Steady state analysis of the CLMS and augmented CLMS algorithms for noncircular complex signals

The recently introduced augmented complex least mean square (ACLMS) algorithm is shown to be suitable for the processing of both second order circular (proper) and noncircular (improper) signals, by virtue of the underlying widely linear model. In theory, both the linear CLMS and widely linear ACLMS achieve the same mean square error for propers signals, whereas the ACLMS exhibits lower mean square error for improper signals. However, improperness can arise due to the system noise, input, or channel model and to shed more light on the convergence and steady state properties of ACLMS and CLMS in these cases we here employ the energy conservation principle. Simulations in adaptive prediction and system identification settings for signals with different probability distributions and degrees of noncircularity support the the analysis.