ROYAL INSTITUTE OF TECHNOLOGY KUNGL TEKNISKA HÖGSKOLAN. Department of Signals, Sensors & Systems Signal Processing S STOCKHOLM

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1 Optimal Array Signal Processing in the Presence of oherent Wavefronts P. Stoica B. Ottersten M. Viberg December 1995 To appear in Proceedings ASSP{96 R-S3-SB-9529 ROYAL NSTTUTE OF TEHNOLOGY Department of Signals, Sensors & Systems Signal Processing S STOKHOLM KUNGL TEKNSKA HÖGSKOLAN nstitutionen för Signaler, Sensorer & System Signalbehandling STOKHOLM

2 OPTMAL ARRAY SGNAL PROESSNG N THE PRESENE OF OHERENT WAVEFRONTS Petre Stoica Bjorn Ottersten Mats Viberg Systems and ontrol Group Uppsala University, P.O. Box 27 S{ Uppsala, Sweden Signal Processing/S3 Royal nstitute of Technology S{ Stockholm, Sweden Department of Applied Electronics halmers University of Technology S{ Gothenburg, Sweden ABSTRAT The problem of estimating the parameters of several wavefronts from the measurements of multiple sensors is often referred to as array signal processing. The maximum likelihood (ML) estimator in array signal processing for the case of non-coherent signals has been studied extensively. The focus here is on the ML estimator for the case of stochastic coherent signals which arises due to, for example, specular multipath propagation. We show the very surprising fact that the ML estimates of the signal parameters obtained by ignoring the information that the sources are coherent, coincide in large samples with the ML estimates obtained by exploiting the coherent source information. Thus, the ML signal parameter estimator derived for the non-coherent case (or its large-sample realizations such as MODE or WSF) asymptotically achieves the lowest possible estimation error variance (corresponding to the coherent ramer-rao bound). 1. NTRODUTON n several array signal processing applications, the presence of coherent emitter signals is not uncommon. Specular multipath, in for example, radar or wireless communications, may give rise to coherent signals at the array. The stochastic maximum likelihood (ML) estimator has been derived under the stochastic emitter signal model for non-coherent signals. t is known to achieve the ramer-rao bound (RB) in large samples [1, 2, 3]. This important property of the stochastic ML estimator depends on the assumption that the emitter signals are non-coherent (i.e. no two signals are fully correlated). When coherent emitter signals are present, the maximum likelihood estimator must be reformulated and a new ramer-rao lower bound on the estimation error variance is also obtained. Note that there are methods for consistently detecting the presence of coherent signals [4, 5]. n the section below, the stochastic ML problem for coherent emitter signals is formulated, see also [6]. The emitter covariance matrix is constrained to have low rank and by making use of an appropriate parameterization, we show that explicit large-sample expressions for the ML estimates of the noise power and the signal covariance matrix can be obtained. Furthermore, we show the intriguing result that the asymptotic estimation accuracy of the signal parameters (such as the angles-of-arrival) is not aected by the knowledge of the rank of the emitter signal covariance matrix. Thus, the original stochastic ML estimator that ignores the rank information, or its large-samples realizations (such as MODE and WSF [7, 4, 2]) achieves the lowest possible estimation error variance for the signal parameters. 2. PROBLEM FORMULATON onsider the following narrowband array model y(t) = A()x(t) + n(t) : (1) The measurement vector, y(t), represents the m sensor outputs. The n emitter signals are collected in the vector x(t) and the additive noise is denoted n(t). The array response, A(), is a known function of the unknown signal parameters,, (for example, the anglesof-arrival). The stochastic model assumes that the complexvalued observation vector is zero-mean and circularly Gaussian distributed with Efy(t)y (s)g = R ts = (APA + 2 ) ts (2) where the superscript () denotes the conjugate transpose, 2 is the power of the (spatially white) sensor noise, and P = Efx(t)x (t)g is the emitter signal covariance matrix. The objective is to estimate the signal parameters, the emitter signal covariance matrix, and the noise variance based on the sensor measurements. t is assumed that the number of signals, n, is known.

3 When discussing the coherent ML estimator and the corresponding RB, we also assume that the rank of P is known. The maximum likelihood (ML) estimator based on the model above, consists of solving the following problem ([8, 9, 6, 2]): min [ln jrj + Tr(R 1 ^R)] (3) ; 2 ;P where j j and Tr() denote the determinant and the trace of a matrix respectively and ^R is the sample covariance matrix of the array output vector ^R = 1 N NX t=1 y(t)y (t) : (4) Hereafter, N denotes the number of available snapshots. The problem in (3) is separable, see [8, 9] ^ 2 ( ^) = 1 h m n Tr ^P( ^) = A y () (5) A() ^R i h i ^R ^ 2 () A () y (6) A() ^P()A () + ^ 2 () (7) A y = (A A) 1 A A = AA y (8) By concentrating the likelihood function, the dimension of the optimization problem is reduced by (n 2 + 1). On the other hand, the result has a drawback: equations (5){(8) are derived by considering the minimization of the normalized negative log-likelihood function in (3) with respect to P over the set of Hermitian matrices, and not over the set of Hermitian positive (semi)denite matrices as it should. As a consequence, the minimizing ^P( ^) so obtained in (6) is not guaranteed to be positive (semi)denite, and indeed it is known [6] that ^P( ^) may be indenite. When P is positive denite, (5){(7) still provide a large-sample realization of the ML estimator since ^P( ^) will be a valid estimate for suciently large N. n the case of coherent sources, however, the covariance matrix, P, is singular, whereas ^P(^) given by (6) is nonsingular (w.p.1) for any nite N. As a consequence the ML estimator must be reformulated to exploit the knowledge that the sources are coherent. n fact, according to the parsimony principle of parameter estimation (see, e.g., [10]), incorporation of any a priori knowledge into the ML problem should lead to improved accuracy compared with the situation where that knowledge is not incorporated (such as is the case in (5){(7)). This means that ^ given by (7) might be expected to be less accurate than the ML signal parameter estimate based on the information that P is singular. nterestingly enough, numerical evaluations of the ramer-rao bounds derived by ignoring and, respectively, exploiting the information that rank(p) < n have shown that the asymptotic signal parameter bounds corresponding to the two cases are identical [2]. Below, we derive explicit large-sample expressions, comparable with (5){(7), for the estimators that minimize (3) in the case of coherent sources. Also, it is shown that (7) still provides a large-sample realization of the ML signal parameter estimate even in scenarios where it is known a priori that P is rank decient. 3. ML ESTMATON FOR OHERENT SOURES Let the rank of the emitter signal covariance matrix be p = 4 rank(p) n. When there are coherent emitter signals present, p < n. t is assumed that the problem is \parameter identiable", i.e., the array manifold is unambiguous and m > 2n p, [11]. The signal covariance matrix, P, can uniquely be written as P = S [ ] (9) where S is p p, is (n p) p, and is p p. Note that this square root parameterization requires p 2 + 2(n p)p = n 2 (n p) 2 real parameters whereas the full parameterization (3) requires n 2 parameters. n what follows it is assumed that S is a nonsingular matrix. As the source signals can be arbitrarily ordered, the previous assumption on S does not introduce any restriction. Dene A c = A (10) With this notation, the matrix R in (2) can be written as R = A c SA c + 2 (11) Let denote the parameter vector composed of the elements of along with the real and imaginary parts of the elements of. We parameterize the negative log-likelihood function in (3) by, S and 2. Using this parameterization, the matrix P is guaranteed to be positive (semi)denite and to satisfy the rank constraint. Since S in the representation (11) of R is positive denite, it follows from the discussion on (5){(7) that

4 large-sample realizations of the ML estimates of 2, S and are given by ^ 2 ( ^) = 1 m p Tr h A c () ^R i ^S( ^) = A y c()h^r ^ 2 () i A y c () (12) (13) A c ()^S()A c() + ^ 2 () : (14) Next we derive an explicit expression for the ML estimate of the part of corresponding to. To this end, we note the following result proved in [1, 7, 3, 2]: Theorem 1 A large-sample realization of ^ in (14) (i.e., an estimate of having the same asymptotic distribution as ^) is given by the maximizer of the function where f() = Tr [ Ac ()W] (15) Ac = A c A y c ; W E s ^~ 2 ^ 1 ^E s (16) and where ^E s ; ^ and ^~ are dened as follows. Let f^k g p and f^e k=1 kg p k=1 denote the p principal (largest) eigenvalues of ^R and their associated orthonormal eigenvectors, respectively. Then ^E s = [^e 1 ; ; ^e p ] (17) ^ = diag (^ 1 ; ; ^ p ) (18) ^~ = diag (^ 1 ^ 2 ; : : : ; ^ p ^ 2 ) (19) with ^ 2 being a consistent estimate of 2. A straightforward calculation shows that f(), de- ned in the previous theorem, can be rewritten as where f() = Tr [X (A A)X] 1 [X (A WA)X] X = (20) : (21) Let f k g p and k=1 fv kg p k=1 denote the p principal eigenvalues and their associated eigenvectors of the matrix (A A) 1 (A WA), and dene ^V() = [v 1 ; : : : ; v p ] : (22) Observe that the function f() is invariant to the postmultiplication of X by any nonsingular matrix. t then follows from the extended Rayleigh quotient result in [12] or from the Poincare separation theorem (see, e.g., [13]) that f() px k=1 where the upper bound is obtained for k (23) XG V() (24) and where G is an arbitrary nonsingular matrix. f ^V() is partitioned as ^V1 () g p ^V() = (25) ^V 2 () g n p then (24) implies G = ^V1 () ^V 2 () (26) from which we immediately obtain the following explicit expression for the minimizing matrix: ^() V 2 () ^V 1 1 () : (27) To arrive at a concentrated form for f(), observe that rank[(a A) 1 (A WA)] = rank[a () ^E s ] (28) where now denotes the indeterminate signal parameter vector. Let E s denote the limit of ^E s as N! 1 (E s is dened from the p principal eigenvectors of R, similarly to ^E s in (17)). t is well known that the subspace generated by E s is included in the range of the true array matrix A. This means that there exists an n p matrix Q such that which implies that E s = AQ =) (E sa)q = (29) rank(e sa) = p : (30) Under a weak analyticity condition on the matrix A(), viewed as a function of, it then follows that rank h A () ^E s i = p (31) for nite but large N values and for almost any vector in a compact set including the true parameters (see, e.g., [10]). Hence, for suciently large N and generically in, we have rank (A A) 1 (A WA) = p (32)

5 which implies that the function of, left after the concentration of f() with respect to, is given by (cf (23)): px k=1 k = Tr (A A) 1 A WA : (33) n other words, a large-sample realization of the ML estimator of, under the a priori information that rank(p) = p, is given by Trh A()W i : (34) This estimate, known as the MODE/WSF estimate, was shown in [7, 3] to be a large-sample realization of the stochastic ML estimator in (7) (also see Theorem 1 above). We summarize the previously derived results in the following theorem. Theorem 2 The ML signal parameter estimator (7), derived by ignoring any a priori information about the matrix P, is a large-sample realization of the ML signal parameter estimator that does exploit the information that P is a positive (semi)denite matrix of rank p. A large-sample realization of (7) can be obtained by minimizing the MODE/WSF function in (34). Once is estimated, from either (7) or (34), largesample solutions to the ML estimation problem for 2 and P, under the condition that P is positive semidefinite and its rank is p, are given by (12) and, respectively, (9), (13) where ^ is composed from ^ in either (7) or (34) and ^( ^) in (27). The asymptotic properties of the ML estimates of the noise power and the signal covariance matrix are derived in [14]. The ML estimator of these parameters, which makes use of the rank information, outperforms the ML estimator which ignores that information. Thus, as opposed to the situation for the signal parameters, there is an improvement in estimation accuracy for the noise power and the signal covariance matrix estimates. 4. ONLUSONS The maximum likelihood estimator is formulated for the case of stochastic coherent signals impinging on an antenna array. t is shown that the ordinary ML signal parameter estimator, which does not use a priori knowledge about the rank of the emitter covariance matrix, is a large-sample realization of the ML signal parameter estimator exploiting the fact that P is a positive (semi)denite matrix of rank p. The implication of this result, and the main contribution of this paper, is that the knowledge of the rank of the signal covariance matrix does not have to be incorporated in the ML estimation procedure of the signal parameters for large samples. Rather, the signal parameters may be estimated by solving the much simpler unconstrained problem. Once the signal parameters are estimated, large-sample closed-form solutions to the ML estimation problem for 2 and P, under the condition that P is positive semidenite and its rank is p, are given. 5. REFERENES [1] P. Stoica and A. Nehorai, \Performance Study of onditional and Unconditional Direction-of-Arrival Estimation", EEE Trans. ASSP, ASSP-38:1783{1795, October [2] B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, n Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99{151. Springer-Verlag, Berlin, [3] B. Ottersten, M. Viberg, and T. Kailath, \Analysis of Subspace Fitting and ML Techniques for Parameter Estimation from Sensor Array Data", EEE Trans. on SP, SP-40:590{600, March [4] M. Viberg, B. Ottersten, and T. Kailath, \Detection and Estimation in Sensor Arrays Using Weighted Subspace Fitting", EEE Trans. SP, SP-39(11):2436{ 2449, Nov [5] M. Wax and. Ziskind, \Detection of the Number of oherent Signals by the MDL Principle", EEE Trans. on ASSP, ASSP-37(8):1190{1196, Aug [6] Y. Bresler, \Maximum Likelihood Estimation of Linearly Structured ovariance with Application to Antenna Array Processing", n Proc. 4th ASSP Workshop on Spectrum Estimation and Modeling, pages 172{175, Minneapolis, MN, Aug [7] P. Stoica and K. Sharman, \Maximum Likelihood Methods for Direction-of-Arrival Estimation", EEE Trans. ASSP, ASSP-38:1132{1143, July [8] J.F. Bohme, \Estimation of Spectral Parameters of orrelated Signals in Waveelds", Signal Processing, 11:329{337, [9] P. Stoica and A. Nehorai, \On the oncentrated Stochastic Likelihood Function in Array Signal Processing", Technical report, No 9214, Yale University, New Haven, 1992, To appear in ircuits, Systems, and Signal Processing. [10] T. Soderstrom and P. Stoica, System dentication, Prentice-Hall nternational, London, U.K., [11] M. Wax and. Ziskind, \On Unique Localization of Multiple Sources by Passive Sensor Arrays", EEE Trans. on ASSP, ASSP-37(7):996{1000, July [12] P. Stoica, \Extension of Rayleigh Quotient with Application", Bul. of Polytechnic nst., Automatic ontr. Ser., XLV:49{53, [13].R. Rao, Linear Statistical nference and its Applications, John Wiley & Sons, New York, [14] Petre Stoica, Bjorn Ottersten, Mats Viberg, and Randy Moses, Maximum likelihood array processing for stochastic coherent sources, n EEE Trans. on Signal Processing, February 1996.