Complementary Mean Square Analysis of Augmented CLMS for Second-Order Noncircular Gaussian Signals

A novel physical insight is provided into the behavior and performance of the augmented complex least mean square (ACLMS) algorithm for widely linear adaptive estimation of general second-order noncircular (improper) Gaussian signals, whereby the off-diagonal elements of the covariance and complementary covariance matrices are nonzero. This is achieved through a novel complementary mean square analysis, a counterpart to the standard mean square analysis, and which focuses on the behavior of the complementary second-order statistics of the output error and the augmented weight error vector. We next establish the effect of the degree of input noncircularity on the evolution for these two key parameters that govern the ACLMS. Both transient and steady-state performances are addressed and a stability bound on the step-size for their convergence is established. Simulations in the system identification setting support the analysis.