Performance analysis of the deficient length augmented CLMS algorithm for second order noncircular complex signals

Abstract

The augmented complex LMS (ACLMS) algorithm deals with second order noncircular (improper) input signals, based on widely linear modelling and the use of full second order statistical information. In current analyses of ACLMS, it is implicitly or explicitly assumed that the length of the adaptive filter is equal to that of the unknown system’s impulse response (optimal model order). In many applications, however, the length of the adaptive filter is smaller than required, the so called deficient length case, which renders the analysis for a ‘sufficient length’ ACLMS inadequate. To this end, we examine the statistical behaviour of the ACLMS algorithm in undermodelling situations. Exact expressions are developed to completely characterise both the transient and steady-state mean and mean square performances of the deficient length ACLMS for general second order noncircular Gaussian input signals. This is achieved using the recently introduced approximate uncorrelating transform (AUT), in order to jointly diagonalise the covariance and pseudo-covariance matrices with a single singular value decomposition (SVD), which both simplifies the analysis and enables a link between the degree of input noncircularity and the steady state mean square error (MSE) performance of the deficient length ACLMS. Simulations in system identification settings support the analysis.