IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002 1027 Detection of Signals by Information Theoretic Criteria: General Asymptotic Performance Analysis Eran Fishler, Member, IEEE, Michael Grosmann, and Hagit Messer, Fellow, IEEE Abstract—Detecting the number of sources is a well-known and a well-investigated problem. In this problem, the number of sources impinging on an array of sensors is to be estimated. The common approach for solving this problem is to use an infor- mation theoretic criterion like the minimum description length (MDL), or the Akaike information criterion (AIC). Although it has been gaining much popularity and has been used in a variety of problems, the performance of information theoretic criteria-based estimators for the unknown number of sources has not been sufficiently studied, yet. In the context of array processing, the performance of such estimators were analyzed only for the special case of Gaussian sources where no prior knowledge of the array structure, if given, is used. Based on the theory of misspecified models, this paper presents a general asymptotic analysis of the performance of any information theoretic criterion-based estimator, and especially of the MDL estimator. In particular, the performance of the MDL estimator, which assumes Gaussian sources and structured array when applied to Gaussian sources, is analyzed. In addition, it is shown that the performance of a certain MDL estimator is not very sensitive to the actual distribution of the source signals. However, appropriate use of prior knowledge about the array geometry can lead to significant improvement in the performance of the MDL estimator. Simulation results show good fit between the empirical and the theoretical results. Index Terms—Array processing, asymptotic analysis, MDL. I. INTRODUCTION M OST parametric bearing estimation techniques assume that the number of sources impinging on the array is known or pre-estimated using some technique. The problem of estimating the number of sources impinging on a passive array of sensors has received a considerable amount of attention during the last two decades (see, among many others, [1]–[6]). The most common approach for estimating this number is to apply information theoretic criteria, like the minimum descrip- tion length (MDL) or Akaike information criterion (AIC) [7]. Since 1985 [3], when first suggested for estimating the number of narrowband sources impinging on an array of sensors, the MDL estimator practically became the standard tool for accom- plishing this task. Manuscript received March 20, 2001; revised January 2, 2002. The associate editor coordinating the review of this paper and approving it for publication was Dr. Alex B. Gershman. E. Fishler was with the Department of Electrical Engineering—Systems, Tel Aviv University, Tel Aviv, Israel. He is now with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: efishler@ee.princeton.edu). M. Grossman was with the Department of Electrical Engineering—Systems, Tel Aviv University, Tel Aviv, Israel. He is now with the Advanced Technology Group, AudioCodes Ltd., Tel Aviv, Israel. H. Messer is with the Department of Electrical Engineering—Systems, Tel Aviv University, Tel Aviv, Israel (e-mail: messer@eng.tau.ac.il). Publisher Item Identifier S 1053-587X(02)03222-1. A. Problem Formulation Assume an array of sensors, and denote by the re- ceived, -dimensional, signal vector at time instance . In addi- tion, denote by the number of signals impinging on the array. A common model for the received signal vector is (1) where is a matrix com- posed of -dimensional vectors, where lies on the array manifold . is called the array response vector or the steering vector, and is referred to as the steering matrix. is a signals vector, and is a vector of the additive noise. is the unknown parameter vector associated with the th source. Finally, assume also that and are ergodic, independent random vector processes. Many problems may be formulated using this simple, linear model (see [8], [9], and references therein). These problems differ by the structure of the mixing matrix by the assumed knowledge about the unknown parameters or by the statistical modeling. For example, in bearing estimation [9], is a scalar (namely the bearing of the th source). In other problems, are all the elements of the th column of , which represent a com- plete lack of knowledge about the steering matrix [3]. Denote by the complete unknown parameter vector, given that sources exist. represents the vector of un- known parameters that do not belong to any specific source, for example the noise level. Denote by the parameter space of . The problem is to estimate the number of sources , given independent snapshots of the array output . B. MDL Approach The information theoretic criteria approach is a general ap- proach for choosing a model that fits the data mostly from a family of possible models [7], [10]. That is, given a parameter- ized family of probability densities, for various , select such that (2) where is the log-likelihood of the measurements denoted by , is a gen- eral penalty function associated with the th family, and . is usually referred to as the maximum likelihood (ML) estimate of the unknown parameters given the th family of distributions. 1053-587X/02$17.00 © 2002 IEEE