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

\({\mathbb{T}}\)-Proper Hypercomplex Centralized Fusion Estimation for Randomly Multiple Sensor Delays Systems with Correlated Noises

“…Note that, depending on the vanishing of the different pseudo correlation functions R xx ν (t, s), ν = * , η, η , various kinds of tessarine properness can be defined. In particular, the following properness conditions in the tessarine domain have recently been introduced in [23,24]. Definition 4.…”

“…Remark that, T 1 properness is more restrictive than T 2 properness. Statistical tests to experimentally check whether a signal is T k -proper, k = 1, 2, or improper have been proposed in [23,24].…”

“…Nevertheless, in the case of joint T k -properness, k = 1, 2, the optimal linear processing is reduced to a T k linear processing, with the corresponding decrease in the dimension of the problem. In particular, T 1 linear processing is based on the tessarine random signal itself, and T 2 linear processing considers the augmented vector given by the signal and its conjugate [24].…”

“…These properties make each algebra more appropriate for every specific problem. With this in mind, in [21][22][23][24] the application of these two isodimensional algebras is compared with the objective of showing how the choice of a particular algebra may determine the proposed method performance.…”

“…In this sense, ref. [23,24] introduced the concept of T 1 and T 2 -properness and provided a statistical test to determine whether a signal presents one of these properness properties. According to the type of properness, the most suitable form of processing is T 1 linear processing, which supposes to operate on the signal itself, or T 2 linear processing, based on the augmented vector given by the signal and its conjugate.…”