IEEE TRANSACTIONS ON SIGNAL PROCESSING, …nehorai/paper/Gogineni_Sparse...5316 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011 In this paper, we propose compressive

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011 5315

Target Estimation Using Sparse Modeling forDistributed MIMO Radar

Sandeep Gogineni, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—Multiple-input multiple-output (MIMO) radar sys-tems with widely separated antennas provide spatial diversityby viewing the targets from different angles. In this paper, weuse a novel approach to accurately estimate properties (position,velocity) of multiple targets using such systems by employingsparse modeling. We also introduce a new metric to analyzethe performance of the radar system. We propose an adaptivemechanism for optimal energy allocation at the different transmitantennas.We show that this adaptive energy allocationmechanismsignificantly improves in performance over MIMO radar systemsthat transmit fixed equal energy across all the antennas. We alsodemonstrate accurate reconstruction from very few samples byusing compressive sensing at the receivers.

Index Terms—Adaptive, compressive sensing, multiple-inputmultiple-output (MIMO) radar, multiple targets, optimal design,sparse modeling, widely separated antennas.

I. INTRODUCTION

M mULTIPLE-INPUT multiple-output (MIMO) radar[1]–[7] has attracted a lot of attention recently due to

the improvement in performance it offers over conventionalsingle antenna systems. MIMO radar is typically used in twoantenna configurations, namely distributed and colocated. IndistributedMIMO radar [1]–[3], [8], [9] the antennas are widelyseparated. This enables viewing the target from different angles(see Fig. 1). Hence, if the target returns between a particulartransmitter and receiver are weak, then it is highly likely thatthey will be compensated by the returns between other antennapairs. In [10], Cramér–Rao lower bound is derived for thetarget localization accuracy while using MIMO radar. Further,[11] shows the derivation of the Ziv–Zakai lower bound ontarget localization estimation. These papers address the esti-mation problem in a single target scenario. While distributedMIMO radar exploits the spatial diversity, colocated MIMOradar [4]–[7] exploits the waveform diversity. In colocatedconfiguration, all the antennas are closely spaced and hencethe target radar cross section (RCS) values are the same for

Manuscript received September 13, 2010; revised March 25, 2011 and June29, 2011; accepted July 31, 2011. Date of publication August 18, 2011; date ofcurrent version October 12, 2011. The associate editor coordinating the reviewof this manuscript and approving it for publication was Prof. Konstantinos I.Diamantaras. This work was supported by the Department of Defense under theAir Force Office of Scientific Research MURI Grant FA9550-05-1-0443, ONRGrant N000140810849, and NSF Grant CCF-1014908.The authors are with the Department of Electrical and Systems Engineering,

Washington University in St. Louis, t. Louis, MO 63130 USA (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2011.2164070

Fig. 1. MIMO radar with widely separated antennas.

all transmitter–receiver pairs. RCS denotes the transformationundergone by the transmitted signal while reflecting from thesurface of the target. Since all the antennas are closely spacedin the colocated configuration, each antenna pair will havethe same RCS value as all the signals bounce off the targetsurface at the same angle. This is contrary to the distributedantenna configuration that we consider in this paper where eachpair has a different RCS value. In this paper, we study MIMOradar with widely separated antennas in the context of sparsemodeling and compressive sensing for estimating the positionsand velocities of multiple targets.Compressive sensing allows us to accurately reconstruct

data from significantly fewer samples than the Nyquist rateif the received signal is sparse in some basis representa-tion [13]–[16]. With the improvement in the capabilities ofthe computational resources, it has become more feasible touse compressive sensing for different medical and engineeringapplications [17]–[22]. Since the number of targets in a radarscene is often limited, we can use sparse modeling to representthe radar data. Therefore, compressive sensing is applicableto the field of radar [19]–[22]. So far, compressive sensinghas been used for MIMO radar only in the context of closelyspaced antennas [21]–[23]. To the best of our knowledge,the important configuration of distributed MIMO radar hasnot been approached from a sparse modeling or compressivesensing perspective. This configuration is important since itprovides spatial diversity. In [22], the authors call their systemdistributed MIMO radar even though they are actually usingcolocated MIMO radar. This is evident from the fact that theyuse the same RCS value for all transmitter–receiver pairs.

1053-587X/$26.00 © 2011 IEEE

5316 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 2011

In this paper, we propose compressive sensing for MIMOradar with widely separated antennas and demonstrate that thesampling rates can be significantly reduced (see also [24]).Adaptive radar design has been an active topic for a number

of years. Implementing the radar system in a closed loop byadaptively choosing the properties of the transmitted waveformsbased on the knowledge of the environment gives significant im-provement in performance. In [3] and [25]–[35], the advantagesof adaptive design are demonstrated under different configura-tions. In [26]–[30], adaptive polarization design is considered inthe context of single-input single-output (SISO) radar systems.References [3] and [31]–[35] discuss the problem of waveformdesign for MIMO radar. In [3], the problem of optimal transmitwaveform polarization selection is considered for distributedMIMO radar in the context of target detection. In [31]–[33],waveform design has been addressed from an information the-oretic perspective also. Apart from this, [33] also deals with thepresence of multiple targets. In [31] and [32], the problem ofwaveform design is presented for MIMO radar using mutual in-formation and minimum mean-square error estimation. In thispaper, the parameters that we optimally design are the transmitwaveform energies. We propose an optimal adaptive energy al-location mechanism for distributed MIMO radar by making useof the estimates of the complex target attenuations from the pre-vious processing interval. We show that this adaptive mecha-nism gives significant improvement in performance overMIMOradar systems that transmit fixed equal energy across all the an-tennas (see also [36]).This paper is organized as follows. In Section II, we derive the

signal model for MIMO radar with widely separated antennasby using sparse representation. In Section III, we deal with theproblem of sparse support recovery to infer about the propertiesof the targets (position, velocity). We present two well knownalgorithms for the recovery of the sparse vectors. In Section IV,we propose a new metric to judge the performance of the radarsystem. We propose also an optimal energy allocation mecha-nism to improve the performance of this MIMO radar system.In Section V, we introduce compressive sensing to reduce thesampling rates at the receiver side. In Section VI, we use nu-merical results to show the performance of the sparse recoveryalgorithms. We demonstrate the improvement offered by adap-tive energy allocation at the transmit antennas. Further, we showthat we can accurately estimate the target properties from signif-icantly fewer number of samples by using compressive sensing.In Section VII, we conclude this paper.

II. SIGNAL MODEL

In this section, we describe the signal model for our MIMOradar system. We assume that there are transmitters,receivers, and targets. Further, we assume that all the targetsare moving in a two dimensional plane. However, without lossof generality, we can extend the analysis in this paper to the threedimensional case. We assume that each of the targets containsmultiple individual isotropic scatterers. The bandwidth of thetransmitted waveform determines the resolution of the system.We require very high bandwidth to resolve each of the individualscatterers of the target. Due to practical bandwidth constraints,however, the system cannot resolve these individual scatterers.

Therefore, this collection of scatterers can be expressed as onepoint scatterer which represents the RCS center of gravity ofthese multiple scatterers [1], [8]. By point target, we refer tothe smallest target that can be resolved by the system. The RCS

center of gravity of the target is located aton a Cartesian coordinate system and it moves with a velocity

. The position and velocity parameters repre-sent the center of gravity of the target during a particular pro-cessing interval. The transmitter and receiver are located

at and , respectively. We transmitorthonormal waveforms from the different transmitters. Hence,the transmitted energy from the antenna and the totaltransmitted energy . Let be the com-plex baseband waveform transmitted from the transmitter.Then, the bandpass signal emanating from the transmit an-tenna is given as

(1)

where denotes the real part of the argument, ,is the carrier frequency. These signals travel in space and

reflect off the surfaces of the targets and are captured by the re-ceivers. Further, we assume that the cross correlations betweenthese waveforms is close to zero for different delays [1]–[3]. Let

denote the attenuation corresponding to the target be-tween the transmitter and the receiver. Note that the at-tenuation is dependent on the transmitter–receiver indices underconsideration. This is a result of the wide separation between theantennas. For a colocated MIMO setup, the RCS value wouldbe the same for all transmitter–receiver indices [4], [5].Under a narrow band assumption on thewaveforms, the band-

pass signal arriving at the receiver can be expressed as

(2)

where and are the delay and Doppler shift corre-sponding to the target.

(3)

(4)

where and denote the unit vector from the trans-mitter to the target and the unit vector from the target tothe receiver, respectively; is the inner product operator,and is the speed of propagation of the wave in themedium. Theterm represents the phase shift and it is also depen-dent on the transmitter–receiver indices under consideration.The received signals at each receiver are first down converted

from the radio frequency and then passed through a bank ofmatched filters, each of which corresponds to a particular trans-mitter. Assume that the target attenuations values do not varywithin a pulse duration and the Doppler shift is small. Therefore,

GOGINENI AND NEHORAI: TARGET ESTIMATION USING SPARSE MODELING FOR DISTRIBUTED MIMO RADAR 5317

varies slowly when compared with the wave-form and is almost constant across a pulse duration. Inother words, it can be taken outside of the integral in thematchedfilter operation. This is a valid assumption for targets whose ve-locity is much smaller than the speed of light in the medium. So,the integral only contains the waveform terms and under the or-thogonality assumption of the waveforms for all delays [1]–[3],the sampled outputs of the matched filter at the receiverare given as

(5)

where is the additive noise at the output of thematched filter of the receiver, represents a set containingall the targets that contribute to the matched filter output at. and denote the sample index and sampling interval,respectively. Note that the waveform term is no longerpresent in this equation as it is integrated out of the matchedfilter due to the orthogonality of the waveforms (see also [2]).We define the target state vector . Hence,

the important properties of the target (position, velocity) arespecified by . The goal is to estimate for all the targets.Now, we discretize the target state space into a grid of possiblevalues . Hence, each of the targets is asso-ciated with a state vector belonging to this grid. If the presenceof a target at would contribute to the matched filter output at, then define

(6)

Otherwise, . Also, if is the state vector of thetarget, we define

(7)

Otherwise, . For each , we stack , , andcorresponding to different transmitters to obtain di-

mensional column vectors , and , respectively.Similarly, we arrange into dimensionaldiagonal matrix .

(8)

(9)

(10)

(11)

where refers to a diagonal matrix whose entries aregiven by and denotes the transpose of . Further, wearrange , , and intodimensional column vectors , , and , respectivelyand , into dimensional diag-

onal matrix .

(12)

(13)

(14)

(15)

Finally, stacking and into di-mensional column vector and dimen-sional matrix, respectively, we obtain

(16)

(17)

Therefore, we can express the received vector at as

(18)

where is a sparse vector with nonzero entries.Note that the nonzero entries of this vector appear in blocksof size . Therefore, we can call as a block sparsevector with nonzero blocks and each block containing

entries. We have expressed our observed data atusing sparse representation. For each matched filter, let the sam-pled output signal for each pulse contain samples. Whenthe velocities of the targets are much smaller than the speed ofwave propagation in the medium, we require multiple pulses toestimate these velocities since the effect of the Doppler with inone pulse duration will be negligible (see [37, Fig. 4.2]). Hence,in each processing interval we consider pulses. Therefore,in each processing interval, we have samplesat the output of each matched filter. We assume that the targetattenuation values do not vary over a period of pulses.Now, we stack , , and into

(19)

(20)

(21)

to obtain

(22)

Note that in the above expression for the measurement vector,is known and only depends on the actual targets present

in the illuminated area. The nonzero entries of represent thetarget attenuation values and the corresponding indices deter-mine the positions and velocities. Further, note that in order toobtain the measurement vector in the above equation, each ofthe receivers sends their measurements to a common processorthat stacks them appropriately to obtain . This common pro-cessor performs the estimation that we describe in the next sec-tion. None of the receivers perform any local estimation becauseany such approach can only be sub optimal. In [38], the authorsshow that the estimation error of a MIMO radar system is in-creased while employing decentralized processing.


III. SPARSE SUPPORT RECOVERY

In the previous section, we have expressed the signal re-ceived across receive antennas over samples usingsparse representation. In order to find the properties of thetargets (position, velocity), we need to recover the sparse vectorfrom the measurements . There are many approaches to

perform the recovery. Two approaches are basis pursuit [39](BP) and matching pursuit [40] (MP). These algorithms arewell known in the field of sparse signal processing. Further,these algorithms recover sparse vectors but do not exploit theknowledge of the block sparsity. However, very recently, in[41], the authors propose an extension of the matching pursuitalgorithm called block-matching pursuit (BMP) that exploitsthe knowledge of block sparsity. In this section, we presentBP and BMP for sparse support recovery. We shall use thesealgorithms in the numerical simulations to demonstrate theperformance of the MIMO radar system. We will use the samealgorithms while employing compressive sensing.

A. Basis Pursuit

Basis pursuit is an optimization principle. It is presentedunder two scenarios; in the absence of noise and in the presenceof noise.1) Absence of Noise: In the absence of noise, BP aims at

minimizing under the constraint . Since usually, there are many different vectors that satisfy the

constraint. We choose the solution that has the least norm.This optimization problem can be modeled as a linear program[39]. There are many existing algorithms to solve this problem.It can be solved using CVX, a package for specifying and solvingconvex programs [42], [43].2) Presence of Noise: Clearly the above approach of basis

pursuit will fail in the presence of noise. Hence, in [39], theauthors propose basis pursuit de-noising (BPDN). This is anunconstrained minimization problem

(23)

When the columns of are normalized, typicallywhere represents the noise level [39].

In this problem, since we consider multiple pulses in each pro-cessing interval, the columns are not normalized by definition.Therefore, we scale the value accordingly to perform thesimulations. We used CVX to implement this algorithm. Wepresent the results in Section VI.

B. Block-Matching Pursuit

Before we describe BMP, we shall first give a description ofthe conventional MP. It is an iterative algorithm [40] that can beused for sparse signal recovery. Since all the columns of arenot necessarily independent, there are infinitely many solutionsfor even when there is no noise. In MP, we first initialize thereconstructed vector and the residual . In eachsubsequent iteration , we project the residual vectoronto all the columns of and pick the column that has the

highest correlation with the residual. We update the estimatedreconstructed vector

(24)

We finally update the residual as

(25)

Even though this algorithm can be used to recover , it does notmake use of the knowledge that the vector is block sparse. In[41], the authors propose BMP which exploits this knowledge.Similar to MP, we initialize the reconstructed vectorand the residual . We divide the columns of intoblocks of size . There are such blocks. In each subse-quent iteration, we project the residual vector onto all of theseblocks and pick the block that gives the highest energy after

projection. Now, the estimated reconstructed vector is updatedby adding the projections onto each of the columns of this block.The residual is also updated accordingly. Note that the main dif-ference when compared with MP is that here the updates aredone one block (columns corresponding to different transmitterand receivers) at a time whereas in MP, the updates are done onecolumn at a time. In [41], the authors analyze the performanceof block sparsity based approaches and show that the improve-ment in using the block sparsity based recovery algorithms ismaximum when all the columns within a block are orthogonal.It can be easily checked that the columns corresponding to dif-ferent transmitter–receiver pairs in the basis matrix are or-thogonal by the definition of in our signal model. Therefore,BMP is well suited for recovering in this problem.The performance of the sparsity based estimation approaches

is determined by the correlations between the columns of thedictionary matrix and the distance between the adjacent gridpoints. More specifically, when the nonzero entries of the sparsevector appear in blocks, a major factor in determining the per-formance of the system is the block coherence measure [41]. Let

denoted the block of the dictionary. Therefore, hascolumns. Define

(26)

The block coherence measure is defined as

(27)

where denotes the spectral norm of . Theblock coherence measure should be small in order to obtaingood performance. However, as the grid points come closer, theresolution is improved but block coherence measure increasesbecause the correlation between the adjacent blocks will in-crease. Therefore, it is a tradeoff between the grid size andthe coherence measure. In Fig. 2, we plot the block coherencemeasure as a function of the distance between adjacent gridpoints (in m and m/s for position and velocity, respectively).


Fig. 2. Block coherence measure as a function of the distance between adjacentgrid points.

We observe the coherence measure increases as the distancereduces.

IV. OPTIMAL ADAPTIVE ENERGY ALLOCATION

Before we propose the energy allocation mechanism, we firstdefine a new performance metric that naturally fits into thismultiple target scenario. Conventional metrics like mean-squareerror (MSE) are commonly used in radar applications and theyare apt in single target scenarios. However, they do not effi-ciently capture the estimation accuracies in multitarget scenario.For example, even if the estimates of the parameters of some ofthe targets are poor, the overall MSE (averaged over all the tar-gets) can still be small if the estimates of majority of the othertargets are very accurate. Hence, the deficiencies in the esti-mates of the weak targets will go unnoticed. To overcome thisproblem, we propose a new performance metric. We will de-scribe this metric in this section.As mentioned earlier, the length vector has

only nonzero entries. Let the reconstructed vectorbe denoted by . We would like to have the most significant

entries of correspond to the same indices as thenonzero entries of the actual sparse vector . If this is not thecase, then we will wrongly map the target states for one ormore targets. We define a length vector

(28)

This vector essentially combines the energies of the componentscorresponding to the different transmit-receiver pairs for eachpoint in the target state space. Further, define as a lengthvector which contains the values that carries at the correctindices. Similarly we define as a length vector that takesa value of 0 at the correct indices and takes the same valuesas at every other index. It is clear that the nonzero entries ofcorrespond to the nontarget states and the zero entries corre-

spond to the correct target states.

We define the metric

(29)

The numerator of this metric denotes the weakest target compo-nent in the reconstructed vector. The denominator denotes thestrongest nontarget component in the reconstructed vector. Ifthis metric has value greater than one, then all the actual (cor-rect) target indices dominate the other indices in and hencethe estimates of position and velocity will exactly match thetrue values. Otherwise, at least one of the nontarget indices willdominate the weakest target and hence, the position and velocityestimates do not match the true values. Note that onlyguarantees exact estimation of the position and velocity. The ac-curacy in the estimates of the target attenuations is determinedby the exact value taken by . If is large, then most of thereconstructed energy is distributed in the correct target indices,thereby giving accurate estimates of the attenuations. Hence, thehigher the value of , the better the performance of the system.In Section VI, we use this metric to analyze the results.Adaptive energy allocation has been shown to provide im-

proved detection performance in distributed MIMO radar sys-tems [44]. In this paper, we will present a novel adaptive en-ergy allocation scheme to improve the estimation performance.Let be the energy of the waveform transmitted from thetransmitter. We initialize the system by transmitting mul-

tiple pulses of unit energy waveforms , from all thetransmitters. Hence, the total energy transmitted per pulse is

. After collecting a vector of outputs at themultiple receiver matched filters, the processor performs thesparse recovery to estimate the attenuations using the al-gorithms mentioned in the previous section. Since the differentantenna pairs view the targets from different angles, these atten-uations and their corresponding estimates will be different fromeach other. Hence, equal energy allocation to all the transmittersdoes not necessarily give the best performance. After the esti-mation, the energy allocation scheme is applied to decide uponthe transmit energies for the next set of transmit pulses whilekeeping the total transmitted energy constant. The goal of thisscheme is to maximize the minimum target returns. This is nat-urally motivated from the performance metric defined earlier inthis section. The numerator in the performance metric denotesthe minimum target returns. We solve the following optimiza-tion problem and find the optimal such that

(30)

We can solve the above optimization problem using CVX [42],[43]. Since this problem depends only on the dimensionality ofthe MIMO radar configuration and the number of targets andnot on the huge dimensionality of the basis dictionary, it canbe solved quickly. This makes it amenable to use in practicalsystems in an online manner. After solving this problem, theprocessor feeds back this information to the transmitters whichsend the next set of pulses with these optimally selected valuesof energies. Hence, the system operates in a closed loop. The


energy allocation mechanism discussed above is not only ap-plicable to the sparsity based estimation method mentioned inthis paper but it is relevant in any multiple target scenario. Notethat it only requires us to have estimates of the attenuations. Inthis paper, these estimates are computed using sparse supportrecovery. In principle, these estimates can be computed usingany other approach and still this energy allocation scheme willbe relevant. We shall show in Section VI that this optimal choiceof waveform energies gives significant improvement in perfor-mance.

V. COMPRESSIVE SENSING

Compressive sensing allows us to accurately reconstruct datafrom significantly fewer samples than the Nyquist rate if thereceived signal is sparse [13]. Nyquist rate sampling assumesthe signals to be bandlimited. Similarly, the requirement for ap-plying compressive sampling is that the signals must have asparse representation using some basis. In Section II, we sawthat the measurement vector has dimensions . Sinceour measurement vector is sparse in the space spanned by thecolumns of the matrix , the theory of compressive sensingsays that we can reconstruct the vector from far fewer samplesthan contained in the vector . If the sensing basis is representedby , then the coherence measure between and measuresthe largest correlation between them. must be such that it hasas little coherence with as possible [13]. Since random ma-trices satisfy low coherence properties, we generate the entriesof the dimensional sensing matrix fromindependent Gaussian distribution, where .Since the entries of are independent from each other, eachsensor will project its received data separately using Gaussiansequences of appropriate lengths and send the compressed datato the fusion center which will combine the data from differentsensors appropriately. The new measurement vector at the fu-sion center in the presence of noise is

(31)

For reconstruction of , we use the same algorithms as pre-sented in Section III. Define as the percentageof samples used in compressive sensing. In Section VI, we willshow the performance of the MIMO radar system for differentlevels of compression. The adaptive energy allocation mecha-nism presented in the previous section can also be applied forthe case of compressive sensing. We use the estimates of the

target attenuations to select the energy allocation for the nextprocessing interval.

VI. NUMERICAL RESULTS

We begin with a description of the simulated scenario. Wesimulated a 2 2 MIMO radar system. We denote the positionsof all the transmitters, targets and receivers on a common Carte-

sian coordinate system. The transmitters are located at

m and m, respectively. The receiver loca-tions are m and m, respectively. Thecarrier frequency of the transmitted waveforms is 1 GHz.Within each processing interval, we consider three pulses that

are transmitted 33.3 ms apart. We choose for the simu-lation results. Therefore, has 972 entries. We divide the targetposition space into 9 9 grid points and the target velocityspace into 5 5 grid points. Therefore, the total number of pos-sible target states . We considered the presence of 3targets. Hence, the 8100 dimensional sparse vector has only

nonzero entries corresponding to the targets.The positions and the velocities of the targets are given as

m (32)

m/s (33)

m (34)

m/s (35)

m (36)

m/s (37)

The complex attenuations corresponding to the three targets are

(38)

(39)

(40)

where .The entries of are generated independently from Gaussian

distribution. We assume each of these samples has the samevariance . We define the signal-to-noise ratio (SNR) for theMIMO radar system as

(41)

First, we compare the performances of the two algorithmsbasis pursuit de-noising and block-matching pursuit that we pre-sented in Section III. We performed these simulations at anSNR of 3.7 dB. For BMP, we used ten iterations. Since it isnot possible to plot the position and velocity on the same plot,we plotted the estimates of position and velocity separately. Forcomputing the estimate at a particular grid point on the posi-tion plot, we average over all 5 5 velocity grid points corre-sponding to that position grid point. Similarly, we average overall the 9 9 position grid points in order to obtain the velocityplot. We do this only to be able to plot position and velocity es-timates separately.From Fig. 3 and Fig. 4, we can see that both the algorithms are

able to estimate the positions and velocities of the 3 targets at anSNR of 3.7 dB but the performance of BP is poor especially forthe velocity estimates. We can observe this from Fig. 3 becausethe grid points surrounding the correct velocity points also havesignificant energies. However, it is important for us to analyzethe performances of the two algorithms by evaluating the per-formance metric . Fig. 5 plots as a function of the SNR andwe can clearly see that BMP outperforms BP. The value ofremains above 1 for much lower SNR for BMP when comparedwith MP. This clearly shows the improvement in performanceas a result of exploiting the block sparse structure of the vector .


Fig. 3. Reconstructed vectors using basis pursuit de-noising at: (a) position estimates and (b) velocity estimates.

We used 25 independent Monte Carlo runs to generate these re-sults. When , then some of the nontarget states dominatethe reconstructed vector and hence estimates of the target po-sitions and velocities are incorrect for at least one target. SinceBMP outperforms BP, for all further simulation results, we shalluse only BMP.Now, we shall demonstrate the advantages of having adaptive

energy allocation. We assume we have estimates of the targetattenuations from the estimation of the previous processinginterval. We apply the optimization principle we described inSection IV. The reconstructed vectors for an SNR of 3.7 dBare plotted in Fig. 6. We can clearly see that an equalizationeffect has been achieved when compared with Fig. 4. This isa result of the optimization. Now, we quantitatively comparethe performances of the MIMO radar system with and withoutoptimal energy allocation. We solve the optimization problempresented in Section IV to obtain the optimal and

. Note that the total transmitted energy is the same. As we see from Fig. 7, the adaptive energy

allocation gives significant improvement in performance. Thevalue of for the optimal energy scheme is higher when com-pared with the equal energy transmission. Even at an extremelylow SNR of 10.7 dB, the value of remains greater than

Fig. 4. Reconstructed vectors using block-matching pursuit at: (a) position estimates and (b) velocity estimates.

Fig. 5. Performance metric for basis pursuit de-noising and block-matchingpursuit as a function of SNR.

1 for the proposed energy allocation scheme whereas it fallsbelow 1 with equal transmit energies.Next, we demonstrate the improvement offered by theMIMO

system over conventional SISO systems. This improvement is


Fig. 6. Reconstructed vectors with optimal energy allocation at: (a) position estimates and (b) velocity estimates.

Fig. 7. Performance metric with and without adaptive energy allocation.

a result of the spatial diversity provided by distributed MIMOradar. We get multiple views of the target in MIMO radar. InFig. 8, we see that MIMO system significantly outperforms theSISO system. For the SISO system, we considered transmitter

Fig. 8. Performance metric for MIMO and SISO systems as a function ofthe noise level .

Fig. 9. Performance metric for different percentages of samples.

and receiver to be present at the locations and , respec-tively. For fairness of comparison, we increased the numberof samples per each pulse by a factor of for the SISOsystem.Now, we shall present the results for compressive sensing. As

we defined earlier, the percentage of samples used is given by

(42)

We plot the performance of MIMO radar for different percent-ages of samples used. As expected, we observe from Fig. 9 thatthe performance degrades as the percentage of samples reduces.However, even while using just 25% of the samples, we can ob-tain for SNR as low as 3.5 dB. In other words, thereconstructed estimates of the position and velocity match thetrue values at an SNR of 3.5 dB while using only 25% of thesamples. We used 25 independent Monte Carlo runs to producethese results.We show the advantages of optimal energy allocation even

for the compressive sensing scenario. From Fig. 10, we observe


Fig. 10. Performance metric with and without adaptive energy allocationwith 25% of samples.

Fig. 11. Reconstructed vectors with optimal energy allocation atusing 25% of the samples: (a) position estimates and (b) velocity

estimates.

that the optimal choice of transmit energies gives significantimprovement in performance even when we have just 25% ofthe samples. The reconstructed vectors for an SNR of 3.7 dB

Fig. 12. Reconstructed vectors with optimal energy allocation atwith 20% modeling errors in all the targets: (a) position estimates and

(b) velocity estimates.

are plotted in Fig. 11. We can see an equalization effect evenhere.Finally, we wish to investigate the performance of the sparse

recovery algorithm when there are modeling errors. Morespecifically, the targets may not fall exactly on the grid points.This can be a result of the grid size not being small enough.Also, the movement of the targets within the processing inter-vals can also lead to these modeling errors. We quantify themodeling error in each dimension as a percentage of the max-imum possible error in that dimension. The maximum possibleerror is half the grid size in that dimension. Fig. 12 shows thereconstructed vector in the presence of 20% modeling errorin both the and dimensions for each of the 3 targets. Weobserve that at an SNR of 3.7 dB, the target parameters aremapped to the nearest grid points even in the presence of 20%modeling error. Note that we considered simultaneous errors inall the targets in both the dimensions. The system can handlelarger errors when we consider the modeling errors separately.Fig. 13 shows the reconstructed vector in the presence of 95%modeling error in only the dimension for one of the 3 targets.We observe that at an SNR of 3.7 dB, the target parametersare mapped to the nearest grid points even in the presence of


Fig. 13. Reconstructed vectors with optimal energy allocation atwith 95% modeling errors in one of the targets in one dimension:

(a) position estimates and (b) velocity estimates.

95% modeling error. Further, by including more pulses withina processing interval, the system can increase its robustness.

VII. CONCLUSION

In this paper, we used a novel approach to estimate the posi-tions and velocities of multiple targets using MIMO radar sys-tems with widely separated antennas by employing sparse mod-eling and compressive sensing. We also proposed a new metricto analyze the performance of these systems.We then developedan adaptive optimal energy allocation mechanism to get signif-icant improvement in performance. We used numerical simula-tions to demonstrate this improvement. We demonstrated thatby employing compressive sensing, we can accurately recon-struct the target properties from very few samples. Finally, weshowed that the proposed system is robust to modeling errorsthat may arise due to the discretization of the target state space.In future work, we shall extend our results in this paper to the

case of extended targets. In such a scenario, the multiple targetswill have impulse responses as opposed to a single reflectioncoefficient that we use for point targets. Further, we will modelthe grid mismatch error using scaled von Mises distribution andanalyze the estimation performance. Uniform distribution is a

special case of von Mises distribution. Von Mises distributionis commonly used for modeling phase errors in radar problems[44] since the phase is bounded between . Since the griderror is bounded by half the grid size, scaled von Mises distri-bution fits this problem well.

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Sandeep Gogineni (S’08) received the B.Tech. de-gree in electronics and communications engineering(with Hons. in signal processing and communica-tions) from the International Institute of InformationTechnology, Hyderabad, India, in 2007, and the M.S.degree in electrical engineering from WashingtonUniversity in St. Louis (WUSTL), MO, in 2009.He is currently working towards the Ph.D. degree

in electrical engineering from WUSTL. His researchinterests are in statistical signal processing, radar, andcommunications systems.

Mr. Gogineni was selected as a finalist in the student paper competitions at the2010 International Waveform Diversity and Design (WDD) Conference and the2011 IEEE Digital Signal Processing and Signal Processing Education Work-shop.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees from the Technion,Haifa, Israel, and the Ph.D. from Stanford Univer-sity, Stanford, Ca.iPreviously, he was a faculty member at Yale Uni-

versity and the University of Illinois at Chicago. He iscurrently the Eugene and Martha Lohman Professorand Chair of The Preston M.Green Department ofElectrical and Systems Engineering at WashingtonUniversity in St. Louis (WUSTL). He serves as theDirector of the Center for Sensor Signal and Infor-

mation Processing at WUSTL.Dr. Nehorai served as Editor-in-Chief of the IEEE TRANSACTIONS ON

SIGNAL PROCESSING during the years 2000 to 2002. In the years 2003 to2005, he was Vice-President (Publications) of the IEEE Signal ProcessingSociety (SPS), Chair of the Publications Board, and member of the ExecutiveCommittee of this Society. He was the Founding Editor of the special columnson Leadership Reflections in the IEEE Signal Processing Magazine from 2003to 2006. He received the 2006 IEEE SPS Technical Achievement Award andthe 2010 IEEE SPS Meritorious Service Award. He was elected DistinguishedLecturer of the IEEE SPS for the term 2004 to 2005. He was corecipient ofthe IEEE SPS 1989 Senior Award for Best Paper, coauthor of the 2003 YoungAuthor Best Paper Award, and corecipient of the 2004 Magazine Paper Award.In 2001, he was named University Scholar of the University of Illinois. He hasbeen a Fellow of the Royal Statistical Society since 1996.

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