Journal of Communications and Networks

The Korean Institute of Commucations and Information Sciences (한국통신학회)
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Estimation of the Number of Sources Based on Hypothesis Testing

  • Xiao, Manlin (Department of Electronic Engineering, University of Electronic Science and Technology of China) ;
  • Wei, Ping (Department of Electronic Engineering, University of Electronic Science and Technology of China) ;
  • Tai, Heng-Ming (Department of Electrical Engineering, University of Tulsa)
  • Received : 2011.10.27
  • Accepted : 2012.06.11
  • Published : 2012.10.31
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Abstract

Accurate and efficient estimation of the number of sources is critical for providing the parameter of targets in problems of array signal processing and blind source separation among other such problems. When conventional estimators work in unfavorable scenarios, e.g., at low signal-to-noise ratio (SNR), with a small number of snapshots, or for sources with a different strength, it is challenging to maintain good performance. In this paper, the detection limit of the minimum description length (MDL) estimator and the signal strength required for reliable detection are first discussed. Though a comparison, we analyze the reason that performances of classical estimators deteriorate completely in unfavorable scenarios. After discussing the limiting distribution of eigenvalues of the sample covariance matrix, we propose a new approach for estimating the number of sources which is based on a sequential hypothesis test. The new estimator performs better in unfavorable scenarios and is consistent in the traditional asymptotic sense. Finally, numerical evaluations indicate that the proposed estimator performs well when compared with other traditional estimators at low SNR and in the finite sample size case, especially when weak signals are superimposed on the strong signals.

Keywords

References

  1. X. Mestre and M.A. Lagunas, "Modified subspace algorithms for DOA estimation with large arrays," IEEE Trans. Signal Process., vol. 56, pp. 598-614, 2008. https://doi.org/10.1109/TSP.2007.907884
  2. A. J. Bell, "An information-maximization approach to blind separation and blind deconvolution," Neural Comput., vol. 7, pp. 1129-1159, 1995. https://doi.org/10.1162/neco.1995.7.6.1129
  3. A. P. Liavas, P. A. Regalia, and J. P. Delmas, "Blind channel approximation: Effective channel order determination," IEEE Trans. Signal Process., vol. 47, pp. 3336-3344, 1999. https://doi.org/10.1109/78.806077
  4. M. Wax and T. Kailath,"Detection of signals by information theoretic criteria," IEEE Trans. Acoustics, Speech, Signal Process., vol. 33, pp. 387-392, 1985. https://doi.org/10.1109/TASSP.1985.1164557
  5. H. Akaike, "A new look at the statistical model identification," IEEE Trans. Automatic Control, vol. 19, pp. 716-723, 1974. https://doi.org/10.1109/TAC.1974.1100705
  6. P.-J. Chung, J. F. Bohme, A. O. Hero, and C. F. Mecklenbräuker, "Detection of the number of signals using a multiple hypothesis test," in Proc. IEEE SAM, Barcelona, Spain, 2004, pp. 221-224.
  7. E. Fishler and H. Messer, "Order statistics approach for determining the number of sources using an array of sensors," IEEE Trans. Signal Process. Lett., vol. 6, pp. 179-182, 1999. https://doi.org/10.1109/97.769363
  8. E. Fishler and H. Messer, "On the use of order statistics for improved detection of signals by the MDL criterion," IEEE Trans. Signal Process., vol. 48, pp. 2242-2247, 2000. https://doi.org/10.1109/78.852005
  9. H.-T. Wu, J.-F. Yang, and F.-K. Chen, "Source number estimator using transformed Gerschgorin radii," IEEE Trans. Signal Process., vol. 43, pp. 1325-1333, 1995. https://doi.org/10.1109/78.388844
  10. P. Stocia and Y. Selen, "Model-order selection: A review of information criterion rules," IEEE Signal Process. Mag., vol. 21, pp. 36-47, 2004. https://doi.org/10.1109/MSP.2004.1311138
  11. Q. T. Zhang, K. M. Wong, P. C. Yip, and J. P. Reilly, "Statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing," IEEE Trans. Acoustics, Speech, Signal Process., vol. 37, pp. 1557-1567, 1989. https://doi.org/10.1109/29.35394
  12. R. R. Nadakuditi, "Applied stochastic eigen-analysis," Ph.D. dissertation, Massachusetts Institute of Technology, MA, 2007.
  13. L. C. Zhao, P. R. Krishnaiah, and Z. D. Bai, "On detection of numbers of signals in prensence of white noise," J. Multivariate Analysis, vol. 20, pp. 1-25, 1986. https://doi.org/10.1016/0047-259X(86)90017-5
  14. B. Nadler, "Onparametric detection of signals by information theoretic criteria: Performance analysis and an improved estimator," IEEE Trans. Signal Process., vol. 58, pp. 2746-2756, 2010. https://doi.org/10.1109/TSP.2010.2042481
  15. E. Fishler, M. Grosmann, and H. Messer, "Detection of signals by information theoretic criteria: General asymptotic performance analysis," IEEE Trans. Signal Process., vol. 50, pp. 1027-1036, 2002. https://doi.org/10.1109/78.995060
  16. R. R. Nadakuditi and A. Edelman, "Sample eigenvalue based detection of high-dimensional signals in white noise using relatively few samples," IEEE Trans. Signal Process., vol. 56, pp. 2625-2638, 2008. https://doi.org/10.1109/TSP.2008.917356
  17. J. R. Schott, "A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix," J. Multivariate Analysis, vol. 97, pp. 827-843, 2006. https://doi.org/10.1016/j.jmva.2005.05.003
  18. S. Kritchman and B. Nadler, "Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory," IEEE Trans. Signal Process., vol. 57, pp. 3930-3941, 2009. https://doi.org/10.1109/TSP.2009.2022897
  19. J.Wishart, "The generalized product moment distribution in samples from a normal multivariate population," Biometrika, vol. 20, pp. 32-52, 1928.
  20. V. A. Marcenko and L. A. Pastur, "Distribution of eigenvalues in certain sets of random matrices," Mat. Sb. (N.S.), vol. 72, pp. 507-536, 1967.
  21. D. Jonsson, "Some limit theorems for the eigenvalues of a sample covariance matrix," J. Multivariate Analysis, vol. 12, pp. 1-38, 1982. https://doi.org/10.1016/0047-259X(82)90080-X
  22. B. A. Johnson, Y. I. Abramovich, and X. Mestre, "Music, g-music, and maximum-likelihood performance breakdown," IEEE Trans. Signal Process., vol. 56, pp. 3944-3958, 2008. https://doi.org/10.1109/TSP.2008.921729
  23. S. Kritman and B. Nadler, "Determining the number of components in a factor model from limited noisy data," Chem. Int. Lab. Syst., vol. 94, pp. 19-32, 2008. https://doi.org/10.1016/j.chemolab.2008.06.002