2962 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 Further Results and Insights on Subspace Based Sinusoidal Frequency Estimation Martin Kristensson, Magnus Jansson, Member, IEEE, and Björn Ottersten, Senior Member, IEEE Abstract—Subspace-based methods for parameter identifica- tion have received considerable attention in the literature. Starting with a scalar-valued process, it is well known that subspace-based identification of sinusoidal frequencies is possible if the scalar valued data is windowed to form a low-rank vector-valued process. MUSIC and ESPRIT-like estimators have, for some time, been applied to this vector model. In addition, a statistically attrac- tive Markov-like procedure for this class of methods has been proposed. Herein, the Markov-like procedure is reinvestigated. Several results regarding rank, performance, and structure are given in a compact manner. The large sample equivalence with the approximate maximum likelihood method by Stoica et al. is also established. Index Terms—Correlation, eigenvalues and eigenfunctions, frequency estimation, maximum likelihood estimation, multi- dimensional signal processing, singular value decomposition, spectral analysis. I. INTRODUCTION M ODEL-based parameter estimation using sub- space-based methods can be an attractive alternative to maximum likelihood estimation (MLE). In many cases, subspace methods provide accurate estimates at a reasonable computational cost. To apply subspace methods, a low-rank model of the system at hand must be available. In some cases, like in array signal processing, this structure is present directly in the received data. In other cases, e.g., sinusoidal frequency estimation [3], [18], [19], system identification [10], [27], and blind channel identification [13], [26], the low-rank vector-valued data structure can be obtained by applying a window to the received data. Vector-valued data models obtained from an underlying scalar valued process are, in this paper, referred to as windowed data models. Intuitively, the statistical properties of subspace methods when applied to windowed data models are different from models where the low-rank structure is physically present in the system. In this paper, the statistical properties of sub- space-based estimators applied to windowed data models are explained using a subspace-based sinusoidal frequency estimator as an example. The focus is thus not on obtaining a new estimator but on gaining insight on the behavior of the Manuscript received August 25 1998; revised August 30, 2001. The associate editor coordinating the review of this paper and approving it for publication was Prof. Lang Tong. M. Kristensson is with Nokia Networks, Kista, Sweden (e-mail: martin.kris- tensson@nokia.com). M. Jansson and B. Ottersten are with the Department of Signals, Sensors, and Systems, Royal Institute of Technology, Stockholm, Sweden (e-mail: mag- nusj@s3.kth.se). Publisher Item Identifier S 1053-587X(01)10486-1. studied class of methods. The estimator is close to the algorithm presented in [3], which was proposed as a statistically attractive alternative to MUSIC [19] and ESPRIT [18]. Several variations of these methods exist, and the area is still progressing; see, e.g., [5]. For an introduction and for a more complete list of references concerning frequency estimation, see [16] and [21]. The complicated statistical structure of the vector valued process is evident by studying the analysis in [3]. However, we show here that by carefully exploiting the structure, compact expressions for the estimation error covariance can, in fact, be obtained. In addition, these expressions enable further analysis of the rank properties of certain weighting and residual covariance matrices. These rank properties were left as an open question in [3], but they are, in fact, essential when determining optimal weighting matrices. The rank properties also make it possible to establish the large sample equivalence of the Markov estimator of [3] and the approximate maximum likelihood (AML) approach in [22]. This relation shows that the subspace approach provides the minimum asymptotic error covariance in the class of all estimators based on a given set of covariance estimates. The outline of the paper is as follows. In Section II, the data model is presented, followed by a description of the subspace- based estimator in Section III. Next, the large sample equiv- alence of different sample estimates of covariance matrices is discussed. The central parts of the paper are Sections V and VI, where the statistical results are derived. The paper is concluded with some implementational aspects and a simulation example in Section VII. II. DATA MODEL AND DEFINITIONS The samples of the scalar-valued observed signal are assumed to be the sum of complex-valued sinusoids in additive zero-mean white Gaussian noise (1) Here real-valued amplitude; frequency; phase of th sinusoid. The amplitudes and the frequencies are modeled as deterministic quantities. The frequencies are assumed to be distinct. The phases are uniformly distributed on 1053–587X/01$10.00 © 2001 IEEE