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A Generalized Contrast Function and Stability Analysis for Overdetermined Blind Separation of Instantaneous Mixtures

xlzhu_dau{at}tsinghua.edu.cn

Xian-Da Zhang

zxd-dau{at}tsinghua.edu.cn National Laboratory for Information Science and Technology, Department of Automation, Tsinghua University, Beijing 100084, China

Ji-Min Ye

jinhh{at}xsyu.edu.cn School of Science, Xidian University, Xi'an 710071, China

In this letter, the problem of blind separation of n independent sources from their m linear instantaneous mixtures is considered. First, a generalized contrast function is defined as a valuable extension of the existing classical and nonsymmetrical contrast functions. It is applicable to the overdetermined blind separation (m > n) with an unknown number of sources, because not only independent components but also redundant ones are allowed in the outputs of a separation system. Second, a natural gradient learning algorithm developed primarily for the complete case (m = n) is shown to work as well with an n x m or m x m separating matrix, for each optimizes a certain mutual information contrast function. Finally, we present stability analysis for a newly proposed generalized orthogonal natural gradient algorithm (which can perform the overdetermined blind separation when n is unknown), obtaining an expectable result that its local stability conditions are slightly stricter than those of the conventional natural gradient algorithm using an invertible mixing matrix (m = n).