An algebraic real translation of hypercomplex linear systems and its application to adaptive filtering

The hypercomplex (e.g., complex, quaternion) valued linear model often arises in the signal processing field and attract increasing attention recently. In this paper, we present an algebraic translation of a hypercomplex valued linear systems into a real valued linear model. This translation is designed by taking advantage of isomorphism between hypercomplex numbers and multi-dimensional real vectors and enables us to straightforwardly apply real valued optimization frameworks to various estimation problems for the hypercomplex linear model. We also clarify the useful algebraic properties of the translation. As an application to hypercomplex valued adaptive filtering problems, we derived A_m-adaptive projected subgradient method (A_m-APSM) for hypercomplex valued system identification problems, and show that many hypercomplex adaptive filtering algorithms can be viewed as a special case of this algorithm. Numerical example shows that a new algorithm derived from proposed algorithm outperforms existing hypercomplex adaptive algorithms.