A Full Mean Square Analysis of CLMS for Second-Order Noncircular Inputs

A full mean square transient and steady-state analysis of the complex least mean square (CLMS) algorithm is provided for strictly linear estimation of general second-order noncircular (improper) Gaussian inputs. To this end, we also consider the performance assessment in terms of the evolution of the complementary mean square error (CMSE) and the complementary covariance (pseudocovariance) matrix of the weight error vector of CLMS. This makes it possible to measure the degrees of noncircularity of the output error and the weight error vector, which arise due to second-order noncircularity (improperness) of the system input and system noise. The recently introduced approximate uncorrelating transform, which allows for joint direct diagonalization of both the input covariance and complementary covariance matrices with a single singular value decomposition, is then employed in order to derive a unified bound on the step-size, which guarantees the convergence of both the standard MSE and the proposed CMSE. A joint consideration of the standard mean square performance analysis and the proposed complementary performance analysis is shown to provide full second order, closed form, statistical descriptions of both the transient and steady state performances of CLMS for second-order noncircular (improper) Gaussian input data. Simulations in the system identification setting support the analysis.