Covariance Matrix Estimation Under Degeneracy for Complex Elliptically Symmetric Distributions

In several adaptive algorithms, the data covariance matrix must be frequently estimated, particularly in modern wireless systems where, due to high mobility, signal statistics change in time. When the underlying distribution is degenerate, likelihood-based estimators of the covariance matrix and/or scatter matrix may not exist for non-Gaussian signals. A well-known class of non-Gaussian distributions is the family of complex elliptically symmetric (CES) distributions, which includes several well-known distributions as special cases. In this paper, we give some results on the degeneracy of CES distributions and derive the maximum likelihood (ML) estimator and M-estimators of the scatter matrix for degenerate CES vectors. Estimation with insufficient data is addressed by a regularization approach. We also formulate the Cramér-Rao lower bound (CRLB) on the elements of the scatter matrix for degenerate CES vectors. Our results coincide with those of the nondegenerate CES vectors whenever the scatter matrix is full rank. Numerical results are presented to justify the efficiency of the proposed method.