Asymptotically Optimal Recovery of Gaussian Sources from Noisy Stationary Mixtures: The Least-noisy Maximally-separating Solution

We address the problem of source separation from noisy mixtures in a semi-blind scenario, with stationary, temporally-diverse Gaussian sources and known spectra. In such noisy models, a dilemma arises regarding the desired objective. On one hand, a maximally separating solution, providing the minimal attainable Interference-to-Source-Ratio (ISR), would often suffer from significant residual noise. On the other hand, optimal Minimum Mean Square Error (MMSE) estimation would yield estimates which are the least distorted versions of the true sources, often at the cost of compromised ISR. Based on Maximum Likelihood (ML) estimation of the unknown underlying model parameters, we propose two ML-based estimates of the sources. One asymptotically coincides with the MMSE estimate of the sources, whereas the other asymptotically coincides with the (unbiased) least-noisy maximally-separating solution for this model. We prove the asymptotic optimality of the latter and present the corresponding Cramér-Rao lower bound. We discuss the differences in principal properties of the proposed estimates and demonstrate them empirically using simulation results.