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.

The filtering estimation problem under uncertainty conditions is addressed for a class of improper quaternion signals, called widely factorizable, characterized because their augmented correlation function is a factorizable kernel. From the knowledge of the correlation functions involved, a recursive algorithm is designed for the computation of the widely linear (WL) filtering estimate and its associated mean squared error. The main advantage of the proposed solution is that it can be applied in situations where a state-space model is not readily at hand. The benefits of the proposed WL filtering algorithm is analyzed through a simulation example where WL filtering errors are compared with respect to the strictly linear (SL) counterparts, showing the superior behavior of the former over the latter.

A series representation for continuous-time quaternion random signals is given. The series expansion is based on augmented statistics and provides uncorrelated scalar real-valued random variables. The proposed technique implies a dimension reduction of the four-dimensional original problem to a one-dimensional problem. As a particular case, the quaternion Karhunen-Loève expansion is obtained. Finally, two illustrative applications to the quaternion widely linear detection and estimation problems are presented.

In this paper, the problem of estimating an improper complex-valued random signal in colored noise with an additive white part is addressed. We tackle the problem from a mathematical perspective and emphasize the advantages of this rigorous treatment. The formulation considered is very general in the sense that it permits us to estimate any functional of the signal of interest. Finally, the superiority of the widely linear estimation with respect to the conventional estimation techniques is both theoretically and experimentally illustrated.

Series representations of the more usual linear operations in weak sense on a second-order stochastic process are studied. The starting point of this analysis is the optimal Cambanis expansion of the stochastic process considered. Likewise, the extensions of the approximate series expansions based on the Rayleigh-Ritz method are presented for such linear operations on the process. The main advantages of these extensions are that they are computationally feasible and entail a significant reduction in the computational burden. Finally, their applicability as a practical simulation tool is examined.

The linear least mean-square (LLMS) error estimation problem of a nonstationary signal corrupted by additive white noise is studied. The formulation of the problem is very general, in the sense that it deals with different estimation problems (smoothing, filtering, and prediction) involving correlation between the signal and the white noise and the possibility of estimating a linear operation (in quadratic mean) of the signal. The obtained solution is in the form of a suboptimum estimate and is derived by using the approximate series expansions for stochastic processes with the aim of solving the Wiener-Hopf equation in the general (nonstationary) case. The main characteristic of this new solution is that it can be computed efficiently using a recursive algorithm similar to the Kalman filter without requiring the signal to obey a state-space model.

Two distinct approximate series representations are obtained for the entire class of measurable, second-order stochastic processes defined on any interval of the real line. They include as particular cases all earlier approximate representations based on the Rayleigh-Ritz method. It is also shown that each of them converges with a different type of convergence. Finally, two applications in statistical communication theory are presented.