Wide-sense Markov signals on the tessarine domain. A study under properness conditions

• Published in: Signal processing Vol. 183; p. 108022
• Main Authors: Navarro-Moreno, Jesús, Ruiz-Molina, Juan Carlos
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.06.2021
• Subjects: [formula omitted]-properness; Estimation; Intermittent observations; Tessarine processing; Wide-sense Makov signals; Intermittent observations; Tessarine processing; Wide-sense Makov signals; Estimation; Ti-properness
• ISSN: 0165-1684; 1872-7557
• DOI: 10.1016/j.sigpro.2021.108022

•Quaternions are not always the best algebra for processing 4D hypercomplex signals.•Properness properties should be analyzed for deciding between either quaternion or tessarine processing.•Tessarines provide a suitable framework for addressing the estimation problem. The quaternion algebra is not always the best choice for processing 4D hypercomplex signals. This paper aims to explore tessarines as an alternative algebra to solve the estimation problem. More concretely, wide-sense Markov signals in the tessarine domain are introduced and their properties under properness properties are analyzed. Firstly, the T2-properness condition in the tessarine setting is defined and then, the linear estimation problem under tessarine processing is addressed. The equivalence between the optimal estimator based on tessarine widely linear processing and the one based on tessarine T2 processing is proved, thus attaining a notable reduction in computational burden. Next, the Ti-proper wide-sense Markov signals, i=1,2, are defined and a forwards representation for modeling them is suggested. Finally, the estimation problem with intermittent observations for this class of signals is tackled. Specifically, based on the forwards representation, two algorithms for the problems of filtering, prediction and fixed-interval smoothing are devised. Numerical simulations are developed where the superiority of the Ti estimators, i=1,2, over their counterparts in the quaternion domain is shown.