Split quaternion nonlinear adaptive filtering

A split quaternion learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of three- and four-dimensional signals is proposed. The derivation takes into account the non-commutativity of the quaternion product, an aspect neglected in the derivation of the existing learning algorithms. It is shown that the additional information taken into account by a rigorous treatment of quaternion algebra provides improved performance on hypercomplex processes. A rigorous analysis of the convergence of the proposed algorithms is also provided. Simulations on both benchmark and real-world signals support the approach.