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.

The repercussions of quaternion C ^η -properness on the continuous-time simulation and estimation problems are studied. As a first step, a semi-widely linear quaternion Karhunen-Loève expansion is derived. Then this series representation is used in both problems to give the solutions. The simulation technique proposed is useful for transformations of nonstationary Gaussian quaternion signals. The estimation problem is also analyzed in a continuous-discrete setting and by considering two different formulations: the Cambanis formulation and the signal plus noise model. Initially, the estimators are developed on the basis of a semi-widely linear processing. Afterwards, the solutions are improved by incorporating the information supplied by the square quaternion observations and by using a semi-widely nonlinear processing. In this last scenario, we assume Gaussian observations and unknown probability distributions of the signal of interest.

An approach to the simulation of continuous-time complex-valued random signals under widely linear processing is investigated. The technique is based on a version of the Karhunen-Loève expansion for improper stochastic signals defined on any interval of the real line which takes into account the full statistical information of the signal, i.e., the correlation and complementary correlation functions. It is especially designed for simulating approximately a broad class of signals including complex Gaussian or transformation of complex Gaussian signals and it can be accommodated to simulate any widely linear operations of the signal of interest. Its feasibility and accuracy is examined numerically by considering an example in which the technique devised is compared with the classical approach that ignores the complementary correlation information.

This paper addresses the fusion estimation problem in tessarine systems with multi-sensor observations affected by mixed uncertainties when under ${\mathbb{T}}_k$-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 ${\mathbb{T}}_k$-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.