An adaptive nonlinear function controlled by kurtosis for blind source separation

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 κ₄. It was suggested that tanh y and y³, where y is the output, are useful nonlinear functions for super-Gaussian (κ₄>0) and sub-Gaussian (κ₄<0), respectively. In this paper, an adaptive nonlinear function is proposed. It has a form of f(y)=a tanh y+(1-a)y 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(κ₄). In a learning process, κ₄(n) of the output signal is calculated at each sample n, and a(n) is determined by a(n)=q(κ₄(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 y³ based on polarity of κ₄(n)