IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 4, APRIL 1999 945 A Subspace Method for Direction of Arrival Estimation of Uncorrelated Emitter Signals Magnus Jansson, Member, IEEE, Bo G¨ oransson, Student Member, IEEE, and Bj¨ orn Ottersten, Member, IEEE Abstract— Herein, a novel eigenstructure-based method for direction estimation is presented. The method assumes that the emitter signals are uncorrelated. Ideas from subspace and co- variance matching methods are combined to yield a noniterative estimation algorithm when a uniform linear array is employed. The large sample performance of the estimator is analyzed. It is shown that the asymptotic variance of the direction estimates coincides with the relevant Cram´ er–Rao lower bound (CRB). A compact expression for the CRB is derived for the case when it is known that the signals are uncorrelated, and it is lower than the CRB that is usually used in the array processing literature (assuming no particular structure for the signal co- variance matrix). The difference between the two CRB’s can be large in difficult scenarios. This implies that in such scenarios, the proposed method has significantly better performance than existing subspace methods such as, for example, WSF, MUSIC, and ESPRIT. Numerical examples are provided to illustrate the obtained results. Index Terms—Algorithms, array signal processing, eigenvalues and eigenfunctions, linear algebra, maximum likelihood estima- tion, matrix decomposition, parameter estimation, singular value decomposition, spectral analysis, statistics. I. INTRODUCTION T HE PROBLEM of estimating the directions of arrival (DOA’s) of signals from array data is well documented in the literature. A number of high-resolution algorithms or eigenstructure methods have been presented and analyzed (see, e.g., [1]–[8]). In some applications, such as radio as- tronomy, communications, etc., it is reasonable to assume that the signals are spatially uncorrelated. One disadvantage with eigenstructure or so-called subspace-based methods is that it is difficult to incorporate prior knowledge of the signal correlation into the eigendecomposition. Hence, it is difficult to use this prior information to increase the estimation accuracy. Herein, we propose an estimator that combines ideas from subspace and covariance matching methods [9], [10], which makes it possible to incorporate prior knowledge of the signal correlation into the estimator. It should be noted that the Cram´ er–Rao lower bound (CRB) usually used when compar- ing estimators in array signal processing is not the proper one in this scenario. As will be shown here, the CRB that should be used to compare estimators that use priors on the source Manuscript received August 13, 1997; revised March 6, 1998. The associate editor coordinating the review of this paper and approving it for publication was Dr. Yingbo Hua. The authors are with the Department of Signals, Sensors, and Systems, Royal Institute of Technology (KTH), Stockholm, Sweden (e-mail: mag- nus.jansson@s3.kth.se). Publisher Item Identifier S 1053-587X(99)02135-2. correlation is, in general, lower than that presented in, e.g., [1] and [11]. We derive the relevant CRB for the case under study and give a compact matrix expression for the CRB on the DOA parameters. It is also shown that the proposed method yields estimates that asymptotically attain this CRB on the estimation error covariance. The proposed estimator is based on a weighted least squares fit of certain elements in the eigendecomposition of the output covariance of the array. In general, this leads to a multimodal cost function that has to be minimized with some multidi- mensional search technique. This will, of course, lead to a computationally rather unattractive problem. However, for the common special case of a uniform linear array (ULA), the cost function can be reparameterized in a similar manner as in MODE and IQML [4], [6], [12] (see also [13]). In this case, the estimates of the directions can be found from the roots of a certain polynomial. This leads to a significant computational simplification. Note that a similar reparameterization and “non- iterative” scheme has yet to be found for maximum-likelihood (ML) or covariance matching techniques. Thus, the presented method has the important property of yielding a noniterative solution with the same asymptotic accuracy as ML. This paper is organized as follows. In Section II, the model and assumptions are given. In Section III, we present the DOA estimator. In Section IV, the CRB for uncorrelated sources is derived. The proposed estimator is analyzed in Section V, where the large sample properties of the estimates are derived. In Section V, we also derive the optimal weighting matrix, leading to minimum variance estimates of the DOA. The numerical implementation of the method when a ULA is employed is discussed in Section VI. Numerical examples are provided in Section VII to demonstrate the performance of the proposed algorithm. Finally, some conclusions are given in Section VIII. II. DATA MODEL AND ASSUMPTIONS Consider uncorrelated, narrowband, plane waves imping- ing on an unambiguous array consisting of sensors. The spatial response of the array is assumed to be parameterized by the directions of arrival . The functional form of the array response vector is assumed to be known. This scenario is described by the model where . The source signals and the additive noise are assumed to be zero-mean Gaussian 1053–587X/99$10.00  1999 IEEE