A class of widely linear quaternion blind equalisation algorithms

Quaternion adaptive filters have been applied extensively to model three- and four-dimensional phenomena in signal processing, but most of them require a known reference signal. In this paper, a class of widely linear quaternion-valued Godard (WL-QGodard) algorithms is derived, which include the widely linear quaternion-valued constant modulus algorithm (WL-QCMA) as a special case. The derived filter allows for signal recovery operations in the absence of reference signals to be performed directly in the quaternion domain, eliminating the need for transformation to real-valued vector algebras and preserving the advantages of the quaternion division algebra. Compared to state-of-the-art quaternion blind equalisation algorithms, the proposed algorithm models the signal transmission channel using the widely linear quaternion framework, which has more extensive applicability and can better represent real-world scenarios. Furthermore, aided by GHR calculus, for the first time, we present a performance analysis framework for the QGodard algorithm and WL-QGodard algorithms, which depicts the dynamic and their static convergence behaviours, overcoming the challenges posed by the noncommutative quaternion algebra and non-isomorphism between the quaternion equalisers and real-valued equalisers. Finally, simulation results over physically meaningful wireless communication signals indicate the effectiveness and superiority of the proposed WL-QCMA, and the validity of the theoretical performance analysis.