A class of quaternion valued affine projection algorithms

The strictly linear quaternion valued affine projection algorithm (QAPA) and its widely linear counterpart (WLQAPA) are introduced, in order to provide fast converging stochastic gradient learning in the quaternion domain, for the processing of both second order circular (proper) and second order noncircular (improper) signals. This is achieved based on the recent advances in augmented quaternion statistics, which employs all second order information available, together with the associated widely linear models and through performing rigorous gradient calculation (HRHR-calculus). Further, mean square error analysis is performed based on the energy conservation principle, which provides a theoretical justification for the WLQAPA offering enhanced steady state performance for quaternion noncircular (improper) signals, a typical case in real world scenarios. Simulations on benchmark circular and noncircular signals, and on noncircular real world 4D wind and 3D body motion data support the analysis.

► HR-calculus and quaternion widely linear model is used to derive the quaternion affine projection.
► The widely linear quaternion affine projection algorithm (WLQAPA) is second order optimum.
► MSE analysis is performed on both the WLQAPA and its strictly linear version.
► The analysis confirms the superiority of the WLQAPA for noncircular signals.
► Simulations on real world data (wind fields) support the analytical results.