A new proof of the generalized Hamiltonian–Real calculus. Issue 9 (September 2016)

• Record Type: Journal Article
• Authors: Xu, Dongpo; Gao, Hua; Mandic, Danilo P.
• Is Part Of: Royal Society open science. Volume 3:Issue 9(2016)
• Publication Date: 2016-09
• Subjects: GHR calculus -- product and chain rules -- duality relationship -- widely linear quaternion least mean square; Science -- Periodicals; 500
• DOI: 10.1098/rsos.160211
• Languages: English
• ISSNs: 2054-5703

Abstract

Abstract : The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain.