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 Separation (BSS) and the properties of the classic adaptive algorithms when either the scoree function or a general (non-score) nonlinearity is employed in the algorithm. In new ndings it is shown that the separating solution for both sub-and super-Gausss ian signals can be stabilized by an algorithm employing any given nonlinearity. For these separating solutions the steady-state error levels are also given in terms of the nonlinearity and the pdf.s of the source signals. These results show that a transs pose symmetry exists between the nonlinear algorithms for sub-and super-Gaussian signals. The behavior of the algorithm is then detailed when the ideal score-funcc tion nonlinearity 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 single-and multii-channel deconvolution and shown by analysis and extensive simulation to hold for mixed and convolved source signals. The results allow the design of stable algoo rithms for multichannel blind deconvolution with a general nonlinearity when sub-and super-Gaussian source signals are present. der Quellensignale angepasste Nichtlinearittt, die sogenannte Score-Funktion, durch eine allgemeine Nichtlinearittt (z.B. Begrenzungsfunktion oder kubische Funktion)