Adaptive MIMO Radar Design and Detection in Compound-Gaussian Clutter

MIMO Radar Detection and Adaptive Design inCompound-Gaussian Clutter

Murat Akcakaya, Student Member, IEEE and Arye Nehorai∗, Fellow, IEEE

Electrical and Systems Engineering Department.Washington University in St.louis

St.louis, Missouri 63130Emails: {makcak2, nehorai}@ese.wustl.edu

Abstract—Multiple-input multiple-output (MIMO) radars withwidely separated transmitters and receivers are useful to dis-criminate a target from clutter using the spatial diversity of thescatterers in the illuminated scene. We consider the detectionof targets in compound-Gaussian clutter. Compound-Gaussianclutter describes heavy-tailed distributions fitting high-resolutionand/or low-grazing-angle radars in the presence of sea or foliageclutter. First, we introduce a data model using the inverse gammadistribution to represent the clutter texture. Then, we applythe parameter-expanded expectation-maximization (PX-EM) al-gorithm to estimate the clutter texture and speckle as well as thetarget parameters. We develop a statistical decision test usingthese estimates and approximate its statistical characteristics.Based on the approximation of the statistical characteristics ofthis test, we propose an algorithm to adaptively distribute thetotal transmitted energy among the transmitters. We demonstratethe advantages of MIMO and adaptive energy allocation usingMonte Carlo simulations.

I. INTRODUCTION

Multiple-input multiple-output (MIMO) radars with widelyseparated antennas exploit spatial diversity and hence thespatial properties of the targets’ radar cross section (RCS).The RCSs of complex radar targets are quickly changingfunctions of the angle aspect. These target scintillations causesignal fading, which deteriorates the radar performance. Whenthe transmitters are sufficiently separated, the multiple signalsilluminate the target from de-correlated angles, and hence eachsignal carries independent information. This spatial diversityimproves the radar performance by mitigating these scintilla-tions. These systems have the ability to support high resolutiontarget localization; improve the target parameter estimationand detection performance; and handle slow moving targetsby exploiting Doppler estimates from multiple directions, see[1] and references therein.

We assume that the clutter reflections at the receiver followthe compound-Gaussian model. Target detection for MIMOsystems has been addressed with white and colored Gaussiannoise in [2] and [3], respectively. However, real clutter oftendeviates from the complex Gaussian model. Therefore, thisconventional model cannot represent the heavy-tailed clutter

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.

statistics that are distinctive of several scenarios, e.g., high-resolution and/or low-grazing-angle radars in the presence ofsea or foliage clutter [4], [5]. In our model, the compound-Gaussian clutter e =

√uX , where u and X are the texture

and speckle components of the compound model, respectively.The fast-changing X is a realization of a stationary zeromean complex Gaussian process, and the slow-changing u ismodeled as a nonnegative real random process [6].

Clutter modeling with compound-Gaussian distribution re-quires selection of the texture characteristics. Gamma distri-bution for the texture is investigated in [7] for MIMO radarsystems, leading to the well-known K clutter model. In thiswork, we specifically consider the inverse gamma distributionfor texture component u, since similar to its gamma distributedcounterpart inverse gamma fits well with real clutter data [8].Moreover this choice of distribution results in a closed-formmaximum likelihood solution for the joint target and clutterestimation [9].

The applications investigated for MIMO radar assume thatthe total energy is divided equally among the transmitters (see[1, Chapters 8 and 9]). We believe that this assumption maynot be optimal, since MIMO radar systems are sensitive toRCS variations of the target w.r.t. angle and since transmittingsignals with different energies from different transmitters maychange the total received power under the same environmentalconditions. Therefore, we extend our analysis in [10] anddevelop an adaptive algorithm that distributes the total energyamong the transmitters, exploiting the RCS sensitivity of thesystem and optimizing detection performance.

The rest of the paper is organized as follows. In SectionII, we introduce our parametric measurement model underthe generalized multivariate analysis of variance (GMANOVA)framework [11] for the MIMO system. In Section III, we firstpresent a parameter-expanded expectation-maximization (PX-EM) algorithm [12] with the use of the powerful GMANOVAtools to estimate the target and clutter parameters. Using theseestimates, we then formulate a statistical decision test basedon the generalized likelihood ratio (GLR) [13]. In Section IV,we develop the adaptive algorithm, and in Section V, we useMonte Carlo simulations to analyze the detection performanceand improvements due to adaptive design. Finally we provideconcluding remarks in Section VI.

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II. RADAR MODEL

In this section, we develop measurement and statisticalmodels for a MIMO radar system to detect a target in the rangecell of interest (COI). Our goal is to present an algorithm,within a GMANOVA framework when the signal and noiseparameters are unknown.

A. Measurement Model

We consider a two dimensional (2D) system with M trans-mitters and N receivers. Define (xTxm

, yTxm), m = 1, ...,M ,

and (xRxn, yRxn

), n = 1, ..., N , as the locations of thetransmitters and receivers, respectively. We also assume astationary point target located at (x0, y0) and having radarcross section (RCS) values changing w.r.to the angle aspect(e.g., multiple scatterers, which cannot be resolved by thetransmitted signals, with (x0, y0) as the center of gravity)[1]. Define the complex envelope of the signal from the mth

transmitter is βmsm(t), m = 1, . . . , M , such that |βm|2 asthe transmitted energy with

∑Mm=1 |βm|2 = E (E is constant

for any M ) and∫Ts

|sm(t)|2dt = 1, m = 1, . . . , M , withTs as the signal duration. We write the lowpass equivalent ofthe received signal at the nth receiver following [1]:

rn(t) =M∑m=1

αnmσnmβmsm(t− τnm)e−jψnm + en(t), (1)

where

• σnm is the square root of the target RCS seen by the mth

transmitter and nth receiver pair

• αnm =

√GtxGrxλ

2

(4π)3R2mR

2n

is the channel parameter from

the mth transmitter to the nth receiver, with Gtx andGrx as the gains of the transmitting and receiving an-tennas, respectively; λ as the wavelength of the incom-ing signal; Rm =

√(xTxm

− x0)2 + (yTxm− y0)2 and

Rn =√

(xRxn− x0)2 + (yRxn

− y0)2 as the distancesfrom transmitter and receiver to target, respectively

• τnm = (Rm + Rn)/c, and c is the speed of the signalpropagation in the medium

• ψnm = 2πfcτnm, with fc as the carrier frequency• e(t) is additive clutter noise.

To enable the data separation at the receiver side arriv-ing from the different transmitters, we assume low–cross-correlation transmitted signals. The design of signals withthese properties is a challenging research subject [14], butfor simplification of the problem and demonstration of ourmethods and analysis, we assume that the required signalcriteria are met (this assumption is commonly made for MIMOradar, see [1, Chapters 8 and 9] and references therein.) Hence,we apply matched-filtering and obtain the output of the nth

receiver corresponding to the ith transmitter :

rni = βiαniσnie−jψni + eni, (2)

where rni =∫ τni+Ts

τnirn(t)s∗i (t − τni)dt and eni =∫ τni+Ts

τnien(t) s∗i (t− τni)dt.

Then, combining the received data corresponding to thetransmitted signal si(t) for one pulse, we obtain

ri = Aixi + ei, (3)

where• ri = [r1i, . . . , rNi]T

• Ai = βidiag(α1ie−jψ1i , . . . , αNie

−jψNi)• xi = [σ1i, . . . , σNi]T

• ei = [e1i, . . . , eNi]T .We stack the receiver outputs corresponding to all the signalsinto an NM × 1 vector

y = Ax + e, (4)

where• y = [rT1 , . . . rTM ]T

• A = blkdiag(A1, . . . , AM )• x = [xT1 , . . . xTM ]T

• e = [eT1 , . . . eTM ]T .We transmit K pulses and assume that the target is stationaryduring this observation time; then

Y = [y(1) y(2) · · · y(K)]NM×K

= Axφ + E, (5)

where φ = [1, . . . 1]1×K , and E = [e(1) e(2) · · ·e(K)]NM×K

is the additive noise.

B. Statistical Model

In (5), we assume that X (target RCS values) is unknowndeterministic.

We consider the compound-Gaussian distributione(k)=

√u(k)X (k), k = 1, . . . , K, to model the clutter

with u(k) and X (k) as the texture and speckle components,respectively; see [9] and references therein. The realizationsof the fast-changing component, X (k), k = 1, . . . , K, areindependent and identically distributed (i.i.d.) and follow acomplex Gaussian distribution with zero mean and covarianceΣ. The texture is the slow-changing component [6]; thuswe consider it to be constant during the pulse duration Ts.However, we also assume the texture to be an independentrealization of the same random process from pulse to pulse;i.e., Cov(u(k), u(k

′)) = 0 for k �= k

′. Therefore, e(k)

k = 1, ...,K, are i.i.d., and we can write the conditionaldistribution for the observation Y in (4) asK∏k=1

py|u(y(k)|u(k)) =K∏k=1

1|πu(k)Σ| exp

{− [y(k) − Axφ(k)]H

· [u(k)Σ]−1 [y(k) − Axφ(k)]}, (6)

where“(·)H” denotes the Hermitian transpose.Observe that (5) , conditioned on u(k), k = 1, . . . , K,

with known A and φ and unknown x and Σ, is a GMANOVAmodel. We assume that w(k) = 1/u(k) follows the gammadistribution (consequently u(k) follows the inverse gammadistribution) with unit mean and unknown shape parameterv > 0 as in [9]; i.e.,

pw(w(k); v) =1

Γ(v)vvw(k)v−1 exp [−vw(k)] , (7)

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where Γ(·) is the gamma function. Therefore, we consider x,Σ, and v as the unknown parameters.

III. DETECTION AND ESTIMATION ALGORITHMS

We present in this section the maximum likelihood estima-tion (MLE) of the unknown parameters and target detectiontest. We derive a statistical decision test based on GLR usingthe observed data likelihood function to determine the presenceof a target in the COI. We choose between two hypotheses inthe following parametric test:{ H0 : x = 0,Σ, v

H1 : x �= 0,Σ, v , (8)

with the speckle covariance Σ and the inverse texture shapeparameter v as the nuisance parameters. We compute the GLRtest by replacing the unknown parameters with their MLEs inthe likelihood ratio test. Then, we reject H0 (target-free case)in favor of H1 (target-present case) when

GLR =p1(Y ; x1, Σ1, v1)p0(Y ; Σ0, v0)

> η, (9)

where p0(·) and p1(·) are the observed data likelihood func-tions under H0 and H1. Moreover, Σ0 and Σ1 are the MLEs ofΣ, and v0 and v1 are the MLEs of the shape parameter v underH0 and H1; x1 is the MLE of x under H1; η is the detectionthreshold. For this special case of compound-Gaussian modelwith inverse gamma texture, it is easy to obtain a closed-formexpression for the marginal pdf py(·) of y(k), which is acomplex multivariate t distribution [9].

We compute the MLEs of the vector x, speckle covariancematrix Σ, and texture distribution shape parameter v using thehierarchical data model presented in (6) and (7). We considertwo iterative loops for the MLE computations: (i) inner loopand (ii) outer loop. In the inner loop, first we introduce the PX-EM algorithm to obtain the MLEs x and Σ for a fixed v. ThePX-EM algorithm has the same convergence properties as theclassical EM algorithm, but it outperforms the EM algorithm inglobal rate of convergence [12]. In the outer loop, we estimatev using the MLEs from the inner loop [9].

1) Inner Loop: PX-EM algorithm for inverse gammatexture.-Recall that x, Σ, and v are the unknown parameters. We firstestimate θ = {x,Σ}, assuming that v is known. We imple-ment the PX-EM algorithm by adding a new unknown param-eter μw, the mean of w(k), to this set; i.e., θ∗ = {x,Σ∗, μw}.In this model, since the maximization step performs a moreefficient analysis by fitting the expanded parameter set, thePX-EM algorithm has a rate of convergence at least as fast asthe EM algorithm [12]. Under this expanded model the pdf ofw(k) is

pw(w(k); v, μw) =1

Γ(v)

(v

μw

)vw(k)v−1 exp [−vw(k)/μw] .

(10)Consider θ = R(θ∗) = {x,Σ∗/μw}, where R(·) is thereduction function (many-to-one) from the expanded to the

original space. Moreover, μ0w = 1 is the null value such that

the complete-data model is preserved.

Since the complete-data likelihood function belongs to anexponential family, we simplify the PX-EM algorithm fol-lowing [15]. Thus the PX-E step reduces to calculating theconditional expectation of the natural complete-data sufficientstatistics given the observed data and expanded unknownparameters with μw = μ0

w. Then, the PX-M step simplyreplaces the natural complete-data sufficient statistics in theMLE expressions of x, Σ∗ and μw with their conditionalexpectations obtained in the PX-E step.

We find the expressions for the MLEs of x, Σ∗, and μwusing the complete data log-likelihood function. These MLEscan be shown to be functions of the natural complete-datasufficient statistics. Since (5) is a GMANOVA model, we usethe results of [11] for the MLEs of x and Σ∗ and the resultsof [9, Appendix A] for the PX-EM algorithm.

We define i and j as the inner and outer loop iterationindexes, respectively. We summarize the algorithm:

PX-E Step: Calculate the conditional expectation of thesufficient statistics under H1, concentrated at v(j), the jth

iteration step estimate of v

Using the properties of the compound-Gaussian modelwith inverse gamma distributed texture [16], we observe thatw(k)|y(k) follows a gamma distribution with

w(i)1 (k) = Ew|y[w(k)|y(k); θ

(i)

∗ ]

= (v(j) +MN) ·{v(j) + d(k, θ

(i)

∗ )}−1

, (11)

where θ(i)

∗ = {x(i), Σ(i)

∗ , μ(i)w = μ0

w = 1} is the

estimate of θ∗ at the ith iteration and d(k, θ(i)

∗ ) =[y(k) − Ax(i)φ(k)

]H [Σ

(i)]−1 [

y(k) − Ax(i)φ(k)]. Then,

T(i)1 =

1K

K∑k=1

y(k)φ(k)Hw(i)1 (k) (12a)

T(i)2 =

1K

K∑k=1

y(k)y(k)Hw(i)1 (k) (12b)

T(i)3 =

1K

K∑k=1

φ(k)φ(k)Hw(i)1 (k) (12c)

T(i)4 =

1K

K∑k=1

w(i)1 (k) (12d)

For the PX-M step, we first define S(i) =

T(i)2 − T

(i)1

(T

(i)3

)−1 (T

(i)1

)Hand Q(i) =

A[AH

(S(i)

)−1A

]−1

AH .

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PX-M Step:

x(i+1)1 =

[AH

(S(i)

)−1

A

]−1

AH

·(S(i)

)−1

T(i)1

(T

(i)3

)−1

, (13a)

Σ(i+1)

∗ =(S(i)

)−1

+[I

MN−Q(i)

(S(i)

)−1]T

(i)1

·(T

(i)3

)−1 (T

(i)1

)H·[I

MN−Q(i)

(S(i)

)−1]H

, (13b)

μ(i+1)w = T

(i)4 , (13c)

Σ(i+1)

1 = Σ(i+1)

∗ /μ(i+1)w , (13d)

Under H0, we calculate Σ0, with x = 0, and update thesufficient statistics accordingly.

2) Outer Loop: MLE of the shape parameter of the inversegamma texture.We compute v(j+1) by maximizing the concentrated observeddata (y(k), k = 1, . . . ,K) log-likelihood function using the

estimates from the PX-EM step. We denote x(∞), Σ(∞)

0 ,

and Σ(∞)

1 as the estimates of x and Σ obtained upon theconvergence of the inner loop and compute

v(j+1)1 = arg max

v

K∑k=1

ln py

(x1,y(k), Σ

(∞)

1 , v)

(14)

Under H0, we calculate v(j+1)0 using x = 0 and Σ

(∞)

0 in (14).The GLR test computed upon convergence of (12), (13)

and (14) in (9) under H0 and H1 , results in a complicatedform which is impossible to statistically analyze. Therefore,we simplify it to the ratio of determinants of the covarianceestimates under different hypotheses, (see (15)), which is alsosimilar to the general form of GLR test presented in [11],to analyze its statistical characteristics (see Section IV). Firstwe assume known texture components and compute the GLRtest accordingly. Then using the data from the target-freeneighboring cells as the secondary data, we run the innerand outer loops of the estimation algorithm to compute theconditional mean of the texture components in (11) given thesecondary data. We replace the texture components with theircorresponding conditional mean values reducing the GLR testto

λ =

∣∣∣T (∞)2

∣∣∣∣∣∣∣T (∞)2 −Q(∞)

(S(∞)

)−1T

(∞)1

(T

(∞)3

)−1 (T

(∞)1

)−1∣∣∣∣> η

′,

(15)where | · | is the determinant operator, and T (∞)

1 , T (∞)2 , T (∞)

3 ,S(∞) and Q(∞) are obtained in one step using (12) with (11)computed using the secondary data as just explained.

IV. ADAPTIVE DESIGN

In this section, we first demonstrate the asymptotical sta-tistical characteristics of the detection test derived in Section

III. Based on this result, we then construct a utility functionfor adaptive energy allocation to improve the detection per-formance. We determine the optimum transmitted energy byeach transmitter according to this utility function.

We investigate the statistical properties of the test in (15),λ assuming that the texture components are known; that is,w(k), k = 1, . . . , K, are known. With this assumption thetest is the complex version of Wilks’ lambda, and under thenull hypothesis (H0) it is shown to follow the probabilitydensity function of multiplication of complex beta randomvariables. However the exact statistical analysis of this testunder alternative hypothesis (H1) is very difficult, exceptfor some special cases; thus the distributions are mostlyapproximated [17]. Therefore, following our work in [18], weemploy asymptotical approximations for the distributions ofthe tests, and we find, applying a similar approach used forreal Gaussian random variables in [19], [20], that under H0,as K → ∞, K lnλ has a complex chi-square distributionwith NM degrees of freedom. For known texture components,the distribution under H0 does not depend on the specklecovariance; hence, in the limit it is a constant false-alarm rate(CFAR) test.

Under H1, as K → ∞, K lnλ has a non-central complexchi-square distribution with NM degrees of freedom. That is,K lnλ ∼ Cχ2

NM (δ) [19] , [21]. The non-centrality parameteris δ = tr

(Σ−1(Axφ)(Axφ)H

)In practice the texture components are unknown and we

replace them by their conditional mean values computed giventhe secondary data. Since the unknown parameters Σ, andv of the secondary data belong to a canonical exponentialfamily (since the complete-data likelihood function belongsto an exponential family and could be written in canonicalform), their estimates are consistent and hence the conditionalmean value in (11), computed given the secondary data,converges to the minimum mean square error estimate of thetexture component in probability [15]. Then, in the mean,the detection performance of the detector with known textureprovides an upper bound for the one with unknown texture.Optimizing the upper bound provides a way to optimize thedetection performance. Therefore, we use the performancecharacteristics of the detector with known texture componentscomputing a utility function for adaptive energy allocation tooptimize the detection performance.

We observe that detection performance is optimized bymaximizing the detection probability for a fixed value ofprobability of false alarm. It is shown in [21] that, underasymptotic approximation, the non-centrality parameter andprobability of detection are positively proportional. Thereforewe maximize the non-centrality parameter with respect to theenergy parameters, βm, m = 1, . . . , M (see (3) and (4) forthe relation between the non-centrality parameter, and β’s).We also incorporate an energy constraint in the maximization,∑Mm=1 |βm|2 = E, such that the total transmitted energy

is the same, independent of the system configuration andenergy distribution. We define β = [β1, . . . , βM ]T , then the

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

optimization problem reduces to

β =argmax

β

[tr(Σ−1(Axφ)(Axφ)H) − μ(

M∑m=1

|βm|2 − E)

],

(16)where μ is the Lagrange multiplier. Without loss of generality,we assume E = 1, then after some algebraic manipulationsusing the structure of matrix A from (4), we show that thisoptimization problem further reduces to

β =argmax

β s.t βT β = 1

[βTQβ

], (17)

where Q is computed using Σ, A and x. This equation has aunique solution such that β is the eigenvector correspondingto the largest eigenvalue of the matrix Q.

Note that Σ, x are unknown in practice and we replacethem with their estimates.

V. NUMERICAL EXAMPLES

We present numerical examples using Monte Carlo (MC)simulations to illustrate our analytical results. We show thereceiver operating characteristics and improvement in detec-tion performance due to adaptive energy allocation for theMIMO system. The results are obtained from 2 ∗ 104 MCruns. We follow the scenario shown in Fig. 1. We assume thatour system is composed of M transmitters and N 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; the antennagains (Gtx and Grx) are 30dB; the signal frequency (fc) is1GHz. The angle between the transmitted signals a1 = a2 =... = aM = 10◦ and similarly between the received signalsb1 = ... = bN = 10◦. Hence Rm, m = 1, . . . , M , andRn, n = 1, . . . , N , in (1) are calculated accordingly. Inthis scenario, all the transmitters and receivers see the targetfrom different angles. Throughout the numerical examples, wechoose M = 2 and K = 40 pulses for each transmitted signal.

We choose the spatial covariance of the speckle componentsin a block diagonal form (Σ = blkdiag[Σ1, . . . , ΣM ]) due tothe assumption of low–cross-correlation signal transmission,see eqns. (2) and (3). Σm, m = 1, . . . , M , are positivedefinite N ×N matrices with entries Σm[i, j] = ρ

|i−j|s , with

i, j = 1, . . . , N . This form of covariance (ρs = 0.3) forMIMO radar is used in [7] to account for the correlationbetween the received signals at different receivers due to thesame transmitter. The target parameters x are chosen randomlyfor simulation purposes; that is, the entries are assigned as therealizations of a zero mean complex Gaussian random variablewith unit variance. Later, x is scaled to meet the desired signal-to-clutter ratio (SCR) conditions. We define the SCR similarto [9] in (18). Moreover, the shape parameter of the texturecomponent is chosen to be v = 4 (values between 3 and 9 areoften good choices for heavy tail fitting [16]).

SCR =1K

∑Kk=1(Axφ(k))H(Axφ(k))

E{u(k)}trΣ . (18)

We compare the receiver operating characteristics (ROC)of MIMO radar with conventional phased-array (Conv.) radarin Fig. 2(a). MIMO M × N and Conv. M × N stand forthe MIMO and conventional radar systems, respectively, withM transmitters and N receivers. The model of Conv. radar isobtained from (1) using the fact that all the channel coefficientsof the system (target RCS and distances of the radars to thetarget) are the same, since each transmitter and receiver pairsees the target from the same angle and distance. For fairnessof comparison, the total transmitted energy, E, is kept thesame for both Conv. and MIMO systems.

In Fig. 2(a), we assume for that spatial correlation ρs = 0.3,SCR=−10 dB, and the total energy is equally divided amongthe transmitters. In MIMO radar applications, the use of mul-tiple orthogonal waveforms results in 10log10(M) dB loss inSCR [1, Chapter8]. Then, for fair comparison, we set SCR=−7dB for Conv. system. As expected when the number of thereceivers, N , increases, the performances of both MIMO andConv. radar systems improve. However, MIMO radar alwaysoutperforms Conv. radar. The observed advantage of MIMOover Conv. radar stems from the diversity gain obtained bymultiple looks at the target. That is, MIMO radar systems havethe ability to exploit the spatial diversities, gaining sensitivityabout the RCS variations of the target to enhance systemperformance.

In Fig. 2(b), we demonstrate the improvement in the de-tection performance due to the adaptive energy allocation.We compute the receiver operating characteristics for MIMOradar when the total energy (E) is equally divided among thetransmitters (MIMO M ×N on the figure) and subsequentlywhen E is adaptively distributed among the transmitters usingour algorithm (MIMO M × N Adap. on the figure). Weassume SCR=-13dB (calculated for the equally distributedenergy scenario) which is different than the value chosen forFig. 2(a) to clearly demonstrate the effect of our adaptivealgorithm. The adaptive method optimally allocates the totalenergy to transmitters depending on the target RCS valuessuch that the signal-to-clutter ratio increases for the sametotal energy, E, and environment conditions. Increasing theSCR under the same target and environment conditions alsoincreases the performance.

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10−2

10−1

100

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Probability of false alarm (PFA)

Pro

bab

ility

of

det

ecti

on

(P

D)

ROC MIMO vs. Conv. Radar K=40

MIMO 2X6MIMO 2X4Conv. 2X6Conv. 2X4

(a)

10−2

10−1

100

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Probability of false alarm (PFA)

Pro

bab

ility

of

det

ecti

on

(P

D)

ROC MIMO with Adaptive Energy Distribution K=40 SCR=−13dB

MIMO 2X6 MIMO 2X4MIMO 2X6 Adap.MIMO 2X4 Adap.

(b)

Fig. 2. (a) Receiver operating characteristics of MIMO and conventionalphased-array radar (Conv.). (b) Receiver operating characteristics of MIMOradar with and without adaptive energy allocation.

VI. CONCLUDING REMARKS

We developed a statistical detector based on GLR fora MIMO radar system in compound-Gaussian clutter withinverse gamma distributed texture when the target and clutterparameters are unknown. First, we introduced measurementand statistical models within the GMANOVA framework andapplied the PX-EM algorithm to estimate the unknown pa-rameters. Using these parameters, we developed the statisticaldecision test detector. Moreover, we asymptotically approxi-mated the statistical characteristics of this decision test andused it to propose an algorithm to adaptively distribute thetotal transmitted energy among the transmitters. We usedMonte Carlo simulations and demonstrated the advantageof MIMO over conventional radar for target detection andthe detection performance enhancement due to our adaptiveenergy distribution algorithm. Our future work will focus onrobust MIMO detectors.

REFERENCES

[1] J. Li and P. Stoica, MIMO Radar Signal Processing. Wiley-IEEE Press,Oct. 2008.

[2] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valen-zuela, “Spatial diversity in radars - models and detection performance,”IEEE Trans. Signal Process., vol. 54, no. 3, pp. 823–838, Mar. 2006.

[3] A. Sheikhi and A. Zamani, “Temporal coherent adaptive target detec-tion for multi-input multi-output radars in clutter,” Radar Sonar andNavigation IET, pp. 86–96, Apr. 2008.

[4] K. Ward, C. Baker, and S. Watts, “Maritime surveillance radar. I. Radarscattering from the ocean surface,” in IEE Proc. F, vol. 137, Apr. 1990,pp. 51–62.

[5] F. Gini, A. Farina, and G. Foglia, “Effects of foliage on the formation ofK-distributed SAR imagery,” Signal Processing, vol. 75, pp. 161–171,Jun. 1999.

[6] M. Greco, F. Bordoni, and F. Gini, “X-band sea-clutter nonstationarity:Influence of long waves,” IEEE J. Ocean. Eng., vol. 29, no. 2, pp. 269–283, Apr. 2004.

[7] P. Sammartino, C. Baker, and H. Griffiths, “Adaptive MIMO radarsystem in clutter,” Radar Conference 2007 IEEE, pp. 276–281, Apr.2007.

[8] A. Balleri, A. Nehorai, and J. Wang, “Maximum likelihood estimationfor compound-Gaussian clutter with inverse gamma texture,” IEEETrans. Aerosp. Electron. Syst., vol. 43, no. 2, pp. 775–779, 2007.

[9] J. Wang, A. Dogandzic, and A. Nehorai, “Maximum likelihood estima-tion of compound-Gaussian clutter and target parameters,” IEEE Trans.Signal Process., vol. 54, no. 10, pp. 3884–3898, Oct. 2006.

[10] M. Akcakaya, M. Hurtado, and A. Nehorai, “MIMO radar detection oftargets in compound-Gaussian clutter,,” in Proc. 42nd Asilomar Conf.Signals, Syst. Comput., Pacific Groove, CA, USA, Oct. 2008 (invited).

[11] A. Dogandzic and A. Nehorai, “Generalized multivariate analysis ofvariance: A unified framework for signal processing in correlated noise,”IEEE Signal Process. Mag., vol. 20, pp. 39–54, Sep. 2003.

[12] C. Liu, D. Rubin, and Y. Wu, “Parameter expansion to accelerate EM:The PX-EM algorithm,” Biometrika, vol. 85, pp. 755–770, Dec. 1998.

[13] S. M. Kay, Fundamentals of Statistical Signal Processing: DetectionTheory. Upper Saddle River, NJ: Prentice Hall PTR, 1998.

[14] Y. Abramovich and G. Frazer, “Bounds on the volume and heightdistributions for the MIMO radar ambiguity function,” IEEE SignalProcessing Letters, vol. 15, pp. 505–508, 2008.

[15] P. Bickel and K. Doksum, Mathematical Statistics: Basic Ideas andSelected Topics, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2000.

[16] K. Lange, R. Little, and J. Taylor, “Robust statistical modeling using thet distribution,” Journal of the American Statistical Association, vol. 84,no. 408, pp. 881–896, 1989.

[17] E. J. Kelly and K. M. Forsythe, “Adaptive detection and parameterestimation for multidimensional signal models,” Lincoln Laboratory,MIT, Lexington, MA, Tech. Rep. 848, Apr. 1989.

[18] S. Sen and A. Nehorai, “Target detection in clutter by an adaptive OFDMradar,” IEEE Signal Processing Letters, to be published.

[19] T. W. Anderson, An Introduction to Multivariate Statistical Analysis,2nd ed. Wiley, Sep. 2003.

[20] E. F. Vonesh and V. Chincihilli, Linear and Nonlinear Models for theAnalysis of Repeated Measurements, 1st ed. Dekker, 1996.

[21] Y. Fujikoshi, “Asymptotic expansions of the non-null distributions ofthree statistics in GMANOVA,” Annals of the Institute of StatisticalMathematics, vol. 26, no. 1, pp. 289–297, Dec. 1974.

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