1MIMO Radar Sensitivity Analysis for TargetDetection

Murat Akcakaya, Student Member, IEEE and Arye Nehorai, Fellow, IEEEThe Preston M. Green Electrical and Systems Engineering Department

Washington University in St. LouisSt. Louis, Missouri 63130

Emails: muratakcakaya@wustl.edu, nehorai@ese.wustl.edu

AbstractFor multi-input multi-output (MIMO) radar withwidely separated antennas, we consider the effect of imperfectorthogonality of the received signals on detection performance. Inpractice, mutual orthogonality among the received signals cannotbe achieved for all Doppler and delay pairs. We introduce a datamodel considering the correlation among the data from differenttransmitter-receiver pairs as unknown parameters. Based on theexpectation-maximization algorithm, we propose a method toestimate the target, correlation, and noise parameters. We thenuse these estimates to develop a statistical decision test. Usingthe asymptotical statistical characteristics and the numericalperformance of the test, we analyze the sensitivity of the MIMOradar with respect to changes in the cross-correlation levels ofthe measurements. In our numerical examples, we demonstratethe effect of the increase in the correlation among the receivedsignals from different transmitters on the detection performance.

Index TermsMIMO Radar Signal Processing, Target Detec-tion, Sensitivity Analysis

I. INTRODUCTION

MIMO radars with widely separated antennas exploit spatialproperties of the targets radar cross section (RCS). This spatialdiversity provides the radar systems with the ability to supporthigh resolution target localization [1], [2]; to improve thetarget parameter estimation [3]-[6], detection in homogeneousand inhomogeneous clutter [7]-[10], and tracking performance[11]; and to handle slow moving targets by exploiting Dopplerestimates from multiple directions [8], [12]. In spite of theadvantages demonstrated in the above mentioned works, inpractice MIMO radar suffers from the non-orthogonality ofthe received signals.

Previous work on MIMO radar assumes signal transmissionwith insignificant cross-correlation to separate the transmittedwaveforms from each other at each receiver. However, for aMIMO radar, since the waveform separation is limited bythe Doppler and time delay resolution [13] (see also [14],[15]), the absent or low cross-correlation of the waveformfor any Doppler and time delay is not only important butalso challenging. In our work, to realistically model the radarmeasurements, we also consider the non-zero cross-correlationamong the signals received from different transmitters. Wemodel these parameters as deterministic unknowns, and then

This work was supported by the Department of Defense under the Air ForceOffice of Scientific Research MURI Grant FA9550-05-1-0443 and ONR GrantN000140810849.

we analyze the sensitivity of the MIMO radar target detectionwith respect to changes in the cross-correlation levels (CCLs)of the received signals. To the best of our knowledge, thisissue has never been addressed before. We here show thatan increase in the CCL decreases the detection performance.Moreover, we observe that radar systems with more receiversand/or transmitters have better detection performance, but suchsystems are more sensitive to changes in the CCL. Therefore,the performance analysis that was made under an assumptionof no or lowcross-correlation signal might be too optimistic.To simplify the analysis and better demonstrate our results, wefocus on stationary target scenarios; however, we will extendour results to moving target detection in future work. Notethat by MIMO radar we refer to a MIMO radar with widelyseparated antennas.

The rest of the paper is organized as follows. In Section II,we introduce our parametric measurement and statistical mod-els. In Sections III-A and III-B, respectively, we developa method based on the expectation-maximization (EM) al-gorithm [16] to estimate the target, correlation, and noiseparameters, and then we use these estimates to formulatea Wald target detection test [17], [18]. In Section III-C,we compute the Cramer-Rao bound (CRB) on the error ofparameter estimation, and in Section III-D, using the CRBresults, we analyze the asymptotic statistical characteristics ofthe Wald test [17]. In Section IV, using Monte Carlo simu-lations and theoretical results, we analyze the changes in theperformance of the target detection for different CCLs amongthe received signals, and hence demonstrate the sensitivity ofthe MIMO radar target detection to the imperfect separationof different transmitted signals at each receiver. Finally, weprovide concluding remarks in Section V.

II. RADAR MODELIn this section, we develop measurement and statistical

models for a MIMO radar system in the presence of non-zerocross-correlation among the transmitted waveforms. We usethese models to develop a statistical decision test and obtainits asymptotical statistical characteristics.

A. Measurement ModelWe consider a two dimensional (2D) spatial system with

transmitters and receivers. We define (Tx , Tx ),

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2 = 1, ..., , and (Rx , Rx ), = 1, ..., , as the locationsof the transmitters and receivers, respectively. We assume astationary point target (the same analysis directly extends tomoving targets with known Doppler shift) located at (0, 0)and having RCS values changing w.r.t. the angle aspect (e.g.,multiple scatterers, which cannot be resolved by the transmit-ted signals, with (0, 0) as the center of gravity) [1]. Definethe complex envelope of the narrow-band signal from the transmitter as (), = 1, . . . , , such that 2 isthe transmitted energy with

=1 2 = ( is constant

for any ) and s()2d = 1, = 1, . . . , , with

s as the signal duration. We write the complex envelope ofthe received signal at the receiver as follows [1]:

() =

=1

( ) + (), (1)

where is the complex target reflection coefficient seen by

the transmitter and receiver pair

=

2

(4)322

is the channel parameter from

the transmitter to the receiver, with and as the gains of the transmitting and receiving an-tennas, respectively; is the wavelength of the incomingsignal; and =

(

Tx 0)2 + (Tx 0)2 and

=(Rx 0)2 + (Rx 0)2 are the distances

from transmitter and receiver to target, respectively = ( + )/, and is the speed of the signal

propagation in the medium = 2c, with c as the carrier frequency, and () is additive measurement noise.

We will apply matched filtering to (1) and obtain the measure-ment at the receiver corresponding to the transmitterfor a single pulse as

= ()+

=1, =

()+,

(2)where

= +s

() ( )d

= +s

( ) ( )d, self correlation ofthe signal

=min (,)+smax (,)

( ) ( )d is thecross-correlation between and signals at the receiver, and

= +s

() ( )d.

Note here that since perfect signal separation is not possiblefor all delay and Doppler values, unlike previous approaches,we do not ignore the cross-correlation terms .

Then, we collect the data at the receiver correspondingto different transmitters for one pulse in an 1 columnvector

= + , (3)where

= [1, . . . , ] , () corresponds to the trans-pose of a matrix

[] = ([])

= for () as the

complex conjugate, , = 1, . . . , and = ,where [] corresponds to () element of a matrix

for = , [] = = diag(11(1), . . . , ( ))

is an diagonal matrix with entry as

()

= [1, . . . , ]

, and = [1, . . . , 2].We stack the receiver outputs corresponding to all the

signals into an 1 vector: = + , (4)

where = [1 , . . . ,

]

= blkdiag(1, . . . , ) is an blockdiagonal matrix with as the block diagonal entry

= blkdiag(1, . . . , ) = [1 , . . . ,

]

, and

= [1 , . . . , ]

.

Note that this analysis is for a specific range gate and Dopplershift.

We assume that pulses are transmitted from each trans-mitter; then

= [(1) (2) ()] =+, (5)

where = [(1) ()] , and = [(1) (2) ()] is the additive noise.

B. Statistical ModelWe now introduce our statistical assumptions for the mea-

surement model. We assume, in (5), is the matrix of the deterministic unknown

correlation parameters () is the 1 vector of the complex Gaussian

distributed target reflection coefficients, E[()] = 0,E[()() ] = 2s and E[()() ] = 0, with2s as the unknown variance, and for , = 1, . . . , = 1 when = , and zero otherwise

() is the 1 vector of the complex Gaussiandistributed additive noise, E[()] = 0, E[()() ] =2e and E[()() ] = 0, such that 2e is theunknown variance, and

() and () are uncorrelated for all and .Here () corresponds to the Hermitian transpose of a matrix.

To sum up, we consider the reflection coefficient variance2s , noise variance 2e , and the correlation terms as thedeterministic unknown parameters. We use the deterministicunknown parameter assumption for to demonstrate thesensitivity of the system to changes in the level of the cross-correlation values in . In practice, the matched-filter outputis sampled at discrete delay values [19] (see also [20]), andwe assume each range gate is represented by a single sample.However a target in one range gate, even though represented

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3by a delay , might actually be located at a delay such that /2, and is the pulse width. Moreover, theMIMO ambiguity function demonstrates that the zero cross-correlation can not be achieved for all delay values [13].Therefore, even though we know the delay that representsthe range gate of interest, we assume we do not know theexact value of the matched-filter output (cross-correlationand self correlation terms are unknowns). To simplify theproblem, we also assume that the unknown correlation termsare deterministic. This assumption is reasonable, because inour model, the target is stationary or moving with a knownconstant speed (since we apply our detector for a specificrange and Doppler pair). Then, under these assumptions, thedifference between the time delays of the received signals,corresponding to different transmitter and receiver pairs, duethe target (located in the far field) does not change significantlyduring the processing intervals (this is reasonable since thetarget speed is much slower than the speed of light). Ourdetector works for a specific Doppler shift and range gate pair,and the correlation terms are deterministic unknowns.

Due to the distributed nature of the MIMO radar system,we assume that the target returns for different transmitter andreceiver pairs are independent from each other. We also assumethat the target returns for different pulses are independentrealizations of the same random variable. Under these assump-tions, we write the distribution of the data in (5) as=1

(();2s , 2 ,) =

1

exp(

=1 ()), (6)

where () = ()1() = blkdiag(1, . . . , ), and = 2s

+

2e .

III. STATISTICAL DECISION TEST FOR TARGETDETECTION

In this Section, we propose a Wald test for the detection ofa target located in the range cell of interest (COI). This testdepends on the maximum likelihood estimates (MLEs) of theunknown parameters as well as on the CRB on the estimationerror under the alternative hypothesis. Therefore, we also de-velop a method for the estimation of the unknown parametersbased on the EM algorithm, then accordingly compute theCRB on the estimation error to derive the statistical test.

A. Wald TestWe choose between two hypotheses in the following para-

metric test: { 0 : 2s = 0,, 2e1 : 2s = 0,, 2e , (7)

where the correlation and the noise variance 2e are thenuisance parameters. This is a composite hypothesis test,therefore a uniformly most powerful (UMP) test does notexist for the problem. As a sub-optimum approximation, ageneralized likelihood ratio test (GLRT) is the most commonly

used solution. Even though there is no optimality associatedwith the GLRT solution [17], it is known to work wellin practice [18]-[23]. However, in (7), since the MLEs ofthe nuisance parameters cannot be obtained under 0, wedo not use the GLRT; instead, we propose to use a Waldtest. The Wald test depends only on the estimates of theunknown parameters under 1. Moreover, to demonstrate theresults of our analysis, we focus on the asymptotic statisticalcharacteristics of the decision test. Because the Wald test andGLRT were shown to have the same asymptotic performance[17], we choose a Wald test instead of the GLRT.

We define the set of unknown variables as ={2s ,

2e ,

}, and compute the Wald test as

w =(2s1 2s0

)([1(1)

]2s

2s

)1 (2s1 2s0

), (8)

where

2s1 and 2s0 are the estimates of 2s under 1 and 0,respectively (2s0 = 0 under 0)

1(1) is the inverse of the Fisher information matrix(FIM) calculated at the estimate of under 1, and

the subscript 2s2s of the inverse of the FIM is the valueof the inverse FIM corresponding to 2s , that is, the CRBon the 2s estimation error.

We reject 0 (the target-free case) in favor of 1 (the target-present case) when w is greater than a preset threshold value.

B. Estimation Algorithm

The Wald test proposed in (8) requires estimation of theunknown parameters under the alternative hypothesis 1.Since the number of the measurements is the same as thenumber of the random reflections from the target, there is noclosed-form solution to the estimates of unknown parameters,so we cannot use the concentrated likelihood methods pro-posed in [24]-[27] to estimate . Instead we propose to developan estimation method based on the EM algorithm.

We consider , , and ( ,) as the observed, unobserved,and complete data, respectively.

The complete-data likelihood function belongs to an ex-ponential family; hence we simplify the EM algorithm [28].In the estimation (E) step, we first calculate the conditionalexpectation of the natural complete-data sufficient statisticsgiven the observed data [using (()();s, 2e ,)].Then, in the maximization (M) step, we obtain the MLEexpressions for the unknown parameters using the complete-data log-likelihood function, and simply replacing the naturalcomplete-data sufficient statistics, obtained in the E step, inthe MLE expressions.

E Step: We assume that the iteration estimatesof the set of the unknown parameters as () ={(s2)(), (e2)(), () }, and we compute the conditionalexpectation w.r.t.

(()();()

)of the sufficient statis-

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4tics under 1:

()1 =

1

=1

()() , (9a)

()2 =

1

=1

()

[()

+ () ()

(() ()

)]

(),(9b)

()3 =

1

=1

()(() ()

)

(), (9c)

()4 =

1

=1

()

+ () ()

(() ()

), (9d)

where

() () = (s2)()

(

()

) (()

)1(), and

()

= (s2)() (s2)()

(

()

)(()

)1

()

, where from (6), () =(s

2)()()

(

()

)+ (e

2)() .

Thus, () () and ()

are the mean and the covariance ofthe conditional distribution (()();() )).

M Step: We replace the natural complete-data sufficientstatistics with their conditional expectations from (9) in theMLE expressions. We first apply the results of the general-ized multivariate analysis of variance framework [29] for theMLE of , for = 1, . . . , . After concentrating thecomplete data log-likelihood function w.r.t. the MLE of ,we compute the MLEs of 2s and 2 .

(+1)

= ()3

(()2

)1, (10a)

(2e )(+1) =

1

=1

(tr[()1

]

2Re(tr[(3)

()(+1)

])+tr

[()(+1)

(+1) 2

]),(10b)

(2s )(+1) =

1

=1

tr[()4

]. (10c)

The above iteration is performed until (2s )(), (2e )(), and

()converge. tr() is the trace of a matrix, and Re() is the

real part of a complex number.

C. Computation of the Cramer-Rao BoundIn this section, to obtain the Wald test in (8), we com-

pute the CRB on the error of the 2 estimation. We de-fine = [2s , 2e ,Re{vech(1)} , Im{vech(1)} , . . . ,Re{vech()} , Im{vech()} ] , such that vech createsa single column vector by stacking elements on and below themain diagonal. Recall that for = 1, . . . , is Hermitiansymmetric. Therefore, estimating is the same as estimating in Section III-A. Here Im() stands for the imaginary partof a complex number.

Fig. 1. MIMO antenna system with transmitters and receivers.

Considering the statistical assumptions in Section II-B, weobtain the elements of the FIM [30]:

[()] = tr

=1

=1

(1

1

). (11)

Then [ (1)()]2s 2s = [(1)()]11 is the CRB on the 2s

estimation error.

D. Detection PerformanceIn this section, we analyze the asymptotic statistical char-

acteristics of the Wald test proposed in (8). In Section IV, weuse these asymptotic characteristics to demonstrate the changein detection performance due to changes in the level of thecross-correlation terms.

When we apply the Wald test in (8) to the hypothesistesting problem formulated in (7), following the results in [17,Chapter 6 and Appendix 6C], we can show that

W

{ 21 under 0 21 () under 1 , (12)

where 21 is a central chi-square distribution with one degree of

freedom 21 () is a non-central chi-square distribution with one

degree of freedom and a non-centrality parameter , and =

(2s) (

CRB2s)1 (

2s).

Here 2s is the true value under 1, and following thediscussions in Section III-C, CRB2s = [

(1)()]2s 2s is theCRB on 2s estimation error, and it is computed using the truevalues of under 1.

IV. NUMERICAL EXAMPLESWe present numerical examples to illustrate our analytical

results on the sensitivity of MIMO radar target detection tochanges in the cross-correlation levels of multiple signalsreceived from different transmitters. Using the asymptotictheoretical results from Section III-D, we show the effect ofthe changes in CCL on the receiver operating characteristics(ROC), and the detection probability of the statistical test. Wealso compare the asymptotic and actual ROCs of the Walddetector. We use the EM algorithm from Section III-B tonumerically compute the actual ROC curve of the decision

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5test in Section III-A. The numerical results are obtained from2 103 Monte Carlo simulation runs.

We follow the scenario shown in Fig. 1. We assume thatour system is composed of transmitters and receivers,where the antennas are widely separated. The transmitters arelocated on the y-axis, whereas the receivers are on the x-axis;the target is 10km from each of the axes (i.e., (0, 0) =(10 km, 10 km)); the antenna gains ( and ) are 30dB;the signal frequency (c) is 1GHz. The angles between thetransmitted signals are 1, 2, ..., and similarly betweenthe received signals are 1, ..., . We consider three differentMIMO setups in our examples.

= 2 and = 3 (MIMO 2 3); 1 = 10, and2 = 20

; 1 = 10, 2 = 10, and 3 = 25; = 3 and = 3 (MIMO 3 3); 1, 2 are the same

as MIMO 2 3, and 3 = 35; 1, 2, and 3 are thesame as MIMO 2 3;

= 3 and = 5 (MIMO 3 5); 1, 2, and 3 arethe same as MIMO 3 3; 1, 2, and 3 are the same asMIMO 2 3, 4 = 20, and 5 = 20.

Then , = 1, . . . , , and , = 1, . . . , , in (1)are calculated accordingly. In this scenario, all the transmittersand receivers see the target from different angles.

We define the signal-to-noise ratio (SNR) as the ratio be-tween the traces of the signal covariance and noise covariance:

SNR =2s2e

=1 (

)

.

We define the average CCL (ACLL) as the ratio between thetotal power of the non-zero cross-correlation terms and theself correlation of the individual signals, then ACCL becomes

ACCL = 10 log10[ 1

2

( =1

=1(

)(

)

=1

=2

=1(

)(

)

)].

For example ACCL = 10 dB means that the ACCL is 10dB below the average self-correlation values. As the ACCLdecreases, separation of the transmitted signals for differentdelays gets easier. In the following we investigate the effectof the changes in the ACCL on detection performance.

In Fig. 2, for fixed SNR = 5 dB and for different MIMOconfigurations, MIMO 23 (Fig. 2(a)), MIMO 33 (Fig. 2(b))and MIMO 35 (Fig. 2(c)) at different ACCL values (-5, -10and -20 dB), we demonstrate both the asymptotic and numeri-cal receiver operating characteristics of the statistical decisiontest. For a large number of transmitted pulses, = 500, weobtain the numerical ROC using the EM algorithm proposedin Section III-B in (8). We show that for sufficiently large, the actual ROC of the Wald test is very close to theasymptotic one. We observe that as the ACCL decreases, thedetection performance improves. This improvement is due tothe fact that a decrease in the ACCL results in a decrease inthe CRB of the 2s estimation error, causing an increase inthe non-centrality parameter in Section III-D, and hence anincrease in . Moreover, a system with more transmittersand/or receivers has better detection performance, but alsomore sensitivity to changes in ACCL.

In Fig. 3, for fixed

= 0.01, we plot the

as

a function of the SNR for different MIMO configurations

101 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of False Alarm (PFA)

Prob

abili

ty o

f Det

ectio

n (P

D)

MIMO 2x3 Asymptotic ACCL=20dBMIMO 2x3 Asymptotic ACCL=10dBMIMO 2x3 Asymptotic ACCL=5dBMIMO 2x3 Numerical ACCL=20dBMIMO 2x3 Numerical ACCL=10dBMIMO 2x3 Numerical ACCL=5dB

(a)

101 1000.4

0.5

0.6

0.7

0.8

0.9

1

Probability of False Alarm (PFA)

Prob

abili

ty o

f Det

ectio

n (P

D)

MIMO 3x3 Asymptotic ACCL=20dB MIMO 3x3 Asymptotic ACCL=10dBMIMO 3x3 Asymptotic ACCL=5dBMMO 3x3 Numerical ACCL=20dBMIMO 3x3 Numerical ACCL=10dBMIMO 3x3 Numerical ACCL=5dB

(b)

101 1000.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Probability of False Alarm (PFA)

Prob

abili

ty o

f Det

ectio

n (P

D)

MIMO 3x5 Asymptotic ACCL=20dBMIMO 3x5 Asymptotic ACCL=10dBMIMO 3x5 Asymptotic ACCL=5dBMIMO 3x5 Numerical ACCL=20dBMIMO 3x5 Numerical ACCL=10dBMIMO 3x5 Numerical ACCL=5dB

(c)Fig. 2. Receiver operating characteristics of the target detector for differentACCL values and (a) MIMO 2 3 (b) MIMO 3 3 (c) MIMO 3 5configurations.

and different ACCL values. This figure also supports ourargument on the relationship between changes in the ACCLand detection performance: a decrease in the ACCL improvesthe detection performance. In this figure, we can also observethe effect of the number of the transmitters and/or receivers onthe detection performance. Systems with more antennas havebetter performance, but the increase in performance comeswith a price: such a system becomes more sensitive to changesin the ACCL.

V. CONCLUSION

We analyzed the detection sensitivity of MIMO radar tochanges in the cross-correlation levels of the signals at eachreceiver from different transmitters. We formulated a MIMO

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610 9 8 7 6 5 4 3 2 1 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SignaltoNoise Ratio (in dB)

Prob

abili

ty o

f Det

ectio

n (P

D)

MIMO 3x5 ACCL=20dBMIMO 3x5 ACCL=10dBMIMO 3x5 ACCL=5dBMIMO 3x3 ACCL=20dBMIMO 3x3 ACCL=10dBMIMO 3x3 ACCL=5dBMIMO 2x3 ACCL=20dBMIMO 2x3 ACCL=10dBMIMO 2x3 ACCL=5dB

MIMO 3x5

MIMO 3x3

MIMO 2x3

PFA=0.01

Fig. 3. Probability of detection vs. SNR for different ACCL values ( =0.01).

radar measurement model considering the correlation terms asdeterministic unknowns. We proposed to use an EM basedalgorithm to estimate the target, correlation, and noise pa-rameters. We then developed a Wald test for target detection,using the estimates obtained from the EM estimation step. Wealso computed the CRB on the error of parameter estimation,and used these results to obtain an asymptotical statisticalcharacterization of the detection test. Using the asymptoticalresults and Monte Carlo simulations, we demonstrated thesensitivity of the MIMO radar target detection performance tochanges in the cross-correlation levels of the received signals.We showed that as the level of the correlation increases,the detection performance deteriorates. We also observed thatMIMO systems with more transmitters and/or receivers havebetter detection performance, but they are more sensitive tochanges in the correlation levels. In our future work, we willextend this analysis to moving target detection and estimation.

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978-1-4244-8902-2/11/$26.00 2011 IEEE 638

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