The HC calculus, quaternion derivatives and caylay-hamilton form of quaternion adaptive filters and learning systems

We introduce a novel and unifying framework for the calculation of gradients of both quaternion holomorphic functions and nonholomorphic real functions of quaternion variables. This is achieved by considering the isomorphism between the quaternion domain H and the bivariate complex domain C×C, and by exploiting complex calculus to simplify the quaternion gradient calculation. The validation of the proposed HC calculus is performed against the existing HR calculus, and its convenience is illustrated in the context of gradient-based quaternion optimisation as well as in adaptive learning systems. Quaternion adaptive filtering algorithms and a dynamical perceptron update are next derived based on the bivariate complex representation of quaternions and the HC calculus. Simulations on both synthetic and real-world multidimensional signals support the analysis.