Global convergence towards statistical independence for noisy mixtures of stationary and non-stationary signals

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

“…We can obtain orthogonal signals in the same way used in [19,23] by a simple Cholesky decomposition of the covariance matrix of mixed signals by writing it as follows:…”

“…The authors in [19] proved that by means of optimization techniques, the statistical independence of the set of orthogonal signals can be obtained by the simple generalization of the separation of the case (2×2). This means that after whitening the observations by applying (4), and to achieve statistical independence for the case (3×3).…”

“…Via the ICA and to complete the separation by designing a system called the separation system, the operation must be based on criteria called in the BSS by the separation criteria. As an example, we can find in the literature proposals which achieve the separation by minimizing the mutual information [16] and others which rely on the central limit theorem by maximizing the kurtosis of the signals [17][18][19], where there are also solutions that take the minimization of crosscumulants as a separation criterion [20].…”

“…Whatever the criterion used, most of the proposals that deal with the problem of BSS, are iterative methods that rely on adaptation or optimization techniques that aim to adapt the parameters of the separation system in order to converge towards the optimal separation [19,21,22], while, we find few solutions that deal with the problem, algebraically or analytically to build a separation system that directly completes the separation [20,23].A common problem with the use of adaptive techniques to solve the BSS problem is that the completion of the separation depends on several adaptation factors to reach the optimal solution, which generates an unfavorable computational cost in many applications. On the other hand, the direct solutions reach the objective of separation in a very fast way but only for specific case of the problem, but for other cases and even if the problems are similar, a new analytical study must be carried out.…”

“…By optimization via a genetic algorithm (GA), the authors in [19] arrived at a solution that solves both problems of the geometric solution, and their solution gives good results in terms of source retrieval, but as we mentioned, the execution time by applying this type of optimization techniques is considerable and the accuracy of the results depends on the accuracy of the algorithm itself. Therefore, and in order to solve the geometric problem of the BSS by a fast and efficient method, in this document and for the instantaneous linear case of statistically independent sources, we present a simple analytical solution which is based on the calculation of the phase shift of a sinusoidal objective function based on the maximization of kurtosis, and which leads to the resolution of the geometric problem, whatever the nature of the source signals (stationary and non-stationary).…”