Transpose properties in the stability and performance of the classic adaptive algorithms for blind source separation and deconvolution. (Q1575762)

This paper presents a tutorial review of the problem of blind source separation and the properties of the classic adaptive algorithms when either the score-function or a general (nonscore) nonlinearity is employed in the algorithm. In new findings it is shown that the separating solution for both sub- and super-Gaussian 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 pdfs of the source signals. These results show that a transpose symmetry exists between the nonlinear algorithms for sub- and super-Gaussian signals. The behavior of the algorithm is then detailed when the ideal score-function 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 multi-channel deconvolution and shown by analysis and extensive simulation to hold for mixed and convolved source signals. The results allow the design of stable algorithms for multichannel blind deconvolution with a general nonlinearity when sub- and super-Gaussian source signals are present.