A second-order statistics method for blind source separation in post-nonlinear mixtures

Abstract

• Derivation of the conditions and the constraints on the Post-Nonlinear (PNL) model for Second-Order Statistics (SOS)-based separation. • Proposition of an SOS-based method for separation using a time-extended formulation. • Formulation considers the analytical computation of the time-extended covariance matrices. • Promising perspectives for the extension to any polynomial mixing functions in the PNL model. In the context of nonlinear Blind Source Separation (BSS), the Post-Nonlinear (PNL) model is of great importance due to its suitability for practical nonlinear problems. Under certain mild constraints on the model, Independent Component Analysis (ICA) methods are valid for performing source separation, but requires use of Higher-Order Statistics (HOS). Conversely, regarding the sole use of the Second-Order Statistics (SOS), their study is still in an initial stage. In that sense, in this work, the conditions and the constraints on the PNL model for SOS-based separation are investigated. The study encompasses a time-extended formulation of the PNL problem with the objective of extracting the temporal structure of the data in a more extensive manner, considering SOS-based methods for separation, including the proposition of a new one. Based on this, it is shown that, under some constraints on the nonlinearities and if a given number of time delays is considered, source separation can be successfully achieved, at least for polynomial nonlinearities. With the aid of metaheuristics called Differential Evolution and Clonal Selection Algorithm for optimization, the performances of the SOS-based methods are compared in a set of simulation scenarios, in which the proposed method shows to be a promising approach.