A class of fast quaternion valued variable stepsize stochastic gradient learning algorithms for vector sensor processes

We introduce a class of gradient adaptive stepsize algorithms for quaternion valued adaptive filtering based on three- and four-dimensional vector sensors. This equips the recently introduced quaternion least mean square (QLMS) algorithm with enhanced tracking ability and enables it to be more responsive to dynamically changing environments, while maintaining its desired characteristics of catering for large dynamical differences and coupling between signal components. For generality, the analysis is performed for the widely linear signal model, which by virtue of accounting for signal noncircularity, is optimal in the mean squared error (MSE) sense for both second order circular (proper) and noncircular (improper) processes. The widely linear QLMS (WL-QLMS) employing the proposed adaptive stepsize modifications is shown to provide enhanced performance for both synthetic and real world quaternion valued signals. Simulations include signals with drastically different component dynamics, such as four dimensional quaternion comprising three dimensional turbulent wind and air temperature for renewable energy applications.