Source recovery by analytical maximization of phase-shifted kurtosis for the mixtures of noisy and noiseless signals

Abstract

This manuscript presents a work that provides a study as well as a simple analytical solution for solving the blind source separation problem (BSS) for noiseless and noisy linear mixing of statistically independent stationary and nonstationary signals. The study is based on the exploitation of the probabilistic characteristics of the mixed signals by using the statistics of the second order and the fourth order for the completion of the separation. The proposed solution consists mainly of two steps based on the concept of the geometric solution. For the case of the mixture of two sources (2×2), the first step aims to transform the dependent signals into orthogonal signals (whitening) via the principal component analysis (PCA) principle. After the application of the PCA and in order to complete the statistical independence of the two uncorrelated signals, the second step aims to determine an adequate rotating angle that leads directly to the separation, and this angle is determined in this work analytically by the simple calculation of the phase shift of a sinusoidal objective function based on the sum of the kurtosis of the whitened signals. In the case of several sources (n×n), the solution (2×2) can be applied by a simple generalization which leads to the global separation. Whether for the noisy or noiseless case, the results obtained prove the reliability and efficiency by applying this analytical solution to achieve the desired objective, in particular by comparing the proposed algorithm with the application of two other separation algorithms, one of which involves the application of optimization techniques