On the Convergence and Stability in Second-Order Statistics BSS
In Chap. 6, we have derived natural gradient SOS-BSS algorithms (NG-SOS-BSS) to adapt the separation matrix W. This chapter addresses the question of the convergence of these algorithms, that is, the convergence of W to a separating matrix. An examination of the global convergence is of practical interest because it may indicate the range of admissible step-sizes μ for which these NG-SOS-BSS algorithms globally converge. This issue is discussed in Sect. 7.1, which explains why the analysis of the global convergence is hardly tractable for NG-SOS-BSS algorithms, even in the case of instantaneous mixing. However, in the instantaneous case, an analysis is possible for a simplified decorrelation algorithm that is closely related to NG-SOS-BSS. Then, we propose an interpretation of the results of this analysis for convolutive NG-SOS-BSS. In contrast to the global behavior, the local behavior of convolutive NG-SOS-BSS may be analyzed. This is discussed in Sect. 7.2 where sufficient local stability conditions are given.
KeywordsEquilibrium Point Global Convergence Global Stability Local Stability Separation Matrix
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