Proper Processing of $\beta$-Quaternion Wide-Sense Markov Signals From Randomly Lost Observations

This letter addresses the problem of estimating wide-sense Markov signals within the
$\beta$
-quaternion domain, under conditions of first-order properness. This property offers a significant advantage that cannot be achieved in the real fields: a reduction in the dimensionality of the state-space model, resulting in lower computational load when running estimation algorithms. Once the equation of the wide-sense Markov signal model has been obtained, and assuming that the observations used to estimate the signal are affected by packet dropouts, a filtering algorithm based on first-order
$\beta$
-quaternion processing is devised for the optimal estimator. The effectiveness of the estimators proposed is illustrated through numerical comparison with those obtained by using traditional quaternion processing.