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Multi-sensor Distributed Fusion Estimation for $\mathbb{T}_k$-proper Factorizable Signals in Sensor Networks with Fading Measurements

Rosa M. Fernández-Alcalá, José D. Jiménez-López, Jesús Navarro-Moreno, Juan C. Ruiz-Molina

TL;DR

This paper addresses distributed fusion estimation for $\mathbb{T}_k$-proper factorizable 4D tessarine signals in multisensor networks with fading measurements. It introduces a two-stage approach that first computes local linear MMSE estimators at each sensor using $\mathbb{T}_k$-proper processing based on second-order statistics, and then fuses these local estimates with MMSE weights to form a global estimator. The key contributions include recursive algorithms for local filtering, prediction, and smoothing, plus closed-form updates for cross-sensor pseudo-covariances that enable efficient distributed fusion in both $\mathbb{T}_1$ and $\mathbb{T}_2$ settings. Numerical results show that $\mathbb{T}_k$-proper estimators outperform quaternion-domain counterparts and require significantly less computation than full 4D widely linear processing, highlighting their practical value for real-time, robust multi-sensor data fusion under fading.

Abstract

The challenge of distributed fusion estimation is investigated for a class of four-dimensional (4D) commutative hypercomplex signals that are $\mathbb{T}_k$-proper factorizable, within the framework of multiple-sensor networks with different fading measurement rates. The fading effects affecting each sensor's measurements are modeled as a stochastic variables with known second-order statistical properties. The estimation process is conducted exclusively based on these second-order statistics. Then, by exploiting the $\mathbb{T}_k$-properness property within a tessarine framework, the dimensionality of the problem is significantly reduced. This reduction in dimensionality enables the development of distributed fusion filtering, prediction, and smoothing algorithms that entail lower computational effort compared with real-valued approaches. The performance of the suggested algorithms is assessed through numerical experiments under various uncertainty conditions and $T_k$-proper contexts. Furthermore, simulation results confirm that $\mathbb{T}_k$-proper estimators outperform their quaternion-domain counterparts, underscoring their practical advantages. These findings highlight the potential of $\mathbb{T}_k$-proper estimation techniques for improving multi-sensor data fusion in applications where efficient signal processing is essential.

Multi-sensor Distributed Fusion Estimation for $\mathbb{T}_k$-proper Factorizable Signals in Sensor Networks with Fading Measurements

TL;DR

This paper addresses distributed fusion estimation for -proper factorizable 4D tessarine signals in multisensor networks with fading measurements. It introduces a two-stage approach that first computes local linear MMSE estimators at each sensor using -proper processing based on second-order statistics, and then fuses these local estimates with MMSE weights to form a global estimator. The key contributions include recursive algorithms for local filtering, prediction, and smoothing, plus closed-form updates for cross-sensor pseudo-covariances that enable efficient distributed fusion in both and settings. Numerical results show that -proper estimators outperform quaternion-domain counterparts and require significantly less computation than full 4D widely linear processing, highlighting their practical value for real-time, robust multi-sensor data fusion under fading.

Abstract

The challenge of distributed fusion estimation is investigated for a class of four-dimensional (4D) commutative hypercomplex signals that are -proper factorizable, within the framework of multiple-sensor networks with different fading measurement rates. The fading effects affecting each sensor's measurements are modeled as a stochastic variables with known second-order statistical properties. The estimation process is conducted exclusively based on these second-order statistics. Then, by exploiting the -properness property within a tessarine framework, the dimensionality of the problem is significantly reduced. This reduction in dimensionality enables the development of distributed fusion filtering, prediction, and smoothing algorithms that entail lower computational effort compared with real-valued approaches. The performance of the suggested algorithms is assessed through numerical experiments under various uncertainty conditions and -proper contexts. Furthermore, simulation results confirm that -proper estimators outperform their quaternion-domain counterparts, underscoring their practical advantages. These findings highlight the potential of -proper estimation techniques for improving multi-sensor data fusion in applications where efficient signal processing is essential.

Paper Structure

This paper contains 18 sections, 9 theorems, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic Knowledge
  3. Notation
  4. Tessarine algebra
  5. Problem statement
  6. $\mathbb{T}_k$-proper distributed fusion methodology
  7. $\mathbb{T}_k$-proper local estimation algorithms
  8. $\mathbb{T}_k$-proper distributed estimation algorithms
  9. Numerical simulations
  10. Conclusions
  11. Proof of Proposition \ref{['prop1']}
  12. Proof of Proposition \ref{['prop2']}
  13. Proof of Theorem \ref{['ThLocalFilter']}
  14. Proof of Theorem \ref{['LocalPredictor']}
  15. Proof of Theorem \ref{['LocalSmoother']}
  16. ...and 3 more sections

Key Result

Proposition 1

If $\mathbf{x}(t)\in\mathbb{T}^n$ is $\mathbb{T}_k$-proper, for $k=1,2$, and widely factorizable then, the augmented pseudo-autocorrelation function of $\bar{\mathbf{x}}(t)$ takes the following form $\boldsymbol{\Gamma}_{\bar{\mathbf{x}}}(t,s)=\mathop{\mathrm{diag}}\nolimits\left(\boldsymbol{\Gamma} where $\mathbf{A}_k(t)=\left[ \boldsymbol{I}_{kn}, \boldsymbol{0}_{kn\times (4-k)n} \right]\bar{\m

Figures (5)

  • Figure 1: $\mathbb{T}_k$-proper distributed filtering (solid line), prediction (dashed line) and smoothing (dotted line) error pseudo-variances.
  • Figure 2: $\mathbb{T}_1$-proper vs QSL local and distributed filtering error pseudo-variances in a $\mathbb{T}_1$-proper setting (a), and $\mathbb{T}_2$-proper vs QSWL local and distributed filtering error pseudo-variances in a $\mathbb{T}_2$-proper setting (b).
  • Figure 3: Means of the error variances of the $\mathbb{T}_k$-proper distributed filtering estimates and their counterparts in a $\mathbb{T}_1$-proper setting (a), and in a $\mathbb{T}_2$-proper setting (b), in terms of the number of observations.
  • Figure 4: Difference between QSL and $\mathbb{T}_1$-proper filtering and smoothing error variances in a $\mathbb{T}_1$-proper setting (a), and between QSWL and $\mathbb{T}_2$-proper filtering and smoothing error variances in a $\mathbb{T}_2$-proper setting
  • Figure 5: Computing time (in seconds) using a $\mathbf{T}_k$-proper and a WL distributed fusion filtering algorithm in a $\mathbb{T}_1$-proper setting (a), and in a $\mathbb{T}_2$-proper setting (b), and varying the number of observations from $50$ to $500$.

Theorems & Definitions (20)

  • Definition 1: $\mathbb{T}_k$-properness
  • Remark 1
  • Definition 2: Widely factorizable signals
  • Remark 2
  • Proposition 1
  • Definition 3: $\mathbb{T}_k$-proper factorizable signals
  • Definition 4
  • Remark 3
  • Proposition 2
  • Proposition 3
  • ...and 10 more