Performance Analysis of Minimum<tex>$ell_1$</tex>-Norm Solutions for Underdetermined Source Separation

Abstract

Results of the analysis of the performance of minimum /spl lscr//sub 1/-norm solutions in underdetermined blind source separation, that is, separation of n sources from m(<n) linearly mixed observations, are presented in this paper. The minimum /spl lscr//sub 1/-norm solutions are known to be justified as maximum a posteriori probability (MAP) solutions under a Laplacian prior. Previous works have not given much attention to the performance of minimum /spl lscr//sub 1/-norm solutions, despite the need to know about its properties in order to investigate its practical effectiveness. We first derive a probability density of minimum /spl lscr//sub 1/-norm solutions and some properties. We then show that the minimum /spl lscr//sub 1/-norm solutions work best in a case in which the number of simultaneous nonzero source time samples is less than the number of sensors at each time point or in a case in which the source signals have a highly peaked distribution. We also show that when neither of these conditions is satisfied, the performance of minimum /spl lscr//sub 1/-norm solutions is almost the same as that of linear solutions obtained by the Moore-Penrose inverse. Our results show when the minimum /spl lscr//sub 1/-norm solutions are reliable.