A Quaternion Widely Linear Adaptive Filter

A quaternion widely linear (QWL) model for quaternion valued mean-square-error (MSE) estimation is proposed. The augmented statistics are first introduced into the field of quaternions, and it is demonstrated that this allows for capturing the complete second order statistics available. The QWL model is next incorporated into the quaternion least mean-square (QLMS) algorithm to yield the widely linear QLMS (WL-QLMS). This allows for a unified approach to adaptive filtering of both-proper and-improper signals, leading to improved accuracies compared to the QLMS class of algorithms. Simulations on both benchmark and real world data support the analysis.