Model Order Selection for Short Data: An Exponential Fitting Test (EFT)
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Research Article Model Order Selection for Short Data: An Exponential Fitting Test (EFT) Angela Quinlan,1 Jean-Pierre Barbot,2 Pascal Larzabal,2 and Martin Haardt3 1 Department
of Electronic and Electrical Engineering, University of Dublin, Trinity College, Ireland ´ Laboratory, Ecole Normale Sup´erieure de Cachan, 61 avenue du Pr´esident Wilson, 94235 Cachan Cedex, France 3 Communications Research Laboratory, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany 2 SATIE
Received 29 September 2005; Revised 31 May 2006; Accepted 4 June 2006 Recommended by Benoit Champagne High-resolution methods for estimating signal processing parameters such as bearing angles in array processing or frequencies in spectral analysis may be hampered by the model order if poorly selected. As classical model order selection methods fail when the number of snapshots available is small, this paper proposes a method for noncoherent sources, which continues to work under such conditions, while maintaining low computational complexity. For white Gaussian noise and short data we show that the profile of the ordered noise eigenvalues is seen to approximately fit an exponential law. This fact is used to provide a recursive algorithm which detects a mismatch between the observed eigenvalue profile and the theoretical noise-only eigenvalue profile, as such a mismatch indicates the presence of a source. Moreover this proposed method allows the probability of false alarm to be controlled and predefined, which is a crucial point for systems such as RADARs. Results of simulations are provided in order to show the capabilities of the algorithm. Copyright © 2007 Angela Quinlan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
In sensor array processing, it is important to determine the number of signals received by an antenna array from a finite set of observations or snapshots. A similar problem arises in line spectrum estimations. The number of sources has to be determined successfully in order to obtain good performance for high-resolution direction finding estimates. A lot of work has been published concerning the model order selection problem. Estimating the number of sources is traditionally thought of as being equivalent to the determination of the number of eigenvalues of the covariance matrix which are different from the smallest eigenvalue [1]. Such an approach leads to a rank reduction principle in order to separate the noise from the signal eigenvalues [2]. Anderson [3] gave a hypothesis testing procedure based on the confidence interval of the noise eigenvalue, in which a threshold value must be assigned subjectively. He showed [3] that the log-likelihood ratio to the number of snapshots is asymptotic to a χ 2 distribution. For a small number of snapshots, James introduced the idea of “modified statistics” [4]. In [5], Chen et al.
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