Separating Mixed Signals in a Noisy Environment Using Global Optimization
SIAM Undergraduate Research Online · 2015
In this paper, we propose and analyze a class of blind source separation (BSS) methods to recover mixed signals in a noisy environment.Blind source separation aims at recovering source signals from their mixtures without detailed knowledge of the mixing process.Motivated by the work presented in [23], we propose a new optimization method based on second order statistics that considers the impact of Gaussian noise.By treating the Gaussian noise as a separate source signal and using an extra measurement of the mixed signals, we formulate the source separation problem as a global optimization problem that minimizes the cross-correlation of the recovered signals.In the case when the cross-correlation of the source signals is exactly zero, we give precise solvability conditions and prove that our global optimization method gives an exact recovery of the original source signals up to a scaling and permutation.In the case when the cross-correlation is small but nonzero, we perform stability and error analysis to show that the global optimization method still gives an accurate recovery with a small error.We also analyze the solvability for the two-signal case when the mixing matrix is degenerate.To the best of our knowledge, this is the first error analysis of BSS methods.The numerical results using realistic signals confirm our theoretical findings and demonstrate the robustness and accuracy of our methods.