Widely Linear Complex-Valued Estimated-Input LMS Algorithm for Bias-Compensated Adaptive Filtering With Noisy Measurements

In this paper, a novel widely linear complex-valued estimated-input adaptive filter (WLC-EIAF) is first proposed for processing noisy input and output data in the complex domain. The WLC-EIAF consists of two steps: (i) estimation of noise-free input and (ii) update of the weight vector, which is realized by alternating theminimization of an instantaneous perturbation with both input and output data. Based on theWLC-EIAFmethod and adopting the least mean-square (LMS) scheme, a widely linear complex-valued estimated-input LMS (WLC-EILMS) algorithm is developed. It is able to achieve an unbiased parameter estimation and, thus, outperforms the widely linear complex-valued LMS (WL-CLMS) algorithm in the presence of noisy input and output. In particular, for Gaussian signals, closed-form expressions are derived for its steady-state excessmean-square error performance. Furthermore, the linear complex-valued estimated-input LMS and linear realvalued estimated-input LMS algorithms are presented, which are two simplified versions of the WLC-EILMS for circular and realvalued signals, respectively. Simulation results demonstrate that the proposed methods achieve significantly improved performance in terms of mean-square deviation and mean-square error when compared to the WL-CLMS and CLMS algorithms.