Analysis of Least Stochastic Entropy Adaptive Filters for Noncircular Gaussian Signals

Complex noncircularity is the manifestation of a rotation-dependent probability distribution, and can arise from the properties of the input signal and/or system noise. In contrast to the standard mean-square error (MSE), it is now well understood that the minimum Gaussian entropy criterion provides statistically optimal complex-valued adaptive filters, even when the estimation error is noncircular Gaussian. However, a rigorous analysis of the class of information-theoretic adaptive filtering algorithms based on the entropy cost function is still under investigation. To this end, we provide physical insights into the full second-order performance of the least stochastic entropy (LSE) adaptive filtering algorithm. This is achieved by considering the coupled evolution of both the weight error covariance matrix and complementary covariance matrix. A mean-square stability bound on the step-size is established to guarantee convergence, and to allow for a closed-form evaluation of the weight error variance of the LSE at the steady-state. Simulations in the system identification setting support the analysis.