Optimal Pairwise Fourth-Order Independent Component Analysis

Blind source separation (BSS) aims at the reconstruction of unknown mutually independent signals, so-called sources, from their mixtures observed at the output of a sensor array. The BSS of instantaneous linear mixtures, which finds application in numerous fields, can be solved through the statistical tool of independent component analysis (ICA). This paper concentrates on the analytic solutions for the fundamental two-signal ICA scenario. A novel estimation class, so-called general weighted fourth-order estimator (GWFOE), is put forward, which is based on the fourth- order statistics of the whitened sensor output. By means of a weight parameter, the GWFOE is able to unify a variety of apparently disparate estimation expressions previously scattered throughout the literature, including the well-known JADE method in the two- signal case. A theoretical asymptotic performance analysis is carried out, resulting in the GWFOE large-sample mean square error and the source-dependent weight value of the most efficient estimator in the class. To extend the pairwise estimators to the general scenario of more than two sources, an improved Jacobi-like optimization technique is proposed. The approach consists of calculating the necessary sensor-output fourth-order statistics at the initialization stage of the algorithm, which can lead to significant computational savings when large sample blocks are processed. Based on this idea, adaptive algorithms are also devised, showing very satisfactory convergence characteristics. Experiments illustrate the good performance of these optimal pairwise ICA strategies, in both off- and on-line processing modes.