Abstract
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.
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.
| Original language | English |
|---|---|
| Journal | Signal Processing |
| DOIs | |
| Publication status | Published - 24 Dec 2024 |