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

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.