Abstract
The recently proposed HR-calculus has enabled rigorous derivation of quaternion-valued adaptive filtering algorithms, and has also introduced several equivalent forms of the quaternion least mean square (QLMS). This work aims to address the uniqueness of the solutions to the stochastic gradient optimisation problems, and to provide a unified framework for the derivation and analysis of quaternion least mean square algorithms. In doing so, we assess and compare the properties of the adaptive algorithms in the context of their convergence and steady state performances. For generality, the convergence properties of both QLMS and its widely linear extension, the WL-QLMS are illuminated.