A perspective on CLMS as a deficient length augmented CLMS: Dealing with second order noncircularity

Mean and mean square performance analyses of the strictly linear complex least mean square (CLMS) algorithm are addressed for widely linear estimation (WLE) of second order noncircular (improper) Gaussian inputs, for which both the covariance and pseudo-covariance matrices contain nonzero off-diagonal elements. A detailed performance analysis of standard CLMS in this 'suboptimal' context is not trivial but is important for practical applications. To this end, we here consider the strictly linear CLMS as a 'deficient length' version of the widely linear augmented CLMS (ACLMS) algorithm, which is second order optimal for improper inputs. Rigorous performance analysis is provided, which also statistically quantifies the suboptimality of CLMS in both the transient and steady state stages. The recently introduced approximate uncorrelating transform (AUT) is employed to derive closed-form expressions for the mean square stability and the steady-state performance of CLMS. In addition, since WLE with second order noncircular inputs is a general linear estimation problem in the complex domain C, these results provide a generalised framework from which current statistical descriptions of CLMS for strictly linear estimation (SLE) can be deduced as special cases. Simulations in system identification settings validate the findings.