Multi-Step Knowledge-Aided Iterative ESPRIT for Direction Finding

1 Multi-Step Kno wledge-Aided Iterati ve ESPRIT for Direction Finding Silvio F . B. Pinto 1 and Rodrigo C. de Lamare 1 , 2 Center for T elec ommunications St u d ies (CETUC) 1 Pontifical Catholic University of Rio de Ja neiro, RJ, Brazil. 2 Department of Electronics, University of Y ork, UK Emails: silviof@cetuc.puc -rio.br , delamare@ce tuc.puc-rio.br Abstract —In this work, we propose a subspace-based algo- rithm fo r DO A estimation which iteratively reduces the dis- turbance factors o f t he estima ted data co variance matrix and incorporates prior kno wledge which is gradually obtained on line. An a n alysis of the MSE of the reshaped d ata cov ariance matrix is carried out along with comparisons between computational complexities of the proposed and existing algorithms. Simulation s fo cusi ng on closely-spaced sources, where they are uncorrelated and correlated, illu strate the improv ements achieved. I . I N T RO D U C T I O N In array signal processing, d ir ection-of- arriv al (DO A) esti- mation is a key tas k in a b road range o f importan t app lications including radar an d son ar systems, wireless commun ications and seismology [1]. T raditional high-r esolution metho ds for DO A estimation such as the multiple signal classification (MUSIC) method [2], the roo t-MUSIC algo rithm [3], th e estimation of signal param eters via rotatio nal in variance tech- niques (ESPRIT) [4] and subsp ace techniques [5], [ 6], [7], [8], [9], [10], [11 ], [12], [26], [14], [15 ], [16], [17], [18], [19], [20], [21], [22], [23], [24], [ 25], [26], [27], [ 2 8], [29], [ 30], [3 1], [32], [33], [37], [35], [ 36], [37], [38], [ 3 9], [40], [ 41], [4 2], [43], [44], [45], [46], [ 47], [48], [49], [ 5 0], [51], [ 52], [5 3], [54], [5 5], [56],[5 7], [58], [5 9] exploit the eigenstructure of the input d ata ma trix. These techn iques may fail for red u ced data sets or lo w signal-to-no ise ratio (SNR) le vels where th e expected estimation e r ror is not asymptotic to the Cram ´ er- Rao b ound (CRB) [ 6 0]. The accuracy of the estimates of the covariance matrix is of f undamen tal impo rtance in p arameter estimation. Low lev els of SNR or short data r ecords can result in significant divergences between th e tru e and the sample data covariance matrices. In p ractice, only a modest number o f data snapshots is av ailab le and when the nu mber of snap sh ots is similar to the numb e r of sensor arr ay elements, the estimate d and the true subspaces can differ significantly . Sev eral approaches h av e b een de veloped with the aim of enhancin g the com putation of the c ovariance m a trix [6 1]-[70] and for dealing with large sensor-array systems large [ 71], [72], [73], [74], [75], [ 76], [77], [78], [ 7 9], [80], [ 81], [8 2], [83], [84], [85], [86], [ 87], [97], [88], [ 8 9], [90], [ 91], [9 2], [93], [9 4], [95], [96], [98], [99],[100], [1 01], [102], [103], [104], [10 5], [106], [1 11], [1 0 8], [109], [ 1 10], [111], [112], [113], [11 4], [1 1 5], [118], [11 7], [11 8], [119], [120]. Diagonal loading [61] and shrinkage [62], [63], [6 4] tech- niques can en hance the estimate o f the data covariance matrix by weighin g and individually increasing its dia g onal by a real constan t. Nevertheless, the eigenvectors remain the same, which leads to unaltered estimates of the signal an d noise projection matrices ob tained fr om the enhan ced covariance matrix. Add itionally , an improvement o f the estimates of the covariance matrix can be achieved by employing for- ward/backward a verag in g and spatial smoo th ing approa c hes [65], [66]. The former lead s to twice the nu mber o f the original samp les and its corre sp onding enhancem ent. The latter extracts the array cov arianc e matrix as the av erag e of all covariance matrices fr om its sub- arrays, resulting in a greater nu mber of samples. Both techniqu e s ar e employed in signal d ecorrelation . An approac h to improve MUSI C dealing with the con dition in which the nu mber of snapshots and the sen so r eleme n ts approac h in finity was presented in [6 7]. Nev er th eless, this technique is no t that effectiv e f or r educed number of snapshots. Oth er ap proach e s to deal with redu ced data sets or low SNR levels [68], [70] con sist o f reiterating th e proced u re of adding pseudo-n oise to the observations which results in new estimates of the covariance matrix. Th en, th e set of solution s is co m puted from pr eviously stored DOA estimates. In [ 1 21], two aspects resulting f rom the com putation of DO As for redu ced data sets or low SNR levels hav e been studied using the roo t-MUSIC techniq ue. The first aspect dealt with the probab ility o f estimated signal ro ots taking a smaller magnitude than th e estimated noise roots, wh ich is an anomaly th at lead s to wron g cho ices of the closest roots to the un it circ le . T o mitigate this pro blem, different group s of roots are con sidered as p otential so lu tions for the signal sour ces and the most likely one is selected [122]. The second aspect previously me n tioned, sho wn in [12 3], refers to the fact that a reduced part of the tru e sign al eig e nvectors exists in the sam p le noise subspace (an d vice -versa). Such coexistence h as b een expressed by a Frob enius n orm of the related irregularity m a tr ix an d in troduced its mathematical found ation. An iterative techniqu e to enhance the ef ficacy of root-MUSIC by reducing this anomaly makin g use of th e grad - ual reshaping of the sample d ata cov arian ce matrix has been reported . Inspired by the work in [ 121], we have developed an ESPRIT -based meth od known as T wo-Step KAI- ESPRIT (TS-ESPRIT) [ 124], which co mbines that modification s of the sample d a ta covariance matrix with the use of prior knowledge [125]-[131] a bout the covariance matrix of a set of imp inging signals to en hance the estimatio n accur a cy in th e finite sample size r egion . In p ractice, this prior k nowledge cou ld be from the signals co ming fro m known base stations o r fro m static users in a system. TS- ESPRIT deter mines the value of a correction factor that reduces the undesirable ter m s in the estimation of 2 the signal and noise subspaces in an iter ati ve process, resulting in better estimates. In this w ork [132], [133], we p resent the Multi-Step KAI ESPRIT (MS-KAI-ESPRIT) appr o ach that re fin es the covariance matrix of the input data via mu ltiple steps o f reduction of its u n desirable terms. This work pr esents the MS-KAI-ESPRIT in further detail, an analysis of th e mean squared error (MSE) o f the data covariance matrix free of undesired ter ms (side effects), a more accurate study of the computatio nal co mplexity and a compreh e nsi ve stud y of MS- KAI-ESPRIT and othe r com peting techniqu es for scenar ios with b oth uncor related and correlated signals. U n like TS- ESPRIT , which makes use of only one iteratio n and available known DO As, M S-KAI-ESPRIT employs mu ltiple iteratio ns and ob tains prior knowledge on line. At each iteration of MS- KAI-ESPRIT , the initial V andermo nde matr ix is upd ated by replacing an increasing numb er of steering vector s of initial estimates with their correspo nding refined versions. In other words, at each iter ation, the knowledge obtained on line is updated , allowing th e d ir ection finding algor ithm to co rrect th e sample covariance matrix estimate, wh ich y ields more accur ate estimates. In summary , this work has the following contributions: • The pr oposed MS- KAI-ESPRIT techniqu e. • An MSE analy sis o f the covariance matr ix o btained with the propo sed MS-KA I -ESPRIT alg orithm. • A comp rehensive performance study o f MS-KAI-ESPRIT and comp eting techn iques. This paper is organized as follows. Section I I describ es the system model. Section I II p resents the p roposed MS- KAI-ESPRIT algorithm. In section I V , an analytical study of the MSE o f the data covariance matrix free o f side- e ffects is carr ied o ut to gether with a study of the co mputation a l complexity o f th e prop o sed a n d competing a lg orithms. In Section V , we present a n d discuss th e simulation r e sults. Section VI co ncludes the paper . I I . S Y S T E M M O D E L Let us assum e that P nar rowband signals from far-field sources imp inge on a un iform linear array (ULA) of M ( M > P ) sensor elements f r om dir ections θ = [ θ 1 , θ 2 , . . . , θ P ] T . W e also consider th at the sensors are spaced fr om each other by a distance d ≤ λ c 2 , where λ c is the signal wa velength , and that without loss of g enerality , we h av e − π 2 ≤ θ 1 ≤ θ 2 ≤ . . . ≤ θ P ≤ π 2 . The i th d ata snapshot of the M -dimension al array ou tput vector can be modeled as x ( i ) = A s ( i ) + n ( i ) , i = 1 , 2 , . . . , N , (1) where s ( i ) = [ s 1 ( i ) , . . . , s P ( i )] T ∈ C P × 1 represents the zero- mean source data vector, n ( i ) ∈ C M × 1 is th e vector of white circular co mplex Gaussian n oise with z e r o mean and variance σ 2 n , and N den otes the number of available snapsh ots. The V andermond e matrix A ( Θ ) = [ a ( θ 1 ) , . . . , a ( θ P )] ∈ C M × P , kn own as the arra y manifold, contains the array steering vectors a ( θ j ) correspo nding to the n th source, which can be expr essed as a ( θ n ) = [1 , e j 2 π d λ c sin θ n , . . . , e j 2 π ( M − 1) d λ c sin θ n ] T , (2) where n = 1 , . . . , P . Using the fact that s ( i ) and n ( i ) are modeled as uncor related linearly indepen dent variables, th e M × M sign al cov ar ia n ce matrix is calcu lated by R = E  x ( i ) x H ( i )  = A R ss A H + σ 2 n I M , (3) where th e superscrip t H an d E [ · ] in R ss = E [ s ( i ) s H ( i )] an d in E [ n ( i ) n H ( i )] = σ 2 n I M denote the Hermitian transposition and the expectation operator and I M stands fo r the M - dimensiona l iden tity matrix. Since the true signal covariance matrix is unk n own, it must be estimated and a widely -adopted approa c h is the samp le a verag e formula given by ˆ R = 1 N N X i =1 x ( i ) x H ( i ) , (4) whose estimation accu racy is dep endent o n N . I I I . P R O P O S E D M S - K A I - E S P R I T A L G O R I T H M In this section, we present the proposed MS-KAI-ESPRIT algorithm and detail its main fe a tures. W e star t by expan ding (4) using (1) as derived in [121]: ˆ R = 1 N N X i =1 ( A s ( i ) + n ( i )) ( A s ( i ) + n ( i )) H = A ( 1 N N X i =1 s ( i ) s H ( i ) ) A H + 1 N N X i =1 n ( i ) n H ( i ) + A ( 1 N N X i =1 s ( i ) n H ( i ) ) + ( 1 N N X i =1 n ( i ) s H ( i ) ) A H | {z } ” undesirableterm s ” (5) The first two terms of ˆ R in (5) can be con sidered as estimate s of the two summan ds of R given in (3), which represent the sign al and the n oise com ponents, respectively . The last two terms in (5) are un desirable side effects, which can be seen as estimates fo r the cor relation between the signal and the no ise vectors. The system model un der study is based on noise vectors which ar e zero-mean and also ind ependen t of the signal vectors. Thus, the signal an d n oise componen ts are un correlated to each other . As a co nsequence , for a large enoug h number of samples N , the last two ter ms of (5) tend to zero. Nevertheless, in p ractice the numb er of available samp les can be limited. In such situation s, the last two terms in (5) may have significan t values, wh ich causes the deviation of th e estimates of the signal and the noise subspaces fro m the true signal and no ise sub sp aces. The key poin t of the propo sed MS-K A I -ESPRIT algorith m is to mod if y the sample d ata cov ar iance matrix estimate at e ach iteration by gradually incorp o rating the knowledge pr ovid ed by the n ewer V andermond e matr ices w h ich pro gressiv ely em- body the refine d estimates f rom the pr eceding iteration . Based on these u pdated V and ermonde matr ices, refined estimates of the projection matrices of the signal and n oise sub spaces ar e calculated. These estimates of projection m a tr ices associated with the initial sample covariance matrix estimate and the reliability factor are employed to r educe its side effects and allow the a lg orithm to choose the set of estimates that has the h ighest likelihoo d of being the set of the true D OAs. Th e modified covariance matrix is c o mputed by comp u ting a scaled version o f the und esirable terms of ˆ R , as p ointed out in ( 5). 3 The steps o f the pr oposed algor ithm are listed in T ab le I. The algo rithm starts by comp uting the sample data covariance matrix (4). Next, the DO As are estimated using the ESPRIT algorithm . T he su perscript ( · ) (1) refers to the estimation task perfor med in the first step. N ow , a proced ure consisting of n = 1 : P iter ations starts b y for ming the V an dermond e matrix using the DOA estima tes. Then , the amp litudes of th e sou r ces are estimated such that the sq u are n orm of th e differences between the observation vector and the vector containin g estimates and the av ailable known DOAs is m inimized. This problem can be formulated [121] as: ˆ s ( i ) = ar g min s k x ( i ) − ˆ A s k 2 2 . (6) The min imization of (6) is achieved using the least squ ares technique and th e solu tion is described by ˆ s ( i ) = ( ˆ A H ˆ A ) − 1 ˆ A x ( i ) (7) The noise com ponent is then estimated as th e dif fer ence between the estimate d signal and the observations mad e b y the arra y , as given by ˆ n ( i ) = x ( i ) − ˆ A ˆ s ( i ) . (8) After estimating th e sig nal an d no ise vectors, the thir d term in (5) c a n be co mputed as: V , ˆ A ( 1 N N X i =1 ˆ s ( i ) ˆ n H ( i ) ) = ˆ A ( 1 N N X i =1 ( ˆ A H ˆ A ) − 1 ˆ A H x ( i ) × ( x H ( i ) − x H ( i ) ˆ A ( ˆ A H ˆ A ) − 1 ˆ A H ) o = ˆ Q A ( 1 N N X i =1 x ( i ) x H ( i )  I M − ˆ Q A  ) = ˆ Q A ˆ R ˆ Q ⊥ A , (9) where ˆ Q A , ˆ A ( ˆ A H ˆ A ) − 1 ˆ A H (10) is an estimate of th e projection m atrix o f the signal subspace, and ˆ Q ⊥ A , I M − ˆ Q A (11) is an estimate of the projection matrix o f the noise sub sp ace. Next, as par t of the proced ure consisting of n = 1 : P iterations, the modified data covariance m a trix ˆ R ( n +1) is obtained by comp uting a scaled version of the estimated ter ms from the initial sample d ata covariance matrix as g iv en by ˆ R ( n +1) = ˆ R − µ ( V ( n ) + V ( n ) H ) , (12) where the sup erscript ( · ) ( n ) refers to the n th iteration pe r- formed . The scaling or reliability facto r µ in creases from 0 to 1 incrementa lly , r esulting in modified data covariance matrices. Each of them gives origin to new estimated DO As also denoted by the superscript ( · ) ( n +1) by using th e ESPRIT algor ith m, as briefly describ e d ahe ad. In this work, the rank P is assumed to be known, wh ich is an assumption frequ ently found in the literature. Alternatively , the rank P could be estimated by model-ord er selection schemes such as Akaike´s Info rmation Theoretic Criterion ( A I C) [14 4] and the Minim u m Descriptive Length (MDL) Criterion [1 45]. In or der to estimate the signal and the o rthogo nal subspac es from the data records, we may co n sider two ap proach e s [146], [ 147]: the direct data a p proach an d the cov arian ce approa c h . The direct data appro ach makes use of singular value dec omposition( SVD) of th e data m atrix X , com p osed of the i th d ata snapsho t (1) of the M -dimensiona l array data vector: X =[ x (1) , x (2) , . . . , x ( N )] = A [ s (1) , s (2 ) , . . . , s ( N )] + [ n (1) , n (2) , . . . , n ( N )] = A (Θ) S + N ∈ C M × N (13) Since the num ber of th e sources is a ssum ed k nown or can b e estimated by A I C[144] o r MDL[14 5] , as p reviously mentioned , we can wr ite X as: X =  ˆ U s ˆ U n   ˆ Γ s 0 0 ˆ Γ n   ˆ U H s ˆ U H n  , (14) where the diago nal ma tr ices ˆ Γ s and ˆ Γ n contain th e P largest singular values an d the M − P smallest singular values, respectively . The estima ted sig n al subspace ˆ U s ∈ C M × P consists of the sing ular vectors correspo nding to ˆ Γ s and the orthog onal su bspace ˆ U n ∈ C M × ( M − P ) is related to ˆ Γ n . I f the signal subspace is estimated a rank- P ap proximatio n of the SVD can be ap plied. The covariance approach applies the eig en value decompo- sition (EVD) of th e samp le covariance matrix (4 ), which is related to the data ma trix (13): ˆ R = 1 N N X i =1 x ( i ) x H ( i ) = 1 N X X H ∈ C M × M (15) Then, the EVD o f (15) can be car ried o ut as fo llows: ˆ R =  ˆ U s ˆ U n   ˆ Λ s 0 0 ˆ Λ n   ˆ U H s ˆ U H n  , (16) where the diagonal matrices ˆ Λ s and ˆ Λ n contain th e P largest and the M-P smallest e igen values, respectively . The estimated signal subspace ˆ U s ∈ C M × P correspo n ding to ˆ Γ s and the orthog onal subspace ˆ U n ∈ C M × ( M − P ) complies with ˆ Γ n . If the signal subspace is estimated a ra n k-P appro x imation of the EVD can be applied . W ith in fin ite precision arithmetic, both SVD an d EVD c a n b e consider ed eq uiv alent. Howe ver, as in practice, finite p recision ar ithmetic is emp loyed, ’ squarin g’ the data to o btain the Gramian X X H (15) can result in rou n d-off errors an d overflow . These a r e po tential p r oblems to be aware when using the covariance ap proach. Now , we can briefly revie w ESPRIT . W e start by for ming a twofold subarr ay config u ration, as ea c h row of the array steering matrix A ( Θ ) cor r esponds to one sensor element of the antenna array . The sub arrays ar e sp ecified b y two ( s × M ) - dimensiona l selection matrices J 1 and J 2 which ch oose s elements of the M existing sensors, respe cti vely , where s is in th e rang e P ≤ s < M . For max im um overlap, the matr ix J 1 selects the first s = M − 1 elements a n d the matrix J 2 selects the last s = M − 1 rows o f A ( Θ ) . 4 Since the m atrices J 1 and J 2 have now been co mputed, we can estimate the operato r Ψ by solving the approximatio n of the shift inv arian c e equ ation ( 17) giv en by J 1 ˆ U s Ψ ≈ J 2 ˆ U s . (17) where ˆ U s is ob tained in (16). Using the least sq uares (LS) metho d, wh ich yields ˆ Ψ = arg min Ψ k J 2 ˆ U s − J 1 ˆ U s Ψ k F =  J 1 ˆ U s  † J 2 ˆ U s , ( 1 8) where k · k F denotes the Frobenius no rm and ( · ) † stands for the pseud o-inv er se. Lastly , the eig en values λ i of ˆ Ψ con tain the estimates o f the spatial freq uencies γ i computed as: γ i = ar g ( λ i ) , (19) so that the DO As can b e calculated as: ˆ θ i = ar csin  γ i λ c 2 π d  (20) where for (1 9) and (20) i = 1 , · · · , P . Then, a new V and ermond e matrix ˆ B ( n +1) is for m ed by the steering vectors of th ose refin e d estimates o f the DOAs. By using this u pdated matrix , it is possible to comp ute the refined estimates of th e p rojection matrices of the signal ˆ Q ( n +1) B and the no ise ˆ Q ( n +1) ⊥ B subspaces. Next, emp loying the refined estima te s of the p rojection matrices, the in itial samp le data matrix , ˆ R , and the number of sensors an d sources, the stoch astic m aximum likelihood objective function U ( n + 1 ) ( µ ) [122] is c o mputed for each value of µ at the n th iteration, as follows: U ( n + 1 ) ( µ ) = ln det  ˆ Q ( n +1) B ˆ R ˆ Q ( n +1) B + T race { ˆ Q ⊥ ( n +1) B ˆ R } M − P ˆ Q ( n +1) ⊥ B  . (21) The previous compu tatio n selects the set of unavailable DO A estimates th at have a higher likelihood at each itera- tion. Then , th e set o f estimated DOAs correspo n ding to the optimum value o f µ that minimizes (2 1) also at ea ch n th iteration is de termined. Finally , th e output of the pr oposed MS- KAI-ESPRIT alg orithm is fo r med by the set of the estimates obtained at the P th iteration, as describe d in T able I. I V . A N A L Y S I S In this sectio n , we carry out an analy sis of the MSE o f the data c ovariance ma tr ix fr ee of side effects along with a study of th e com putational complexity of the p roposed MS-KAI- ESPRIT and existing dir ection fin ding algorithms. A. MSE An alysis In this sub section we show that at the fir st of the P iterations, the MSE of the d ata covariance matrix fr ee of side effects ˆ R ( n +1) is less than or eq u al to the MSE of th at of the original one ˆ R . Th is can be formulated as: MSE  ˆ R ( n +1)  ≤ MSE  ˆ R  (22) or , alter n ativ ely , as MSE  ˆ R ( n +1)  − MSE  ˆ R  ≤ 0 (23) The proo f of th is inequality is provided in the Appendix. T ABLE I P R O P O S E D M S - K A I - E S P R I T A L G O R I T H M Inputs: M , d , λ , N , P Recei ved vectors x (1) , x (2) , · · · , x ( N ) Outputs: Estimates ˆ θ ( n + 1 ) 1 ( µ opt ) , ˆ θ ( n + 1 ) 2 ( µ opt ) , · · · , ˆ θ ( n + 1 ) P ( µ opt ) First step: ˆ R = 1 N N P i =1 x ( i ) x H ( i ) { ˆ θ (1) 1 , ˆ θ (1) 2 , · · · , ˆ θ (1) P } E S P R I T ← − − − − − − ( ˆ R , P, d, λ ) ˆ A (1) = h a ( ˆ θ ( 1 ) 1 ) , a ( ˆ θ ( 1 ) 2 ) , · · · , a ( ˆ θ ( 1 ) P ) i Second step : fo r n = 1 : P ˆ Q ( n ) A = ˆ A ( n ) ( ˆ A ( n ) H ˆ A ( n ) ) − 1 ˆ A ( n ) H ˆ Q ( n ) ⊥ A = I M − ˆ Q ( n ) A V ( n ) = ˆ Q ( n ) A ˆ R ˆ Q ( n ) ⊥ A fo r µ = 0 : ι : 1 ˆ R ( n +1) = ˆ R − µ ( V ( n ) + V ( n ) H ) { ˆ θ ( n +1) 1 , ˆ θ ( n +1) 2 , · · · , ˆ θ ( n +1) P } E S P R I T ← − − − − − − ( ˆ R ( n +1) , P , d, λ ) ˆ B ( n +1) = h a ( ˆ θ ( n + 1 ) 1 ) , a ( ˆ θ ( n + 1 ) 2 ) , · · · , a ( ˆ θ ( n + 1 ) P ) i ˆ Q ( n +1) B = ˆ B ( n +1) ( ˆ B ( n +1) H ˆ B ( n +1) ) − 1 ˆ B ( n +1) H ˆ Q ( n +1) ⊥ B = I M − ˆ Q ( n +1) B U ( n + 1 ) ( µ ) = ln det ( · ) , ( · ) = ˆ Q ( n +1) B ˆ R ˆ Q ( n +1) B + T race { ˆ Q ⊥ ( n +1) B ˆ R } M − P ˆ Q ( n +1) ⊥ B ! µ ( n +1) opt = arg min U ( n + 1 ) ( µ ) DOAs ( n +1) = ( ∗ ) , ( ∗ ) = { ˆ θ ( n + 1 ) 1 ( µ opt ) , ˆ θ ( n + 1 ) 2 ( µ opt ) , · · · , ˆ θ ( n + 1 ) P ( µ opt ) } ˆ A ( n +1) = n a ( ˆ θ ( n + 1 ) { 1 , ··· , n } ) o S n a ( ˆ θ ( 1 ) { 1 , ··· , P } − { 1 , ··· , n } ) o end for end for B. Compu tational Comp lexity An alysis In this section, w e ev aluate the computatio nal cost of the pro posed MS-KAI-ESPRIT algo r ithm which is compare d to the f ollowing classical subsp ace m e thods: ESPRIT [4], MUSIC [2], Root-M USIC [ 3], Conjug ate Gradient ( CG) [ 138], Auxiliary V ector Filterin g (A VF) [13 9] and TS-ESPRIT [ 124]. The ESPRIT and MUSIC-based m ethods use the Sing ular V alue Decomp o sition ( SVD) of the sample covariance matr ix (4). The co mputationa l complexity o f MS-KAI-E SPRIT in terms of num ber of multiplications and addition s is d epicted in T able II, where τ = 1 ι + 1 . The incremen t ι is defined in T able I. As c an b e seen, for th is specific configu ration u sed in the simulatio n s V MS-KAI-ESPRIT shows a relatively high computatio nal burden with O ( P τ ( 3M 3 + 8MN 2 )) , where τ is ty pically a n integer that ranges from 1 to 20 . It can be noticed th at fo r the configur ation used in th e simu latio ns 5 ( P = 4 , M = 40 , N = 2 5 ) 3 M 3 and 8 M N 2 are compar able, resulting in two domin ant terms. It can also be seen that the number of multiplicatio ns requir e d by the pr oposed algo r ithm is more significan t than the numbe r o f ad ditions. For this reason, in T able II I, we comp uted only the co mputation a l burden of the previously mention e d algorith m s in terms o f multiplications for th e p u rpose o f co mparisons. In th a t table, ∆ stands fo r the search step. Next, we will evaluate the influence of the number of sensor elemen ts on the number of mu ltiplications based on the specific configur ation described in T able II. Supposing P = 4 n arrowband signals imp inging a ULA of M sensor elements an d N = 2 5 av ailable sn apshots, we ob tain Fig. 1. W e can see the ma in trend s in terms of computational cost measured in mu ltip lications of the pr oposed and an alyzed algorithm s. By examining Fig. 1, it can be n oticed that in th e range M = [20 7 0] sensors, the curves describing th e exact number o f multiplication s in MS-KAI-ESPRIT and A VF ten d to merge. For M = 40 , this ratio ten ds to 1 , i.e. the num ber of multiplicatio n s a r e almost equiv alent. T ABLE II C O M P U TA T I O NA L C O M P L E X I T Y - M S - K A I - E S P R I T Multipl ications P τ [ 10 3 M 3 + M 2 (3P + 2) + M ( 5 2 P 2 + 1 2 P + 8N 2 ) MS-KAI +P 2 ( 17 2 P + 1 2 )] -ESPRIT (Proposed) +P [2M 3 + M 2 (P) + M( 3 2 P 2 + 1 2 P) + P 2 ( P 2 + 3 2 )] +2M 2 (P) + M (P 2 − P + 8N 2 ) + P 2 (8P − 1) Addition s P τ [ 10 3 M 3 + M 2 (3P − 1) + M ( 5 2 P 2 − 9 2 P + 8N 2 ) +P(8P 2 − 2P − 5 2 )] +P [2M 3 + M 2 (P − 2) + M ( 3 2 P 2 − 1 2 P) − P(P + 1 2 )] +2M 2 (P) + M (P 2 − 4P + 8N 2 ) + P(8P 2 − P − 2) T ABLE III C O M P U TA T I O NA L C O M P L E X I T Y - O T H E R A L G O R I T H M S Algorith m Multipl ications MUSIC [2] 180 ∆ [M 2 + M(2 − P) − P] + 8MN 2 root-MUSIC[3] 2M 3 − M 2 P + 8MN 2 A VF [139] 180 ∆ [M 2 (3P + 1) + M (4P − 2) + P + 2] +M 2 N CG [138] 180 ∆ [M 2 (P + 1) + M (6P + 2) + P + 1] + M 2 N ESPRIT[4] 2M 2 P + M(P 2 − 2P + 8N 2 ) + 8P 3 − P 2 τ [ 3M 3 + M 2 (3P + 2) + M ( 5 2 P 2 − 3 2 P + 8N 2 ) +P 2 ( 17 2 P + 1 2 ) + 1] TS-ESPRIT [124]* +[ 2M 3 + M 2 (3P) + M( 5 2 P 2 − 3 2 P + 8N 2 ) +P 2 ( 17 2 P + 1 2 )] V . S I M U L A T I O N S In this section, we examin e the p erform a nce o f the pro posed MS-KAI-ESPRIT in terms o f pro bability of resolution and RMSE and compare them to th e standard ESPRIT [4], the Iterative ESPRIT (I ESPRIT), wh ich is also dev elop ed here by combin ing the app r oach in [121] that exploits k nowledge of the structure of the c ovariance m atrix and its p e rturbation terms, the Con jugate Gradient (CG) [138], the Roo t-MUSIC 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 10 4 10 5 10 6 10 7 10 8 10 9 Number of sensors Number of multiplications as powers of 10 MUSIC Root−MUSIC[3] ESPRIT[4] TS−ESPRIT[20]* MS−KAI−ESPRIT CG[33] AVF[34] Fig. 1. Number of multiplic ations as powers of 10 versus number of sensors for P = 4 , N = 25 . [3], and the MUSI C [2] algorithm s. Despite TS- ESPRIT is based o n the knowledge of available known DOAS and the propo sed MS- K AI -ESPRIT d oes not have access to prior knowledge, TS-ESPRIT is plotted with the aim o f illustrating the co mparisons. For a fair comparison in terms of RMSE and p robability of resolu tion of all studied algorithm s, we suppose that we do no t have prio r knowledge, that is to say that although we have available known DOAs, we compute TS-ESPRIT as th ey were unavailable. W e em ploy a ULA with M=40 sensors, inter-element spacing ∆ = λ c 2 and assume ther e ar e four u ncorrelate d com plex Gaussian signals with equal power imp inging o n the array . Th e clo sely-spaced sources ar e separated by 2 . 4 o , at (10 . 2 o , 12 . 6 o , 15 o , 17 . 4 o ) , and the number of a vailable snap sh ots is N =25. For TS- ESPRIT , as p reviously mentio ned, we presume a priori kn owl- edge of the last true DO AS (1 5 o , 17 . 4 o ) In Fig. 2, we show the pro bability o f resolutio n versu s SNR. W e take into account the criterion [141], in which two sources with DOA θ 1 and θ 2 are said to be r esolved if their respective estimates ˆ θ 1 and ˆ θ 2 are such that both    ˆ θ 1 − θ 1    and    ˆ θ 2 − θ 2    are less th an | θ 1 − θ 2 | / 2 . The p roposed MS- KAI-ESPRIT alg orithm ou tperform s IESPRIT developed her e , based on [121], an d the standard ESPRIT [4] in the range between − 6 and 5 dB an d M USIC [2] from − 6 to 8 . 5 dB . MS- KAI-ESPRIT also outper f orms CG [13 8] and Root-Music [3] throug hout the whole rang e of values. The p o or perfo rmance of the latter co uld be expected from the r esults for two closed signals obtained in [121]. When compar e d to TS-ESPRIT , which as previously discussed, was supposed to ha ve the best perfo rmance, the pro posed MS-KAI-ESPRIT algorithm is ou tperform ed b y the for mer o nly in the range between − 6 and − 2 dB . From this last po int to 20 dB its perfor m ance is superior or equ al to the other algor ithms. In Fig. 3, it is sh own the RMSE in d B versus SNR, where the term CRB r e fers to the squ are roo t of the determ inistic Cram ´ er - Rao bound [142]. The RMSE is defined as: RMSE = v u u t 1 L P L X l =1 P X p =1 ( θ p − ˆ θ p ( l )) 2 , (24) where L is the n umber of trials. 6 The results show th e superior perfor mance of MS-KAI- ESPRIT in th e rang e b etween − 2 . 5 an d 5 dB. Fr om this last point to 20 dB, MS-KAI-ESPRIT , IESPRIT , ESPRIT and TS- ESPRIT have similar perfo rmance. The only rang e in which MS-KAI-ESPRIT is outper formed lies in the r a nge between − 6 an d − 2 . 5 dB. From this last p oint to 20 dB its perf ormance is better o r similar to the others. −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR(dB) Probability of Resolution MS−KAI−ESPRIT ESPRIT [4] Root−MUSIC [3] MUSIC [2] CG [33 ] IESPRIT [17] TS−ESPRIT[20]* Fig. 2. Probability of resolut ion versus SNR with P = 4 uncorrela ted sources, M = 40 , N = 25 , L = 100 runs −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 −18 −15 −12 −9 −6 −3 0 3 6 9 12 15 18 SNR(dB) RMSE (dB) CRB MS−KAI−ESPRIT ESPRIT [4] Root−MUSIC [3] MUSIC [2] CG [33] IESPRIT [17] TS−ESPRIT [20]* Fig. 3. RMSE and the square root of CRB versus SNR with P = 4 uncorrel ated sources, M = 40 , N = 25 , L = 100 runs Now , we focus on the perfo rmance o f MS-KAI-ESPRIT under more severe cond itio ns, i.e ., we analyze it in term s o f RMSE when at least two o f the four equ al-powered Gaussian signals a r e stro n gly co r related, as shown in the following signal correlatio n matr ix R ss (25): R ss = σ 2 s    1 0 . 9 0 . 6 0 0 . 9 1 0 . 4 0 . 5 0 . 6 0 . 4 1 0 0 0 . 5 0 1    . (25) The signal-to-no ise ratio ( S N R ) is defined as S N R , 10 log 10  σ 2 s σ 2 n  . In Fig . 4, we can see the perfor mance of the same algo rithms p lo tted in Fig. 3 in terms o f RMSE(dB) versus SNR computed afte r 250 runs, when the sig n al co rre- lation matrix is given by (25). As can be seen, the supe rior perfor mance of MS-KAI -ESPRIT occur s in the wh ole ra nge −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 −18 −15 −12 −9 −6 −3 0 3 6 9 12 15 18 SNR (dB) RMSE (dB) CRB MS−KAI−ESPRIT ESPRIT [4] Root−MUSIC [3] MUSIC [2] CG [33] IESPRIT [17] TS−ESPRIT [20]* Fig. 4. RMSE and the square root of CRB versus SNR with P = 4 correlated sources, M = 40 , N = 25 , L = 250 runs between 4 . 0 and 12 dB , which can be considere d a small but consistent gain . From 12 dB to 20 d B MS-KAI-ESPRIT , TS-ESPRIT , I ESPRIT and E SPRIT have similar per formanc e. The values fo r which MS-KAI-E SPRIT is outperform ed are in the ran ge b e tween − 6 . 0 and 4 . 0 dB. In Fig. 5, we hav e provid ed fur th er simulation s to illustra te the perfor mance of each iteratio n of MS-KAI ESPRIT in terms of RMSE. The resulting iter ations can be compar e d to each oth e r and to the the orig inal ESPRIT , which corre sp onds to the first step of M S-KAI ESPRIT . For this purpo se, we have co n sidered the same scen ario emp loyed before, except for the num ber of th e trials, which is L = 200 run s for all simulations. In particular, we have con sidered the case of correlated sou rces. Fro m Fig. 6, which is a magn ified detail of Fig. 5, it can be seen th at the estimates be c o me more accura te with the incr ease o f iterations. −6 −4 −2 0 2 4 6 8 10 12 14 16 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 SNR(dB) RMSE(dB) CRB 4th iteration(n=P=4) 3rd iteration(n=3) 2nd iteration(n=2) 1st iteration(n=1) ESPRIT− first step Fig. 5. RMSE for each iterati on of MS-KAI ESPRIT ,origina l ESPRIT and CRB versus SNR with P = 4 correlat ed sources, M = 40 , N = 25 , L = 200 runs V I . C O N C L U S I O N S W e have propo sed the MS-KAI-ESPRIT algo rithm which exploits the knowledge of sou rce signals obtain ed on line and the struc tu re of the covariance matrix and its p erturba- tions. An an alytical study o f th e MSE of this matrix free of side effects has shown th at it is less or equal th an the 7 4 5 6 7 8 9 10 11 12 13 −9 −8 −7 −6 −5 −4 SNR(dB) RMSE(dB) CRB 4th iteration(n=P=4) 3rd iteration(n=3) 2nd iteration(n=2) 1st iteration(n=1) ESPRIT−first step Fig. 6. RMSE for each iteration of MS-KAI ESPRIT ,origina l ESPRIT and CRB versus SNR with P = 4 correlat ed sources, M = 40 , N = 25 , L = 200 runs -magnificat ion MSE o f th e o r iginal matrix, resulting in better performan ce of MS-KAI -ESPRIT esp ecially in scen a rios wh ere limited number of samples are available. The proposed M S- KAI- ESPRIT alg orithm can obtain significan t gains in RMSE o r probab ility of resolution pe r forman c e over previously repo rted technique s, an d has e x cellent potential for applications with short data reco r ds in large-scale antenn a systems for wir eless commun ications, radar an d other large sensor arr ays. The rela- ti vely hig h comp utational burden requ ired, wh ich is associated with extra matrix multiplication s, the incremen t ι ap plied to reduce the undesirab le side effects an d the iteration s needed to progr essi vely inco rporate th e knowledge obtain e d on lin e a s newer estimates can b e justified for the sup erior perform ance achieved. Futu re work will consider approaches to reducing the com putational co st. A P P E N D I X Here, we pr ove the inequality (23) descr ib ed in Section IV -A. W e start by expressing the MSE of the orig inal d ata covariance m atrix (4) as: MSE  ˆ R  = E h k ˆ R − R k 2 F i . (26) where R is the true covariance matrix . Similarly , the MSE of the data cov ar ia n ce matrix free of side effects ˆ R ( n +1) can be expressed f or the first iteration n = 1 by making u se o f (12), as follows MSE  ˆ R ( n +1)    n =1 = MSE  ˆ R (2)  = E h k ˆ R (2) − R k 2 F i = E h k ˆ R − µ ( V (1) + V (1) H ) − R k 2 F i = E h k  ˆ R − R  − µ ( V (1) + V (1) H ) k 2 F i (27) where for the sake of simplicity , fro m now on we o mit the superscript (1) , which refers to the first iter ation. In order to exp and the result in (27), we make use of the fo llowing propo sition: Lemma 1 : The squared Frobenius norm o f the difference between any two m atrices A ∈ C m × m and B ∈ C m × m is giv en by k A − B k 2 F = k A k 2 F + k B k 2 F −  T r A H B + T r AB H  (28) Proof of Le m ma 1 : The Frobeniu s nor m of any D ∈ C m × m matrix is defined [1] as k D k F =   m X i =1 m X j =1 | d ij | 2   1 2 =  T r  D H D  1 2 (29) W e exp r ess D as a d ifference between two matrice s A and B , both also ∈ C m × m . M aking use of Lemma1 a n d the prop erties of the tra c e, we obtain k A − B k 2 F = T r h ( A − B ) H ( A − B ) i = T r  A H − B H  ( A − B )  = T r  A H A  − T r  A H B  − T r  B H A  + T r  B H B  = k A k 2 F + k B k 2 F −  T r A H B + T r AB H  , (30) which is the desired result. Now , a ssuming that the true R [13 4] and the data covariance matrices ˆ R [13 4] are Hermitian an d using ( 2 7) combin ed with Lemma1, the cyclic [1 3 5] property of the trace and th e linearity [13 6] pro perty of the expected value, we get MSE  ˆ R (2)  = E n k ˆ R − R k 2 F + µ 2 k V + V H k 2 F − T r   ˆ R − R  H µ  V + V H   − T r h µ  V + V H  H  ˆ R + R io = E n k ˆ R − R k 2 F + µ 2 k V + V H k 2 F − µ T r   ˆ R − R  H  V + V H   − µ T r h  V + V H  H  ˆ R + R io = E n k ˆ R − R k 2 F + µ 2 k V + V H k 2 F − µ T r h ˆ R − R   V + V H  i − µ T r h  V H + V   ˆ R + R io = E n k ˆ R − R k 2 F + µ 2 k V + V H k 2 F − µ T r h ˆ R − R   V + V H  i − µ T r h ˆ R + R   V + V H  io = E n k ˆ R − R k 2 F o + µ 2 E  k V + V H k 2 F  − 2 µ E n T r h ˆ R − R   V + V H  io = MSE  ˆ R  + µ 2 E  k V + V H k 2 F  − 2 µ E n T r h ˆ R − R   V + V H  io (31) By m oving th e first summand of ( 31) to its first element, we obtain the inten ded expression f o r the difference b etween the M S E s o f the d a ta covariance matrix fr ee of pertu rbations a n d the original o n e, i.e. : 8 MSE  ˆ R ( n +1)    n =1 − MSE  ˆ R  = µ 2 E  k V + V H k 2 F  − 2 µ E n T r h ˆ R − R   V + V H  io . (32) Now , we expand the expressions inside brace s of the secon d member of (32) in d i v idually . W e start with the first summan d k V + V H k 2 F = k V k 2 F + k V H k 2 F + T r  V H V H  + T r  ( V H ) H V  = k V k 2 F + k V H k 2 F + T r  V H V H  + T r ( VV ) . (33) The equation (33) can be com puted by using the projection matrices of th e signal an d the no ise sub spaces and th e data covariance matrix by using (9), ( 11), the idempo tence [1] [1 3 5] of ˆ Q A and the cyclic p roperty [135] of the trace. Startin g with the com putation o f its fourth summan d, we hav e T r ( VV ) = T r h ˆ Q A ˆ R ˆ Q ⊥ A   ˆ Q A ˆ R ˆ Q ⊥ A i = T r h ˆ Q A ˆ R  I M − ˆ Q A  ˆ Q A ˆ R  I M − ˆ Q A i = T r h ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A   ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A i = T r h ˆ Q A ˆ R ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A ˆ R ˆ Q A − ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R + ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R ˆ Q A i = T r  ˆ Q A ˆ R ˆ Q A ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R ˆ Q A  − T r  ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R  + T r  ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R ˆ Q A  = T r  ˆ Q A ˆ R ˆ Q A ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R  + T r  ˆ Q A ˆ R ˆ Q A ˆ R  = 0 . (34) T aking into account that th e data covariance matr ix ˆ R and the estimate of the p rojection matrix of the noise subspace ˆ Q ⊥ A are Herm itian, we can e valuate the third sum mand of (3 3) as follows: T r  V H V H  = T r   ˆ Q A ˆ R ˆ Q ⊥ A  H  ˆ Q A ˆ R ˆ Q ⊥ A  H  = T r   ˆ Q ⊥ A  H ˆ R H ˆ Q H A    ˆ Q ⊥ A  H ˆ R H ˆ Q H A  = T r nh ˆ Q ⊥ A ˆ R ˆ Q A i h ˆ Q ⊥ A ˆ R ˆ Q A io = T r nh I M − ˆ Q A  ˆ R ˆ Q A i h  I M − ˆ Q A  ˆ R ˆ Q A io = T r nh ˆ R ˆ Q A − ˆ Q A ˆ R ˆ Q A i h ˆ R ˆ Q A − ˆ Q A ˆ R ˆ Q A io = T r n ˆ R ˆ Q A ˆ R ˆ Q A − ˆ R ˆ Q A ˆ Q A ˆ R ˆ Q A − ˆ Q A ˆ R ˆ Q A ˆ R ˆ Q A + ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R ˆ Q A o = T r  ˆ R ˆ Q A ˆ R ˆ Q A  − T r  ˆ R ˆ Q A ˆ Q A ˆ R ˆ Q A  − T r  ˆ Q A ˆ R ˆ Q A ˆ R ˆ Q A  + T r  ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R ˆ Q A  = T r  ˆ R ˆ Q A ˆ R ˆ Q A  − T r  ˆ R ˆ Q A ˆ R ˆ Q A  − T r  ˆ Q A ˆ R ˆ Q A ˆ R  + T r  ˆ Q A ˆ R ˆ Q A ˆ R  = 0 . (35) By using (29), we can expand the first and the second summand s of (33) as f ollows: k V k 2 F + k V H k 2 F = T r  V H V  + T r   V H  H V H  = T r  V H V  + T r  VV H  = T r  VV H  + T r  VV H  = 2 T r  VV H  . (36) Equation ( 36) can be expre ssed in terms of the p rojection matrices of th e signal an d the no ise sub spaces and th e data covariance, in a similar way as for the third and fourth summand s of (33), as follows: 2 T r  VV H  = 2 T r   ˆ Q A ˆ R ˆ Q ⊥ A   ˆ Q A ˆ R ˆ Q ⊥ A  H  = 2 T r  ˆ Q A ˆ R  I M − ˆ Q A  h ˆ Q A ˆ R  I M − ˆ Q A i H  = 2 T r   ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A   ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A  H  = 2 T r n ˆ Q A ˆ R ˆ R ˆ Q A − ˆ Q A ˆ R ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A ˆ R ˆ Q A + ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R o = 2 n T r  ˆ Q A ˆ R ˆ R ˆ Q A  − T r  ˆ Q A ˆ R ˆ Q A ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R ˆ Q A  + T r  ˆ Q A ˆ R ˆ Q A ˆ Q A ˆ R o = 2 n T r  ˆ Q A ˆ Q A ˆ R ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R  + T r  ˆ Q A ˆ R ˆ Q A ˆ R o = 2 n T r  ˆ Q A ˆ Q A ˆ R ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R o (37) From (33), (3 4), (35), (36) and (37), we ob tain the first summand of (3 2), a s follows: µ 2 E  k V + V H k 2 F  = 2 µ 2 E n T r  ˆ Q A ˆ Q A ˆ R ˆ R  − T r  ˆ Q A ˆ R ˆ Q A ˆ R o (38) 9 In orde r to finish the expansion of the expression s inside braces of the seco nd member of (3 2), now we deal with its second summand , in which we make use of the cyclic pro perty [135] of the tra c e and the idem potence [ 1] [135] of ˆ Q A . T r h ˆ R − R   V + V H  i = n T r  ˆ R − R   ˆ Q A ˆ R ˆ Q ⊥ A +  ˆ Q A ˆ R ˆ Q ⊥ A  H  = T r n ˆ R − R  h ˆ Q A ˆ R  I M − ˆ Q A  +  ˆ Q A ˆ R  I M − ˆ Q A  H  = T r n ˆ R − R  h ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A +  ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A  H  = T r n ˆ R − R  h ˆ Q A ˆ R − ˆ Q A ˆ R ˆ Q A + ˆ R ˆ Q A − ˆ Q A ˆ R ˆ Q A io = T r n ˆ R ˆ Q A ˆ R + ˆ R ˆ R ˆ Q A − 2 ˆ R ˆ Q A ˆ R ˆ Q A − R ˆ Q A ˆ R − R ˆ R ˆ Q A + 2 R ˆ Q A ˆ R ˆ Q A o = T r ˆ R ˆ Q A ˆ R + T r ˆ R ˆ R ˆ Q A − 2 T r ˆ R ˆ Q A ˆ R ˆ Q A − T r R ˆ Q A ˆ R − T r R ˆ R ˆ Q A + 2 T r R ˆ Q A ˆ R ˆ Q A = T r ˆ Q A ˆ R ˆ R + T r ˆ Q A ˆ R ˆ R − 2 T r ˆ Q A ˆ R ˆ Q A ˆ R − T r R ˆ Q A ˆ R − T r ˆ Q A R ˆ R + 2 T r ˆ Q A R ˆ Q A ˆ R = 2 T r ˆ Q A ˆ R ˆ R − 2 T r ˆ Q A ˆ R ˆ Q A ˆ R − T r R ˆ Q A ˆ R − T r ˆ Q A R ˆ R + 2 T r ˆ Q A R ˆ Q A ˆ R = 2 T r ˆ Q A ˆ Q A ˆ R ˆ R − 2 T r ˆ Q A ˆ R ˆ Q A ˆ R − T r R ˆ Q A ˆ Q A ˆ R − T r ˆ Q A ˆ Q A R ˆ R + 2 T r ˆ Q A R ˆ Q A ˆ R (39) By using (39), we can straigh tforwardly write the second summand o f the second member of (3 2) in terms of the projection matrices of the signal an d the no ise su b spaces an d the data covariance matr ix as f ollows: − 2 µ E n T r h ˆ R − R   V + V H  io = − 2 µ E n 2 T r ˆ Q A ˆ Q A ˆ R ˆ R − 2 T r ˆ Q A ˆ R ˆ Q A ˆ R − T r R ˆ Q A ˆ Q A ˆ R − T r ˆ Q A ˆ Q A R ˆ R + 2 T r ˆ Q A R ˆ Q A ˆ R o = − 4 µ E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 2 µ n − T r E h R ˆ Q A ˆ Q A ˆ R i − T r E h ˆ Q A ˆ Q A R ˆ R i +2 T r E h ˆ Q A R ˆ Q A ˆ R io = − 4 µ E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 2 µ n − T r R ˆ Q A ˆ Q A E h ˆ R i − T r ˆ Q A ˆ Q A R E h ˆ R i +2 T r ˆ Q A R ˆ Q A E h ˆ R io (40) Now , by u sing (38) and (40), an d assuming th at E h ˆ R i is an unbiased estimate o f ˆ R , i. e., E h ˆ R i = R , we can rewrite (3 2) as follows: MSE  ˆ R ( n +1)    n =1 − MSE  ˆ R  = µ 2 E  k V + V H k 2 F  − 2 µ E n T r h ˆ R − R   V + V H  io = 2 µ 2 E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 4 µ E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 2 µ n − T r R ˆ Q A ˆ Q A R − T r ˆ Q A ˆ Q A RR +2 T r ˆ Q A R ˆ Q A R o = 2 µ 2 E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 4 µ E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 2 µ n − 2 T r R ˆ Q A ˆ Q A R + 2 T r ˆ Q A R ˆ Q A R o = 2 µ 2 E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 4 µ E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 4 µ n T r ˆ Q A ˆ Q A RR − T r ˆ Q A R ˆ Q A R o =  2 µ 2 − 4 µ  E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o − 4 µ n T r ˆ Q A ˆ Q A RR − T r ˆ Q A R ˆ Q A R o (41) Next, we will discuss equation (41). For this purpo se, we assume that the estimate of the pro jection matrix o f the signal subspace ˆ Q A [1], the true R [134] and the data cov arian ce matrices ˆ R [13 4] a r e Hermitian. For the next steps we will make use of the following Th eorem which is proved in [137]: Theorem 1 : For two Her m itian matrices A and B of the same order, T r ( AB ) 2 k ≤ T r  A 2 k B 2 k  , (42) where k is in integer . By rep lacing A with ˆ Q A and B with ˆ R in (42) and also considerin g k = 1 , we ha ve T r  ˆ Q A ˆ R  2 ≤ T r  ˆ Q 2 A ˆ R 2  ∴ T r ˆ Q A ˆ R ˆ Q A ˆ R ≤ T r ˆ Q A ˆ Q A ˆ R ˆ R ⇒ T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R ≥ 0 (43) Similarly , makin g A = ˆ Q A and B = R fo r k = 1 , we obtain T r  ˆ Q A R  2 ≤ T r  ˆ Q 2 A R 2  ∴ T r ˆ Q A R ˆ Q A R ≤ T r ˆ Q A ˆ Q A RR ⇒ T r ˆ Q A ˆ Q A RR − T r ˆ Q A R ˆ Q A R ≥ 0 (44) Next, we an alyze the behavior of the expressions − 4 µ and  2 µ 2 − 4 µ  based o n th e reliability factor µ ∈ [0 1] , as d efined in (12). In orde r to illustrate the case bein g studied , we assum e that both expressions are co ntinuous functions as dep icted in Fig. 7. It can be seen in it that in the rang e [0 1 ] both 10 0 0.2 0.4 0.6 0.8 1 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 X: 1 Y: −2 µ f ( µ ) X: 1 Y: −4 X: 0 Y: 0 (2 µ 2 − 4 µ ) − 4 µ Fig. 7. Beha vior of  2 µ 2 − 4 µ  and − 4 µ for µ ∈ [0 1] expressions assum e values f( µ ) ≤ 0 , i.e. : F or µ ∈ [0 1 ] :   2 µ 2 − 4 µ  ≤ 0 − 4 µ ≤ 0 (45) Now , we can consider the trac e s which for m the subtraction in (43) as d ifferent rand om variables y ( ω ) and x ( ω ) , i.e.: T r ˆ Q A ˆ Q A ˆ R ˆ R = y ( ω ) T r ˆ Q A ˆ R ˆ Q A ˆ R = x ( ω ) ) , ∀ ω ∈ Ω . (46) In addition , we can sup pose th at there is a ran dom variable z ( ω ) always greater th an zero, i.e., z ( ω ) ≥ 0 , so that z ( ω ) = y ( ω ) − x ( ω ) ≥ 0 , ∀ ω ∈ Ω (47) T aking the expectation o f (4 7) an d ap plying its properties of linearity and m onotonicity [136], [140], we obtain E [ z ( ω )] = E [ y ( ω ) − x ( ω )] ≥ 0 , (48) which, by ma king use of (4 6), r e sults in E [ z ( ω )] = E [ y ( ω ) − x ( ω )] = E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o ≥ 0 (49) Next, we can combine th e inequalities (45) with (4 9) to compute the seco nd member of (41), for µ ∈ [0 1] . For its first summand , we combin e ( 45) and ( 49), as f ollows: ( E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o ≥ 0  2 µ 2 − 4 µ  ≤ 0 , µ ∈ [0 1] , (50) to obta in in a straightforward way  2 µ 2 − 4 µ  E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o ≤ 0 (51) Similarly , we can com p ute its second member, by comb ining (45) and (44), as described by ( T r ˆ Q A ˆ Q A RR − T r ˆ Q A ˆ R ˆ Q A ˆ R ≥ 0 − 4 µ ≤ 0 , µ ∈ [0 1] , (52) to obta in also stra ightforwardly the expression giv en by − 4 µ n T r ˆ Q A ˆ Q A RR − T r ˆ Q A R ˆ Q A R o ≤ 0 (53) By com bining the inequalities (5 1) and (53) with ( 41), we have MSE  ˆ R ( n +1)    n =1 − MSE  ˆ R  =  2 µ 2 − 4 µ  E n T r ˆ Q A ˆ Q A ˆ R ˆ R − T r ˆ Q A ˆ R ˆ Q A ˆ R o | {z } ≤ 0 − 4 µ n T r ˆ Q A ˆ Q A RR − T r ˆ Q A R ˆ Q A R o | {z } ≤ 0 (54) ∴ MSE  ˆ R ( n +1)    n =1 − MSE  ˆ R  ≤ 0 (55) which is the desired result. R E F E R E N C E S [1] H. L. 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