Amultiple-input multiple-output (MIMO) radar uses multiple

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1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 6, JUNE Iterative Generalized-Likelihood Ratio Test for MIMO Radar Luzhou Xu Jian Li, Fellow, IEEE Abstract We consider a multiple-input multiple-output (MIMO) radar system both the transmitter receiver have multiple well-separated subarrays with each subarray containing closely spaced antennas. Because of this general antenna configuration, both the coherent processing gain the spatial diversity gain can be simultaneously achieved. We compare several spatial spectral estimators, including Capon APES, for target detection parameter estimation. We introduce a generalizedlikelihood ratio test (GLRT) a conditional generalizedlikelihood ratio test (cglrt) for the general antenna configuration. Based on GLRT cglrt, we then propose an iterative GLRT (iglrt) procedure for target detection parameter estimation. Via several numerical examples, we show that iglrt can provide excellent detection estimation performance at a low computational cost. Index Terms Adaptive arrays, block-diagonal growth-curve (BDGC) model, detection, generalized-likelihood ratio test (GLRT), growth-curve (GC) model, localization, multiple-input multipleoutput (MIMO) radar, parameter estimation. I. INTRODUCTION Amultiple-input multiple-output (MIMO) radar uses multiple antennas to simultaneously transmit several linearly independent waveforms. It also uses multiple antennas to receive the reflected signals. It has been shown that by exploiting this waveform diversity, MIMO radar can overcome performance degradations caused the radar cross section (RCS) fluctuations [1] [4], achieve flexible spatial transmit beampattern design [5], [6], provide high-resolution spatial spectral estimates [7] [17], significantly improve the parameter identifiability [18]. The statistical MIMO radar, studied in [1] [4], aims at resisting the scintillation effect encountered in radar systems. It is well known that the RCS of a target, which represents the amount of energy reflected from the target toward the receiver, changes rapidly as a function of the target aspect [19], the locations of the transmitting receiving antennas. The Manuscript received May 3, 2006; revised August 10, The associate editor coordinating the review of this manuscript approving it for publication was Dr. Erchin Serpedin. This work was supported in part by the Office of Naval Research under Grant No. N , the Defense Advanced Research Projects Agency under Grant No. HR , the National Science Foundation under Grant No. CCF Opinions, interpretations, conclusions, recommendations are those of the authors are not necessarily endorsed by the United States Government. The authors are with the Department of Electrical Computer Engineering, University of Florida, Gainesville, FL USA ( xuluzhou@dsp.ufl. edu; li@dsp.ufl.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TSP target scintillation causes severe degradations in the target detection estimation performance of the radar. By spacing the transmit antennas, which transmit linearly independent signals, far away from each other, a spatial diversity gain can be obtained as in the MIMO wireless communications to this scintillation effect [1] [4]. Flexible transmit beampattern designs are investigated in [5] [6]. Different from the statistical MIMO radar, the transmitting antennas are closely spaced. The authors in [5] [6] show that the waveforms transmitted via closely spaced antennas can be optimized to obtain several transmit beampattern designs with superior performance. For example, the covariance matrix of the waveforms can be optimized to maximize the power around the locations of interest also to minimize the cross correlation of the signals reflected back to the radar by these targets, thereby significantly improving the performance of the adaptive MIMO radar techniques. Due to the significantly larger number of degrees-of-freedom of a MIMO system, improved transmit beampatterns can be achieved with a MIMO radar than with its phased-array counterpart. In [9], a MIMO radar technique is suggested to improve the radar resolution. The idea is to transmit orthogonal coded waveforms by antennas to receive the reflected signals by antennas. At each receiving antenna output, the signal is matched-filtered using each of the transmitted waveforms to obtain channels, the dataadaptive Capon beamformer [20] is applied. It is proven in [9] that the beampattern of the proposed MIMO radar is obtained by the multiplication of the transmitting receiving beampatterns; hence, it has high resolution. However, [9] considers only single-target scenarios. A MIMO radar scheme is considered in [16] [17] that can deal with the presence of multiple targets. Similar to some of the MIMO radar approaches [5] [15], linearly independent waveforms are transmitted simultaneously via multiple antennas. Due to the different phase shifts associated with different propagation paths from transmitting antennas to targets, these independent waveforms are linearly combined at the targets with different phase factors. As a result, the signal waveforms reflected from different targets are linearly independent, allowing the direct application of many adaptive techniques to achieve higher resolution interference rejection capability. Several adaptive nonparametric algorithms, some of which also model steering vector errors, are presented in [16] [17]. The MIMO radars discussed above can be grouped into two classes according to their antenna configurations. One class is the conventional radar array, in which both the transmitting receiving antennas are closely spaced for coherent transmission detection [5] [17]. The class other is the diverse antenna X/$ IEEE

2 2376 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 6, JUNE 2007 configuration, the antennas are separated far away from each other to achieve spatial diversity gain [1] [4]. To exploit the benefits of both schemes, we consider a general antenna configuration in this paper, i.e., both the transmitting receiving antenna arrays consist of several well-separated subarrays with each subarray containing closely spaced antennas. We establish the growth curve models [21] [23] devise several estimators for the proposed MIMO radar system. The remainder of this paper is organized as follows. Section II presents the MIMO radar signal model. In Section III, we discuss two adaptive spatial spectral estimators, including Capon [20] APES [24]. In Section IV, we introduce a generalizedlikelihood ratio test (GLRT) a conditional generalized-likelihood ratio test (cglrt) then propose an iterative GLRT (iglrt) procedure for target detection parameter estimation. Numerical examples are provided in Section V. II. SIGNAL MODEL Consider a narrowb MIMO radar system with subarrays for transmitting receiving, respectively. The th transmit th receive subarrays have, respectively, closely spaced antennas. We assume that the subarrays are sufficiently separated, hence, for each target, its RCSs for different transmit receive subarray pairs are statistically independent of each other. Let be the steering vectors of the th transmitting subarray the th receiving subarray, respectively, denotes the target location parameter, for example, its angular location. Let the rows of be the waveforms transmitted from the antennas of the th transmit subarray. We assume that the arrival time is known. Then, the signal received by the th subarray due to the reflection of the target at can be written as are the total numbers of receive transmit antennas, respectively, is the number of data samples of the transmitted waveforms, denotes the transpose operator, is a block-diagonal matrix with being its diagonal submatrices. Then, (1) can be rewritten in the growth-curve (GC) model [21], [22] the th element of the matrix is, is defined similarly to in (2), the rows of are the reflected waveforms by the target at location, i.e., Note that when, the signal model in (6) reduces to the MIMO radar model in [16] [17], as when, it reduces to the diversity data model in [1] [3]. Based on this data model, we hereafter derive two classes of nonparametric methods, i.e., spatial spectral estimation GLRT, for target detection localization. III. SPATIAL SPECTRAL ESTIMATORS We discuss two spatial spectral estimators for the proposed MIMO radar system. We use these methods to estimate the complex amplitudes in for each of interest from the observed data matrix. The Frobenius norm of the estimated forms a spatial spectrum in the 1-D case or a radar image in the 2-D case. We can then estimate the number of targets their locations by searching for the peaks in the estimated spectrum (or image). A simple way to estimate in (6) is via the least-squares (LS) method, as follows: (6) (7) (8) is the complex amplitude proportional to the RCS for the th receive transmit subarray pair for the target at the location. The matrix denotes a residual term for the unmodeled noise, e.g., interferences from targets other than at other range bins, intentional or unintentional jamming. For notational simplicity, we will not show explicitly the dependence of on. Let (1) (2) (3) (4) (5) denotes the conjugate transpose. However, as any other data-independent beamforming-type method, the LS method suffers from high sidelobes low resolution. In the presence of strong interference jamming, the method completely fails to work. Hence, we discuss two robust adaptive spatial spectral estimation approaches that offer higher resolution interference suppression capabilities. A. Capon The Capon estimator for in (6) consists of two main steps [20], [22], [25]. The first step is a generalized Capon beamforming step. The second step is an LS estimation step, which involves a matched filter to the known waveform. The generalized Capon beamformer can be formulated as follows: subject to (9) is the weighting matrix used to achieve noise, interference, jamming suppression while keeping the

3 XU AND LI: ITERATIVE GENERALIZED-LIKELIHOOD RATIO TEST FOR MIMO RADAR 2377 desired signal undistorted, denotes the trace of a matrix, (10) is the sample covariance matrix with being the number of data samples. Solving the optimization problem in (9), we have (11) By using (6) (11), the output of the Capon beamformer can be written as By applying the LS method to (12), the Capon estimate of follows: (12) (13) B. APES The generalized APES method is a straightforward extension of the APES method [24], [26], which can be formulated as subject to (14) denotes the Frobenius norm, is the weighting matrix. Minimizing the cost function in (14) with respect to yields Then, the optimization problem reduces to with (15) subject to (16) IV. GENERALIZED-LIKELIHOOD RATIO TEST GLRT has been used widely for target detection localization. We derive below a GLRT cglrt for the proposed MIMO radar, then propose an iglrt procedure for improved performance. A. Generalized-Likelihood Ratio Test Throughout this section, we assume that the columns of the interference noise term in (6) are independently identically distributed (i.i.d.) circularly symmetric complex Gaussian rom vectors with mean zero an unknown covariance matrix. Consider the following hypothesis test problem: (20) i.e., we want to test if there exists a target at location or not. Similarly to [27] [28], we define a generalized-likelihood ratio (GLR) as follows: (21) (, 1) is the pdf of under the hypothesis. From (21), we note that the value of the GLR,, lies between 0 1. If there is a target at a location of interest, we have, i.e., ; otherwise. Under Hypothesis,wehave (22) denotes the determinant of a matrix. Maximizing (24) with respect to yields: (23) (17) For notional simplicity, we have omitted the dependence of on. Solving the optimization problem of (16) gives the generalized APES beamformer weighting matrix: Inserting (18) in (15), we readily get the APES estimate of as (18) is defined in (10). Similarly, under Hypothesis Maximizing (24) with respect to, we have yields (24) (19) (25) Interestingly, we note that (19) has the same form as the ML estimate in [21] [22]. However, the APES estimate is derived based on the beamforming method,, unlike the ML in [21] [22], it does not need a probability density function (pdf) of. Hence, the optimization problem in the denominator of (21) reduces to (26)

4 2378 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 6, JUNE 2007 Following [21] [22] dropping the dependence of,, on for notional convenience, we have Substituting (23) (30) into (21) yields (31) We remark that when there are multiple targets, the number of targets (say ) are known a priori, the GLRT in (31) can be extended to a multivariate counterpart by considering the following hypothesis testing problem: (32) that when (27) (28) is defined in (17). To get (27), we have used the fact [29], the equality in (28) holds equates to the APES estimate in (19). Note that As a parametric method, this multivariate GLRT can provide better target detection parameter estimation performance than its univariate counterpart. However, the multivariate GLRT is computationally intensive because it needs to search in the -dimensional parameter space. Moreover, the number of targets is hardly known a priori in practice. We propose below an iglrt, which requires only a 1-D search (like the univariate GLRT) but provides a target detection parameter estimation performance close to the multivariate GLRT. B. Conditional Generalized-Likelihood Ratio Test Before we describe the iglrt procedure, we first consider the following hypothesis testing problem, referred to as the cglrt. Suppose that we know that there are targets at the locations, we want to determine if there are any additional targets. This problem can be formulated in the following hypothesis testing problem: (33) Note that both the equations in (33) are in the form of the blockdiagonal growth-curve (BDGC) model studied in [23]. For convenience, we rewrite (33) as (34) From (25), (28), (29), it follows that (29) (30) (35) (36) (37) (38) (39) (40)

5 XU AND LI: ITERATIVE GENERALIZED-LIKELIHOOD RATIO TEST FOR MIMO RADAR 2379 Similarly, we define a cglr as follows: (41) is the pdf of under the hypothesis, is the covariance matrix of the columns of. We first consider the optimization problem of the numerator in (41). Maximizing with respect to yields Therefore, we get (49) Hence, the optimization problem reduces to (42) be the eigenvalues of the matrix in (50), which sat-. Through some matrix manipulations, we Let isfy that obtain (50) with (43) The optimization problem in (43) does not appear to admit a closed-form solution because is a block-diagonal matrix. Herein, we adopt a technique used in [30] to approximate a closed-form solution. Note that (51) (44) (45) denotes the vectorization operator (stacking the columns of a matrix on top of each other), respectively, is the Hermitian square root of. In (51), we have omitted the high-order terms of for the approximation. Hence, for a large number of data samples, the optimization problem in (43) can be approximated as: with (52) (46) with denoting the generalized matrix inverse. Consider the idempotent matrices. Assume that the number of data samples is large enough, i.e.,. Note that is an matrix. Hence, we have rank rank (47) with rank denoting the rank of a matrix. Then, we have (48) To solve the above optimization problem, we introduce below two partitioned matrix operations two lemmas without proof (see [23] for the detailed proofs). Definition 1: Let be a partitioned matrix with being the th submatrix of. Then, the blockdiagonal vectorization operation is defined by (53) Definition 2: Let be two partitioned matrices with conformal partitioning with being the th

6 2380 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 6, JUNE 2007 submatrices of, respectively. Then, the generalized Khatri Rao product [31] is defined by we have used Lemmas 1 2, the equality holds when (54) denotes the th submatrix of the given partitioned matrix denotes the Kronecker matrix product. Note that both the block-diagonal vectorization the generalized Khatri Rao product are defined based on a particular matrix partitioning, i.e., different matrix partitionings will lead to different results. Throughout this paper, the partitioned matrix operation are all based on the partitionings in (35) (40). It is also worth pointing out that matrix partitioning may be inherited through matrix operations. For example, for the partitioned matrix given by (37), is a partitioned matrix with the th submatrix being. Lemma 1: Let be two partitioned matrices with block rows block columns, let be a block-diagonal matrix with compatible dimensions conformal partitioning with. Then (55) By using (42), (44), (51), (60), it follows that Similarly, we have (61) (62) (63) Lemma 2: Let be two partitioned matrices with one block row block columns, let be two partitioned matrices with one block row block columns with compatible dimensions with, respectively. Then Now, let (56) (64) are defined similarly to in (46) in (63), respectively. Substituting (62) (64) into (41) yields the conditional GLR (65) i.e., By using Lemma 1 (with ), we obtain Hence (57) (58) (59) C. Iterative Generalized-Likelihood Ratio Test The basic idea of the iglrt is to detect localize targets sequentially. In each step of the iteration, the results from the previous iterations steps are exploited for the detection localization of new targets by calculating cglr. Specifically, we first perform GLRT to get the location of the dominant target, the following targets are detected localized by using cglrt conditioned on the most recently available estimates. The detailed steps of iglrt are described in Table I. Once the locations of the targets are determined, the amplitudes of the reflected signals can be estimated by using the AML estimator in [23] (60) (66),, are defined similar to,, in (37), (38), (46), respectively. We note that Step III of the above iglrt algorithm actually minimizes the function with respect to by using the cyclic minimization (CM) technique [32]. Under a mild condition, i.e.,, wehave.

7 XU AND LI: ITERATIVE GENERALIZED-LIKELIHOOD RATIO TEST FOR MIMO RADAR 2381 TABLE I IGLRT ALGORITHM Fig. 1. Cumulative density functions of the Cramér Rao bounds for (a) (b). Furthermore, we know that the CM algorithm monotonically decreases the cost function. Hence, the iglrt algorithm is convergent. When is the true number of targets, iglrt reduces to an approximate (parametric) maximum-likelihood estimator. As we will show via numerical examples, the mean-square error (MSE) of the estimate of iglrt approaches the corresponding Cramér Rao bound (CRB) for a large number of data samples. On the other h, we note that iglrt needs only a 1-D search, hence, is computationally efficient. V. NUMERICAL EXAMPLES In this section, we first compare the CRBs for MIMO radars with different antenna configurations then present the detection localization performance of the proposed methods. A. Cramér Rao Bound We first study the CRB under various antenna configurations. Consider a MIMO radar system with antennas for transmitting receiving. We assume that the receiving transmitting antennas are grouped into multiple subarrays (each being a uniform linear array with half-wavelength spacing between adjacent elements): MIMO Radar A: one subarray with eight antennas for transmitting receiving; MIMO Radar B: two subarrays each with four antennas for transmitting, one subarray with eight antennas for receiving; MIMO Radar C: eight subarrays each with one antenna for transmitting, one subarray with eight antennas for receiving; MIMO Radar D: two subarrays each with four antennas for transmitting receiving. We assume that the transmitted waveforms are linearly orthogonal to each other the total transmitted power is fixed to be 1, i.e.,. We consider a scenario in which targets are located at 40, 4 0, the elements of are i.i.d. circularly symmetric complex Gaussian rom variables with zero mean unit variance. There is a strong jammer at 10 with amplitude 100, i.e., 40 db above the reflected signals. The received signal has 128 snapshots is corrupted by a zero-mean spatially colored Gaussian noise with an unknown covariance matrix. The th element of the unknown noise covariance matrix is SNR. Fig. 1(a) (b) shows the cumulative density functions (CDFs) of the CRBs for MIMO radar with various antenna configurations when SNR 20 db. (The CRB of is similar to that of, hence, is not shown.) The CDFs are obtained by 2000 Monte Carlo trials. In each trial, we generate the elements of romly then calculate the corresponding CRBs using (85) given by in the Appendix. For comparison purposes, we also provide the CDF of the phased-array (single-input multiple-output) counterpart, i.e., the special case of the above MIMO radar when, with the same total transmission power. As expected, the MIMO radar provides much better performance than the phased-array counterpart. Due to the fading effect of the elements of, the CRB of MIMO Radar A varies within a large range. Within a 95% confidence interval (i.e., when CDF varies from 2.5% to 97.5%), its CRB for varies approximately from 5 to 5. The CRBs for MIMO Radar C varies within a small range. To evaluate the CRB performance, we define an outage CRB [4] for a given probability, denoted by CRB,as CRB CRB (67) Fig. 2(a) (d) shows the outage CRB CRB of, as functions of SNR. As expected, the SNR gains depend on the probability. As we can see, when, MIMO Radar C outperforms the other radar configurations provides around 20-dB 12-dB improvements in SNR compared with the phased-array MIMO Radar A, respectively. On the other h, Fig. 2(d) shows that MIMO Radars A B outperform others when. B. Target Detection Localization We focus next on MIMO Radar B, i.e., a MIMO radar system with two subarrays (each with four antennas) for transmitting one subarray (with eight antennas) for receiving. We first consider a scenario in which three targets are located at 40, 20 0 with the corresponding

8 2382 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 6, JUNE 2007 Fig. 2. Outage CRB versus SNR. (a) CRB for, (b) CRB for, (c) CRB for, (d) CRB for. Fig. 4. Spatial spectra, GLR cglr pseudospectra, when = 040, = 04, = 0. (a) Capon, (b) APES, (c) GLRT, (d) iglrt. Fig. 3. Spatial spectra, GLR cglr pseudospectra,when = 040, = 020, = 0. (a) LS, (b) Capon, (c) APES, (d) GLRT, (e) iglrt. elements in,, being fixed to 2, 2, 1, respectively. The other simulation parameters are the same as for Fig. 1. The Frobenius norm of the spatial spectral estimates of versus, obtained by using LS, Capon, APES, are given by Fig. 3(a) (c). For comparison purposes, we show the true spatial spectrum via dashed lines in these figures. As seen from Fig. 3, the LS method suffers from high sidelobes poor resolution problems. Due to the presence of the strong jamming signal, the LS estimator fails to work properly. Capon APES possess excellent interference jamming suppression capabilities. The Capon method gives very narrow peaks around the target locations. However, the Capon estimates of,, are biased downward. The APES method gives more accurate estimates around the target locations but its resolution is worse than that of Capon. Note that in Fig. 3(a) (c), a false peak occurs at 10 due to the presence of the strong jammer. Although the jammer waveform is statistically independent of the waveforms transmitted by the MIMO radar, a false peak still exists since the jammer is 40 db stronger than the weakest target the number of data samples is finite. Fig. 3(d) (e) gives the GLRT, the iglrt results, as functions of the target location parameter. For convenience, in Fig. 3(e), we have included all cglr functions obtained by iglrt, each indicating one target. As expected, we get high GLRs (cglrs) at the target locations low GLRs (cglrs) at other locations, including the jammer location. By comparing the GLR with a threshold, the false peak due to the strong jammer can be detected rejected, a correct estimate of the number of the targets can be obtained by both methods. Next, we consider a more challenging example, is 4, while all the other simulation parameters are the same as before. As shown in Fig. 4(c), the APES, Capon, GLRT methods fail to resolve the two closely spaced targets at 4 0. On the other h, iglrt gives well-resolved peaks around the true target locations. To illustrate the procedure of the iglrt algorithm, we give the GLR cglrs obtained in Steps I II of iglrt in Fig. 5(a) (d). Fig. 5(a) (b) shows the GLR the cglr, respectively, is the estimated location of target 1 from. As we can see, there is no peak at around 0 in both figures. Yet, a clear peak is shown in in

9 XU AND LI: ITERATIVE GENERALIZED-LIKELIHOOD RATIO TEST FOR MIMO RADAR 2383 Fig. 8. Outage CRB MSE versus number of data samples for (a) (b). Fig. 5. GLR cglr Pseudo-Spectra obtained in Steps I II of iglrt, when = 040, = 04, = 0. (a) (), (b) (j^ ), (c) (j^ ; ^ ), (d) (j^ ; ^ ; ^ ). Fig. 6. Cumulative density functions of the CRBs MSEs for (a) (b). MSEs are very close to the corresponding the CRBs decrease almost linearly as SNR increases. Fig. 8 gives the outage MSE CRB as functions of when SNR db. As expected, the outage MSE approaches the corresponding CRB as increases. VI. CONCLUSION We have considered a multiple-input multiple-output (MIMO) radar system with a general antenna configuration that can be used to achieve both the coherent processing gain the spatial diversity gain. We have first introduced several spatial spectral estimators, including Capon APES, for target detection parameter estimation. By using our results on the growth curve models, we have provided a generalized-likelihood ratio test (GLRT) a conditional generalized-likelihood ratio test (cglrt), then proposed an iterative GLRT (iglrt) procedure for the MIMO radar system. Via several numerical examples, we have shown that the iglrt method can provide excellent target detection parameter estimation performance at a low computational cost. APPENDIX CRAMÉR RAO BOUND Consider a MIMO radar system with received signal can be written as targets. Then, the Fig. 7. Outage CRB MSE versus SNR for (a) (b). Fig. 5(c), which indicates the existence location of target 3. The cglr in Fig. 5(d) shows that no additional target exists other than the targets at,,. In other words, the iglrt method correctly estimates the number of targets to be 3. Now, we consider the elements in,, as i.i.d complex Gaussian rom variables with mean zero unit variance. The other parameters are the same as those in Fig. 6. Fig. 6(a) (b) presents the CDFs of the MSEs of as well as the CRBs, when SNR 20 db 128. As we can see, the MSEs of the iglrt are very close to the corresponding CRBs. Fig. 7(a) (b) shows the outage MSE CRB when as functions of SNR when 128. Again, the Let (68) (69) (70) (71) denote the real imaginary parts, respectively. Assume that the columns of are independent identically distributed (i.i.d.) circularly symmetric complex Gaussian rom vectors with zero-mean an unknown covariance matrix.

10 2384 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 6, JUNE 2007 Using the same argument as in [22, App. A], we know that the unknowns in will not affect the CRBs of. Hence, we need only to calculate the following Fisher information matrix (FIM) with respect to,,, as follows: is a matrix with its element being FIM (72) denotes the FIM with respect to. Note that (73) (74) (75) Inserting (74) into (73) after some matrix manipulations, we obtain Similarly, we have (78) (79) (80) (81) (82) are both partitioned matrices with blocks with their submatrices being, respectively (83) (84) with denoting the Kronecker product. Substituting (76) (82) into (73), after some matrix manipulations, we get is the covariance matrix of the trans- mitted waveforms. Hence (76) CRB (85) (77) CRB CRB (86)

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Kelly, Finite-sum expressions for signal detection probabilities, Lincoln Laboratory, MIT, Lexington, MA, Tech. Rep. 566, [29] D. A. Harville, Matrix Algebra From a Statistician s Perspective. New York: Springer, [30] J. Li, B. Halder, P. Stoica, M. Viberg, Computationally efficient angle estimation for signals with known waveforms, IEEE Trans. Signal Process., vol. 43, no. 9, pp , Sep [31] S. Liu, Matrix results on the Khatri Rao Tracy Singh products, Linear Algebra Its Appl., vol. 289, pp , [32] P. Stoica Y. Selén, Cyclic minimizers, majorization techniques, expectation-maximization algorithm: A refresher, IEEE Signal Process. Mag., pp , Jan Luzhou Xu received the B.Eng. M.S. degrees in electrical engineering from Zhejiang University, Hangzhou, China, in , respectively. He is currently working towards the Ph.D. degree with the Department of Electrical Computer Engineering, University of Florida, Gainesville. From 1999 to 2001, he was with the Zhongxing Telecommunication Company, China, he was involved in the system algorithm design of mobile communication equipment. From 2001 to 2003, he worked at the Wireless Communication Group, Philips Research, Shanghai. His research interests include statistical array signal processing, their applications in wireless communications biomedical engineering. Jian Li (S 87 M 91 SM 97 F 05) received the M.Sc. Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in , respectively. From April 1991 to June 1991, she was an Adjunct Assistant Professor with the Department of Electrical Engineering, The Ohio State University, Columbus. From July 1991 to June 1993, she was an Assistant Professor with the Department of Electrical Engineering, University of Kentucky, Lexington. Since August 1993, she has been with the Department of Electrical Computer Engineering, University of Florida, Gainesville, she is currently a Professor. Her current research interests include spectral estimation, statistical array signal processing, their applications. Dr. Li received the 1994 National Science Foundation Young Investigator Award the 1996 Office of Naval Research Young Investigator Award. She was an Executive Committee Member of the 2002 International Conf. on Acoustics, Speech Signal Processing, Orlo, FL, May She was an Associate Editor of the IEEE TRANS. SIGNAL PROCESS. from 1999 to 2005 an Associate Editor of the IEEE Signal Processing Magazine from 2003 to She has been a member of the Editorial Board of Signal Processing, a publication of the European Association for Signal Processing (EURASIP), since She is presently a member of two of the IEEE Signal Processing Society technical committees: the Signal Processing Theory Methods (SPTM) Technical Committee the Sensor Array Multichannel (SAM) Technical Committee. She is a Fellow of the Institution of Electrical Engineers (IEE). She is a member of Sigma Xi Phi Kappa Phi.