An adaptive nonlinear function controlled by kurtosis for blind source separation

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Abstract
In blind source separation, convergence and separation performances are highly dependent on a relation between probability density functions (pdf) of signal sources and nonlinear functions used in updating coefficients of a separation block. This relation was analyzed based on kurtosis /spl kappa//sub 4/. It was suggested that tanh y and y/sup 3/, where y is the output, are useful nonlinear functions for super-Gaussian (/spl kappa//sub 4/>0) and sub-Gaussian (/spl kappa//sub 4/<0), respectively. In this paper, an adaptive nonlinear function is proposed. It has a form of f(y)=a tanh y+(1-a)y/sup 3//4, where a is controlled by the kurtosis of the output signal yk(n). It is assumed that the pdf p(y) of the output signal satisfies the stability condition f(y)=-(dp(y)/dy)/p(y). Based on this assumption, the parameter a and the kurtosis is related. This relation is approximated by a function a=q(/spl kappa//sub 4/). In a learning process, /spl kappa//sub 4/(n) of the output signal is calculated at each sample n, and a(n) is determined by a(n)=q(/spl kappa//sub 4/(n)). Then, the nonlinear function f (y) is adjusted. Blind separation of music signals of 2-5 channels were simulated. The proposed method is superior to a method, which switches tanh y and y/sup 3/ based on polarity of /spl kappa//sub 4/(n).

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