Wide-Sense Markov Signals on the Tessarine Domain. A Study under Properness Conditions

Abstract

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 $\mathbb{T}_2$-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 $\mathbb{T}_2$ processing is proved, thus attaining a notable reduction in computational burden. Next, the $\mathbb{T}_i$-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 $\mathbb{T}_i$ estimators, $i=1,2$, over their counterparts in the quaternion domain is shown.