Correspondence. Adaptive MIMO Radar Design and Detection in Compound-Gaussian Clutter

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1 Correspondence Adaptive MIMO Radar Design and Detection in Compound-Gaussian Clutter Multiple-input multiple-output (MIMO) radars with widely separated transmitters and receivers are useful to discriminate a target from clutter using the spatial diversity of the scatterers in the illuminated scene. We consider the detection of targets in compound-gaussian clutter, describing heavy-tailed clutter distributions fitting high-resolution and/or low-grazing-angle radars in the presence of sea or foliage clutter. First we introduce a data model using the inverse gamma distribution to represent the clutter texture. Then we apply the parameter-expanded expectation-maximization (PX-EM) algorithm to estimate the clutter texture and speckle as well as the target parameters. We develop a statistical decision test using these estimates and demonstrate its statistical characteristics. Based on the statistical characteristics of this test, we propose an algorithm to adaptively distribute the transmitted energy among the transmitters and maximize the detection performance. We demonstrate the advantages of the MIMO setup and adaptive energy allocation in target detection in the presence of compound-gaussian clutter using Monte Carlo (MC) simulations. I. INTRODUCTION Multiple-input multiple-output (MIMO) radars with widely separated antennas exploit spatial diversity such as the spatial properties of the targets radar cross section (RCS). The RCSs of complex radar targets are quickly changing functions of the angle aspect. These target scintillations cause signal fading, which deteriorates the radar performance. When the transmitters and receivers are sufficiently separated, the target reflections are de-correlated, and hence each signal carries independent information. This spatial diversity improves the radar performance by mitigating these scintillations. These systems have the ability to improve the target parameter estimation and detection performance see [] and references therein. We model the clutter reflections at the receiver with a compound-gaussian model. This model represents the heavy-tailed clutter statistics that are distinctive of several scenarios, e.g., high-resolution Manuscript received January 5, 00; revised June 4, 00; released for publication September, 00. IEEE Log No. T-AES/47/3/ Refereeing of this contribution was handled by Y. Abramovich. This work was supported by the Department of Defense under the Air Force Office of Scientific Research MURI Grant FA , and ONR Grant N //$6.00 c 0 IEEE 00 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 3 JULY 0

2 and/or low-grazing-angle radars in the presence of sea or foliage clutter [, 3]. The compound-gaussian clutter e = p ux,whereu and X are the texture and speckle components of the compound model, respectively. The fast-changing X is a realization of a stationary zero mean complex Gaussian process, and the slow-changing u is modeled as a nonnegative real random process [4]. Gamma distribution for the texture is investigated in [5] for MIMO radar systems, leading to the well-known K clutter model. In this work, we specifically consider the inverse gamma distribution for texture component u, sincesimilarto its gamma distributed counterpart inverse gamma fits well with real clutter data [6]. Moreover this choice of distribution simplifies in a closed-form maximum likelihood solution for the joint target and clutter estimation as it follows a complex multivariate-t distribution [7]. We extend our analysis in [8] and develop an adaptive algorithm that distributes the total energy among the transmitters, exploiting the RCS sensitivity of the system and optimizing detection performance. Transmitting signals with different energies from different transmitters may change the total received power under the same environmental conditions. This is in contrast to the applications investigated for MIMO radar that distribute the total energy equally among the transmitters (see [, ch. 8 and 9]). Note also that in [8], we focus on the applications which require low pulse-repetition frequency (low-prf) radar. In this paper, we consider radar setup with higher PRF and formulate the statistical properties of the clutter accordingly. We developed a maximum likelihood estimation (MLE) algorithm for target and clutter parameter estimation under compound-gaussian clutter for single radar [7]. In this work we extend this method for MIMO setup and develop a statistical decision test based on generalized likelihood ratio (GLR) [9]. The rest of the paper is organized as follows. In Section II, we introduce our parametric measurement model under the generalized multivariate analysis of variance (GMANOVA) framework [0] for the MIMO system. In Section III, we first present a parameter-expanded expectation-maximization (PX-EM) algorithm [] to estimate the target and clutter parameters. Using these estimates, we then formulate a statistical decision test based on the GLR. In Section IV, we develop the adaptive algorithm, and in Section V, we use Monte Carlo (MC) simulations to analyze the detection performance and improvements due to adaptive design. Finally, we provide concluding remarks in Section VI. II. RADAR MODEL In this section, we develop measurement and statistical models for a MIMO radar system to detect a target in the range cell of interest (COI). Our goal is to present an algorithm, within a GMANOVA framework when the signal and noise parameters are unknown. A. Measurement Model We consider a two-dimensional (D) system with M transmitters and N receivers. Define (x Txm,y Txm ), m =,:::,M, and(x Rxn,y Rxn ), n =,:::,N, as the locations of the transmitters and receivers, respectively. We also assume a stationary point-like target located at (x 0,y 0 ) and having reflection coefficient values changing w.r.t. the angle aspect (e.g., multiple scatterers, which cannot be resolved by the transmitted signals, with (x 0,y 0 ) as the center of gravity) []. Define the complex envelope of the signal from the mth transmitter as ms m (t), m =,:::,M, such that j mj is the transmitted energy with P M R m= j mj = E (E is constant for any M) and T s js m (t)j dt =,m =,:::,M, witht s as the signal duration. We write the lowpass equivalent of the received signal at the nth receiver following []: MX r n (t)= nm ¾ nm ms m (t nm )e jã nm + e n (t) () m= where ¾ nm is the complex target reflection coefficient seen by the mth transmitter and nth receiver pair, such that the amplitude of ¾ nm corresponds to the RCS; nm = p G tx G rx =(4¼) 3 Rm R n is the channel parameter from the mth transmitter to the nth receiver, with G tx and G rx as the gains of the transmitting and receiving antennas, respectively; as the wavelength of the incoming signal; q R m = (x Txm x 0 ) +(y Txm y 0 ) and q R n = (x Rxn x 0 ) +(y Rxn y 0 ) as the distances from transmitter and receiver to target, respectively; nm =(R m + R n )=c, andc is the speed of the signal propagation in the medium; Ã nm =¼f c nm,withf c as the carrier frequency; e(t) is additive clutter noise. To enable the data separation at the receiver side due to the reflection of the multiple transmitted signals from the target, we assume low cross-correlation transmitted signals. The design of signals with these properties is a challenging research subject [], but to simplify the problem and demonstrate our methods and analysis, we assume that the assumed signal characteristics are met (this assumption is commonly made in MIMO radar, see [, ch. 8 and 9] and references therein.) Hence, we apply matched filtering and range gating, then obtain the output of the nth receiver corresponding to the ith CORRESPONDENCE 0

3 transmitter: r ni = i ni ¾ ni e jã ni + e ni () are IID, and we can write the conditional distribution for the observation Y in (4) as where r ni = R ni +T s ni r n (t)s i (t ni )dt, ande ni = R ni +T s ni e n (t) s i (t ni )dt. Since we apply range gating, we represent the range COI using the delay ni observed by the nth receiver and ith transmitter. Note that different transmitter receiver pairs have different delays corresponding to the same range COI. However since we know the array configuration and the range COI, we assume we have the knowledge of these delays. Moreover ni might be interpreted as the sampling time after the match filtering for the signal transmitted by the ith transmitter and received by the nth receiver representing the range COI, see for example [3]. Then, combining the received data corresponding to the transmitted signal s i (t) for one pulse, we obtain r i = A i x i + e i (3) where r i =[r i,:::,r Ni ] T, A i = idiag( i e jã i,:::, Ni e jã Ni ), x i =[¾ i,:::,¾ Ni ] T, e i =[e i,:::,e Ni ] T. We stack the receiver outputs corresponding to all the signals into an NM vector y = Ax + e (4) where y =[r T,:::rT M ]T, A = blkdiag(a,:::,a M ), x =[x T,:::xT M ]T, e =[e T,:::eT M ]T. We transmit K pulses and assume that the target is stationary during this observation time; then Y =[y() y() y(k)] NM K = AxÁ + E (5) where Á =[,:::] K,andE =[e() e() e(k)] NM K is the additive noise. B. Statistical Model In (5), we assume that x (target RCS values) is unknown deterministic. We consider the compound-gaussian distribution e(k)= p ux (k), k =,:::,K, to model the clutter with u and X (k) as the texture and speckle components, respectively; see [7] and references therein. The realizations of the fast-changing component X (k), k =,:::,K, are independent and identically distributed (IID) and follow a complex Gaussian distribution with zero mean and covariance. The texture is the slow-changing component; thus we consider it to be constant during the coherent processing interval (CPI), but changing from CPI to CPI according to a probability density function (pdf) of a nonnegative random variable [4, 4]. Therefore, e(k) j uk=,:::,k, KY p yju (y(k) j u) k= = KY expf [y(k) AxÁ(k)]H j¼u j k= [u ] [y(k) AxÁ(k)]g (6) where ( ) H denotes the Hermitian transpose. Observe that conditioned on u, with known A and Á and unknown x and, (5) is a GMANOVA model. We assume that w ==u follows the gamma distribution (consequently u follows the inverse gamma distribution) with unit mean and unknown shape parameter v>0asin[7];i.e., p w (w;v)= (v) vv w v exp[ vw] (7) where ( ) is the gamma function. Therefore, we consider x,, andv as the unknown parameters. III. DETECTION AND ESTIMATION ALGORITHMS We present in this section the MLE of the unknown parameters and the target detection test. We derive a statistical decision test based on GLR using the observed data likelihood function to determine the presence of a target in the COI. We choose between two hypotheses H 0 (target-free case) and H (target-present case) with the speckle covariance and the inverse texture shape parameter v as the nuisance parameters. We compute the GLR test by replacing the unknown parameters with their MLEs in the likelihood ratio test. Then, we reject H 0 in favor of H when GLR = p (Y; ˆx, ˆ, ˆv ) p 0 (Y; ˆ > (8) 0, ˆv 0 ) where p 0 ( ) andp ( ) are the observed data likelihood functions under H 0 and H ; ˆ 0 and ˆ are the MLEs of, andˆv 0 and ˆv are the MLEs of the shape parameter v under H 0 and H ; ˆx is the MLE of x under H ; is the detection threshold. We compute the observed data likelihood function (v + KNM) p (Y;x,,v)= j¼ j K (v)v KNM Ã + [y(k) AxÁ(k)] H [ ] k= [y(k) AxÁ(k)]=v! v KNM (9) 0 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 3 JULY 0

4 and under H 0, p 0 (Y;,v) isthesameas(9)with x = 0. Note that y, v, and(y,v) correspond to the observed, unobserved, and complete data, respectively. We compute the MLEs of the vector x, speckle covariance matrix, and texture distribution shape parameter v using the hierarchical data model presented in (6) and (7). Similar to our work in [7], we apply two iterative loops for the MLE computations: ) inner loop and ) outer loop. In the inner loop, first we introduce the PX-EM algorithm to obtain the MLEs ˆx and ˆ for a fixed v. ThePX-EM algorithm has the same convergence properties as the classical EM algorithm, but it outperforms the EM algorithm in global rate of convergence []. In the outer loop, we estimate v using the MLEs from the inner loop [7]. ) Inner Loop: PX-EM algorithm for inverse gamma texture. Recall that x,, andv are the unknown parameters. We first estimate μ = fx, g, assuming that v is known. We implement the PX-EM algorithm by adding a new unknown parameter ¹ w, the mean of w, tothisset;i.e.,μ = fx,,¹ w g. In this model, the maximization step performs a more efficient analysis by fitting the expanded parameter set []. Under this expanded model the pdf of w is p w (w;v,¹ w )= μ v v w v exp[ vw=¹ (v) ¹ w ]: w (0) Consider μ = R(μ )=fx, =¹ w g,wherer( ) isthe reduction function (many-to-one) from the expanded to the original space. Moreover, ¹ 0 w = is the null value such that the complete-data model is preserved. We define i and j as the inner and outer loop iteration indexes, respectively. Since the complete-data likelihood function belongs to an exponential family, the PX-EM algorithm reduces (see also [7], [5]) to first obtaining the conditional mean of the sufficient statistics using the unobserved data given the observed data, and then plugging these sufficient statistics values in the MLE expressions of the unknown variables. a) PX-E Step: Calculate the conditional expectation of the sufficient statistics under H, concentrated at ˆv (j),thejth iteration step estimate of v. Using the properties of the compound-gaussian model with inverse gamma distributed texture [6], we observe that w j Y follows a gamma distribution with ŵ (i) =E (i) wjy [w j Y; ˆμ ] ( ) =(ˆv (j) + KMN) ˆv (j) (i) + d(k, ˆμ ) ˆμ (i) k= () where = fˆx (i) (i), ˆ, ˆ¹(i) w = ˆ¹0 w =g is the estimate of (i) μ at the ith iteration and d(k, ˆμ )=[y(k) Aˆx (i) Á(k)] H [ ˆ (i) ] [y(k) Aˆx (i) Á(k)]. Then, T (i) = K T (i) = K T (i) 3 = K k= k= k= y(k)á(k) H ŵ (i) (a) y(k)y(k) H ŵ (i) (b) Á(k)Á(k) H ŵ (i) (c) T (i) 4 = ŵ (i) : (d) For the PX-M step, we define S (i) = T (i) T (i) (T(i) 3 ) (T (i) )H and Q (i) = A[A H (S (i) ) A] A H and obtain the maximum likelihood estimates similar to [0]. b) PX-M Step: ˆx (i+) =[A H (S (i) ) A] A H (S (i) ) T (i) (T(i) 3 ) (3a) ˆ (i+) =(S (i) ) +[I MN Q (i) (S (i) ) ]T (i) (T(i) 3 ) ˆ¹ (i+) w (T (i) )H [I MN Q (i) (S (i) ) ] H (3b) = T (i) 4 (3c) ˆ (i+) (i+) = ˆ = ˆ¹ w (i+) : (3d) Under H 0, we calculate ˆ 0 and ŵ 0 with x = 0 and update the sufficient statistics accordingly. ) Outer Loop: MLE of the shape parameter of theinversegammatexture. We compute ˆv (j+) by maximizing the concentrated observed data (y(k), k =,:::,K) log-likelihood function using the estimates from the PX-EM step. We denote ˆx (), ˆ () () 0,and ˆ as the estimates of x and obtained upon the convergence of the inner loop and compute ] T =argmax[lnp (Y, ˆx () [ˆv (j+) v, ˆ (),v)]: (4) ˆ () 0 Under H 0, we calculate ˆv (j+) 0 using x = 0 and in (4). The GLR test (8), computed upon convergence of (), (3), and (4) under H 0 and H,results in a complicated form which is difficult to analyze statistically. Therefore, we simplify it to the ratio of determinants of the covariance estimates under different hypotheses, (see (5)), which is also similar to the general form of the GLR test presented in [0], to analyze its statistical characteristics (see Section IV). First, for a fixed texture component, we compute the GLR test. Then we assume that the target is present only in the range COI and the texture is completely correlated over a few neighboring range cells. Since the texture is the slow changing component, this assumption is reasonable for high CORRESPONDENCE 03

5 resolution radar (see also [5]). Next, using the data from the target-free neighboring cells as the secondary data, we run the inner and outer loops of the estimation algorithm to compute the conditional mean of the texture component in () given the secondary data. We replace the texture component with its corresponding conditional mean value reducing the GLR test to jt () j = > 0 jt () Q () (S () ) T () (T () 3 ) (T () ) j (5) where j j is the determinant operator, and T (), T (), T () 3, S (),andq () are obtained using () in () in one step. That is, using the secondary data and the PX-EM algorithm the conditional mean of the texture component is computed as in (). Then using () in (), (5) is computed in one step. IV. ADAPTIVE DESIGN In this section we first demonstrate the asymptotic statistical characteristics of the detection test derived in Section III. Based on this result, we then construct a utility function for adaptive energy allocation to improve the detection performance. We determine the optimum transmitted energy by each transmitter according to this utility function. We define ŵs as the conditional mean value of the texture component obtained from () upon the convergence of () and (4) using the target-free secondary data. Note that since the unknown parameters and v of the secondary data belong to a canonical exponential family (since the complete data likelihood function belongs to an exponential family and could be written in canonical form), their estimates are consistent and hence the conditional mean value in (), computed given the secondary data converges to the minimum mean square error estimate (MMSE) of the texture component in probability (converges in probability) as the number of the observations increases (asymptotically) [5, Theorem 5..]. Moreover from [5, Theorem 5.5. and Theorem 5.5.3], we know that the MMSE asymptotically converges to MLE with probability (almost surely). Therefore, since MLE is consistent in probability, ŵs asymptotically converges to the true texture value w 0 in probability. The test in (5) is a function of ŵs y(k) for k =,:::,K. We observe that ŵs! w 0 and ŵs y(k)! w 0 y(k)=z(k) in probability, such that z(k)»n(0, ) and N (AxÁ, ) under H 0 and H, respectively. Then from [5, Appendix A.4.8], (ŵs y(k)) asymptotically converges to (z(k)) for k =,:::,K in distribution. Moreover following a similar approach taken for real Gaussian random variables in [7], [8], we find that (5) as a function of z(k) (complex version of Wilks lambda) as K!, K ln has a complex chi-square distribution with NM degrees of freedom under H 0. Since this distribution does not depend on the speckle covariance, in the limit (5) is a constant false-alarm rate (CFAR) test. Under H,asK!, K ln has a noncentral complex chi-square distribution with NM degrees of freedom. That is, K ln» CÂ NM (±) [7 9]. The noncentrality parameter is ± =tr( (AxÁ)(AxÁ) H ): (6) We observe that detection performance is optimized by maximizing the detection probability for a fixed value of probability of false alarm. It is shown in [9] that, under asymptotic approximation, the noncentrality parameter and probability of detection are positively proportional. Therefore we maximize the noncentrality parameter with respect to the energy parameters m, m =,:::,M (see (3) and (4) for the relation between the noncentrality parameter, and s). We also incorporate an energy constraint in the maximization, P M m= j mj = E, such that the total transmitted energy is the same, independent of the system configuration and energy distribution. We define =[,:::, M] T, then the optimization problem reduces to " Ã M!# X ˆ =argmax tr( (AxÁ)(AxÁ) H ) ¹ j mj E m= (7) where ¹ is the Lagrange multiplier. Without loss of generality, we assume E =, then after some algebraic manipulations using the structure of matrix A from (4), we show that this optimization problem further reduces to ˆ = argmax [ TQ ] (8) s.t T = where Q is computed as in the Appendix such that Q = Kdiag(q,:::,q MM ) = Kdiag((Ā x )H (Ā x ),:::,(ĀM x M )H M (ĀM x M )) (9) where (recalling from (5)) A = blkdiag(a,:::,a M ) A i = idiag( i e jã i,:::, Ni e jã Ni )= iā i x =[x T,:::xT M ]T x i =[¾ i,:::,¾ Ni ] T = blkdiag(,:::, M ); see also Section V for the covariance matrix assumption. Here q mm corresponds to the total received power at all the receivers due to the mth transmitter. We solve (8) to obtain the optimum power allocation. This equation has a unique solution such that ˆ is the eigenvector corresponding to the largest eigenvalue of the matrix Q. SinceQ is diagonal, the maximum eigenvalue is the maximum of q mm, 04 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 3 JULY 0

6 Fig.. MIMO antenna system with M transmitters and N receivers. m =,:::,M, (maximum total received power at all the receivers, maximum diagonal entry of Q). If q ii is the maximum eigenvalue, the eigenvector is u i = [ ], all zeros but at the ith location. This suggests that for optimum power allocation we transmit all the power from the ith transmitter. However, we modify this result and put minimum and maximum power constraints for each transmitter, this means that for adaptive design we transmit the maximum available power from the transmitter having the maximum eigenvalue, and transmit minimum power from the rest of the transmitters. This approach also provides the positivity constraint. Note that, x are unknown in practice and we replace them with their estimates for the adaptive design. V. NUMERICAL EXAMPLES We present numerical examples using MC simulations to illustrate our analytical results. We show the receiver operating characteristics and improvement in detection performance due to adaptive energy allocation for the MIMO system. The results are obtained from 0 4 MC runs. We follow the scenario shown in Fig.. We assume that our system is composed of M transmitters and N receivers, where the antennas are widely separated. The transmitters are located on the y-axis, whereas the receivers are on the x-axis; the target is 0 km from each of the axes; the antenna gains (G tx and G rx ) are 30 db; the signal frequency (f c ) is GHz. The angle between the transmitted signals a = a = = a M =0 ± and similarly between the received signals b = = b N = 0 ±.HenceR m, m =,:::,M, andr n, n =,:::,N, in () are calculated accordingly. In this scenario, all the transmitters and receivers see the target from different angles. Throughout the numerical examples, we choose M =andk = 40 pulses for each transmitted signal. We choose the spatial covariance of the speckle components in a block diagonal form ( = blkdiag[,:::, M ]) due to the assumption of low cross-correlation signal transmission; see () and (3). m, m =,:::,M, are positive definite N N matrices with entries m [i,j]=½ ji jj s,withi,j =,:::,N. This form of covariance for MIMO radar is used in [5] to account for the correlation between the received signals at different receivers due to the same transmitter. The target parameters x are chosen randomly for simulation purposes; that is, the entries are assigned as the realizations of a zero mean complex Gaussian random variable with unit variance. Later, x is scaled to meet the desired signal-to-clutter ratio (SCR) conditions. We define the SCR similar to [7] in (0). Moreover, the shape parameter of the texture component is chosen to be v =4(values between 3 and 9 are often good choices for heavy tail fitting [6]). SCR = P K k= (AxÁ(k))H (AxÁ(k)) : (0) K Efu(k)gtr In Fig. (a), MIMO M N and Conv. M N stand for the MIMO and conventional radar systems, respectively, with M transmitters and N receivers. The model of Conv. radar is obtained from () similar to [0] using the fact that all the channel coefficients of the system (target RCS and distances of the radars to the target) are the same, since each transmitter and receiver pair sees the target from the same angle and distance. For fairness of comparison, the total transmitted energy E is kept the same for both Conv. and MIMO systems. In Fig. (a), we assume that spatial correlation ½ s =0:0 (low correlation due to widely separated setups), SCR = 0 db, and the total energy is equally divided among the transmitters. In MIMO radar applications, the use of multiple orthogonal waveforms results in 0log 0 (M) dblossinscr [, ch. 8]. Then, for fair comparison, we set SCR = 7 db for Conv. system. The observed advantage of MIMO over Conv. radar stems from the diversity gain obtained by multiple looks at the target. That is, MIMO radar systems have the ability to exploit the spatial diversities, gaining sensitivity about the RCS variations of the target to enhance system performance. In Fig. (b), we demonstrate the improvement in the detection performance due to the adaptive energy allocation. We compute the receiver operating characteristics for MIMO radar when the total energy (E) is equally divided among the transmitters (MIMO M N on the figure) and subsequently when E is adaptively distributed among the transmitters using our algorithm (MIMO M N Adap. on the figure). The adaptive method optimally allocates the total energy to transmitters depending on the target RCS values such that the SCR increases for the same total energy E and environment conditions. Increasing the SCR under the same target and environment conditions also increases the performance. VI. CONCLUDING REMARKS We developed a statistical detector based on GLR for a MIMO radar system in compound-gaussian CORRESPONDENCE 05

7 Fig.. (a) Receiver operating characteristics of MIMO and conventional phased-array radar (Conv.). (b) Receiver operating characteristics of MIMO radar with and without adaptive energy allocation. clutter with inverse gamma distributed texture when the target and clutter parameters are unknown. First we introduced measurement and statistical models within the GMANOVA framework and applied the PX-EM algorithm to estimate the unknown parameters. Using these parameters, we developed the statistical decision test detector. Moreover, we asymptotically approximated the statistical characteristics of this decision test and used it to propose an algorithm to adaptively distribute the total transmitted energy among the transmitters. We used MC simulations and demonstrated the advantage of MIMO over conventional radar for target detection and the detection performance enhancement due to our adaptive energy distribution algorithm. Our future work will focus on robust MIMO detectors. 06 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 3 JULY 0

8 APPENDIX In this Appendix, we compute the matrix Q of (8). Since Á =[,:::] K, tr( (AxÁ)(AxÁ) H )=K(Ax) H (Ax): () Recall from (5) that A = blkdiag(a,:::,a M ) A i = idiag( i e jã i,:::, Ni e jã Ni )= iā i x =[x T,:::xT M ]T x i =[¾ i,:::,¾ Ni ] T = blkdiag(,:::, M ); see also Section V for the covariance matrix assumption. Then K(Ax) H (Ax)=K MX m= m (Ām x m )H m (Ām x m ), = HQ () where Q = Kdiag((Ā x )H (Ā x ),:::,(ĀM x M )H M (ĀM x M )). REFERENCES MURAT AKÇAKAYA ARYE NEHORAI Dept. of Electrical and Systems Engineering Washington University in St. Louis One Brookings Drive St. Louis, MO (muratakcakaya@ese.wustl.edu and nehorai@ese.wustl.edu) [] Li, J. and Stoica, P. MIMO Radar Signal Processing. Hoboken, NJN: Wiley IEEE Press, Oct [] Ward, K., Baker, C., and Watts, S. Maritime surveillance radar. I. Radar scattering from the ocean surface. IEE Proceedings Pt. F, 37 (Apr. 990), 5 6. [3] Gini, F., Farina, A., and Foglia, G. Effects of foliage on the formation of K-distributed SAR imagery. Signal Processing, 75 (June 999), 6 7. [4] Greco, M., Bordoni, F., and Gini, F. X-band sea-clutter nonstationarity: Influence of long waves. IEEE Journal of Oceanic Engineering, 9, (Apr. 004), [5] Sammartino, P., Baker, C., and Griffiths, H. Adaptive MIMO radar system in clutter. In Proceedings of the IEEE Radar Conference 007, Apr. 007, [6] Balleri, A., Nehorai, A., and Wang, J. Maximum likelihood estimation for compound-gaussian clutter with inverse gamma texture. IEEE Transactions on Aerospace and Electronic Systems, 43, (007), [7] Wang, J., Dogandzic, A., and Nehorai, A. Maximum likelihood estimation of compound-gaussian clutter and target parameters. IEEE Transactions on Signal Processing, 54, 0(Oct. 006), [8] Akcakaya, M., Hurtado, M., and Nehorai, A. MIMO radar detection of targets in compound-gaussian clutter. In Proceedings of the 4nd Asilomar Conference on Signals, Systems and Computers, PacificGrove,CA,Oct. 008, invited paper. [9] Kay,S.M. Fundamentals of Statistical Signal Processing: Detection Theory. Upper Saddle River, NJ: Prentice-Hall PTR, 998. [0] Dogandzic, A. and Nehorai, A. Generalized multivariate analysis of variance: A unified framework for signal processing in correlated noise. IEEE Signal Processing Magazine, 0 (Sept. 003), [] Liu, C., Rubin, D., and Wu, Y. Parameter expansion to accelerate EM: The PX-EM algorithm. Biometrika, 85 (Dec. 998), [] Abramovich, Y. and Frazer, G. Bounds on the volume and height distributions for the MIMO radar ambiguity function. IEEE Signal Processing Letters, 5 (008), [3] Ward, J. Space-time adaptive processing for airborne radar. Lincoln Laboratory, MIT, Lexington, MA, Technical Report 05, Apr [4] Gini, F. and Greco, M. Texture modelling, estimation and validation using measured sea clutter data. IEE Proceedings Radar, Sonar and Navigation, 49, 3 (June 00), 5 4. [5] Bickel, P. and Doksum, K. Mathematical Statistics: Basic Ideas and Selected Topics (nd ed.). Upper Saddle River, NJ: Prentice-Hall, 000. [6] Lange, K., Little, R., and Taylor, J. Robust statistical modeling using the t distribution. Journal of the American Statistical Association, 84, 408 (989), [7] Anderson, T. W. An Introduction to Multivariate Statistical Analysis (nd ed.). Hoboken, NJ: Wiley, Sept [8] Vonesh, E. F. and Chincihilli, V. Linear and Nonlinear Models for the Analysis of Repeated Measurements (st ed.). New York: Marcel Dekker, 996. [9] Fujikoshi, Y. Asymptotic expansions of the non-null distributions of three statistics in GMANOVA. Annals of the Institute of Statistical Mathematics, 6, (Dec. 974), [0] Fishler, E., et al. Spatial diversity in radars Models and detection performance. IEEE Transactions on Signal Processing, 54, 3(Mar. 006), CORRESPONDENCE 07