Proceedings of the 4th International Symposium on Communications,Control and Signal Processing, ISCCSP 2010, Limassol, ~rus, 3-5 March 2010

MIMO Radar Target Detection with Parametric Scattering CorrelationModel

Tuomas Aittomaki and Visa Koivunen

Abstract-Multiple-input multiple-output (MIMO) radarsutilize various sources of diversity to improve the performanceof the radar. A MIMO radar in which angular diversity isachieved by using widely distributed antennas has been pro­posed to reduce the impact of the fluctuations of the target radarcross-section. This type of system is also known as the statisticalMIMO radar. Typically, it has been assumed that the signalsreceived by different antennas are either fully correlated orindependent depending on the configuration. In this paper, weassume more realistically that the scattering from the target iscorrelated. We show that taking the correlation of the scatteredsignals into account can improve the probability of detectingthe target. This is achieved by using a parametric model for thescattering which allows one to efficiently estimate the scatteringcovariance matrix. The numerical examples demonstrate thatthe probability of detection remains good even when parametersof the model are inaccurate.

I. INTRODUCTION

In a multiple-input multiple-output radar, multiple wave­forms and other sources of diversity are used to improvethe performance of the radar. There exist several differentapproaches to take advantage of the diversity provided bysuch systems. In colocated MIMO radars, the transmittersor receivers function as a coherent array. This approach istaken for example in [1], [2]. An entirely different approachis the MIMO radar with widely separated antennas[3], whichis also referred to as the statistical MIM0 radar[4]. Usingwidely distributed transmitters or receivers provides angulardiversity, as the target is seen from several different angles.

Target detection using widely distributed MIMO radar wasinitially studied in [4]. It was assumed that the noise isspatially white and that the scattering from the target is anindependent process for each transmitter-receiver pair. Usingthese assumptions, the optimal detector in Neyman-Pearsonsense was derived for the different statistical MIMO radarconfigurations in [4]. This result was extended to all theSwerling cases in [5].

We derived the optimal detector for correlated scatteringin spatially colored noise in [6]. However, using the optimallikelihood ratio test for target detection requires the covari­ance matrix of the scattering amplitudes to be known, whichis typically not the case in practice. In order to overcomethis problem, the use the generalized likelihood ratio test(GLRT) was proposed in [7]. The covariance matrix was

This work was supported in part by the Academy of Finland, Center ofExcellence program, in part by Finnish Defence Forces Technical ResearchCentre, and in part by Nokia Foundation

The authors are with Department of Signal Processing and Acoustics,SMARAD CoE, Aalto University, Otakaari 5A, 02150 Espoo, Finland.taittoma@wooster.hut.fi, visa@wooster.hut.fi

978-1-4244-6287-2/10/$26.00 ©2010 IEEE

assumed to have a Toeplitz structure in [7], which followsfrom the assumption that the radar antennas to be positionedon a line with uniform spacing. In this paper, we generalizethe results by using a novel correlation model that does notconstraint the antennas to any particular configuration.

The generalized likelihood ratio test can be formed usingthe maximum likelihood estimates for the unknown param­eters. Using the detectors derived in [6] would thus requireestimation of the covariance matrix of the filtered noiseand that of the scattering amplitudes. Estimating the thelatter is problematic as there is only very limited numberof pulses (samples) that can be used in the estimation.Hence, improvement in performance may not be achieved byusing the direct sample estimate of the covariance matrix. Toachieve better performance, the structure of the covariancematrix of the scattering amplitudes should be exploited.

In a widely distributed MIMO radar, the target RCSdepends on the aspect of the target seen. If the aspectsseen by the antennas of the radar system are sufficientlydifferent, the fluctuations of the RCS are thought to beindependent. In this paper, we assume that the correlationof the fluctuations between the antennas depends on thedifference of the angle to the target. Furthermore, it isassumed that the scattering covariance matrix can be modeledusing the Kronecker model. Using this correlation model, theprobability of detection can be improved even when there iserror in the parameters of the model.

This paper is organized as follows: The signal model isdiscussed in Section II. The derivation of the optimal detectorand GLRT is shown in Section III. Numerical results willbe provided in Section IV. Finally, Sections V gives theconcluding remarks.

II. SIGNAL MODEL

In a signal model for a statistical MIMO radar with Mtransmitters and N receivers, the signal received by the k-thelement can be written as

rk(t) = t j"i;Ckmsm(t - Tkm)el21r! kTnt+ j e!>k", + nk(t),m=l

(1)where P is a power parameter, Ckm a scattering amplitude,8 m the signal transmitted by the m-th transmitter, Tkm thetime delay, fkm the Doppler shift, ¢km a phase term, andnk the receiver noise. We assume that the transmitted signalshave sufficient orthogonality properties so that Tkm and fkm

as well as ¢km can be efficiently estimated and removedusing a bank of matched filters producing a matched filter

where R, is the transmitter covariance matrix and R, is thereceiver covariance matrix. The correlation is modeled to beproportional difference in the angle to the target Om. Theelements of R, are modeled as

where fi is the filtered noise vector. The elements of thematrix C are zero-mean circular complex Gaussian randomvariables with unit variance, as they model the fluctuationsof the target ReS (path losses and other losses are includedin the power parameter P). Letting c = vec(C), we assumethat the covariance matrix of the scattering amplitudes canbe written using the Kronecker model as

(8)

(6)

(7)

£0 = IIII f(r(tk,i)IHo)k

Target detection is typically formulated as hypothesistesting problem with the hypotheses

Ho : no target present

HI: a target is present.

or

The likelihood ratio test (LRT) is known to be an optimaltesting procedure in the Neyman-Pearson sense. Assumingthat scattering is independent from pulse to pulse and noise isi.i.d. for each sample, the likelihood function of the receiversnapshots can be written as

!(r(tk,i)IJio) = (1ra~\MN exp { - :~ Ilr(tk,i)112

} •

The density function under the alternative hypothesis can bewritten as

£1 = IIII f(r(tk,i)IHl)k

where f is the probability density function, k is the pulseindex, and i is the index of the sample in a pulse. Underthe null hypothesis Ho, there is only noise present and thedensity function can be written simply as

(2)

(5)

(4)

(3)

Y = avec(C) + ii(t),

where a is the parameter vector of the function g. Therequirements for 9 are that g(O; a) = 0, ~[g(lxl; a)] :::; 0, andg(-x; a) = g*(x;a). The real part of 9 should also decreasewhen Ix I increases (the larger the angular separation, thesmaller the correlation). A similar model is used also for thereceiver covariance matrix R r .

In the numerical examples in Section IV,

output vector

where

Thus, by using (11), the likelihood ratio can be written as

I:- = IJ det(~W) exp {~yr:W-IYk} · (14)

(10)

(13)

f(r(tk,I), r(tk,2), .. ·IH1 )

1rM N det("W- 1 )= !(r(tk,l), r(tk,2),. · ·IJio) 1rMN det(R

c) (11)

x exp {~yr:W-IYk }

where the matched filter output is

Yk = ~L S*(tk,i - r) 0 r(tk,i) (12)an i

and the weighting matrix

P 1 -1"W = M I 0 2"I +n; .

an

1 {H -1 }!(Ck) = 1rMN det(R

c) exp -c R c c ·

We showed in [7] that using the orthogonality of the wave­forms and the properties of the matrix column stackingoperation vecf-), the joint distribution of r(tk,I), r(tk,2), ...under HI can be written as

where the parameters a, are real and 2 log ao correspondsto P/ M. Examples of the resulting covariance matrix areshown in Section IV. Furthermore, the transmitters functionalso as receivers so that the transmitter covariance matrix R,and the receiver covariance matrix R, are equal.

The angles to the target can be computed from the loca­tions of the radar antennas, which are known, and location ofthe suspected target which can be estimated using the timedelays Tkm. In order to use the GLRT for detection, onethen needs to find the parameter vector a that maximizesthe likelihood of the observations, which typically resultsin better performance than using the covariance matrix ofmatched filter output estimated directly from the samples.The details of the detection are discussed in the next section.

III. TARGET DETECTION

The optimal detectors for statistical MIMO radar configu­rations were derived in [4] assuming independent scatteringand spatially white noise. In [6], we derived the optimaltest statistics for target detection using single pulse withcorrelated scattering and spatially correlated noise. As in [4],we assume that the used waveforms are orthogonal. In thispaper, we show the derivation of the optimal test statisticfor multiple pulses. For simplicity, we assume spatiallywhite noise. Fluctuations of the target radar cross-sectionare assumed to follow Swerling case 2, in which the target isassumed to consist of many independent scatterers of whichfew or none predominate and the fluctuations are independentfrom pulse to pulse[8].

The likelihood ratio test is thus equal to comparing the teststatistic

(15)

to a threshold, but using this test statistic requires requiresa~ and R c to be known.

Assuming that the noise variance is known, !(r(tk,i)I'Ho)is independent of the unknown parameters and maximizing(14) gives the GLRT. We prove next that the ML estimateof R y that maximizes the likelihood function of Y alsomaximizes the likelihood ratio (14).

First, we note that

where R y is the covariance matrix of the matched filteroutput. By using the matrix inversion lemma[9], the inverseof the weighting matrix can be written as

(16)

Therefore, (14) can be written as

c = IJ a;,MN ~et(Ry) exp { -yf{R;lYk + a;IIYkI12}

(17)and we see that Ry that maximizes the likelihood functionof Y also maximizes L.

The MLE of covariance matrix can be obtained simply as

KA 1 ~ Hn, = K LJYkYk,

k=1

where K is the number of pulses. Substituting this into (17)yields

t: = 1 A K exp {-KMN + Ka~tr(Ry)}.[a~MN det(Ry)]

However, this method will not result in good probabilityof detection because typically there is not enough pulsesavailable to estimate the covariance matrix accurately thisway. Better performance can be achieved by exploiting thestructure of the covariance matrix R c discussed in SectionII. One can obtain the ML estimate a by solving

II 1 {H -1 }m:x k det(Ry(a)) exp -Yk Ry (a)Yk

numerically.

IV. NUMERICAL EXAMPLES

In this section, we show numerical examples demonstrat­ing that using the model for the scattering correlation, theGLRT can achieve better probability of detection than whenusing the sample covariance matrix. If the correlation is highenough, this method will also result in better performancethan using the test statistic with identity weights, which is theoptimal detector when the scattering is independent. We com­pare numerically evaluated receiver operating characteristic(ROC) curves of the optimal LRT, the detector with identityweights (W = I in (15)), and two version of the GLRT.In the GLRT, the weighting matrix was either obtained byusing the MLE of the matched filter output covariance matrixor constructed using the numerically solved MLE of theparameter vector a. The MLE was obtained numericallyby optimizing the likelihood function using an active-setalgorithm in MATLAB.

Six antennas were used that functioned as both transmittersand receivers. The SNR was defined as P / a~ = -3dBand number of pulses was 40, which guaranteed that the36 sample covariance matrix was invertible. The differencesin the angle to the target with respect to the first antennawere [0, 0.05027r, 0.07987r, 0.12147r, 0.14007r, 0.19807r]. Thevalues of al were -22, -11.5, and -6.6, whereas a2 was 0.1in all the cases. The ROC curves where evaluated computingthe empirical distributions of the test statistics by the methodof Monte Carlo using 105 samples.

Using the parametric model to form the covariance matrixrequires the angle to the target to be known for each antenna.In order to study how much error in the angle estimates affectthe performance of the GLRT, the ROC curve was evaluatedalso after applying errors [0.10747r, -0.01027r, -0.07677r,-0.04507r, 0.05637r, -0.01847r] to the angles to the target.

The ROC curves are shown in Fig. 1 for the parametervalue al = -22 which corresponds to low correlation. Theabsolute values of the elements of the covariance matricesR, and R, were approximately

1.0000 0.0312 0.0040 0.0002 0.0001 0.00000.0312 1.0000 0.1294 0.0073 0.0020 0.00000.0040 0.1294 1.0000 0.0565 0.0156 0.00030.0002 0.0073 0.0565 1.0000 0.2767 0.00500.0001 0.0020 0.0156 0.2767 1.0000 0.01820.0000 0.0000 0.0003 0.0050 0.0182 1.0000

in this case. It can be seen that there are no big differencesin the performance of the detectors. The detector usingthe identity weights is close to the LRT as most the off­diagonal elements of the covariance matrix are very small.The GLRT detectors that use the correlation suffer fromestimation error and thus do slightly worse than detector withthe identity weights but significantly better than the GLRTusing the sample covariance matrix. Since the off-diagonalelements of the covariance matrix are small, the error in theangle estimates has practically no impact on the detectionperformance.

Fig. 2 shows the ROC curves for al = -11.5. In thiscase, the absolute values of the elements of the transmit and

,-, -, LRT""'" Identity weights- GLRT using sample covar.- - - GLRT using model- - - GLRT using model with error

-3

0.6

0.2

0.8

1.0r------r-------r----~"="==>r.='_.,

ROC

,-, -, LRT""'" Identity weights- GLRT using sample covar.- - - GLRT using model- - - GLRT using model with error

-3

0.8

0.2 '

1.0,--------.--------,------r---='= =---.ROC

Fig. I. ROC curves when SNR is -3 dB, al = -22 and az = 0.1. Whenthe correlation is low, the performance of the detectors is very similar anderrors in the angles have no significant effect.

Fig. 2. ROC curves when SNR is -3 dB, al = -11.5 and az = 0.1. Withsignificant correlation, the GLRT using the correlation model does notablybetter than the detector using identity weights. The GLRT using the samplecovariance matrix performs the worst due to low number of samples.

. ,.....,."

,-, -, LRT""'" Identity weights- GLRT using sample covar.- - - GLRT using model- - - GLRT using model with error

,,

"

,,

,i

-3

0.4 .,..:",..

0.8"",

,.,., '-,-",,'

",0.6 ,,' I ",

0."0 ",

1 .0r-----,-------.-~:_:=""'"'""~===__,

ROC

Fig. 3. ROC curves when SNR is -3 dB, «i = - 6.6 and az = 0.1. Takingthe correlation into account results in significant gain. Using the correlationmodel is seen to be robust to the errors in the estimated angles to the target.

receive covariance matrices were

1.0000 0.1631 0.0560 0.0125 0.0064 0.00080.1631 1.0000 0.3433 0.0764 0.0390 0.00480.0560 0.3433 1.0000 0.2226 0.1137 0.01400.0125 0.0764 0.2226 1.0000 0.5108 0.06290.0064 0.0390 0.1137 0.5108 1.0000 0.12310.0008 0.0048 0.0140 0.0629 0.1231 1.0000

In this case, the LRT and the GLRT using the model do betterthan the detector using the identity weights. With significanterror in the estimated angles to the target, the performance ofthe GLRT only slightly better that the detector with identityweights. If the sample covariance matrix is used, however,the performance is the worst as there are not enough samples(pulses) to form an accurate estimate resulting in decreasedprobability of detection.

In Fig. 3, value of the parameter al was -6.6, whichcorresponds to the absolute values of the element of thecovariance matrix approximately equal to

1.0000 0.3521 0.1903 0.0801 0.0544 0.01630.3521 1.0000 0.5404 0.2275 0.1545 0.04630.1903 0.5404 1.0000 0.4210 0.2860 0.08560.0801 0.2275 0.4210 1.0000 0.6792 0.20330.0544 0.1545 0.2860 0.6792 1.0000 0.29940.0163 0.0463 0.0856 0.2033 0.2994 1.0000

with the given differences in the angles to the target. Whenthe correlation is this high, taking it into account by usingthe GLRT is naturally very beneficial. However, using thesample covariance matrix results in the worst probability ofdetection in this case as well. Although the errors in the angleestimates change the covariance model significantly for highcorrelation, the probability of detection is still significantlylarger than that of the detector with identity weights.

The numerical examples provided in this section demon­strated that the forming the GLRT using the exponential

correlation model can result in improved probability of de­tection compared to the sample covariance matrix or identityweights. This method is effective even when there are errorsin the estimated angles to the target. The tolerance to errorsresults from optimizing the parameters of the model usingthe likelihood function so that best fit to the data is achieved.

V. CONCLUSIONS

In a statistical MIMO radar, angular diversity is achievedby using widely distributed antennas. One aim in this typeof MIMO radar is to see the target from several independentaspects so that the scattering from the target is independent.However, situations in which some of the antennas see asimilar aspect of the target are likely, so some correlation in

the scattering is inevitable.

Using the optimal likelihood ratio test for target detectionwould require the statistics of the received signal to beknown, which is not the case in practice. It was seen thatforming the generalized likelihood ratio test using the directsample covariance matrix resulted in decreased probabilityof detection. In order to achieve an improvement in theperformance, we have proposed forming the GLRT usinga parametric model for the correlation of the scattering.

The numerical examples shown for white noise withknown variance demonstrated that using the correlationmodel to estimate the parameters of the GLRT resulted inimproved performance compared to the detector that does nottake correlation of the scattering into account. The benefitswere seen when the correlation between several antennapairs was more than 0.25. The drawback of the model isthat the angle at which each antenna sees the target needto be estimated. However, estimates of the angles can beobtained using the time differences of arrival and errors inthe estimates are tolerated well.

REFERENCES

[1] D.R. Fuhrmann and G.S. Antonio, "Transmit beamforming for MIMOradar systems using partial signal correlation," in Conference Recordof the Thirty-Eighth Asilomar Conference on Signals, Systems andComputers, 2004, vol. 1, pp. 295-299.

[2] Jian Li and P. Stoica, "MIMO radar with colocated antennas," SignalProcessing Magazine, IEEE, vol. 24, no. 5, pp. 106-114, Sept. 2007.

[3] A.M. Haimovich, R.S. Blum, and L.J. Cimini, "MIMO radar withwidely separated antennas," Signal Processing Magazine, IEEE, vol.25, no. 1, pp. 116-129, 2008.

[4] E. FishIer, A. Haimovich, R.S. Blum, L.J. Cimini, D. Chizhik, andR.A. Valenzuela, "Spatial diversity in radars - models and detectionperformance," IEEE Transactions on Signal Processing, vol. 54, no. 3,pp. 823-838, 2006.

[5] Tuomas Aittomaki and Visa Koivunen, "Performance of mimo radarwith angular diversity under swerling scattering models," IEEE JournalofSpecial Topics on Signal Processing, vol. 4, no. 1, Feb 2010, In press.

[6] Tuomas Aittomaki and Visa Koivunen, "Optimal detector for MIMOradar with angular diversity and correlated scattering," in Proceedingsof International Radar Symposium, 2009, pp. 499-504.

[7] Tuomas Aittomaki and Visa Koivunen, "Exploiting correlation intarget detection using mimo radar with angular diversity," in AsilomarConference on Signals, Systems, and Computers, 2009.

[8] Merrill I. Skolnik, Ed., Radar Handbook, McGraw-Hill, second edition,1990.

[9] W. W. Hager, "Updating the inverse of a matrix," SIAM Rev., vol. 31,no. 2, pp. 221-239, 1989.