Proper Adaptive Filtering in Four-Dimensional Cayley-Dickson Algebras

Abstract

A family of hypercomplex algebras in four dimensions (4D) is proposed to devise adaptive filters. Such a family, called $\beta$-quaternions, has multiplication rules for the complex units that depend on a parameter $\beta$, and this family contains, as particular cases, both standard Hamilton quaternions and split quaternions. In this framework, two notions of properness for random vectors are introduced and their implications on the statistical processing involved are analyzed. Then, statistical tests to check properness in practice and a method to select the best algebra where the properness conditions could hold are provided. Also, proper adaptive filters are suggested and row and column updating problems are studied. The main advantage of the techniques proposed compared with the standard ones is that a notable reduction in the computational burden is achieved. Finally, simulation examples validate the proper adaptive filters and demonstrate that our scheme performs better than the traditional quaternion domain.