Blind Separation of Gaussian Sources With General Covariance Structures: Bounds and Optimal Estimation

We consider the separation of Gaussian sources exhibiting general, arbitrary (not necessarily stationary) covariance structures. First, assuming a semi-blind scenario, in which the sources' covariance structures are known, we derive the maximum likelihood estimate of the separation matrix, as well as the induced Cramér-Rao lower bound (iCRLB) on the attainable Interference to Source Ratio (ISR). We then extend our results to the fully blind scenario, in which the covariance structures are unknown. We show that (under a scaling convention) the Fisher information matrix in this case is block-diagonal, implying that the same iCRLB (as in the semi-blind scenario) applies in this case as well. Subsequently, we demonstrate that the same "semi-blind" optimal performance can be approached asymptotically in the "fully blind" scenario if the sources are sufficiently ergodic, or if multiple snapshots are available.