LARGE INFORMATION PLUS NOISE RANDOM MATRIX MODELS AND CONSISTENT SUBSPACE ESTIMATION IN LARGE SENSOR NETWORKS
Abstract
In array processing, a common problem is to estimate the angles of arrival of K deterministic sources impinging on an array of M antennas, from N observations of the source signal, corrupted by Gaussian noise. In the so-called subspace methods, the problem reduces to estimate a quadratic form (called "localization function") of a certain projection matrix related to the source signal empirical covariance matrix. The estimates of the angles of arrival are then obtained by taking the K deepest local minima of the estimated localization function. Recently, a new subspace estimation method has been proposed, in the context where the number of available samples N is of the same order of magnitude than the number of sensors M. In this context, the traditional subspace methods tend to fail because they are based on the empirical covariance matrix of the observations which is a poor estimate of the source signal covariance matrix. The new subspace method is based on a consistent estimator of the localization function in the regime where M and N tend to +∞ at the same rate. However, the consistency of the angles estimator was not addressed, and the purpose of this paper is to prove this consistency in the previous asymptotic regime. For this, we prove the property that the singular values of M × N Gaussian information plus noise matrix escape from certain intervals is an event of probability decreasing at rate 
for all p. A regularization trick is also introduced, which allows to confine these singular values into certain intervals and to use standard tools as Poincaré inequality to characterize any moments of the estimator. These results are believed to be of independent interest.
References
- Ann. Probab. 26(1), 316 (1998). Google Scholar
- Ann. Probab. 27(3), 1536 (1999). Google Scholar
- F. Benaych-Georges and R. R. Nadakuditi, The singular values and vectors of low rank perturbations of large rectangular random matrices, submitted , arXiv:1103.2221 . Google Scholar
- Indiana Univ. Math. J. 56, 295 (2007), DOI: 10.1512/iumj.2007.56.2886. Google Scholar
- Ann. Probab. 37(1), 1 (2009), DOI: 10.1214/08-AOP394. Crossref, Google Scholar
- J. Multivariate Anal. 12(2), 306 (1982), DOI: 10.1016/0047-259X(82)90022-7. Crossref, Google Scholar
K. R. Davidson and S. J. Szarek , Handbook of the Geometry of Banach Spaces (2001) pp. 317–366. Crossref, Google Scholar- J. Multivariate Anal. 98(6), 1099 (2007), DOI: 10.1016/j.jmva.2006.12.005. Crossref, Google Scholar
- J. Multivariate Anal. 98(4), 678 (2007), DOI: 10.1016/j.jmva.2006.09.006. Crossref, Google Scholar
- IEEE Trans. Information Theory 56(3), 1048 (2010), DOI: 10.1109/TIT.2009.2039063. Crossref, Google Scholar
-
V. L. Girko , Theory of Stochastic Canonical Equations, Vol. I ,Mathematics and its applications 535 ( Kluwer Academic Publishers , Dordrecht , 2001 ) . Crossref, Google Scholar - Ann. Math. 162(2), 711 (2005), DOI: 10.4007/annals.2005.162.711. Crossref, Google Scholar
- Ann. Appl. Probab. 17(3), 875 (2007), DOI: 10.1214/105051606000000925. Crossref, Google Scholar
- W. Hachem, P. Loubaton, J. Najim and P. Vallet, On bilinear forms based on the resolvent of large random matrices, to appear in Ann. Inst. Henri Poincaré . Google Scholar
- J. Appl. Probab. 10(3), 510 (1973), DOI: 10.2307/3212772. Crossref, Google Scholar
- Ann. Math. Stat. 35(2), 475 (1964), DOI: 10.1214/aoms/1177703550. Crossref, Google Scholar
- P. Loubaton and P. Vallet, Almost sure localization of the eigenvalues in a gaussian information plus noise model. Application to the spiked models, to appear in Electronic J. Probab . Google Scholar
- IEEE Trans. Signal Process. 56(2), 598 (2008), DOI: 10.1109/TSP.2007.907884. Crossref, Google Scholar
- Ukrainian Math. J. 57(6), 936 (2005), DOI: 10.1007/s11253-005-0241-4. Crossref, Google Scholar
- J. Multivariate Anal. 52(2), 175 (1995). Google Scholar
- IEEE Trans. Acoustics, Speech Signal Process. 37(5), 720 (1989), DOI: 10.1109/29.17564. Crossref, Google Scholar
-
P. Vallet , An improved MUSIC algorithm based on low rank perturbation of large random matrices , Proc. of the 2011 IEEE Workshop on Statistical Signal Processing ( IEEE Signal Processing Society , 2011 ) . Google Scholar - P. Vallet, P. Loubaton and X. Mestre, Improved subspace estimation for multivariate observations of high dimension: The deterministic signal case, to appear in IEEE Trans. Information Theory , arXiv:1002.3234 . Google Scholar
