well known (see, for exam ple [S, ,3 ,6,7]. Amo ng these advantages are: the lack of any mechan-ical elements in the system, the ability to use advanced sig-nal processing techniques for improving performance, andthe ability to steer multiple beanis at once. In this paperwe are concerned with radars employing multiple antennasboth at the transmitter and at the receiver.

Transmit arrays have been proposed in the form of elec-tronic steered m a y s (ESA). With an ESA, phase shifts at thetransmit antennas form and steer the transmit beam similarto a directional antenna , except that the steering is electronicrather than mechanical. Before introducing a new conce ptfor radar with m ultiple transmit antennas, a fair question toask is whether an ESA has any processing gain (in additionto its mechanical advantages). The ESA essentially mim icsthe scanning operation of a directional antenna. Howev er,as we show next, ESA's have no advantage over systemsthat use a single omnidirectional antenna at the transmit-ter. To that end, we note that the error in that angle of ar-rival estimation is a function of the total received energy.Assume that the total transmitted power is independent ofthe number of transnlit antennas. For the single transnlitantennas case, say that the average transmitted power isP and the duration of the transmitted waveform is T sec-onds. Th e energy received from the target is uPT, where Urepresents the target's'radio cross section (RCS). Now, as-sume an ESA that creates a beam with beamwidth +. Withthe beamwidth 4, the transmitter can realize a gain of y(in linear scale). Howev er, since the transmitter needs toscan the whole space in T seconds, it can illuminate thetarget for only 2 econds. The total received energy isUP?% = OPT. This demonstrates that the amount ofenergy received by the radar is independent of the exactnumber of elements of the ESA. Therefore, the ESA has

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no advantage over a radar systeni with a sing le transmit an-tenna. It follows that in later sections, we are justified tocompare the new radar concept with a single transmit an-tenna system.Every target is characterized by its RCS function [8].A targets RCS function represents the amount of energyreflected from the target toward the receiver as a functionof the target aspect with respect to the transmitterlreceiverpair. It is well known that this function is rapidly chang-ing as a function of the target aspect [8].Both experimentalmeasurements and theoretical results dem onstrate that scin-tillations of 10 dB or more in the reflected energy can occurby ch anging the target aspe ct by as little as one milliradian.These RCS scintillations are responsible for signal fading,which can cause large degradations in the systems detec-tion and estimation p erformances.

ory [10,9], we introd uce the new concept of multiple-inputmultiple-output (MIM O) radar. MIMO com mun ication sys-tems overcome the pr oblem s caused by fading by transmit-ting different streams of information from several decorre-Iated transmitters [10, 111. Since the transmitters are decor-related, different signals undergo independent fading. InMIMO communications systems, the receiver enjoys thefact that the average (over all information streams) signalto noise ratio (SNR) is more or less constant, whereas inconventional systems, which transmits all their energy overa sin gle path, the received SNR varies considerably.

Our proposed MI MO radar enjo ys the sam e benefits en-joye d by MIMO conun unication systems. Specifically, ourpropo sed system over conies arget RCS scintillations by trans-nlitting different signals from several decorrelated transmit-ters. Th e received signal is a superposition of inde pende ntlyfaded signals, and the average SN R of the received signal ismor e or less constant. T his is in marked contrast to conven-tional radar, which un der classical S werling models suffersfrom large variations in the received power.The reader shou ld note that the whole notion of MIMOradar is new, and the main purpose of this paper is to in -troduce this concept. In addition, we present one specificscenario in which our proposed system improves the per-formance considerably. For clarity of presentation, in thispaper, the treatment of MIMO radar focuses on directionfinding (DF), ignorin g range and Doppler effects. In sub-sequent work, we will elaborate on these and many otheraspects associated w ith this concept.The rest of the paper is organized as follows: Section

I1 introduces the MIM O radar signal and channel models,including a classification of various MIMO radar systems.Section I11 presents an analysis of a MIMO DF system.The essence of the analysis is to determine the Cram er-Raobound (CRB) associated with the MIMO radar and coni-pare it with that of a single-antenna system. Th is sectioncontains a specific example with a uniform linear antenna

Motivated by recent developnients in commu nication he-

array at the receiver. Numerical results are provided. Fi-nally, Section IV draws conclusions.

11. MIMO Radar Signal ModelIn this section, we describe a general signal model for theMIMO radar. Th e model focuses on the effect of the tar-get spatial properties ignoring range and Doppler effects.The signal m odel separates the contribu tions of the trans-mit array, target, and receive array, to the received signal.By doing so it provides insight into the principles of MIM Oradar.Not surprisingly, the radar MIMO signal and channelmodel is related to MIMO channel m odels for communica-tions, for example 1121. The signal model can be used todescribe both conventional radar systems and o ur proposedMIMO radar system. Assume a radar system that utilizesan array with M antennas at the transmitter, and N sensorsat the receiver. The transmitter and the receiver are not nec-essarily colloca ted (bistatic radar). Assume also a far fieldconiplex target that co nsists of many (say, Q) independentscatterers with approximately the san ie RCS. This assunip-tion corresponds to a target composed of many small re-flectors. The target is illuminated by narrowband signalswhose amplitude does not change appreciably across thetarget (roughly, that means a bandwidth smaller than c / D ,where c is the speed of light and D is the target length).Each scatterer is assumed to have isotropic reflectivity mod-eled by zero-mean, unit-variance per dimension, indepen-dent and identically distributed (i.i.d.) co mplex random vari-ables c,. The target is then modeled by the diagonal matrix= (l/m)ia g (CO,. ., ~ - 1 ) ,where the normaliza-tion factor makes the target RCS E[aE X * ) ]= 1 (thesuperscript deno tes complex conjuga te) independent of thenumber of scatterers in the model. With RCS fluctuationsthat are fixed during an antenna scan, but vary indepen-dently scan to scan, ou r target model is a classical Swerlingcase I [8].For simpiicity, we assume that the target scatterers arelaid out as a linear array, and that this array and the arraysat the transm itter and receiver are parallel. This scenario isdepicted in Fig. 1 . The signal radiated by the rn-th trans-mit antenna impin ges as a planewave on the Q scatterers atangles Brn,*, q = 0 , . .. Q - 1 (measured with respect tothe normal to the array). This assumption holds when thedistance to target R is much larger than than the transmit-ter array aperture. The signal vector induced by the rn-thtransmit antenna is given by

where A * is the spacing between the first and (q+ 1)-thscatterer, X is the carrier wavelength, and the superscript72


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denotes vec torh atrix transposition. Assume that the tar-get scatterers are uniformly spaced, i.e., A q = qA . Thecommon phase shift between the transmitter and the tar-get has no bearing on perforniance and is left out. WithMIMO radar, we seek to exploit the spatial diversity of thetarget. To ach ieve spatial diversity, it is required that differ-ent transmit antennas see uncorrelated asp ects of the target.Mathem atically, this is expressed as orthogonality betwee nsignal vectors. For an arbitrary antenna element rn, he con-dition for orthogonality with the signal vector induced byelement rn + 1, s given by

where the superscript denotes complex conjuga te and trans-pose. The signal vectors are oroanized in the A4 x Q trans-mit matrix G = [glg2 ~ M F .ssuming that the rangeto target is much larger than the inter-element separationat the transmitter, the range is assumed independent of thetransnlitter antenna. Geom etric conside rations lead to therelation

s inem+l , , - in8,,, x d t / R , ( 2 )where dt is the inter-element spacing at the transmitter. Us-ing this in (l) , and noting that the right hand side of ( 2 ) sindependent of the scatterer index q, we obtain

0-1

q=O, jWdtlR)qA/x = 0 . ( 3 )

A necessary condition for (3) to be met is for the angles tocomplete at least a turn of the unit circ le, i.e.,(4)

This condition obtained solely from geometric considera-tions also has an appealing intuitive physical interpretation.The beamwidth of the energy backscattered from the tar-get towards the transmitter is approxim ately given by AID,where D = (Q - 1)A is the target size. Th e target presentsdifferent aspects to adjacent transmit antennas if the inter-element spacing at the transmitter is greater than the targetbeamw idth coverage at distance R , namelyXRdt 27,

which turns out to be the same a s (4).To complete the transmitter model, we assume phaseshifts imposed on the transnlitted signals represented by th elength-A4 vector b (8') ,where the rn-th element is givenby ej2rr(m-1)dc sin Lowpass equivalents of the trans-mitted waveforms are listed along the d iagonal of the m a-trix S = diag (51,. ~ )The transnlitted waveforms are

normalized such that lsiI2 = 1/M. The normalizing fac-tor ensures that the transmitted power is independ ent of thenumber of transnlit antennas. In case all antennas transnlitthe same waveform, S = SIM, here the subscript deno testhe ord er of the unity matrix.

NBtmcnta

Fig. 1. Bistatic radar scenario. The target consists of niulti-ple scatterers organized in the form of a linear array.The model for the array at the receiver is developed sim-ilar to that at the transmitter, resulting in an Q x N channelmatrix K, where each row a n = 0,. . N - 1 , con-stitutes a signal vector from a scatterer of the target t o thereceiver array. The bearing between scatterer q and antenna

n is denoted en,*.Orthogonality conditions for target signa-tures at the receiver can be developed similar to (4), with theinter-element spacing at the receiver d , replacing dt. A caseof special interest is a receiver array with d , = X / 2 . Thismakes possible unamb iguous DF. Since the range to target isassumed much larger than the array aperture, On,q N en fo ral l q. Consequently, the receive matrix K = 10 @ aT 00 ) ,where the operation is the Kronecker product (each elem entof the first operand is multiplied by the second operand),and the vector 1~ is a Q x 1 vector of ones. Finally, theN x 1 teering vector at the receiver is denoted a e ) and itsdefinition is analogous to b (e') defined earlier.Putting it all together, the MIMO radar channel m odelis given by the M x N matrix

H = GCK. ( 6 )The element hij of the channel matrix provides the comp lex-valued channel gain from transmitter antenna i to receiverantenna j.The signal vector received by the M IM O rada r(after demodulation and matched filtering) is given by

r = HTSb* e') + v, (7)where the superscript * denotes complex conjugate, and theadditive (spatially) white Gaussian noise vector v consistsof i.i.d., zero-mean complex normal random variables with

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variance l/SNR, where SNR is the average signal-to-noiseratio per antenna element at the receiver. T he channel m odelin ( 6 )and the signal model in (7) can be used to represent avariety of scenarios for MIM O radar.Model C lassificationMIM O radar signal models can be classified into three gen-eral groups:

Conventional radar array modeled with an array atthe receiver and a single antenna or an array at thetransmitter. The array elements are spaced at half-wavelengths to enable beamforming and DF.MIMO radar f or DF. Transmit antenna elements arewidely spaced to support spatial diversity aspects ofthe target. Receiver array performs D EMIM O radar for detection of multiple targets. In thisscenario, there are multiple targets in the range cellunder test. Antennas at the transmitter are less sepa-rated than in the previous scenario, such that scatter-ers belonging to the same target are not individuallyresolved. Yet, the separation is sufficient to resolvemultiple targets in the same range cell.

The first two signal models are detailed in the sequel.The third will be discussed in a subsequen t publicatio n.Convent ional Rada r ArrayConventional systems are systems in which the elements ofthe transmitting and receiving arrays are closely spaced . Atthe transmitter, that means that the inter-element spacingdoes not meet (5) or, equivalently, that multiple elementsare contained within one target beamw idth. A t the receiver,the spacing is d, 5 X /2 to enable unam biguous estimationof the angle of arrival.Let the target bearing with respect to the transmit and re-ceive arrays be respectively, 6; an d &. The transmit m atrixis given by G = b (0;) @ 1;. The receive matrix is givenby K = 1~ @ aT 80 ) . t follows that the channel matrix isgiven by 1H = -ab (e:,) aT 0,) , (8)where a = ( l /& ) l T ( , and ( s the vector formed by thetarget gains Cq , (= [CO, . Q-11 .By assumption, Cq arezero-mean, unit-variance per dimen sion, i.i.d.; he nce by thecentral limit theorem a approaches a zero-mean, complexnormal distribution. Subsequently, the' target's RC S laI2,follows a xi chi-square distribution with 2 degrees of free-dom. Note that with this model, there is no diversity 'gain'in the target RCS.With a conventional radar array, all antennas transmitthe same waveform s. Beamforming at the transmitter is

f iT

represented by the vector b' (0') .The signal model in thiscase is given byr = (1/J") a(Oo)bT 0;) b' (0') as +v, (9)

Now, if the receiver uses a beamform er to steer towa rds di-rection 0, then the output of the beamformer isy =0/& a+ eo)r+w'

= ( l / f i ) at (e,) a (e,) bT e;) b* e ') as + 21 (10)bT 0;) b' (0') plays the role of the transmit antenna pat-tern, whereas at (&) a (&) is the receive antenna pattern.Since, E [a] = 0 , E [laI2]= 2, /sI2= l /M, the signalpower in (lo), s given by/at6,) a (eo)bT 0;) b* (8') sI2 5 MN2. Note that theinstantaneous signal power Iat (00 ) a (00) bT 0;) b' (8') ysihas a xi distribution (chi-square with 2 degrees of freedom).

This model represents a bistatic radar where

MIMO Rada r: Direction FindingIn MIM O radar for direction finding (DF), the transmit an-tennas are sufficiently separated to meet the orthogonalitycondition ( 5 ) for targets of interest. The columns of thetransmit matrix G meet the orthogonality condition in (1).In contrast, elements of the receive array are closely sepa-rated to enable D F measurements. A ssume that the target isat angle 80 with respect to the receive array normal. T he re-ceive matrix is given by K = 1~ @ aT &) . Since the goalis to illuminate the target to achieve spatial diversity, phaseshifts at the transmitter are set to zero, b (e ' ) = 1 ~ .ro m(6) , t follows that the channel matrix is given by

1H = -CY @ aT e,),fiwhere the component am, f the M x 1 vector CY , is de-fined a, = (l/m g( , and ( was defined previously.Due to the orthogonality among the transmit vectors g,, thevariates am are uncorrelated. Moreover, for Q --t 00 , therandom variables ( l / f l ) amare zero-mean, unit-variance(per dimension), i.i.d. complex normal.The signal model i s given by

M-I

The signal model, with all normalization factors specifiedso far, ensures that the average transmitted power(l/a)EO1 i s i I 2 ] = 1.Conditionedonthetarget

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vector a , he received vector r is complex, multivariate nor-mal with correlation matrix (2M)-l lla112a (80) at (&) +To gain better insight, conside r a specific case with M =SNR- I ~ .2, N = 1.The signal model is given by

(13)1T =- ails1+ a2s2 )+ 1.4If both antennas transmit the sam e waveform, s1 = sg = s,the received signal is given by

(14)1T =- a1+ a2) + 1.JzSince the channel parameters a1,2 are unknown at the re-ceiver, it is impossible to take advantage of the target spa-tial diversity. This system fails t o achieve the target diversitysought.Conversely, for orthogonal transmitted waveforms suchthat s;sg = 0, Is11 = Is21 = 1/2, the received signal canbe processed to yield the test statistic2 2

5 = I S ; T l 2 + s ;T l2(15)

Since for Q + 00, the random variables lai12, = 1 ,2 ,have a x 2 distribution, and they ar e i.i.d. (due to the orthog-onality between gl and g2 ) , the target component in (15)has a xi (chi-square with 4 degrees of freedom) distribu-tion. This is a consequ ence of the different and uncorrelatedRCS presented by the target to the different elements of thetransmitting antenna. Thus MI MO radar results in a diver-s i 9 gain that manifests itself through a more advantageousdistribution of the target com ponen t in the received signal.

14- ( a1 2 + la2 12) + 1.

111. MIM O D F AnalysisIn a radar DF system, an omnidirectional antenna illumi-nates the space, and based o n the energy reflected from thetarget, the receiver estimates the targets bearing. In thissection we examine the achievable performance of a M IM Oradar when used as a DF system. For simplicity and mathe-matical tractability we make the following assumptions:

The transmitted signal vecto rs is random with a com-plex nornial distribution, and a spatially white, sta-tionary power spectra l density with correlation m atrix(1/M) M .The elements of both the transmitting and the receiv-ing arrays are omnidirectional.Multiple, independent snapshots of the received sig-nal are available for processing.

A common figure of merit for comparing the perfor-mance of different systems is the estimator mean squareerror (MSE). The systems MSE depends on the exact es-timation m ethod, e.g., ML, MUSIC, beamforming, used. Inorder to have a fair comparison between different systems,for each system, w e evaluate a lower bound on the perfor-mance of all unbiased estimators.Cramer-RaoBoundIn what follows, we analyze the performance of a MIMOradar used as an active direction finder with M 3 1 rans-mitting elemen ts. Th e received signal is given by the mod el(12). Conditioned on a,E:= amsi s a zero mean, white,complex nornial random variable with variance (1 M) Ila 1 12 .In ou r model there are three unknown parameters, the direc-tion parameter B , the target parameters a, nd the SNR. Letthe vector denoting th e unknown parameters, @ = [e,S N R ,114121.

The Cramer-Rao bound is probably the best known lowerbound on the MSE of unbiased estimators [131. Denote byp(rl@)he family of distributions of the received signal pa-rameterized by the vector of unknown parameters @. TheCramer-Rao lower bound f or estimating @ is given by,

We are interested only in the direction 8, whereas theothers arenuisance parameters. We denote the correspond-ing bound C RB( BIa), where the notation indicates the con-ditioning of the bound on the unknown parameters a. A salready noted, conditioned on a,r is a complex nornial ran-domv ectorwith correlationmatrix (2A4)- I I L Y I ~ ~ ~ ( ~ ) ~ ~ ( O )+ S N R - ~ I N .The CRB conditioned on a an be computed,and it is given by [14, 131,

where L is the number of snapshots used by the array forestimating 8.We can lower bound the M SE of any unbiased estimatorby averaging the CRB with respect to a. We denote thisbound by ACR B(8) = E, [CRB(OIa)]. By using (16) theACR B(8) is given byACRB(8) =

-

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The following lemm a is essential for deriving a closed formexpression for AC RB (8) .Lemma 1 Ea [&]6 ndEa [ = 2 2 (M - l ) (M - 2 )Proof 1 We first note that llcy112 is distributed as U clii-square random vuriab le with 2 M degrees of reedoms. Thisallows us to write the expe ctation s of interest asE, [ =Ex;, El a n d E a {&} = YxL [ *

Given the density furictiori oft lie x ;M randoni variablex 2 M / 2 - 1 e - z / 2

P X ( =- (18)r ( 2 M / 2 ) 2 M / 2where I? denotes the Ganinia irnction, evahate

05 1 x 2 M / 2 - 1 e - z / 2m , ( 2 M - 2 ) / 2 - l e - 2 / 2- x

1 (19)r ( ( 2 ~ ) / 2 ) 2 ( 2 M - 2 ) / 2-r ( 2 ~ 4 1 2 )2 ~ 1 2 - 2 ( M - )an d

1 ,2M/2-le-x/2[$1 = 2 r ( 2 M / 2 ) 2M/2 dx

.Using the results of the len m a, the ACRB is given by

M4- ( A 4 - ) ( N S N R )It is easy to verify that if the targets RCS is constant,

tha t is ifllcyl12 = 2 M deterministically, the CRB is indepen-dent of M . Having this in mind, it is only natural to definethe systems fading loss as the additional SNR necessaryto achieve the sam e MSE as a system that is not subject tofading.By using the results of Lem ma 1, it is easy to verify thatthe fading loss (in dB ) as a function of the numb er of ele-ments in the transnutting array is lower and upper bounded

as10 oglo- FL(M) 5 lOlog,,, MM - 1 - J(A4 - ) ( M - 2 )

(22)Consider the case M = 1. In this case the fading lossis infinite. Furthermore, the ACRB in (21) is infinite. How-ever, when, the unknown angle parameter 8 is estimated us-ing say, the maximum likelihood estimation, the resultingMSE approaches zero as the SN R approaches infinity. Thisdiscrepancy deserves additional consideration. The CRBbound is a small error bound, that is, it predicts the MSEbased on the behavior of the log-likelihood function in thevicinity of the true parameter vector. Since it is a smallerror bound, this bound ignores the full structure of the pa-rameter space [131, which may result in nonsensical values.For example, in the problem at hand, if lla112 is low, theCRB, (16), nlight be much h igher than 1r2. However, since8 E [-w,T] th e MSE of any estimator is lower than m 2 .Hence in this case, the CRB is useless.The CRB approaches infinity at the rate of &,ha tis , CRB = O(ll~ull-~).herefore in order for the ACRBto yield a finite result, the density function p x ( x ) , whereX = lla112 is a chi-square random variable with 2M de -grees of freedom, should approach zero faster than (Ila1 2 )as IIcy1 + 0. By exanuning the probability density func-tion of X in (1S), it is easy to see that this happens only f orM 2 3. But in reality, since the MSE of any estimator issmaller than 7r2, averaging the perforniance of any estimatorwith respect to cy yields a finite result for any M .Now, consider the case M + W . ere, the fading lossapproaches zero, that is, the variations in the targets SN Rdo not affect the systems MSE. This phenomena can beexplained using the following intuitive argument. Withoutfading, the received signal is r = a(8)s + v, where s szero-mean, (1/M) variance, complex normal. With fad-v. However, according to the central limit theorem, as Mapproaches infinity, a,s approaches a zero-mean,( 2 / M )variance, complex normal random variable. H ence,as M approaches infinity the received signal is equal to thereceived signal of a non-fading system.

ing, the received signal is r = ( l / f i )a(0) M a,s +M

Uniform Linear ArrayIn this section we specifically consider the case of a uni-form linear array (UL A) with om nidirectional antennas atthe receiver. Consider a ULA with N elements with half awavelength spacing. Th e n-th element of the array steeringvector equalsThe n-th element of a(8) is

[a(e)], = ejnTsinB. (23)

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iF rom these relations it follows that the squared norm of thesteering vector is g iven by at(e)a(e)= N , and the squarednoi m of the de rivative of the steering vector is given byN - 1

n= O( N - 1) N ( 2 N - 1)- T2 cos2 e. ( 2 5 )6

Finally, we have

( N - 1) N 24- T 2 c ~ s 2 e (26)

Substituting these results in (21), the ACRB for the case ofa ULA at the receiver is given by,

M2( M - 1) (A4 2) (NSNR)2

Le t us investigate some special cases.8 -+ ~ / 2 : ere, ACRB(0) + 00, confirms that thedirection cannot be estimated at endfire, since the array ha sa zero effective aperture (ze ro resolution).8 = 0: This is the best ca se for estimating the directionparameter. Indeed , at broadside the array has the largesteffective aperture (best resolution).N = 1: The bound is infinite. Indeed, a single omnidi-rectional antenna cannot measure the angle of arrival.Numerical ResultsIn this section, numerical results are provided on the ACRBfo r a ULA with N = 6 elements at the receiver. Perfor-mance is parameterized by the number of transmitting an-tennas M . The transmit antennas are spaced sufficiently toachieve diversity.Fig. 2 depicts the ACRB for various SNRs for largetransmitting arrays with M = 4 an d M = 16 transmittingelements, respectively. The CR B for the case of a Gaussiansource without fading is depicted as well. Empirical resultsare represented through root mean square (RMS) errors ofthe MLE. It is well known that (for spatially and tempo-rally white noise) the MLEfor bearing estimation of a singlesource is given by a conventional beamformer. Th e MLE isthe value of the angle of arrival that maximizes the outputof the beamformer. That number of independent snapshotsused in the estimation was L = 80.

I t is evident from the figure that the ACRE3 values matchthe empirical results for both M = 4 an d M = 16. When

0 2 4 6 8 10 12 14 16 10 20SNR[dS]

Fig. 2. Average Cramer-Rao bound and empirical MSE fo rdirection finding of a Gaussian source. Large transnlit array.

a transnutting array with M = 4 elements is used, the fad-ing loss is about 1.3 dB. This value is consistent with ouranalysis based on (22) predicts that the fading loss will bebetween 1.25 dB and 2.1 dB. When the array with M = 16is used, the fading loss is negligible, as also predicted byour analysis.In Fig. 3, the ACR B and the RMS error of the MLE areshown as functions of SNR for small transmit arrays withM = 1 an d M = 2 elements, respectively. In addition,also shown is the CRB for the case of a Gaussian sourcewithout fading. For the case of M = 2 transmit antennas,we also plot a Modijed ACRB (MACRB). The MACRB isdefined as the lower bound of the CR B at high SN R, suchthat SNR- >> SNR-2, and terms containing the latter areneglected. From (16) we obtain

The MAC RB is obtained by averaging (28) with respect toa. t is easy to see that the MACRB for the ULA at thereceiver equals

6 M= ( ~ 2 p T 2 cos2 e ( M - 1)NSNR

(29)We can observe from the figure that if only on e trans-mitting element is used, the fading loss is very large and itis about 15 dB. However, for large SNR , by adding an ad-ditional transnutting element, the fading loss decreases toabout 2 dB. Moreover, when the SNR is large and (28) istight, the MACR B fits the empirical results quite well.

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1 0

10-1

t I0 2 4 6 8 10 12 14 16 18 20

SNR IdEl

Fig. 3. Average Cramer-Rao bound and empirical MSE fordirection finding of a Gaussian source. Small transmit array.

IV. ConclusionsIn this paper, we introduce MIMO radar, a new conceptin radar that capitalizes on the RC S scintillations with re-spect to the target aspect in order to improve the radarsperformance. We introduced a generalized framework forthe signal model that can accom modate conventional radars,beamformers, and MIMO radar. As demonstration of thepotential advantages that M IMO radar can offer, we evalu-ated the Cramer-Rao bound for bearing estimation. We haveshown that with a few transnlit antennas that illuminate un-correlated aspects of the target, the perform ance (in termsof Cramer-Rao bound) of M IMO radar approaches that ofa steady target. This paper is meant only to introduce theMIM O radar concept. In subsequent work, we will continueto investigate this promising new approach to radar.

1. REFERENCESA. Dogandzic and A. Nehorai, Cramer-Rrao boundsfor estimating range, velocity, and direction with anactive array, lEEE Transaction on Signal Processing ,vol. 49, no. 6, pp . 1122-1 137,June 2001.S . Pasupathy and A. N. Venetsanopoulos, O ptimumactive array processing structure and space-time fac-torability, IEEE Trans. Aerosp. Electron. Syst., vol.10,pp . 770-778,1974.L. Swin delehu rst and P. Stoica, Maxim um likelihoodmeth ods in radar array signal processing, Proc. of theIEEE,vol. 86, no. 2, pp. 421441 , Feb . 1998.S : Haykin, J. Litva, and T. J. Shepherd, Radar Array

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MIMO Radar a Idea Whose Time Has Come

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