On detection of the number of signals when the noise covariance matrix is arbitrary

In this paper, the authors proposed model selection methods for determination of the number of signals in presence of noise with arbitrary covariance matrix. This problem is related to finding the multiplicity of the smallest eigenvalue of [Sigma]2[Sigma]1-1, where [Sigma]2 = [Gamma] + [lambda][Sigma]1, [Sigma]1 and [Sigma]2 are covariance matrices, [lambda] is a scalar, and [Gamma] is non-negative definite matrix and is not of full rank. Also, the authors proposed methods for determination of the multiplicities of various eigenvalues of [Sigma]2[Sigma]1-1. The methods used in these procedures are based upon certain information theoretic criteria. The strong consistency of these criteria is established in this paper.