Transpose properties in the stability and performance of the classic adaptive algorithms for blind source separation and deconvolution

This paper presents a tutorial review of the problem of Blind Source Separa tion (BSS) and the properties of the classic adaptive algorithms when eithe r the score-function or a general (nonscore) nonlinearity is employed in th e algorithm. In new findings it is shown that the separating solution for b oth sub- and super-Gaussian signals can be stabilized by an algorithm emplo ying any given nonlinearity. For these separating solutions the steady-stat e error levels are also given in terms of the nonlinearity and the pdfs of the source signals, These results show that a transpose symmetry exists bet ween the nonlinear algorithms for sub- and super-Gaussian signals. The beha vior of the algorithm is then detailed when the ideal score-function nonlin earity is replaced by a general (hard saturation or u(3)) nonlinearity, The phases of convergence to decorrelated output signals and then to recovery of the source signals are explained. The results are then extended to singl e- and multi-channel deconvolution and shown by analysis and extensive simu lation to hold for mixed and convolved source signals. The results allow th e design of stable algorithms for multichannel blind deconvolution with a g eneral nonlinearity when sub- and super-Gaussian source signals are present . (C) 2000 Elsevier Science B.V. All rights reserved.