Rank-Constrained Maximum Likelihood Estimation ... - Semantic Scholar

I. INTRODUCTION

Rank-Constrained Maximum Likelihood Estimation of Structured Covariance Matrices

BOSUNG KANG, Student Member, IEEE VISHAL MONGA, Senior Member, IEEE The Pennsylvania State University MURALIDHAR RANGASWAMY, Fellow, IEEE Air Force Research Laboratory

This paper develops and analyzes the performance of a structured covariance matrix estimate for the important practical problem of radar space–time adaptive processing in the face of severely limited training data. Traditional maximum likelihood (ML) estimators are effective when training data are abundant, but they lead to poor estimates, degraded false alarm rates, and detection loss in the realistic regime of limited training. The problem is exacerbated by recent advances, which have led to high-dimensional N of the observations arising from increased antenna elements J, as well as higher temporal resolution (P time epochs and finally N = JP). This work addresses the problem by incorporating constraints in the ML estimation problem obtained from the geometry and physics of the airborne phased array radar scenario. In particular, we exploit the structure of the disturbance covariance and, importantly, knowledge of the clutter rank to derive a new rank-constrained maximum likelihood (RCML) estimator of clutter and disturbance covariance. We demonstrate that despite the presence of the challenging rank constraint, the estimation can be transformed to a convex problem and derive closed-form expressions for the estimated covariance matrix. Performance analysis using the knowledge-aided sensor signal processing and expert reasoning data set (where ground truth covariance is made available) shows that the proposed estimator outperforms state-of-the-art alternatives in the sense of a higher normalized signal–to–interference and noise ratio. Crucially, the RCML estimator excels for low training, including the notoriously difficult regime of K ≤ N training samples. Manuscript received June 27, 2012; revised December 5, 2012, and April 29, 2013; released for publication June 24, 2013. IEEE Log No. T-AES/50/1/944805. DOI. No. 10.1109/TAES.2013.120389. Refereeing of this contribution was handled by L. Kaplan. Research was supported by AFOSR Grant No. FA9550-12-1-0333. M. Rangaswamy was supported by the Air Force Office of Scientific Research under Project 2311IN. Authors’ addresses: B. Kang, V. Monga, Department of Electrical Engineering, The Pennsylvania State University, 104 EE East, University Park, PA 16802, E-mail: ([email protected]); M. Rangaswamy, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH. C 2014 IEEE 0018-9251/14/$26.00 

Space–time adaptive processing (STAP), i.e., joint adaptive processing in the spatial and temporal domains [1–3], is the cornerstone of radar signal processing and creates the ability to suppress interfering signals while simultaneously preserving gain on the desired signal. However, for STAP to be successful, interference statistics must be estimated from training samples. Training therefore plays a pivotal role in adaptive radar systems. The radar STAP filter rejects clutter and noise by using an adaptive multidimensional finite impulse response filter structure. which consists of P time taps and J spatial taps. The optimal STAP complex weight vector, which maximizes the return signal from a desired target, requires formation and inversion of the disturbance covariance matrix. Furthermore, this covariance matrix is central to test statistics involved in target detection, such as the normalized matched filter [4] and the generalized likelihood ratio test [5]. The disturbance covariance matrix must be estimated in practice. In the absence of prior knowledge about the interference environment, a large number of homogeneous (target free) disturbance training samples are required to obtain accurate estimates. The quality and quantity of training data are governed by the scale with which the disturbance statistics change with respect to space and time, as well as by systems considerations such as bandwidth. A compelling challenge for radar STAP emerges, because the availability of generous homogeneous training is often unrealistic [6]. This problem is exacerbated because the estimation process must be repeated for each Doppler and range bin of interest. Therefore, recent research in radar STAP has focused on circumventing the lack of generous homogeneous training. One approach to this problem includes the use of a priori information about the radar environment and is widely referred to in the literature as knowledge-based processing [7–13]. A subset of these techniques concerns intelligent training selection [9]. Other methods try to reduce the spatiotemporal degrees of freedom to reduce both the number of required training samples and the computational cost [10, 14, 15]. Enforcing structure on covariance matrices (e.g., Toeplitz) and shrinkage estimation techniques have also been considered [16, 17]. A more thorough review of these techniques is provided in Section II.B. Our work introduces an advance in robust covariance estimation for STAP, known as the rank-constrained maximum likelihood (RCML) estimator of structured covariance matrices. The disturbance covariance matrix exhibits a structure that comprises the sum of noise (white) and clutter covariances, with the clutter component being positive semidefinite and rank deficient. What is of particular interest is that in airborne radar scenarios involving land clutter, this rank can be determined using the Brennan rule [18] under nominal conditions. Incorporation of this rank information in our research is a

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

501

logical evolution of the pioneering work by Steiner and Gerlach [19], wherein they demonstrated that the fast maximum likelihood (FML) estimator, which is cognizant of the eigenstructure of the disturbance covariance, can outperform competing approaches in the literature. From an analytic viewpoint, the contributions of this work are in formulating and solving the RCML estimation problem for structured covariance matrices. The value of the rank has been identified in statistics [20] and in radar signal processing via eigencancelation approaches [2, 21]. Our work overlaps with these techniques in that we use identical eigenvectors for the estimated covariance matrix whose optimal choice can formally be shown to agree with those obtained from a singular value decomposition of the data matrix of training data. We demonstrate in addition that despite the presence of the challenging (nonconvex) rank constraint, our estimation problem can be reduced to a convex optimization problem over the eigenvalues and that closed-form expressions are derived for the estimator. Our central analytical result is the RCML estimator, which like FML and eigencancelation approaches, exploits knowledge of the radar noise floor. The noise power is typically determined by placing the radar in “receive only” mode before going active [22]. For mathematical completeness, we derive another estimator called RCMLLB for when the noise floor is assumed to be unknown and only a lower bound (LB) is available. Our RCMLLB estimator is a generalization of the result reported by Wax and Kailath [23], who quote Anderson [20] for the result. Our experimental contributions include an extensive performance analysis of the RCML method and a performance comparison with a host of competing methods with demonstrated success. Our experimental data were obtained from the Defense Advanced Research Projects Agency (DARPA) knowledge-aided sensor signal processing and expert reasoning (KASSPER) data set, where ground truth covariance is available, which helps in evaluation via well-known figures of merit, such as the normalized signal–to–interference and noise ratio (SINR). The merits of RCML are most pronounced in the low training regime, particularly in the case of K < N training samples, which is a stiff practical challenge for radar STAP. In addition, the proposed RCML estimator is robust to perturbations against the true knowledge of the rank, making it even more appealing from a practical standpoint. The rest of the paper is organized as follows. Section II provides brief background on radar STAP, reviews existing covariance estimation literature, and further motivates our contribution. Section III.A formulates the RCML estimation problem, and subsequent derivation is provided in Section III.B. The solutions of the estimation and optimization problems are derived for the two cases of both known and unknown (known LB) noise levels. Experimental validation of our methods is provided in Section IV, wherein we report on the performance of the proposed estimator and compare it against well-known existing methods in terms of SINR. In addition, rank 502

Fig. 1. Target and interference scenario in airborne radar.

sensitivity of the proposed RCML estimator is evaluated. Finally, concluding remarks with directions for future work are presented in Section V. II. BACKGROUND A. Spatiotemporal Adaptive Processing in Radar

The radar receiver front end consists of an array of J antenna elements, which receives signals from targets, clutter, and jammers. These reflections induce a voltage at each element of the antenna array, which constitutes the measured array data at a given time instant. Snapshots of the measured data collected at P successive time epochs give rise to the spatiotemporal nature of the received radar data. The spatiotemporal product JP = N is defined to be the system dimensionality. Fig. 1 uses the angle-Doppler space to illustrate the need for STAP. The target detection problem can be cast in the framework of a statistical hypothesis test of the form H0 : x = d = c + j + n

(1)

H1 : x = αs(θt , ft ) + d = αs(θt , ft ) + c + j + n

(2)

where x ∈ CJP ×1 is a vector form of the received data under either hypothesis; d represents the overall disturbance, which is the sum of clutter c, jammers j, background white noise n; and s is a known spatio-temporal steering vector [10]. The whiten-and-match filter for detecting a rank-1 signal is the optimum processing method for Gaussian interference statistics. It is given by [5] w= 

R−1 d s sH R−1 d s

⇒ MF =

2 |sH R−1 d x|

sH R−1 d s

H1 ≷ λMF H0

(3)

where Rd is the disturbance covariance matrix. Equation (3) represents the matched filtering of the whitened   −1/2 −1/2 data x = Rd x and whitened steering vector s = Rd s. B. Motivation and Review

Because the covariance matrix plays a crucial role in the detection statistic (see (3)), it is important to estimate it

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

reliably. Widrow et al. and Applebaum proposed leastsquares method [24] and maximum signal-to-noise ratio criterion [25], respectively, using feedback loops. However, these methods were slow to converge to the steady-state solution. Reed, Mallett, and Brennan [26] verified that the sample matrix inverse (SMI) method demonstrated considerably better convergence. In the SMI method, the disturbance covariance matrix can be estimated using K data ranges for training: K  1 ˆd = 1 R XXH xk xH k = K k=1 K

(4)

where K is the number of training data we received, xk ∈ CN , N = J P is the kth vector of training data, and X = [x1 x2 . . . xK ] ∈ CN×K . It is well known that the sample covariance is the unconstrained ML estimator when K ≥ N. Despite this virtue, there remain fundamental problems with the SMI approach. First, typically K > N training samples are needed to guarantee the nonsingularity of the estimated covariance matrix. When K < N, the estimate is singular and therefore precludes implementation of the STAP processor. As much past research has shown [15], the estimate also does quite poorly in the vicinity of K = N training samples. As observed in Section I, a large number of homogeneous training samples are generally not available [6]. To overcome the practical issue of limited training, covariance matrix estimation techniques that enforce and exploit a particular structure have been pursued. Examples of structure include persymmetry [27], the Toeplitz property [16, 28, 29], circulant structure [30], multichannel autoregressive models [31, 32], and physical constraints [33]. The FML method [19], which enforces a special eigenstructure, also falls in this category and is shown to be the most competitive technique experimentally [4, 15]. In particular, the disturbance covariance matrix R represents the following structure R = σ 2 I + Rc

(5)

where Rc denotes the clutter matrix, which has a low rank and is positive semidefinite, and I is an identity matrix. Steiner and Gerlach’s FML technique ensures that the estimated covariance matrix has eigenvalues all greater than σ 2 . Recently, the work by Aubry et al. [34] has also improved upon FML by the introduction of a condition number constraint. Other approaches include Bayesian covariance matrix estimators [9, 35–38] and the use of knowledge-based covariance models [12, 13, 39, 40]. Finally, shrinkage estimation methods have been considered [17, 41–43]. III. RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

explicitly into maximum likelihood (ML) estimation of the disturbance covariance matrix.1 Under ideal conditions (no mutual coupling between array elements and no internal clutter motion), the Brennan rule [18] states that the rank of Rc in the airborne linear phased array radar problem is given by rank Rc = J + γ (P − 1)

(6)

where γ = 2 vp T/d is the slope of the clutter ridge, with vp denoting the platform velocity, T denoting the pulse repetition interval, and d denoting the interelement spacing. Even if there is mutual coupling in practice, Rc has rank r, which is much less than the spatiotemporal product N = JP in many practical airborne radar applications. In addition, powerful techniques have been developed [4] to determine the rank fairly accurately. We first set up the optimization problem to estimate the disturbance covariance matrix with a structural constraint on R and the rank constraint on Rc . The estimation problem when seen as an optimization over R is unfortunately not a convex problem, because neither the cost function nor the constraints (rank) are convex (elaborated upon in Section III.B). However, we show that using a transformation of variables, reduction to a convex form is possible. Furthermore, by invoking Karush–Kuhn–Tucker (KKT) conditions [45] for the resulting convex problems, it is possible to derive a closed-form solution. Akin to FML, we initially assume (Section III.B.1) that the noise power σ 2 is known while setting up and solving the problem. Then we extend our results (Section III.B.2) to the case of unknown noise variance. In that case, we assume that only an LB on the noise power is available. That is, we know cˆ , where the covariance matrix is expressed by R = cI + Rc

(7)

and c is the noise variance (to be estimated or optimized) such that c > cˆ . B. ML Estimation

Let zi ∈ CN be the ith realization of the target-free (stochastic) disturbance vector and K be the number of training samples. That is, i = 1, 2, . . ., K and N = JP. Therefore, under each training sample, zi , under assumption of zero mean, obeys f (zi ) =

1 π N |R|

−1 exp(−zH i R zi )

(8)

which comes from a zero-mean complex circular Gaussian distribution and R is the N × N disturbance covariance matrix. Furthermore, |R| denotes the determinant of R and zH i is the Hermitian (conjugate transpose) of zi . Let Z be the N × K complex matrix whose ith column is the observed vector zi . Because observations zi are

A. Overview of Contribution

The principal contribution of our work was to incorporate the rank of the clutter covariance matrix Rc

1 A preliminary version of the work appeared at the 2012 IEEE Radar Conference [44].

KANG ET AL.: RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

503

independent and identically distributed, the likelihood of observing Z given R is given by f(R) (Z) =

1

π

|R|−K exp (−tr{ZH R−1 Z}) NK

(9)

=

1 |R|−K exp (−tr{R−1 ZZH }) π NK

(10)

=

1 |R|−K exp (−K · tr{R−1 S}) π NK

(11)

where S = K1 ZZH is the well-known sample covariance matrix. Our goal is to find the positive definite matrix R that maximizes the likelihood function f(R) (Z). The logarithm of the likelihood term is log f(R) (Z)= − K · tr{R−1 S}−K log(|R|)−NK log(π). (12) Maximizing the log likelihood as a function of R is equivalent to minimizing the function given by tr{R−1 S} + log(|R|).

(13)

Therefore, (13) is the cost function of our optimization problem. Because the cost function is not a convex function in R, we apply a transformation variables, i.e., let X = σ 2 R−1 and S = σ12 S. Then, the revised cost function in the optimization variable X becomes tr{R−1 S} + log(|R|)   1  = tr{S X} − log  2 X σ 

= tr{S X} − log(|X|) + log σ 2N .

(14) (15)

Because log σ 2N in (15) is a constant, the final cost function to be minimized is tr{S X} − log(|X|). Here, tr{S X} =

N  N  i=1 j =1

(16)

sj i xij is affine and (|X|) is

concave, which implies −log (|X|) is convex. Therefore, the final cost function (16) is convex in the variable X. We now express X and S in terms of their eigenvalue decomposition, i.e., X = H and let S be decomposed as S = VDVH where  and V are orthonormal eigenvector matrices of X and S , respectively, and  and D are diagonal matrices with diagonal entries, which are eigenvalues of X arranged in ascending order and S arranged in descending order, respectively. Using the eigendecompositions, the cost function can be simplified as tr{S X} − log(|X|)

504

= tr{VDVH H } − log(|H |)

(17)

= tr{DVH H V} − log(||)

(18)

≥ tr{D} − log(||)

(19)

= dT λ − 1T log λ

(20)

where d and λ are vectors with entries of eigenvalues of S and X, respectively, and log λ = [log λ1 , log λ2 , · · · , log λN ]T . The equality in (19) holds when  = V. This result is fairly well known from standard unconstrained ML estimation of nonsingular R. That is, over the space of unitary matrices, the optimal  that maximizes the likelihood is the one that matches with the eigenvector matrix of the sample covariance [46]. Therefore, the cost function has the minimum value when  = V. Hence, the optimization may be focused on the vector of eigenvalues λ. Many previous algorithms, such as FML [19], eigencanceler (EigC) [21], and other eigen-based techniques [2], have used the same eigenvectors without eigenvalue optimization. 1) Known Noise Level Case: We first assume the noise power is known. Then the constraints of the optimization problem are ⎧ 2 ⎪ ⎪ R = σ I + Rc ⎨ rank(Rc ) = r . (21) Rc  0 ⎪ ⎪ ⎩ R  σ 2I Because rank(Rc ) = r, Rc has r nonnegative eigenvalues, and the rest of the eigenvalues are all zero. Hence, from (5), R has r eigenvalues, which are greater than or equal to σ 2 , and the rest of the eigenvalues are equal to σ 2 . That is, λ¯ 1 ≥ λ¯ 2 ≥ · · · ≥ λ¯ r ≥ σ 2 (= λ¯ r+1 ) = · · · = σ 2 (= λ¯ N ), where λ¯ i is the ith eigenvalue of R. Hence, the eigenvalues matrix of X = σ 2 R−1 should be λ1 ≤ λ2 ≤ · · · ≤ λr ≤ 1(= λr+1 ) = · · · = 1(= λN ). Now the constraints can be expressed in vector and matrix forms. The first constraint is λ1 ≤ λ2 ≤ · · · ≤ λN , i.e., Uλ 0, where ⎤ ⎡ 1 −1 0 ··· 0 ⎢ .. ⎥ ⎢0 .⎥ 1 −1 · · · ⎥ ⎢ ⎥ ∈ RN×N . . . U=⎢ (22) . . ⎥ ⎢0 . . 0 0 ⎥ ⎢ ⎣0 0 ··· 1 −1 ⎦ 0 0 ··· 0 −1 The second constraint is 0 < λi ≤ 1, which can be expressed by ε λ 1

(23)

where ε is a vector with all entries equal to the same constant ε such that ε is picked as close to zero. This is done to avoid solutions in which any λi exactly equals zero, because that would lead to a singular X. The final constraint is λr+1 = λr+2 = · · · = λN = 1 and is expressed as Eλ = h

(24)

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

where

 E=

0r×r 0(N−r)×r

0r×(N−r) IN−r

 ∈ RN×N

(25)

and h = [0, 0, . . ., 0r , 1, 1, . . ., 1]T . We therefore have the following optimization problem: ⎧ min dT λ − 1T log λ ⎪ ⎪ λ ⎪ ⎪ ⎨ s.t. Uλ 0 (26) −λ −ε ⎪ ⎪ ⎪ λ 1 ⎪ ⎩ Eλ = h The constraint λ 1 forces the maximum rank of Rc to be r. That is, we use a relaxation of the strict inequality, which forces an exact rank constraint. As shown shortly, this relaxation allows us to obtain a closed-form solution, while the optimal solutions to the two problems (original vs. relaxed) are hardly distinguishable [47]. Combining the inequality constraints into one, we have ⎧ dT λ − 1T log λ ⎨ min λ (27) Fλ g ⎩ s.t. Eλ = h  T T  where F = UT −I I , g = 0T −ε T 1T , and E and h are from (24). Here, λ, d, and h ∈ RN ; g ∈ R3N ; and F ∈ R3N×N . Equation (27) is a convex optimization problem, because the cost function is a convex function and feasible constraint sets are convex. A closed-form solution for (27) can be derived using KKT conditions [45] in constrained optimization. The optimal solution λ∗ is    for i = 1, 2, . . . , r min 1, d1i ∗ (28) λi = 1 for i = r + 1, r + 2, . . . , N. The derivation of the closed form is provided in Appendix A. Hence, the optimal solution X∗ is X∗ = V∗ VH

(29)

and the optimal covariance matrix estimator R∗ is R∗ = σ 2 X∗−1 = σ 2 V∗−1 VH

(30)

where V is the eigenvector matrix of the sample covariance matrix S and ∗ is a diagonal matrix with diagonal entries λi . This is a generalization of the FML solution in [19] with the rank-information in addition incorporated. 2) Unknown Noise Level Case: For mathematical completeness, we also derive the case in which σ 2 is estimated as a part of the estimation process. However, we assume that an LB is known and call this estimator RCMLLB . We proceed by defining X = R−1 (instead of X = 2 −1 σ R , as in Section III.B.1), because we assume that the exact noise power is unknown. Using simplifications similar to the ones in Section III.B.1, we have the cost function to be minimized as −1

tr{R S} + log(|R|) = tr{SX} − log(|X|).

(31)

The right-hand side of (31) is also a convex function in X, according to the same reasoning as in Section III.B.1. We no longer use S in this case. Now the constraints are written as ⎧ R = cI + Rc ⎪ ⎪ ⎪ rank(R ⎨ c) ≤ r Rc  0 . (32) ⎪ ⎪ R  cI ⎪ ⎩ c ≥ cˆ The eigenvalues of R and X = R−1 are λ¯ 1 ≥ λ¯ 2 ≥ · · · ≥ λ¯ r ≥ c(= λ¯ r+1 ) = · · · = c(= λ¯ N ) and λ1 ≤ λ2 ≤ · · · ≤ λr ≤ 1c (= λr+1 ) = · · · = 1c (= λN ), respectively. Using reductions similar to the ones in Section III.B.1, we can formulate the following optimization problem: ⎧ min dT λ − 1T log λ ⎪ ⎪ λ,c ⎪ ⎪ ⎪ s.t. Uλ g ⎪ ⎨ −λ −ε . (33) ⎪ λ 1c 1 ⎪ ⎪ ⎪ ⎪ Eλ = h ⎪ ⎩ c ≥ cˆ U and E are same as in Section III.B.1, but h = [0, 0, · · · , 0r , 1c , 1c , · · · , 1c ]T . Again, inequality constraints may be combined into one: ⎧ min dT λ − 1T log λ ⎪ ⎪ ⎨ λ,c s.t. F λg . (34) ⎪ Eλ = h ⎪ ⎩ c ≥ cˆ  where F is same as in Section III.B.1 and g = 0T −ε T  1 T T 1 . c To solve (34), we first fix the noise level c to c¯ ≥ cˆ and solve the optimization problem with respect to λ to obtain ¯ which is a function of c. ¯ By the optimal solution λ∗ (c), ¯ back into the cost function, substituting the optimal λ∗ (c) we can get an auxiliary optimization over the scalar variable c. ¯ the optimization problem (34) can Once we fix c = c, be reduced to a problem in just λ, which is nearly the same problem as (27): ⎧ dT λ − 1T log λ ⎨ min λ (35) F λg ⎩ s.t. Eλ = h where F and E are same as in (34) and g and h are obtained by replacing 1c with 1c¯ . Equations (27) and (35) are the same except that (35) uses 1c¯ 1 and 1c¯ instead of 1 and 1 in g and h, respectively. Using a derivation similar to the one in Appendix A, the optimal solution λ∗ of (35) is ⎧   ⎨ min 1 , 1 for i = 1, 2, . . . , r c¯ di (36) λ∗i = 1 ⎩ for i = r + 1, r + 2, . . . , N. c¯

If we substitute the optimal λ∗ as in (36) back into (34), we get the following problem to solve in the

KANG ET AL.: RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

505

unknown noise level variable c: ⎧ N  ⎨ Gi (c) min c i=1 ⎩ s.t. c ≥ cˆ where

 Gi (c) =

1 + log di di + log c c

if c ≤ di if c > di

TABLE I KASSPER Data Set 1 Parameters

(37)

(38)

for i = 1, 2, . . . , r and di + log c (39) c for i = r + 1, r + 2, . . . , N. This problem also admits a closed form, which is derived in Appendix B. In particular, the optimal c∗ is given by  cˆ if cˆ > dˆ (40) c∗ = ˆ d if cˆ ≤ dˆ Gi (c) =

N

i=r+1 di where dˆ = N−r . Again, the optimal solution λ∗ is  min( 1∗ , 1 ) for i = 1, 2, . . . , r ∗ . (41) λi = 1 c di for i = r + 1, r + 2, . . . , N c∗

Combining the optimal estimated eigenvalue set in (41) (expressed in terms of the optimal noise level variable c∗ ) with the eigenvectors of the sample covariance leads to the optimal estimate of the (inverse of) covariance estimate of R under the rank constraint. This result is a generalization of the result in Wax and Kailath [23], because it includes and exploits an LB on the noise floor when available. We discuss these two algorithms in more detail in Section IV.C. IV. EXPERIMENTAL RESULTS A. Experimental Setup and Method Comparison

Data from the L-band data set of the KASSPER program [48] is used for the performance analysis discussed in this section. The KASSPER data are the result of a significant effort by DARPA to provide a publicly available resource for the evaluation and benchmarking of radar STAP algorithms. As elaborated in [9], the KASSPER data set was carefully captured to represent real-world ground clutter, and it captures variations in underlying terrain, foliage, and urban and man-made structures. Furthermore, the KASSPER data set exhibits two desirable characteristics from the viewpoint of evaluating covariance estimation techniques: 1) the low-rank structure of clutter in KASSPER has been verified by researchers before [4, 9], and 2) the true covariance matrices for each range bin have been made available. This facilitates comparisons via powerful figures of merit in which the theoretical upper bounds and LBs are known. The L-band data set consists of a data cube of 1000 range bins corresponding to the returns from a single coherent processing interval from 11(=J) channels and 32(=P) pulses. Therefore, the dimension of observations 506

Parameter

Value

Carrier frequency Bandwidth No. antenna elements No. pulses Pulse repetition frequency 1000 range bins 91 azimuth angles 128 Doppler frequencies Clutter power No. targets Range of target Doppler frequency

1240 MHz 10 MHz 11 32 1984 Hz 35–50 km 87, 89, . . . 267 deg −992, −976.38, . . ., 992 Hz 40 dB 226 (∼200 detectable targets) −99.2 to 372 Hz

(or the spatiotemporal product) N is 11 × 32 = 352. Other key parameters are detailed in Table I. Finally, a clutter rank2 of r = J + P − 1 = 42 was used by our RCML estimator in all results to follow unless explicitly stated otherwise. We evaluate and compare four covariance estimation techniques. 1) Sample Covariance Matrix: The sample covariance matrix is given in (4). It is well known that the sample covariance is the unconstrained ML estimator under Gaussian disturbance statistics. Consistent with radar literature [26], we refer to the use of this technique as SMI. 2) Fast Maximum Likelihood: The FML [19] uses the structural constraint of the covariance matrix, which is given in (5). The FML method involves calculating the eigenvalue decomposition of the sample covariance and perturbing eigenvalues to conform to the structure in (5). The noise variance σ 2 is assumed to be known or preestimated. FML’s success in radar STAP is widely known [4, 10, 15]. 3) Leave-One-Out Shrinkage Estimator: Shrinkage estimators are powerful estimators of covariance for high-dimensional data that are known to also perturb the eigenstructure of the sample covariance matrix3 [17]— often to ensure nonsingularity of the estimated covariance. While a variety of shrinkage techniques are known [17, 41–43], we choose the leave-one-out covariance matrix estimate (LOOC) shrinkage estimator [49]: R = βdiag(S) + (1 − β)S.

(42)

The value of β is determined via a cross-validation technique so that the average likelihood of omitted samples is maximized. We pick this estimator because it has demonstrated success in the K ≤ N training regime [49]. In addition, the LOOC shrinkage estimator is representative of the diagonal loading techniques applied for nonsingular covariance estimation by Abramovich [50] and Carlson [51]. γ = 1. to this definition, the FML and RCML can also been seen as a special class of shrinkage estimators. 2 Nominally 3 According

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

Fig. 2. Eigenspectra of estimated covariance matrices. Sample covariance matrix (SMI), FML, LOOC shrinkage estimator, EigC, and RCML estimators are compared. (a) K = 352. (b) K = 750.

4) Eigencanceler: The EigC is based on the eigenanalysis, which suggests that a small number of eigenvalues contain all information about interferences (jammers and clutter); therefore, the span of the eigenvectors associated with these significant eigenvalues includes all position vectors that comprise the interference signals [21]. Because we assume that the rank is known a priori, the EigC can be compared with our estimator as we use r dominant eigenvectors as interference eigenvectors. The covariance matrix can be expressed by R=

r 

2 pi vi vH i +σ I

(43)

i=1

where pi and vi are the clutter power and the eigenvector corresponding to r dominant eigenvalues, respectively. For pi σ 2 , it follows from [4, 52] that the estimated inverse covariance matrix can be approximated as r   R−1 ≈ 12 (I − P), where P = vi vH . We apply this σ

i=1

i

inverse covariance matrix in computing the SINR. 5) Rank-Constrained Maximum Likelihood: Our proposed RCML estimator incorporates the structural constraint and the information of the rank of the clutter component. B. Experimental Evaluation

The normalized SINR is used for evaluation of the aforementioned covariance estimation techniques. The SINR is desired to be as high as possible. This figure of merit is plotted against azimuthal angle, as well as Doppler frequency, for distinct training regimes, i.e., low, representative, and generous training. We also show the plot of SINR performance vs. the number of training samples. Finally, we evaluate the robustness of our RCML estimator against perturbations in knowledge of the true rank. 1) Eigenspectrum: First, we provide the eigenspectra plots, i.e., a plot of the eigenvalues of the estimated covariance matrices vs. those of the ground truth covariance. Fig. 2 shows the eigenspectra of various

estimators for K = N = 352 and K = 750 in decibels. The decay of eigenvalues of the true covariance matrix is readily apparent. The first few dominant eigenvalues are well predicted by every method. However, after index 20 or so, RCML shows the tightest overlap with the eigenvalues of the true covariance matrix, and the FML and EigC approaches not far behind. In particular, while FML shows slightly better agreement in capturing the decay, EigC, like RCML, is more accurate in capturing the rank. The SMI, FML, and LOOC approaches are significantly off in their estimate of the clutter rank, which is determined as an outcome for these methods. In the next section, we demonstrate the benefits of incorporating rank and structural information about the disturbance covariance for widely used figures of merit in the radar literature. 2) Normalized SINR vs. Angle and Doppler: The normalized SINR measure [1] is commonly used in the radar literature and is given by η=

|sH  R−1 s|2 R−1 R |sH  R−1 s||sH R−1 s|

(44)

where s is the spatiotemporal steering vector,  R is an estimated covariance matrix, and R is the corresponding true covariance matrix. It is easily seen that 0 < η ≤ 1 and η = 1 if and only if  R = R. Because the steering vector is a function of both azimuthal angle and Doppler frequency, we evaluate the normalized SINR in both angle and Doppler domains. This leads to a SINR surface as a function of azimuthal angle and Doppler, and comparing surface plots across different covariance estimation techniques is cumbersome. We therefore obtain plots as a function of one variable (i.e., just angle or Doppler) by marginalizing (averaging) over the other variable. The SINR is plotted in decibels. In all figures in this paper, SINRdB = 10 log10 η. Therefore, SINRdB ≤ 0. Fig. 3 plots the variation of normalized SINR as a function of the azimuthal angle and the Doppler frequency for a varying number of training samples K. Specifically, Figs. 3a and 3b correspond to K = 300 < N = 352,

KANG ET AL.: RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

507

Fig. 3. Normalized SINR vs. normalized azimuthal angle and Doppler frequency (true ranges of azimuth and Doppler can be seen in Table I). Sample covariance matrix (SMI), FML, LOOC shrinkage estimator, EigC, and RCML estimators are compared. (a) and (b) K = 300. (c) and (d) K = 352. (e) and (f) K = 750. (g) and (h) K = 3000.

Figs. 3c and 3d plot results for K = 352 = N, Figs. 3e and 3f correspond to K = 750 ≈ 2N, and 3g and 3h correspond to K = 3000 N. Figs. 3a–3d report results for the challenging regime of K ≤ N. When K < N, the sample covariance matrix is not invertible; hence, for the results in Figs. 3a and 3b, we used its pseudoinverse as a substitute. Unsurprisingly, the sample covariance technique suffers tremendously when K ≤ N, as is evident from Figs. 3a–3d. LOOC shrinkage does considerably better than SMI, because it forces a reasonably good eigenstructure. The informed estimators, i.e., FML, EigC, and RCML, perform appreciably well, with RCML affording the best overall performance. RCML offers about 1-dB improvement over FML. Even for representative training in Figs. 3e and 3f, the vastly superior performance of the FML, EigC, and RCML techniques is apparent. Again, by incorporating the rank information, the proposed RCML estimator outperforms the competing methods. Finally, Figs. 3g and 3h confirm the intuition that as training becomes close to asymptotic, the gaps between the various methods begin to decrease. Such generous training is typically impossible to obtain in practice. This is because all covariance matrix estimates considered converge to the true covariance matrix with a probability of one in the limit of large training data. 508

3) Performance vs. Number of Training Samples: The results in Section IV.B.2 explore performance against training to some extent. Here, we present bar graphs to explore this issue with a finer granularity. To obtain a single scalar performance measure as a function of training, averaging was carried out over both the angle and the Doppler variables. Fig. 4 presents bar graphs that quantify the normalized SINR (in decibels) as a function of training samples K, where K is varied from as low as 60 to as high as 3000 samples. Two trends are evident from Fig. 4: 1) as intuitively expected, the SINR values increase monotonically with an increase in the number of training samples for all methods (except for the sample covariance technique in the K ≤ N regime, which is a well-known phenomena on observed in past work [19]), and 2) the RCML estimator exhibits remarkably good performance in all training regimes. 4) Rank Sensitivity: With the KASSPER data, the clutter rank conforms to (6) according to the Brennan rule. For the parameters used in our experiments, this leads to a predicted ideal rank of r = J + P − 1 = 42. In a practical situation, departures from the ideal behavior are expected. Hence, we explore the performance our proposed RCML estimator even as incorrect rank information is used.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

Fig. 3. (Continued).

Fig. 4. Normalized SINR vs. number of training samples: 60, 100, 200, 250, 300, 352, 750, and 3000 samples.

Fig. 5. Normalized SINR of RCML for various rank information.

The results in Fig. 5 demonstrate the robustness of RCML to perturbations in the clutter rank. Fig. 5 presents bar graphs that show averaged SINR results for K = 352 and K = 750 training samples. We determined numerically that the “true” rank of the clutter covariance for the range

bin of choice was 43, which is a mild departure from the 42 predicted by the Brennan rule. Comparisons are made between FML and RCML, with the difference that seven cases of RCML are presented—with ranks from 34 to 45. As Fig. 5 reveals, using the true rank of 43 yields the best

KANG ET AL.: RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

509

covariance matrix estimator, but the penalty of the small departure, i.e., using a rank of 40 to 45, which is close to the true rank of 43, leads to a very small performance loss. However, Fig. 5 also shows variants of the RCML result with a somewhat bigger departure, i.e., a rank of 34. In this case, the performance of RCML with a rank of 34 is appreciably lower against using rank values around the true rank of 43. Remarkably, RCML with a rank of 34 is still competitive with FML. Overall, Fig. 5 therefore provides two insights: (1) Because rank information is predicted using the Brennan rule, small departures in practice are possible and our estimator exhibits desirable robustness against such small perturbations to rank, and (2) the value of using the rank information is simultaneously revealed; because RCML with a rank of 34 is competitive with FML, it shows that FML significantly underestimates the true rank in these examples. C. RCML vs. RCMLLB and Wax and Kailath Estimator

Following Anderson’s result [20] in statistics, Wax and Kailath [23] reported an ML estimator of a structured covariance estimator under the rank constraint as follows: λˆ i = λ¯ i

(45)

N  1 σˆ = λ¯ i N − r i=r+1

(46)

ˆ i = Vi 

(47)

2

where λ¯ 1 ≥ λ¯ 2 ≥ · · · ≥ λ¯ N denote the eigenvalues of the sample covariance matrix. It is easy to see that the RCMLLB estimator in (41) is a generalization of this result by employing an LB on the noise floor. To further emphasize that the value of RCML in (28) is the estimator of choice for practical radar STAP, we perform an experimental comparison of RCML, RCMLLB , and the Wax and Kailath [23] estimators in the challenging low training regime. Figs. 6a and 6b show the performance of RCML, RCMLLB , and the Wax and Kailath estimators for K = 30 and K = 300 training samples, respectively. Two versions of RCMLLB are reported, with the LB cˆ set to σ 2 /2 and 0 and the estimators labeled as RCMLLB,σ 2 /2 and RCMLLB,0 , respectively, in the plots. Figs. 6a and 6b clearly reveal that (1) RCML is clearly the best estimator, while the Wax and Kailath estimator is about 30 dB worse for K = 30 and about 3 dB below for K = 300 training samples, and (2) knowledge of an LB helps RCMLLB in that RCMLLB,σ 2 /2 is a better estimator than RCMLLB,0 , which overlaps with the Wax and Kailath estimator.

the RCML estimator, exploits knowledge of the noise floor and delivers significant performance improvements in the low training regime—a stiff practical challenge for STAP. For mathematical completeness, we also derive an estimator for when the noise level is unknown but only an LB is available, which generalizes the estimator developed by Anderson [20] in statistics and employed in signal processing by Wax and Kailath [23]. Future work could consider the incorporation of more constraints on the clutter and disturbance matrix, such as the Toeplitz structure, as well as the use of physically inspired probabilistic priors in a Bayesian setting. APPENDIX A. DERIVATION OF THE CLOSED-FORM λ∗

Before solving the problem using the KKT conditions, we can simplify the problem further. The constraint (24) tells us we already have λr+1 = λr+2 = · · · = λN = 1. This means we do not need to optimize λr+1 , λr+2 , . . . = λN . Therefore, we can arrive at an r variable-optimization problem given by  min dT λ − 1T log λ λ (48) s.t. Fλ g  T T  where F = UT −I I and g = 0T −ε T 1T . Here, λ, d ∈ Rr , g ∈ R3r , U ∈ Rr×r , and F ∈ R3r×r . We can rewrite the Lagrangian and KKT conditions to solve the problem. In Lagrangian L, Rr × R3r × Rr → R associated with the problem is L(λ, μ) = dT λ − 1T log λ + μT (Fλ − g) where μ ∈ R3r . The KKT conditions for any λ to be a minimizer are primal inequality constraints, Fλ g;

510

(50)

dual inequality constraints, μ  0, /  0, ν  0;

(51)

complementary slackness, μ1 (λ1 − λ2 ) = 0 .. . μr−1 (λr−1 − λr ) = 0 μr λr = 0 ν1 (λ1 − ε) = 0 .. . νr (λr − ε) = 0 υ1 (λ1 − 1) = 0 .. . υr (λr − 1) = 0;

V. CONCLUSION

We exploit knowledge of the rank of the clutter subspace to develop and derive a new RCML estimator for STAP. We demonstrate that despite the presence of the challenging rank constraint, the estimation problem can be reduced to a convex optimization problem and admits a closed-form solution. Our central analytical contribution,

(49)

and stationarity, ∇ λ L(λ, μ) = d −

1 + FT μ = 0 λ

(52)

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

Fig. 6. Normalized SINR vs. normalized azimuthal angle (true ranges of azimuth can be seen in Table I). Various RCML and Wax and Kailath (Wax) estimators are compared. (a) K = 30. (b) K = 300.

where λ1 = [ λ11 , λ12 , · · · , λ1r ]T . To make the notation clear, we let μ = [μ1 , μ2 , . . . , μr ]T , ν = [μr+1 , μr+2 , . . . , μ2r ]T , and υ = [μ2r+1 , μ2r+2 , . . . , μ3r ]T . Then the primal inequality constraints mean  λ 1 ≤ λ2 ≤ · · · ≤ λr . (53) ≤ λi ≤ 1 for i = 1, 2, . . . , r First, from complementary slackness, we get ν = 0, because we let ε be a very small number, which is close enough to zero that any λi cannot equal ε. In addition, the slackness means  μi = 0 if λi = λi+1 . (54) υi = 0 if λi = 1

From these two equations, we get μ1 = (d2 − d1 )/2. Because d1 ≥ d2 , this implies μ1 ≤ 0. However, μ1 ≥ 0 from the dual inequality   constraints. Therefore, μ1 = 0 1 and λ1 = min d1 , 1 . Next, the case of λ1 = λ2 = λ3 can be split into two: λ1 = λ2 = λ3 = λ4 and λ1 = λ2 = λ3 = λ4 . By following an approach similar to solving simultaneous equations, we   conclude that μ = 0 and therefore λi = min d1i , 1 . APPENDIX B. DERIVATION OF THE SOLUTION c∗

Let c∗ be an optimal solution to the following optimization problem:

Next, we analyze the stationarity condition:

⎧ ⎨

1 d1 − + μ1 − υ1 = 0 λ1 1 − μ1 + μ2 − υ2 = 0 d2 − λ2 .. . 1 − μr−1 − μr − υr = 0. dr − λr

⎩

This leads to λ1 =

1 , d1

1 = 0. λ1

c

Gi (c)

i=1

(57)

c ≥ σ2

s.t.

where Gi (c) = di λi − log λi , i.e.,  Gi (c) =

We first assume that λ1 = 1, which leads to υ1 = 0. We now have two cases: λ1 = λ2 and λ1 = λ2 . In the former case, we also know μ1 = 0. Therefore, d1 −

N 

min

if c ≤ di if c > di

(58)

for i = 1, 2, . . . , r and

(55)

by the primal inequality constraint   to λ1 ≤ 1, and finally to λ = min d11 , 1 . In the latter case, we can also consider two cases: λ1 = λ2 = λ3 and λ1 = λ2 = λ3 . In the first case, μ2 = υ1 = υ2 = 0. Therefore, the first two equations in the stationarity condition become  d1 − λ11 + μ1 = 0 . (56) d2 − λ11 − μ1 = 0

1 + log di di + log c c

Gi (c) =

di + log c c

(59)

for i = r + 1, r + 2, . . . , N. First, Gi (c) is a constant for c < di and monotonically increasing in c for c ≥ di when i = 1, 2, . . . , r, because the first derivative of the function dci + log c and − cd2i + 1c > 0 for c > di . Then, for i = 1, . . . , r,  Gi (c) =

KANG ET AL.: RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

1 + log di di + log c c

if c ≤ di if c > di

: constant : increasing.

(60) 511

Because we assumed d1 ≥ d2 ≥ · · · ≥ dr , we have r  Gi (c) i=1⎧ r  ⎪ ⎪ (1 + log di ) if c ≤ dr ⎪ ⎪ ⎪ i=1 ⎪ ⎪ r−1 ⎪ ⎪ ⎨  (1 + log di ) + dr + log c if dr ≤ c ≤ dr−1 c = i=1 ⎪ . ⎪ ⎪ .. ⎪ ⎪ ⎪ r ⎪  ⎪ ⎪ if c > d1 ⎩ ( dci + log c)

[4]

[5]

.. .

.

[7]

i=1

(61)

r

In (61), i=1 Gi (c) is a constant when c ≤ dr and an increasing function otherwise. Finally, because r i=1 Gi (c) is continuous at all di , we can conclude that  r  constant if c ≤ dr Gi (c) = (62) increasing if c > dr . i=1

Now, Gi (c) = Hence,

di c

N 1  = di + (N − r) log c. c i=r+1

(63) [13]

 Gi (c) =

decreasing increasing

if c ≤ dˆ if c > dˆ

(64)

i=r+1 di where dˆ = N−r . In addition, it is obvious dˆ < dr ˆ because dr > di for all i = r + 1, r + 2, . . . , N and d, which is the mean value of dr+1 , dr+2 , . . . , dN . From (62) and (65),

N  i=1

 =

[10]

[12]

We can easily see that

i=r+1 N

[9]

[11]

N  di Gi (c) = ( + log c) c i=r+1 i=r+1

N 

[8]

+ log c for i = r + 1, r + 2, . . . , N.

N 

G(c) =

[6]

Gi (c) =

r 

N 

Gi (c) +

i=1

[14]

[15]

[16]

Gi (c)

i=r+1

decreasing if c ≤ dˆ ˆ increasing if c > d.

(65) [17] ∗

Consequently, the optimal solution c , which minimizes the cost function G(c), is determined as  cˆ if cˆ > dˆ ∗ c = ˆ ˆ d if cˆ ≤ d.

[18]

(66)

REFERENCES [1]

[2]

[3]

512

[19]

Monzingo, R. A., and Miller, T. Introduction to Adaptive Arrays, 1st ed. Herndon, VA: SciTech Publishing, 2004. Guerci, J. R. Space–Time Adaptive Processing for Radar. Norwood, MA: Artech House, July 2003. Klemm, R. Principles of Space–Time Adaptive Processing. Piscataway, NJ: IEE Publishing, Apr. 2002.

[20]

[21]

Rangaswamy, M., Lin, F. C., and Gerlach, K. R. Robust adaptive signal processing methods for heterogeneous radar clutter scenarios. Signal Processing, 84, 9 (Sept. 2004), 1653–1665. Robey, F. C., Fuhrmann, D. R., Kelly, E. J., and Nizberg, R. A CFAR adaptive matched filter detector. IEEE Transactions on Aerospace and Electronic Systems, 28, 1 (Jan. 1992), 208–216. Himed, B., and Melvin, W. L. Analyzing space–time adaptive processors using measured data. Conference Record of the 31st Asilomar Conference on Signals, Systems and Computers, 1 (Nov. 1997), 930–935. Capraro, G. T., Farina, A., Griffiths, H., and Wicks, M. C. Knowledge-based radar signal and data processing: A tutorial overview. IEEE Signal Processing Magazine, 23, 1 (Jan. 2006), 18–29. Haykin, S. Cognitive radar: A way of the future. IEEE Signal Processing Magazine, 23, 1 (Jan. 2006), 30–40. Guerci, J. R., and Baranoski, E. J. Knowledge-aided adaptive radar at DARPA: An overview. IEEE Signal Processing Magazine, 23, 1 (Jan. 2006), 41–50. Wicks, M. C., Rangaswamy, M., Adve, R., and Hale, T. B. Space–time adaptive processing: A knowledge-based perspective for airborne radar. IEEE Signal Processing Magazine, 23, 1 (Jan. 2006), 51–65. Miranda, S., Baker, C., Woodbridge, K., and Griffiths, H. Knowledge-based resource management for multifunction radar: A look at scheduling and task prioritization. IEEE Signal Processing Magazine, 23, 1 (Jan. 2006), 66–76. De Maio, A., De Nicola, S., Landi, L., and Farina, A. Knowledge-aided covariance matrix estimation: A MAXDET approach. IET Radar, Sonar and Navigation, 3, 4 (Aug. 2009), 341–356. De Maio, A., Foglia, G., Farina, A., and Piezzo, M. Estimation of the covariance matrix based on multiple a priori models. In Proceedings of the IEEE Radar Conference, Xi”an, China, May 2010, 1025–1029. Wang, H., and Cai, L. A localized adaptive MTD processor. IEEE Transactions on Aerospace and Electronic Systems, 27, 3 (May 1991), 532–539. Gini, F., and Rangaswamy, M. Knowledge Based Radar Detection, Tracking and Classification. Hoboken, NJ: Wiley-Interscience, 2008. Li, H., Stoica, P., and Li, J. Computationally efficient maximum likelihood estimation of structured covariance matrices. IEEE Transactions on Signal Processing, 47, 5 (May 1999), 1314–1323. Chen, Y., Wiesel, A., Eldar, Y. C., and Hero, A. O. Shrinkage algorithms for MMSE covariance estimation. IEEE Transactions on Signal Processing, 58, 10 (Oct. 2010), 5016–5029. Ward, J. Space–time adaptive processing for airborne radar. Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA, Tech Rep., Dec. 1994. Steiner, M., and Gerlach, K. Fast converging adaptive processors or a structured covariance matrix. IEEE Transactions on Aerospace and Electronic Systems, 36, 4 (Oct. 2000), 1115–1126. Anderson, T. W. Asymptotic theory for principal component analysis. The Annals of Mathematical Statistics, 34, 1 (1963), 122–148. Haimovich, A. The eigencanceler: Adaptive radar by eigenanalysis methods.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

[22] [23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

IEEE Transactions on Aerospace and Electronic Systems, 32, 2 (Apr. 1996), 532–542. Skolnik, M. Radar Handbook. New York: McGraw-Hill, 1990. Wax, M., and Kailath, T. Detection of signals by information theoretic criteria. IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP-33, 2 (Apr. 1985), 387–392. Widrow, B., Mantey, P. E., Griffths, L. J., and Goode, B. B. Adaptive antenna systems. Proceedings of the IEEE, 55 (Dec. 1967), 2143–2159. Applebaum, S. P. Adaptive arrays. Syracuse University Research Corporation, North Syracuse, NY, Tech Rep., Aug. 1966. Reed, I. S., Mallett, J. D., and Brennan, L. E. Rapid convergence rate in adaptive arrays. IEEE Transactions on Aerospace and Electronic Systems, AES-10, 6 (1974), 853–863. Nitzberg, R. Application of maximum likelihood estimation of persymmetric covariance matrices to adaptive processing. IEEE Transactions on Aerospace and Electronic Systems, AES-16, 1 (Jan. 1980), 124–127. Fuhrmann, D. R. Application of Toeplitz covariance estimation to adaptive beamforming and detection. IEEE Transactions on Signal Processing, 39, 10 (Oct. 1991), 2194–2198. Abramovich, Y. I., Gray, D. A., Gorokhov, A. Y., and Spencer, N. K. Positive-definite Toeplitz completion in DOA estimation for nonuniform linear antenna arrays. I: Fully augmentable arrays. IEEE Transactions on Signal Processing, 46, 9 (Sept. 1998), 2458–2471. Conte, E., Lops, M., and Ricci, G. Adaptive detection schemes in compound-Gaussian clutter. IEEE Transactions on Aerospace and Electronic Systems, 34, 4 (Oct. 1998), 1058–1069. Roman, J. R., Rangaswamy, M., Davis, D. W., Zhang, Q., Himed, B., and Michels, J. H. Parametric adaptive matched filter for airborne radar applications. IEEE Transactions on Aerospace and Electronic Systems, 36, 2 (Apr. 2000), 677–692. Wang, P., Li, H., and Himed, B. A simplified parametric GLRT for STAP detection. In Proceedings of the IEEE Radar Conference, Pasadena, CA, May 2009, 1–5. Kraay, A. L., and Baggeroer, A. B. A physically constrained maximum-likelihood method for snapshot-deficient adaptive array processing. IEEE Transactions on Signal Processing, 55, 8 (Aug. 2007), 4048–4063. Aubry, A., De Maio, A., Pallotta, L., and Farina, A. Maximum likelihood estimation of a structured covariance matrix with a condition number constraint. IEEE Transactions on Signal Processing, 60, 6 (June 2012), 3004–3021. De Maio, A., and Farina, A. Adaptive radar detection: A Bayesian approach. In Proceedings of the International Radar Symposium, Krak´ow, Poland, May 2006, 1–4. De Maio, A., Farina, A., and Foglia, G. Adaptive radar detection: A Bayesian approach. In Proceedings of the IEEE Radar Conference, Boston, MA, Apr. 2007, 624–629. Besson, O., Tourneret, J. Y., and Bidon, S. Knowledge-aided Bayesian detection in heterogeneous environments.

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

IEEE Signal Processing Letters, 14, 5 (May 2007), 355–358. Wang, P., Li, H., and Himed, B. A Bayesian parametric test for multichannel adaptive signal detection in nonhomogeneous environments. IEEE Signal Processing Letters, 17, 4 (Apr. 2010), 351–354. Gurram, P. R., and Goodman, N. A. Spectral-domain covariance estimation with a priori knowledge. IEEE Transactions on Aerospace and Electronic Systems, 42, 3 (July 2006), 1010–1020. Melvin, W. L., and Showman, G. A. An approach to knowledge-aided covariance estimation. IEEE Transactions on Aerospace and Electronic Systems, 42, 3 (July 2006), 1021–1042. Ledoit, O., and Wolf, M. Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10, 5 (Dec. 2003), 603–621. Stoica, P., Li, J., Zhu, X., and Guerci, J. R. On using a priori knowledge in space–time adaptive processing. IEEE Transactions on Signal Processing, 56, 6 (June 2008), 2598–2602. Won, J. H., Lim, J., Kim, S. J., and Rajaratnam, B. Maximum likelihood covariance estimation with a condition number constraint. Department of Statistics, Stanford University, Stanford, CA, Tech Rep., Aug. 2009. Monga, V., and Rangaswamy, M. Rank constrained ML estimation of structured covariance matrices with applications in radar target detection. In Proceedings of the IEEE Radar Conference, Atlanta, GA, May 2012, 475–480. Boyd, S., and Vandenberghe, L. Convex Optimization, 2nd ed. Cambridge, UK: Cambridge University Press, 2004. Cao, G., and Bouman, C. Covariance estimation for high dimensional data vectors using the sparse matrix transform. Purdue University, Lafayette, IN, Tech Rep., Apr. 2008. Kang, B., and Monga, V. Strict and relaxed rank constrained ML estimator of structured covariance matrices. The Pennsylvania State University, University Park, PA, Tech Rep., Nov. 2012, http://signal.ee.psu.edu/TechReport StrictRCML.pdf. Bergin, J., and Techau, P. High-fidelity site specific radar simulation: KASSPER ‘02 workshop datacube. DAPRA Tech Rep. ISL-SCRD-TR-02-105, Defense Advanced Research Projects Agency, Arlington, VA, May 2002. Hoffbeck, J. P., and Landgrebe, D. A. Covariance matrix estimation and classification with limited training data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18, 7 (July 1996), 763–767. Abramovich, Y. I. A controlled method for adaptive optimization of filters using the criterion of maximum SNR. Radio Engineering and Electronic Physics, 3 (1981), 87–95. Carlson, B. D. Covariance matrix estimation errors and diagonal loading in adaptive arrays. IEEE Transactions on Aerospace and Electronic Systems, 24, 4 (July 1988), 397–401. Kirsteins, I., and Tufts, D. Adaptive detection using low rank approximation to a data matrix. IEEE Transactions on Aerospace and Electronic Systems, 30, 1 (Jan. 1994), 55–67.

KANG ET AL.: RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

513

Bosung Kang (S’12) received B.S. and M.S. degrees from Yonsei University, Seoul, Korea, in 2005 and 2007, respectively. He is pursuing a Ph.D. degree in the Department of Electrical Engineering, The Pennsylvania State University, University Park. Kang worked at LG Electronics as a research engineer, Seoul, Korea, from 2007 to 2011. He is working on covariance matrix estimation in radar applications. His research interests include signal processing, detection and estimation, and convex optimization. Vishal Monga (S’98—M’05—SM’12) received a B.S. degree from the Indian Institute of Technology Guwahati, Guwahati, India, and in 2005 a Ph.D. degree in electrical engineering from the University of Texas, Austin. Monga is an assistant professor of electrical engineering at the main campus of The Pennsylvania State University, University Park. He was with Xerox Research from 2005 to 2009. He established the Information Processing and Algorithms Laboratory at The Pennsylvania State University. His research interests are broadly in statistical signal and image processing. Current research themes in his laboratory include computational color and imaging, multimedia mining and security, and robust image classification and recognition. Monga serves as an associate editor of IEEE Transactions on Image Processing and the SPIE Journal of Electronic Imaging. While with Xerox Research in Rochester, NY, he was selected as the 2007 Rochester Engineering Society Young Engineer of the Year. He was also the recipient of a 2011 Air Force Faculty Fellowship and a Monkowski Early Career Award from the College of Engineering at The Pennsylvania State University. 514

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 50, NO. 1 JANUARY 2014

Muralidhar Rangaswamy (S’89—M’93—SM’98—F’06) received a B.E. degree in electronics engineering from Bangalore University, Bangalore, India, in 1985 and M.S. and Ph.D. degrees in electrical engineering from Syracuse University, Syracuse, NY, in 1992. He is presently employed as the technical advisor for the RF Exploitation Technology Branch within the Sensors Directorate of the Air Force Research Laboratory (AFRL). Before this, he has held industrial and academic appointments. His research interests include radar signal processing, spectrum estimation, modeling non-Gaussian interference phenomena, and statistical communication theory. He has cowritten more than 150 refereed journal and conference record papers in the areas of his research interests. In addition, he is a contributor to seven books and is a coinventor on three U.S. patents. Rangaswamy is the technical editor (associate editor-in-chief) for radar systems in IEEE Transactions on Aerospace and Electronic Systems. He served as the co–editor-in-chief for the Digital Signal Processing journal between 2005 and 2011. Rangaswamy serves on the senior editorial board of the IEEE Journal of Selected Topics in Signal Processing (January 2012–December 2014). He was a two-term elected member of the sensor array and multichannel processing technical committee of the IEEE Signal Processing Society between January 2005 and December 2010 and serves as a member of the Radar Systems Panel (RSP) in the IEEE Aerospace and Electronic Systems Society (IEEE-AESS). He was the general chairman for the 4th IEEE Workshop on Sensor Array and Multichannel Processing, Waltham, MA, July 2006. Rangaswamy has served on the technical committee of the IEEE Radar Conference series in myriad roles (track chair, session chair, special session organizer and chair, paper selection committee member, and tutorial lecturer). He served as the publicity chair for the 1st IEEE International Conference on Waveform Diversity and Design, Edinburgh, U.K., November 2004. He presently serves on the conference subcommittee of the RSP. He is the technical program chairman for the 2014 IEEE Radar Conference. Rangaswamy received the 2012 IEEE Warren White Radar Award, the 2013 Affiliate Societies Council Dayton Outstanding Scientist and Engineer Award, the 2007 IEEE Region 1 Award, the 2006 IEEE Boston Section Distinguished Member Award, and the 2005 IEEE-AESS Fred Nathanson Memorial Outstanding Young Radar Engineer Award. He was elected as a fellow of the IEEE in January 2006 with a citation “for contributions to mathematical techniques for radar space–time adaptive processing.” He received the 2012 and 2005 Charles Ryan Basic Research Award from the Sensors Directorate of AFRL, in addition to more than 40 scientific achievement awards. KANG ET AL.: RCML ESTIMATION OF STRUCTURED COVARIANCE MATRICES

515