A class of widely linear complex-valued adaptive filtering algorithms

Abstract

A large class of signals encountered in communications, biomedical engineering, renewable energy and power systems are conveniently processed in the complex domain C, where traditional adaptive signal processing in C is regarded as a straightforward extension of the corresponding algorithms in the real domain R. However, recent advances in widely linear modelling and augmented complex statistics show the suboptimality of such an assumption. In this work, based on the widely linear model, a class of linear and nonlinear adaptive filtering algorithms have been derived to process the generality of complex-valued signals (both second order circular and noncircular) in both noise-free and noisy environments, and their usefulness in real-world applications is demonstrated through case studies. The focus of this thesis is on the use of augmented second order statistics and widely linear modelling. The so called Augmented Complex Least Mean Square (ACLMS) algorithm has already been extended from the standard CLMS algorithm to perform optimum mean square error(MSE) type of adaptive estimation for the generality of complex-valued signals and has been shown to outperform the CLMS algorithm, however, a theoretical understanding of its performance is still missing. To this end, this thesis first addresses this issue in terms of both convergence analysis and steady state analysis. Next, based on the generalised framework introduced by the derivation of the ACLMS algorithm, a class of widely linear adaptive algorithms have been introduced; these include the Regularised Normalised ACLMS (RNACLMS) algorithm, the Augmented Affine Projection algorithm (AAPA) for linear Finite Impulse Response (FIR) adaptive filters, and also in the context of reservoir computing, for the recently introduced random state space based Echo State Networks (ESNs). Furthermore, the widely linear model has been introduced in the context of distributed networks, where the individual adaptive filters share information with their neighbours to achieve a cooperative estimation. The enhanced performances of the widely linear algorithms are illustrated in renewable energy and power system applications, in particular, for the prediction of wind profiles and frequency estimation of unbalanced three-phase power systems.