I. INTRODUCTION
Array Signal Processing via Sparsity-Inducing Representation of the Array Covariance Matrix ZHANG-MENG LIU, Member, IEEE ZHI-TAO HUANG YI-YU ZHOU National University of Defense Technology China
A method named covariance matrix sparse representation (CMSR) is developed to detect the number and estimate the directions of multiple, simultaneous sources by decomposing the array output covariance matrix under sparsity constraint. In CMSR the covariance matrix elements are aligned to form a new vector, which is then represented on an overcomplete spatial dictionary, and the signal number and directions are finally derived from the representation result. A hard threshold, which is selected according to the perturbation of the covariance elements, is used to tolerate the fitting error between the actual and assumed models. A computation simplification technique is also presented for CMSR in special array geometries when more than one pair of sensors has equal distances, such as the uniform linear array (ULA). Moreover, CMSR is modified with a blind-calibration process under imperfect array calibration to enhance its adaptation to practical applications. Simulation results demonstrate the performance of CMSR.
Manuscript received November 24, 2010; revised June 23 and October 20, 2011, and January 8, 2012; released for publication November 17, 2012. IEEE Log No. T-AES/49/3/944607. Refereeing of this contribution was handled by J. Lee. Authors’ address: School of Electronic Science and Engineering, National University of Defense Technology of China, Deya Road, Changsha 410073, China, E-mail: (
[email protected]).
c 2013 IEEE 0018-9251/13/$26.00 ° 1710
Array signal processing has attracted much attention in the past decades in both commercial and military applications, and various methods have been proposed to deal with the problems in this field. Among them source number detection and direction-of-arrival (DOA) estimation are hot topics, and most of the previous literature focuses on parametric methods to deal with them [1]. Those methods have been demonstrated to perform well in source detection and localization under moderate settings [1—3]. However, most of them rely heavily on accurate covariance matrix estimates, which are hardly available in the most demanding scenarios. Such constraint of the parametric methods leads to much deteriorated performance under low signal-to-noise ratio (SNR) and limited snapshots, and it has blocked their applicability in various practical circumstances [4]. Therefore, further research is required to better address the detection and localization problem of multiple sources in such scenarios. Recently, the technique of sparse representation has attracted much interest in the areas of statistical signal analysis and parameter estimation. Methods falling into this category decompose the observation on overcomplete dictionaries to recover the signal components so as to analyze the signal property and estimate their parameters. By now various methods have been reported to realize such sparse representation; among them, representative examples include matching pursuit (MP) [5], basis pursuit (BP) [6], focal underdetermined system solution (FOCUSS) [7], and sparse Bayesian learning (SBL) [8]. These methods were originally designed for the single measurement vector (SMV) case and then were extended by Cotter, et al. and Wipf, et al. to the multiple measurement vector (MMV) case with the joint-sparsity-enforcing idea. Methods of M-MP, M-FOCUSS [9], and M-SBL [10] were proposed in the MMV case. These joint-sparsity-based methods succeed finding the sparse decomposition of MMV in polynomial time, but the stopping criterion in M-MP and the regularization criterion in M-FOCUSS are difficult to determine, and M-SBL converges slowly in its original form [11]. Among the existing sparse representation methods, some have been introduced to deal with the problem of DOA estimation. The first one, as far as we know, is the global matched filter (GMF) method proposed by Fuchs [12]. GMF is based on uniform circular array and exploits the beam-space samples to realize DOA estimation. Then, Malioutov, et al. proposed the method of L1-SVD (singular valve decomposition) [13, 14], which reduces the model dimension via SVD of the array output. A homotopy technique was later used to optimize the regularization parameter of L1-SVD [15]. L1-SVD contributes much to the
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3
JULY 2013
development of the sparsity-based DOA estimation technique, but we show, in the simulation section, that it does not perform to a satisfying level in DOA estimation precision for spatially adjacent signals. More recently Hyder, et al. introduced their L0-based joint sparse approximation technique [16] to deal with the DOA estimation problem, and they proposed the method of JLZA-DOA (JLZA: joint l0 approximation) [17]. In JLZA-DOA a family of convex Gaussian functions is used to approach the concave L0 function in describing the spatial sparsity of the incident signals. However, as the iteratively updating Gaussian variances are chosen subjectively, it is not guaranteed that JLZA-DOA will converge to the global solution. Moreover, only limited work [13] has been done to study and improve the performance of those sparsity-inducing methods in the presence of array imperfections, such as inaccurate sensor calibration, which is another main concern in practical direction finding systems. This paper develops a source detection and localization method named covariance matrix sparse representation (CMSR) by representing the array output covariance matrix under sparsity constraint. In CMSR the lower left triangular covariance elements are aligned to form a vector, and the signal components are recovered from this vector via sparse representation for source number detection and DOA estimation. In this way the original MMV problem is transformed into an SMV one, and the energy of each signal is aligned better than the joint L2-norm constraint of the MMV representation, which may have led to the higher DOA estimation precision of CMSR than L1-SVD in the simulation section of this paper. Another particularity that differs CMSR from L1-SVD is that the coefficients of the signal components correspond to the signal powers and, thus, are positive real entities, so no model transformation is needed in CMSR to fit the optimization tools [18, 19]. Moreover, CMSR owns an intrinsic characteristic to make good use of the special array geometries, such as minimum redundancy array (MRA) [20], to extend its adaptation to multiple simultaneous signals. Modifications are also made to CMSR by taking imperfect array calibration into consideration to enhance its adaptation to practical applications. Wallace and Jensen have utilized the idea of sparse covariance decomposition to estimate the power angle spectrum (PAS) with spatial continuous bases, and they have also briefly referred (but not explicitly studied) some possible extensions of their model to better fit related problems [21]. Some of the mentioned extensions are also the major concerns in the area of array processing, such as the sparse decomposition of the covariance vector in the element- and beam-spaces, the redundancy reduction by exploiting the shift invariance, and the
spurious arrivals that emerge in the reconstructed spectrum. Nonetheless, only superficial statements were made, and they did not go deep in their study to provide the explicit process for deriving accurate source number and high-precision DOA estimates, which are the major concerns of this paper. To better serve the task of detection and localization of multiple point sources, we consider, in detail, the aspects in sparsity-inducing spatial spectrum estimation and the subsequent detection and estimation procedures. Great efforts are made to analyze the perturbation contaminated in the covariance matrix estimate due to limited snapshots so as to facilitate the selection of the hard threshold during the representation of the inaccurate covariance vector estimate. A 2-norm penalty is used to constrain the fitting error, and the problem is finally solved via L1-L2 regression to keep in line with the rapidly emerging results and toolboxes in quadratic programming [22]. The modifications made to CMSR in the presence of imperfect array calibration is also new in the area of sparsity-promoted array processing, and sufficient simulations are carried out in this paper to compare the performance of CMSR with state-of-the-art methods in both detection and DOA estimation. The rest of this paper mainly consists of seven parts. Section II reviews the idea of the sparsity-based DOA estimation. Section III presents the method of CMSR in general linear arrays in detail. Section IV aims at the simplification of CMSR in arrays with equally-spaced sensors. Section V studies explicitly how to obtain the signal number and high-precision DOA estimates from the CMSR representation result. Section VI proposes a blind-calibration technique to enhance the adaptation of CMSR to imperfect array calibration. Section VII carries out simulations to demonstrate the satisfying performance of CMSR. Section VIII concludes the paper. II.
PROBLEM FORMULATION
When K narrowband signals impinge onto an M-element linear array from directions of # = [#1 , : : : , #K ], the array output at time t is y(t) = A(#)x(t) + v(t)
(1)
where A(#) = [a(#1 ), : : : , a(#K )] is the array responding matrix to the K signals with a(#k ) = [ej'k,1 , : : : , ej'k,M ]T , 'k,m is the phase-delay of the kth signal propagating from the reference to the mth sensor, and x(t) and v(t) are the signal waveform and additive noise, respectively. In this paper we use the functions of a(¢) and A(¢) and their variants to represent the mappings from a direction to an array responding vector and the one from a direction set to an array responding matrix, respectively.
LIU, ET AL.: ARRAY SIGNAL PROCESSING VIA COVARIANCE MATRIX REPRESENTATION
1711
Equation (1) shows that y(t) is a noisy summation of K signal components. If one samples the potential space of the incident signals to obtain a direction set £ and form the corresponding array responding matrix A(£), then the spatial distribution of those signals can be estimated by solving the following sparsity-inducing minimization problem [12—15]
Concatenating those elements column-by-column obtains a new covariance vector r, i.e.
xˆ (t) = min ky(t) ¡ A(£)x(t)k22 + °kx(t)k1
˜ where A(#) = [a˜ (#1 ), : : : , a˜ (#K )], a˜ (#k ) = j('k,2 ¡'k,1 ) [e , : : : , ej('k,M ¡'k,1 ) , : : : , ej('k,M ¡'k,M¡1 ) ]T , ´ = [´1 , : : : , ´K ]T . The estimate of this covariance vector derived ˆ is perturbation-contaminated due to finite from R sampling, i.e. ˜ rˆ = A(#) ´ˆ + "
x(t)
(2)
where ° is the regularization parameter used for balancing the data fitting error and model parsimony, and it is often chosen empirically [23]. The direction set £ is also selected empirically by taking superresolution into consideration, and an equal spatial sampling strategy is always used to simplify notation with a dense enough interval to enhance superresolution and decrease quantization error, which makes L À M and A(£) overcomplete, with L being the cardinality of £. For example, if we sample the [¡90± 90± ] scope with 0:5± interval in a 7-element array, we obtain an overcomplete dictionary of size 7 £ 361. III. DOA ESTIMATION VIA CMSR In this paper we estimate the signal directions by sparsely representing a covariance vector formed with the covariance matrix elements in the MMV scenario, and we name the new method CMSR. Assume that the K incident signals are independent with power ´1 , : : : , ´K , and the covariance matrix is estimated from N snapshots as follows N X ˆ = 1 y(t)yH (t): R N
(3)
t=1
ˆ approaches its When N is adequately large, R perturbation-free counterpart, i.e., ˆ = Efy(t)yH (t)g = R = lim R N!+1
K X
´k a(#k )aH (#k ) + ¾v2 I
k=1
(4) where R is Rp,q =
¾v2
is the noise variance. The (p, q)th element of
K X k=1
´k ej('k,p ¡'k,q ) + ¾v2 ±(p ¡ q),
1 · p,
q·M (5)
where ±(p ¡ q) is the indicator function that equals 1 when p ¡ q = 0 and 0 otherwise. As R is conjugate symmetric, its upper right elements can be represented by the lower left ones, and the diagonal elements are equal and contaminated by unknown noise power, so we extract the M(M ¡ 1) =2 lower left elements of R for DOA estimation. 1712
r = [R2,1 , : : : , RM,1 , : : : , RM,M¡1 ]T :
(6)
Then, it is obvious that r has a K-element summation form as follows ˜ r = A(#)´ (7)
(8)
where " is the perturbation within rˆ that deviates from ˜ the array responding matrix A(#) and ´ˆ consists of the signal power estimates derived from finite-length signal waveforms (see Appendix I for details). Finally, we can recover the signal components from this covariance vector via quadratic programming [22] ˜ 1 min k´k s.t.
(9)
˜ ˜ 2·¯ krˆ ¡ A(£) ´k
˜ where A(£) is an overcomplete dictionary formed by a˜ (μ) with μ 2 £ and ¯ is the fitting error threshold. A more rigorous form of the above objective function is to replace the L1-norm function with an L0-norm one, which accurately stands for the number of signal components contained in the covariance vector. But, as the L0-norm function is concave and NP-hard to optimize, this relaxed convex approximation is more attractive and used more comprehensively. The hard threshold describes the deviation of the estimated covariance vector from the assumed representation model; it depends directly on the perturbation level of the covariance vector estimate. With some straightforward derivation we conclude that the perturbation of the covariance element estimates obeys Proposition 1. PROPOSITION 1 For adequately large N the perturbation of the covariance vector rˆ that deviates ˜ ˆ denoted by ", is approximately from the model A(#) ´, circular complex Gaussian distributed with zero mean, and the variance is given by 0 1 K K K X 1 X BX C Var("p,q ) = f ´k @ ´k0 A + 2¾v2 ´k + ¾v4 g, N 0 k =1 k=1
k 0 6=k
k=1
1·q
