Research Article Augmented Lagrange Based on Modified Covariance Matching Criterion Method for DOA Estimation in Compressed Sensing

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1 e Scientific World Journal, Article ID 24469, page Reearch Article Augmented Lagrange Baed on Modified Covariance Matching Criterion Method for DOA Etimation in Compreed Sening Weijian Si, Xinggen Qu, and Lutao Liu Department of Information and Communication Engineering, Harbin Engineering Univerity, Harbin 5000, China Correpondence hould be addreed to Lutao Liu; Received 27 November 203; Accepted 9 December 203; Publihed February 204 Academic Editor: H. R. Karimi, Z. Yu, and W. Zhang Copyright 204 Weijian Si et al. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. A novel direction of arrival (DOA) etimation method in compreed ening (CS) i preented, in which DOA etimation i conidered a the joint pare recovery from multiple meaurement vector (MMV). The propoed method i obtained by minimizing the modified-baed covariance matching criterion, which i acquired by adding penaltie according to the regularization method. Thi minimization problem i hown to be a emidefinite program (SDP) and tranformed into a contrained quadratic programming problem for reducing computational complexity which can be olved by the augmented Lagrange method. The propoed method can ignificantly improve the performance epecially in the cenario with low ignal to noie ratio (SNR), mall number of naphot, and cloely paced correlated ource. In addition, the Cramér-Rao bound (CRB) of the propoed method i developed and the performance guarantee i given according to a verion of the retricted iometry property (RIP). The effectivene and atifactory performance of the propoed method are illutrated by imulation reult.. Introduction Direction of arrival (DOA) etimation of multiple narrowband ource ha been an intereting reearch topic in array ignal proceing. It application pan many field including radar, communication ytem, and medical imaging [, 2]. Many effective algorithm are propoed for DOA etimation, which are mainly claified into three categorie. The beamforming algorithm uch a MVDR [3]andMEM[4]obtain a nonparametric patial pectrum by optimizing the filter weight. The ubpace algorithm uch a MUSIC [5] and ESPRIT [6] and their derivative [7, 8] exploit the orthogonalityofignalubpaceandnoieubpacefordoaetimation. The ubpace fitting algorithm including Maximum Likelihood (ML) [9] and Weighted Subpace Fitting (WSF) [0] olve a multidimenional nonlinear optimization problem to obtain a precie etimation, but a good initialization i required to enure global convergence and computational complexity i very high. All thee algorithm focu on two important iue: reolution (i.e., the ability to correctly reolve two cloely paced ource) and preciion (i.e., the deviationfromtruedoa)whichareconideredtobethe theoretical bae of evaluating certain algorithm. Many high reolution method uffer from eriou performance degradation in the cenario with low SNR, mall number of naphot, and cloely paced corrected ource. More recently, many reearch application involving compreed ening(cs)[], epecially DOA etimation [2, 3], have become more and more popular in the ignal proceing. Moreover, the retricted iometry property (RIP) baed on the perfect theoretical framework given with modern probability theory by Donoho [] and Candé et al. [4] provide the performance guarantee in CS. Hence, in thi paper, DOA etimation i poed a a joint pare recovery problem where we recover jointly pare ource from multiple meaurement vector (MMV) under CS framework. CS i an emerging area which can break through the limit of Nyquit ampling theorem. On one hand, CS can imultaneouly capture and tore compreed or pare ource at a rate much lower than that of Nyquit ampling. On the other hand, it can recover the original ource uing nonadaptive linear projection meaurement onto a uitable meaurement matrix which atifie the RIP. In CS, the joint pare recovery aim to find a common upport hared by unknown pare

2 2 The Scientific World Journal vector from MMV, which i obtained by the meaurement matrix. Support denote the indice of the nonzero element in the unknown pare vector. A pare olution can be obtained a long a the upport i determined. CS theory ha been widely applied to DOA etimation according to the ource parity, which reult from the fact thattherearemuchfewerourcedirectionthanallpotential direction in the patial domain. The DOA etimation method in CS are attractive ince they have much better etimation performance than conventional etimation method. In [5], Gürbüz et al. firtly formulate the DOA etimation problem under CS framework in the time domain. Wang et al. [6] propoeanewarchitecturetoetimatedoaby exploiting compreive ampling in the patial domain. Stoica et al. [7] make full ue of covariance matching criterion and preent a emiparametric/pare etimation method and it derivative called LIKES [8] for the eparable model. Malioutov et al. [9] preentthel -SVD algorithm for DOA etimation which combine the SVD of the data matrix with a pare recovery method baed on l -norm minimization. A new cla of ubpace-bae algorithm uch a compreive MUSIC (CS-MUSIC) [20] and ubpace-augmented MUSIC (SA-MUSIC) [2]ipropoedinrecentyear. The RIP and variou modified verion of it have been ued a a foundation of performance guarantee [2 24] for the joint pare recovery. The performance guarantee of MUSIC baed on joint pare recovery i given for identifying theuniqueupportinafavorablecae[25]. However, in the unfavorable cae where the number of meaurement vector i maller than the parity or the covariance matrix tend to loe rank due to correlated ource, performance guarantee fail. Lee et al. [2] propoe a new performance guarantee in term of a verion of the RIP under uch unfavorable condition. The performance guarantee of other method uch a greedy algorithm and convex relaxation have been developedtofindthepareolutionin[23, 24]. However, the guarantee of thee method cannot be imply extended to the MMV cae to obtain a better bound for the pare olution. In thi paper, we propoe a novel augmented Lagrange baed on modified covariance matching criterion method called AULMC for DOA etimation in CS. Thi method utilize the minimization of the modified covariance matching criterion which i acquired by uing regularization method to add penaltie in order to obtain the table pare parameter etimation, epecially in the low parity cae. Thi minimization problem i hown to be a emidefinite program (SDP) and tranformed into the contrained quadratic programming problem for the ake of reducing computational complexity. The augmented Lagrange function i formed to olve thi problem by the ue of augmented Lagrange method. AULMC ha a number of advantage over the other method. For example, it provide more precie etimation and higher reolution in the cenario with low SNR, mall number of naphot, and cloely paced correlated ource, anditdoenotneedprioriknowledgeaboutthenumber ofourceortochooetheregularizationparameterofthe l -optimization framework which i very difficult to elect in the DOA etimation. In addition, we give a detailed derivation proce of the cloed-form expreion for the Cramér-Rao bound (CRB) of the new method and dicu an explicit condition that guarantee performance of the new method. Thi performance guarantee i given in term of weaker verion of the RIP which i referred to a weak- RIP. Simulation reult illutrate the performance of the propoed method. It i worth noting that covariance matching criterion ha been ued for DOA etimation [26]. In [26], a pare iterative covariance-bae etimation method, abbreviated a SPICE, i propoed. Our approach i different from SPICE becaue it utilize modified covariance matching criterion which can guarantee the tability of olution even if the ource parity i rather low. In the future work, we will focu on the application of our approach to data-driven deign [27 30]. Now we briefly ummarize the contribution of thi work a follow. (i) A modified covariance matching criterion i propoed by adding penaltie according to the regularization method. (ii) The original SDP problem i tranformed into the contrained quadratic programming problem. The motivation to tranform the problem i that it can reduce computational complexity. (iii) Augmented Lagrange baed on modified covariance matching criterion method i devied to olve the reulting programming problem. (iv) The CRB and performance guarantee of the new method are given in detail. The ret of thi paper i organized a follow. In Section 2, we formulate the DOA etimation problem. Section 3 preent a modified covariance matching criterion. A novel augmented Lagrange baed on modified covariance matching criterion method for DOA etimation i decribed in detail in Section 4; the performance of which i analyzed in Section 5. The performance of the propoed method i evaluated by imulation in Section 6. Concluionare provided in Section Problem Formulation Conider L narrow-band ource, 2,..., L impinging on a enor array that conited of P omnidirectional enor from far field. At time intant t, the array received ource can be given by x (t) = L a (θ k ) k (t) + n (t), () where n(t) C P denote a noie term, θ k Ω i the unknown direction, of the kth ource where Ω denote the entire patial range and a(θ k ) i the P teering vector. Although the DOA etimation of the ingle naphot, which i a typical ingle meaurement vector (SMV) model, ha itvalue,thenumberofnaphotilargerthanoneinthe mot practical application. Correpondingly, the multiple naphot model i a MMV model.

3 The Scientific World Journal 3 Let { θ k } K denoteafinegridwhichcoverω where there exit K(K L)potential direction of the ource, 2,..., K o that the true direction {θ k } L are aligned or arecloetothegrid.thimeanthatif θ k, θ k2,..., θ kl are equal to θ,θ 2,...,θ L,repectively,wehave k ={ l, k=k l (l=,2,...,l), 0, elewhere. Hence,themultiplenaphotmodelcanbewrittenathe following pare form: x (t) = K a ( θ k ) k (t) + n (t) = A (t) + n (t), t=t,t 2...t M, where A =[a( θ ) a( θ 2 ) a( θ K )] i the P Kmanifold matrix correponding to all the potential direction which i alo referred to a an overcomplete dictionary in CS. (t) = [ (t) 2 (t) K (t)] T i a L-pare vector ince it ha at mot L nonzero element in K element, and L i defined a parity where the operator [ ] T denote tranport. {(t i )} M i= are jointly L-pare if they hare a common upport. Hence, the matrix S =[(t ) (t 2 ) (t M )] C K M ha no more than L nonzero row in order to be called row L-pare. The MMV problem i that of identifying the row upport of the unknown matrix S from the matrix Y C N M that conited of MMV which i given by (2) (3) Y = [y (t ) y (t 2 ) y (t M )] = ΦAS + ΦN (4) with a common meaurement matrix Φ of the ize N P with N < P where N ithenumberofnonadaptivelinear projection meaurement, uch a random Gauian matrix or random partial Fourier matrix, and noie matrix N. 3. Modified Covariance Matching Criterion In thi ection, the modified covariance matching criterion i developed in the CS cenario. A conventional covariance matrix of the compreed meaurement ource with the ize N Ni given by R y =E[y (t) y (t)] =ΦAR A Φ + ΦR n Φ, (5) where R = E[(t) (t)] i a K Kcovariance matrix of the pare ource whoe off-diagonal element denote the ource correlation and diagonal element denote the ource power. Since the power are one-to-one correponding to all the potential direction and our focu i on the ource angle parameter etimation, R canbereducedtoadiagonal matrix R ( θ) =diag ( θ θ2 θk ). According to (5), the meaurement matrix can change the covariance matrix of the noie to render the noie colored even if the noie i white. Therefore, thi advere factor mut be conidered before recovering jointly pare ource. In addition, the operator ( ) and E[ ] denote conjugate tranpoe and expectation, repectively. Since ΦR n Φ i a poitive definite Hermitian matrix, a prewhitenedmethodigivenbythecholekydecompoition. Let B be the Choleky factor that atifie (ΦR n Φ ) = B B, (6) where B C N N i an upper triangular matrix of poitive line. Hence, a prewhitened proce i implemented by multiplying y(t) by B in order to obtain a pure ource whoe covariance matrix i written a R y = BΦAR A Φ B + I N = CR C + I N, (7) where C = [c, c 2,...,c K ] and I N i an identity matrix of the ize N N. It i important to note that the unknown covariance matrix of the noie i tranformed into the known identity matrix after the prewhitened proce. Therefore, the prewhitened method improve the robutne to the noie. Then, the covariance matrix of the compreed meaurement ource denoiing i realized by R = R y I N = CR C. (8) The parameter θ can be etimated by a cla of the covariance matching etimation technique (COMET) baed on covariance matching criterion [3]. Thi parameter etimation method i well known to be a large-ample approximation of ML method and provide a more attractive olution than ML etimator. The principle of COMET i that of uing the right data to minimize it data model by the weighted leat-quare (WLS) method. However, the lower the ource parity i, the more likelyititobeill-poedforthecovariancematrixetimation error meaning that the optimal olution obtained directly by minimizing the conventional covariance matching criterion i intable. Hence, we employ regularization method to add penaltie in order to ufficiently exploit prior knowledge to reduce the olution pace for determining the table optimal olution. The modified covariance matching criterion i propoed a the following form: [vec ( R) vec (R ( θ))] Γ P Γ [vec ( R) vec (R ( θ))] +αtr [R ( θ) R ]+βtr [ R ( θ) R ], 4 where parameter α 0 control the olution moothne (guarantee preciion), parameter β 0control the olution cale (guarantee parity), the invere of the matrix Γ, the ample covariance matrix R, andr( θ) are aumed to exit, and the matrix P itheinvereofthecovariancematrix of the reidual, ε = Γ vec( R R( θ)). Since R i equal to R( θ) a M, it follow from [32] thatvec( R R( θ)) atifie the aymptotic normal ditribution with mean zero and variance N R T ( θ) R. In addition, the operator Tr[ ], vec( ), and denote matrix trace, column tacking operation, (9)

4 4 The Scientific World Journal and Kronecker product, repectively. Then, the matrix P can be given by P =E[ ε ( θ) ε ( θ)] = ΓE[vec ( R R ( θ)) vec ( R R ( θ))] Γ = ΓN R T ( θ) RΓ = ΓN DΓ. (0) We conider the normalized data model in (9)and chooe the regularization parameter α=β=nbaed on perceptual criterion [33]. By ubtituting (0) into(9), we have vec ( R R ( θ)) D vec ( R R ( θ)) + Tr [R ( θ) R ]+ Tr [R ( θ) R ]. 4 () By the propertie of vec,, and Tr, the data model can be further implified to vec ( R R ( θ)) D vec ( R R ( θ)) = + Tr [R ( θ)( R R ( θ)) R ]+ Tr [R ( θ) R ] 4 R /2 ( R R ( θ)+ I 2 N 2 ) R /2 ( θ), (2) where denote the Frobeniu norm for matrice or the Euclidean norm for vector. The data model in (2) i conidered to be the modified covariance matching criterion. It can be een from(2) that a poitive definite Hermitian matrix i added to the covariance matrix etimation error by applying penaltie according to regularization method in order to guarantee the tability of olution. 4. DOA Etimation In thi ection, we will utilize the minimization of the modified covariance matching criterion to etimate parameter θ. Let θ = arg min θ R /2 ( R R ( θ)+ I 2 2 ) R /2 ( θ) (3) be the optimal olution of θ in the tructure model of (8). Our goal i to utilize the modified covariance matching criterion for an etimate that i aymptotically equal to θ.bythe propertie of the trace and the Hermitian matrix, a derivation proce i hown a follow, where we omit the dependence on θ for notational convenience: f=tr [ R ( R R + I 2 ) R ( R R + I 2 )] Due to Tr [ R R]= K (c k R c k ) θ k (5) wecandeducethattheminimizationoff i equal to the minimization of h: h= Tr [( R /2 + 2 R /2 ) R /2 ( R /2 + 2 R /2 ) R /2 ] + K (c k R c k ) θ k. (6) Then, we will demontrate that the minimization of h in (6)withrepectto{ θ k } K i an SDP. To prove thi fact, One aume R /2 + 2 R /2 =[r r 2 r N ]. (7) The h in (6)canberewrittena h= N r k R r k + K (c k R c k ) θ k. (8) By the Schur complement, let {x k } N be auxiliary variable atifying [ x k r k] 0. (9) r k R The equivalent form of thi minimization problem i expreed a min x k, θk N x k + K (c k R c k ) θ k.t. θk 0,,2,...,K, [ x k r k] 0,,2,...,N. r k R (20) It i eay to ee that (20) iansdp[34]. Many oftware package can olve an SDP, but olving (6) aansdpi not a good choice becaue SDP olver have generally rather high computational complexity for the value of N, M, and K in the DOA etimation. To olve it effectively, we tranform it into the contrained quadratic programming problem for reducing computational complexity, a decribed next. Since a conitent etimation of θ i given by (5), we can reformulate the minimization of h in (6) a the following contrained minimization by the Schur inequality of the trace: = Tr [ R R R +( R /2 + 2 R /2 ) (4) min θ i 0 Tr [( R /2 + 2 R /2 ) R /2 ( R /2 + 2 R /2 ) R /2 ] R /2 ( R /2 + 2 R /2 ) R /2 ]..t. Wz = K, (2)

5 The Scientific World Journal 5 where W i a K Kdiagonal matrix of [C R C] /2 and z = [z z 2 z K ] T i a K-dimenional vector with z i =θ /2 i, i =,2,...,K.Byubtituting(8) into(2), the objective function in (2)canberewrittena Tr [( R /2 + 2 R /2 )(C ) /2 R /2 C /2 Applying the Newton method, we obtain 2 zz p σ k (z k, μ k )(z k+ z k )= z p σk (z k, μ k ). (29) For notational convenience, we aume that q k = q(z k ), f k = f(z k ), Π k = Π(z k ), b k = b(z k, μ k ), d k = z k+ z k and we get the following equation by (29): ( R /2 + 2 R /2 )(C ) /2 R /2 C /2 ] = Tr [C /2 ( R /2 + 2 R /2 )(C ) /2 R /2 (22) b k d k +σ k q k q k d k + z l(z k, μ k )+σ k q k Π k = 0. (30) Auming that the invere of b k exit, by left-multiplying (30)by(/σ k )q k b k,wehave C /2 ( R /2 + 2 R /2 )(C ) /2 R /2 ]. Baed on the following equation: Tr [MT (d) NT (d)] = d (M T N) d, (23) where denote the Schur-Hadamard product, M and N are both K Kmatrice, T(d) =diag(d d 2 d K ),andd = [d d 2 d K ] T, the objective function (22)canbewritten a Tr [QR /2 QR /2 ] = z Vz, (24) where Q = C /2 ( R /2 +(/2) R /2 )(C ) /2 and V = Q T Q. Hence, the minimization problem in (2)i tranformed into the following form: min z f (z) = z Vz (25).t. Π (z) = Wz K = 0 K which i a typical contrained quadratic programming problem. Itiwellknownthatanimportantclaofmethodfor olving the contrained quadratic programming problem i to form the auxiliary function. To olve (25), we form the following augmented Lagrange function with repect to (25): p σ (z, μ) =f(z) + μ Π (z) + 2 σ Π (z) 2, (26) where μ i the aymptotical olution of the Lagrange multiplier of (25)andσ i a penalty factor. By etting the gradient and the Heian matrix of (26) with repect to z to zero, we have z p σ (z, μ) = z l(z, μ)+σq (z) Π (z), 2 zz p σ (z, μ) = 2 K zz l(z, μ)+σ Π i (z) 2 Π (z) i= +σq (z) q (z), (27) where l(z, μ) = f(z) +μ Π(z) and q(z) = Π (z) = [ Π (z) Π 2 (z) Π K (z)]. One aume that b (z, μ) = 2 K zz l(z, μ)+σ Π i (z) 2 Π (z). (28) i= ( I K σ k + q k b k q k) q k d k = q k b k q kπ k σ k q k b k zl(z k, μ k ). It follow from (3)that where α k atifie (3) q k b k = Π k + α k σ k, (32) ( I K σ k + q k b k q k) α k = q k b k zl(z k, μ k )+Π k. (33) By ubtituting (32)into(30), we have d k = b k (q kα k + z l(z k, μ k )). (34) Both the multiplier factor and the penalty factor are determined with difficulty for utilizing the augmented Lagrange function to olve the contrained quadratic programming problem. Hence, an updated equence for the multiplier factor i given in term of Propoition. Propoition. One aume that μ(z) i the optimal olution of the problem min μ C K q(z)μ + f(z) 2.Then,thefollowing equation hold: where β k =(q k q k) q k b k. Proof. See Appendix A. μ (z k )=μ k + α k + β k d k, (35) It can be deduced from Propoition that μ k + α k i referred to a the next iteration of μ k. We apply a heuritic update equence for the penalty factor to achieve a table olution. If the kth iterative olution z k i cloer to the feaible region than the previou olution z k, the penalty factor i decreaed. Inverely, we increae the penalty factor when z k i not cloer to the feaible region. The pecific tep of olving by the augmented Lagrange method are given a follow. Initialization: et, z C K, σ >0, τ (0, ), and μ = (q q ) q f.

6 6 The Scientific World Journal () Calculate α k and d k in term of (33)and(34). If d k 0, z k i the KKT point of the problem (25); then top iteration. Then, the partial derivative of (39) withrepectto θ, r (t j )=Re{(t j )} and i (t j )=Im{(t j )} are given by (2) We ue Armijo earch method to find the maximum of t k atifying p σk (z k +t k d k, μ k ) p σk (z k, μ k )+τt k p σ k (z k, μ k ) d k. (36) (3) Set z k+ = z k +t k d k and update the multiplier factor and penalty factor, repectively: M ln L θ =2 Re {S j U Φ B n (t j )}, j= ln L r (t j ) =2Re {A Φ B n (t j )}, ln L i (t j ) =2Im {A Φ B n (t j )}, (40) μ k+ = μ k + α k, { 2σ k if Π (z k+) > Π (z k), σ k+ = { { 2 σ k if Π (z k+) Π (z k). (37) where S j = diag((t j )) and U = [da( θ )/d θ da( θ 2 )/ d θ 2 da( θ K )/d θ K ]. Following [35, 37], we can obtain the Fiher information matrix (FIM) a follow: FIM = Λ [Δ r Δ i Δ 2r Δ Mi Δ Mr ] (4) Set k=k+and return to tep (). 5. Performance Analyi 5.. Cramér-Rao Bound. In thi ubection, the cloed-form expreion for the CRB of the propoed method with complex white Gauian noie after the prewhitened proce i illutrated. The bound of the noie variance etimation can be computed eparately a CRB n = /NM (ee [35]). The remaining parameter conit of an unknown vector φ = [ θt T ] T.ItinotaneaytaktogettheCRBoftheunknown parameter. However, fortunately, [36, 37] haveprovideda critical inpiration for the derivation in thi paper. Firt, the likelihood function i given by L( y (t j ), θ) = (2π) MN (/2) MN exp { { { M j= Thu, the log-likelihood function i (y (t j ) BΦA (t j )) (y (t j ) BΦA (t j )) } }. } ln L( y (t j ), θ) =cont (y (t j ) BΦA (t j )) M j= (38) (y (t j ) BΦA (t j )). (39) where G r G i 0 0 G i G r d.. 0 d 0 [. d G r G ] i [ [ 0 0 G i G r ] [ T ln L ln L E{ ( θ θ ) }=Λ M =2 j= Δ r Δ i Δ 2r. Δ Mr Δ Mi, ] ] (4) Re {S j U Φ B BΦUS j }, E { T ln L ln L } { ( { r (t k ) i (t p ) ) } = G i = Im {G } } = Im {2A Φ B BΦA}δ kp, E { T ln L ln L } { ( { r (t k ) r (t p ) ) } =E { T ln L ln L } { ( } { i (t k ) i (t p ) ) } } = G r = Re {G } = Re {2A Φ B BΦA}δ kp, T ln L ln L E{ ( r (t j ) θ ) }=Re {Δ j }=Δ jr = Re {2A Φ B BΦUS j }, j=,2,...,m,

7 The Scientific World Journal 7 T ln L ln L E{ ( i (t j ) θ ) }=Im {Δ j }=Δ ji It i well known that = Im {2A Φ B BΦUS j }, j=,2,...,m. (42) [Δ T r Δ T i ][G r G i ][ Δ r]=re {Δ GΔ}. (43) G i G r Δ i Itcanbededucedfrom(4)and(43)that CRB ( θ) =FIM In thi paper, when the etimation quality i imperfect, epecially in the unfavorable cae, the upport i no longer identified by the algebraic property of R. Hence,a new performance guarantee i given in term of the weak- RIP in the following propoition. Propoition 2. One aume that F = R /2 + (/2) R /2 and R = ΨR ( θ)ψ where Ψ = [ψ ψ 2... ψ K ]. θ i an etimation that i aymptotically equal to θ uch that tr(r ( θ)) tr(r ( θ )) η for η (0, /2 F 2 2 ).LetJ 0 and J ={j i } L i= be the L-dimenional upport that conited of the indexe of L larget element in θ and θ, repectively.if the matrix R atifie α w L+ (R;J 0)>κ (49) = { M 2{ Re {S j U H (I HA(HA) + ) HUS j } } }, j= { } (44) for κ L F 2 2 j J 0 ψ j ψ j 2η F 2 2 (50) where H = BΦ i a N P matrix and ( ) + denote peudoinvere. Note that the CRB in CS i affected not only by the conventional factor, for example, SNR, array tructure andignalrelativelocation,butalobythemeaurement matrix Performance Guarantee. In CS, the RIP ha been deeply tudied for the joint pare recovery by minimizing the l norm. We ay that the matrix C C N K obey the RIP of the order L if there exit a contant δ (0, ) atifying ( δ) 2 2 C 2 2 (+δ) 2 2. (45) Therefore, all ubmatrice of C with L column are uniformly well conditioned. The retricted iometry contant (RIC) of the order L, decribed a δ L (C),ithemalletδ that atifie (45)andδ L (C) atifie δ L (C) = max J =L C J C J I L, (46) where J i a L-dimenional ubupport of J = [,2,...,K] and C J denote the ubmatrix of C with column indexed by J. Note that the condition atifying the RIP i o demanding that it application are limited. Therefore, we hould make ue of a new verion of the RIP, which i called the weak- RIP [38], to control the ize of the recovery error. The weak- RIP i given in the following form: ( α) k 2 2 Ck 2 2 (+α) k 2 2 (47) for all k upported on U, where the cardinality of the et U i L+.IfthematrixC atifie the weak- RIP, it can be deduced that 0 α φ L+ (C U ),whereφ k (C U ) denote the kth larget eigenvalue of C U. The correponding weak- RIC i given by α w L+ (C;U) = min φ L+ (C U ). (48) theupportcanbeidentified. Proof. See Appendix B. It follow from Propoition 2 that the performance guarantee of the propoed method require a mild condition in the unfavorable cae. However, in the favorable cae, the performance guarantee only require a much milder condition, α w L+ (R;J 0)>0, which i an algebraic condition. 6. Simulation Reult In thi ection, the performance of the propoed method i illutrated by everal imulation reult and compared with that of CS-MUISC, SPICE, and CRB under the condition that the number of ource i unknown. We conider the patial ignal impinging on the uniform linear array (ULA) with interpacing λ/2 where λ denote the wavelength of ource. In the ULA cae, the teering vector correponding to the DOA equal to θ k i given by a (θ k )=[ e jπ in(θ k) e j(p )π in(θ k) ] T, (5) where the number of the array element i et to be P=8. In the imulation, the average root mean quare error (RMSE) of the DOA etimation with 50 Monte Carlo run i defined a the ignificant performance index: 50 L (θ RMSE = [ lm θ l ) 2 ] m= 50L l= [ ] /2, (52) where θ lm itheetimateofθ l in the mth run. Thereolutionofthegridicloelyrelatedtothepreciion of the DOA etimation. A coare grid can lead to poor preciion, but a too fine grid increae computational complexity. Therefore, an adaptive grid refinement method i ued to

8 8 The Scientific World Journal 2.6 Amplitude RMSE (deg) Amplitude Amplitude DOA (deg) (a) DOA (deg) (b) DOA (deg) (c) Figure : Superimpoed patial pectra of CS-MUSIC, SPICE, and AULMC in 0 Monte Carlo run, where the red vertical dahed line denote the true DOA. (a) CS-MUSIC, (b) SPICE and (c) AULMC. balance the tradeoff between preciion and computational complexity. In the imulation, we make a coare grid with tep in the range of 90 to 90 andperformalocalfinegrid in the vicinity of location obtained by uing the coare grid. In the firt imulation, we diplay the uperimpoed patial pectra of three algorithm in 0 Monte Carlo run in the cenario with low SNR, mall number of naphot, and five ource impinging from [ ], repectively where two mot cloely paced ource are correlated and the remaining ource are uncorrelated. The patial pectra are hown in Figure with 3 db SNR and 50 naphot. The following fact can be acquired from Figure SNR (db) AULMC SPICE CS-MUSIC CRB Figure 2: RMSE of the DOA etimation veru SNR for 50 naphot. RMSE (deg) AULMC SPICE Number of naphot CS-MUSIC CRB Figure 3: RMSE of the DOA etimation veru number of naphot for 5 db SNR. a follow: the patial pectrum obtained by CS-MUSIC uffer from evere interference at the true direction, epecially at two mot cloely paced correlated ource whoe bia i clearly een from the inert. Two mot cloely paced correlated ource can be reolved by SPICE (note that peak inthepatialpectrumaremuchlargerthan2,buttheyare cut off at 2 to ue the ame cale a the other figure in Figure ), but SPICE can yield fale peak and light bia in the vicinity of the correlated ource and uncorrelated ource, repectively. The propoed method AULMC yield a nearly ideal patial pectrum and provide a precie etimation foralltheource.inummary,aulmcoutperformcs- MUSIC and SPICE in term of the patial pectrum in the cenario with low SNR, mall number of naphot, and cloely paced correlated ource. We analyze the RMSE of three algorithm under different condition in the econd imulation. The ource model i the ame a the firt imulation. Figure 2 how the RMSE a a function of SNR of all the algorithm and CRB in 50 Monte Carlo run for the fixed number of naphot 50, wherea the RMSE veru number of naphot i hown in Figure 3 for thefixedsnr5dbin50montecarlorun.baedonfigure2

9 The Scientific World Journal 9 RMSE (deg) AULMC SPICE CS-MUSIC Angle eperation (deg) Figure 4: RMSE of the DOA etimation veru angle eparation, wherethesnri3dbandthenumberofnaphoti00. performance of AULMC i much better than that of CS- MUSIC or SPICE. 7. Concluion A novel augmented Lagrange baed on modified covariance matching criterion method for DOA etimation i propoed in CS. It i proved that the problem of minimizing the modified covariance matching criterion i an SDP, which can be tranformed into the contrained quadratic programming problem olved by the augmented Lagrange method. A detailed derivation for the CRB and a theoretical performance guarantee for identifying the upport are provided. Simulation reult how that AULMC outperform CS- MUSIC and SPICE in term of the patial pectrum and ha more precie etimation a well a higher reolution, epecially in the cenario with low SNR, mall number of naphot, and cloely paced correlated ource. Table : Comparion of computation time. Time (ec) Number of naphot AULMC CS-MUSIC SPICE SDP olver Appendice A. Proof of Propoition Due to μ(z k )= (q k q k) q k f k,wehave q k q kμ (z k )= q k f k. With (34)and(A.), we get (A.) and 3, we can draw the concluion that the RMSE of AULMC i maller than thoe of other two algorithm and AULMC ha the more ignificant performance advantage than the other two algorithm, epecially in the cenario with low SNR or mallnumberofnaphot.onepoibleexplanationithat AULMC can give the table etimation in every Monte Carlo run.itcanbealoeenthatthermseicloetothecrbwith theincreaeofsnrandthenumberofnaphot. In Figure 4, wediplaytherelationbetweenthermse and angel eparation of correlated ource which can illutrate the reolving ability. Let two correlated ource at angle 20 and 20 +Δθ,wherethetepofΔθ i,beimpingedonthe ULA. The SNR i 3 db and the number of naphot i 00. ItcanbeeenfromFigure 4 that when angle eparation i 2, AULMC fail; however, AULMC can till provide a precie etimation a long a the angle eparation i no le than 3 and ha higher reolution than the other two algorithm. Finally, the computation time of different algorithm verunumber of naphotihown in Table for comparing the efficiency of thee algorithm and SDP olver. Two correlated ource impinge on the ULA at 20 and 25.The SNR i fixed at 3 db. The computation time i obtained by the MATLAB 7.8 (R2009a) on a 2.8 GHz 4 GB PC. For AULMC, the computation time i mainly pent on the iteration of augmented Lagrange. ItcanbeeenfromTable that the computation time of SDP olver i the longet, and although the computation time of AULMC i longer than that of other two algorithm, it i comparable. Moreover, it i worth noting that the q k q k [μ (z k ) μ k α k ]=q k b kd k. (A.2) Therefore, it i implied by (A.2) that(35) hold and the proof of Propoition i completed. B. Proof of Propoition 2 The upport J 0 canbeidentifiedif Tr [FR /2 ( θ) FR /2 ( θ)] max j J 0 Tr [FR /2 ( θ) FR /2 ( θ)] < min. j J\J 0 (B.) By the Cauchy-Schwarz inequality of the trace, we have Tr [(FR /2 ( θ)) 2 ] Tr [(FR /2 ( θ )) 2 ] Tr [F2 ] Tr [R ( θ)] Tr [R ( θ )] = Tr [R ( θ)] Tr [R ( θ )] η. (B.2)

10 0 The Scientific World Journal Then, for all j J 0 we have Tr [(FR /2 ( θ)) 2 ] Tr [(FR /2 ( θ )) 2 ] η+ η+ Tr [F2 ] Tr [R ( θ )] =η+ j J 0 ψ j ψ j θ j η+ L j J 0 ψ j ψ j α w L+ (R;J 0). (B.3) Thu, an upper bound on the left-band ide of (B.) i given by Tr [(FR /2 ( θ)) 2 ] max η+ L j J 0 ψ j ψ j j J 0 αl+ w (R;J 0). Similarly, we can obtain Tr [(FR /2 ( θ)) 2 ] Tr [F2 R ( θ )] = j J\J 0 ψ j F2 ψ j / θ j F 2 η 2 η η (B.4) (B.5) for all j J\J 0 and hence decribe a lower bound on the right-hand ide of (B.)a Tr [(FR ( θ)) 2 ] min j J\J 0 F 2 η. 2 (B.6) Combining (B.4) with (B.6), the upport J 0 can be identified if R atifie for Conflict of Interet α w L+ (R;J 0)>κ (B.7) κ L F 2 2 j J 0 ψ j ψ j 2η F 2. (B.8) 2 The author declare that there i no conflict of interet regarding the publication of thi paper. Acknowledgment Thi work wa upported in part by the National Science Foundation of China under Grant and in part by Fundamental Reearch Fund for the Central Univeritie under Grand no. HEUCF Reference [] J. D. Dai, W. C. Xu, and D. Zhao, Real-valued DOA etimation for uniform linear array with unknown mutual coupling, Signal Proceing,vol.92,no.9,pp ,202. [2] L.Bai,C.-Y.Peng,andS.Biwa, AociationofDOAetimation from two ULA, IEEE Tranaction on Intrumentation and Meaurement,vol.57,no.6,pp.094 0,2008. [3] J. Capon, High-reolution frequency-wavenumber pectrum analyi, Proceeding of IEEE,vol.57, pp ,969. [4] J. P. Burg, Maximum entropy pectral analyi, in Proceeding ofthe37thannualinternationalsegmeetingoklahomacity, 967. [5] R. O. Schmidt, Multiple emitter location and ignal parameter etimation, IEEE Tranaction on Antenna and Propagation, vol.34,no.3,pp ,986. [6] R. Roy and T. Kailath, ESPRIT etimation of ignal parameter via rotational invariance technique, IEEE Tranaction on Acoutic, Speech, and Signal Proceing, vol.37,no.7,pp , 989. [7] B. D. Rao and K. V. S. Hari, Performance analyi of rootmuic, IEEE Tranaction on Acoutic, Speech, and Signal Proceing,vol.37,no.2,pp ,989. [8] R. Bachl, Forward-backward averaging technique applied to TLS-ESPRIT proceing, IEEE Tranaction on Signal Proceing,vol.43,no.,pp ,995. [9] Z. Sun and Z. Yang, Study of nonlinear parameter identification uing UKF and Maximum Likelihood method, in Proceeding of the IEEE International Conference on Control Application (CCA 0), pp , September 200. [0] M. Viberg, B. Otterten, and T. Kailath, Detection and etimation in enor array uing weighted ubpace fitting, IEEE Tranaction on Signal Proceing, vol. 39, no., pp , 99. [] D. L. Donoho, Compreed ening, IEEE Tranaction on Information Theory,vol.52,no.4,pp ,2006. [2] E. T. Northardt, I. Bilik, and Y. I. Abramovich, Spatial compreive ening for direction-of-arrival etimation with bia mitigation via expected likelihood, IEEE Tranaction on Signal Proceing, vol. 6, no. 5, pp , 203. [3] M. Carlin, P. Rocca, G. Oliveri, F. Viani, and A. Maa, Direciton-ofarrival etimation through bayeian compreive ening trategie, IEEE Tranaction on Antenna and Propagation,vol.6,no.7,pp ,203. [4] E. J. Candè, J. Romberg, and T. Tao, Robut uncertainty principle: exact ignal recontruction from highly incomplete frequency information, IEEE Tranaction on Information Theory,vol.52,no.2,pp ,2006. [5] A. C. Gürbüz,J.H.McClellan,andV.Cevher, Acompreive beamforming method, in Proceeding of the IEEE International Conference on Acoutic, Speech and Signal Proceing (ICASSP 08), pp , April [6] Y. Wang, G. Leu, and A. Pandharipande, Direction etimation uing compreive ampling array proceing, in Proceeding of the IEEE/SP 5th Workhop on Statitical Signal Proceing (SSP 09), pp , September [7] P.Stoica,P.Babu,andJ.Li, Newmethodofpareparameter etimationineparablemodelanditueforpectralanalyi of irregularly ampled data, IEEE Tranaction on Signal Proceing,vol.58,no.,pp.35 47,200.

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