The Distributed and Centralized Fusion Filtering Problems of Tessarine Signals from Multi-Sensor Randomly Delayed and Missing Observations under Tk-Properness Conditions

Abstract

This paper addresses the fusion estimation problem in tessarine systems with multi-sensor observations affected by mixed uncertainties when under Tk-properness conditions. Observations from each sensor can be updated, delayed, or contain only noise, and a correlation is assumed between the state and the observation noises. Recursive algorithms for the optimal local linear filter at each sensor as well as both centralized and distributed linear fusion estimators are derived using an innovation approach. The Tk-properness assumption implies a reduction in the dimension of the augmented system, which yields computational savings in the previously mentioned algorithms compared to their counterparts, which are derived from real or widely linear processing. A numerical simulation example illustrates the obtained theoretical results and allows us to visualize, among other aspects, the insignificant difference in the accuracy of both fusion filters, which means that the distributed filter, although suboptimal, is preferable in practice as it implies a lower computational cost.

Multisensor Fusion Estimation for Systems with Uncertain Measurements, Based on Reduced Dimension Hypercomplex Techniques

“…Alternatively, the multisensor fusion estimation problem has also been analyzed by using 4D hypercomplex algebras [16][17][18][19][20][21][22][23]. These algebras appear to be a natural extension of complex algebras comprising a real part and three imaginary parts, which gives rise to ideal structures for describing phenomena in the real physical world.…”

“…In this field, quaternions have been the most common 4D hypercomplex algebra in signal processing, since they have the desirable property of being a normed division algebra. Unlike quaternions, tessarines constitute a commutative algebra which facilitates the extension of the main results obtained in the real and complex fields to the four-dimensional case, and the use of tessarines as a signal processing tool has been gaining popularity in the last few years [22,23,28].…”

“…In the tessarine domain, two types of properness have recently been introduced, namely T 1 and T 2 -properness [24,32], and they have been satisfactorily applied in multisensor fusion estimation problems with uncertainties in the measurements [22,23]. In [22], T k -proper, k = 1, 2, centralized fusion algorithms of reduced dimension are proposed for the computation of the optimal (in the least-squares sense) filter, predictor, and fixedpoint smoother of the state in systems with random one-step delays and correlated noises.…”

“…In [22], T k -proper, k = 1, 2, centralized fusion algorithms of reduced dimension are proposed for the computation of the optimal (in the least-squares sense) filter, predictor, and fixedpoint smoother of the state in systems with random one-step delays and correlated noises. In [23], a more general problem with random sensor delays and missing measurements is analyzed. Under T k -properness conditions, k = 1, 2, the authors have devised computationally efficient centralized and distributed fusion filtering algorithms by considering the LS distributed weighted fusion criterion.…”

“…Therefore, our aim in this paper is to address the prediction and smoothing problems under T k -properness conditions, k = 1, 2, from both centralized and distributed approaches. As in [23], the state to be estimated is assumed to be observed through a state-space model with correlated noises, where measurements may be updated, one-step delayed, or contain only noise according to Bernoulli tessarine random variables. In this setting, both centralized and distributed fusion prediction and smoothing algorithms are provided.…”