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.

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.

The analysis of time series in 4D commutative hypercomplex algebras is introduced. Firstly, generalized Segre’s quaternion (GSQ) random variables and signals are studied. Then, two concepts of properness are suggested and statistical tests to check if a GSQ random vector is proper or not are proposed. Further, a method to determine in which specific hypercomplex algebra is most likely to achieve, if possible, the properness properties is given. Next, both the linear estimation and prediction problems are studied in the GSQ domain. Finally, ARMA modeling and forecasting for proper GSQ time series are tackled. Experimental results show the superiority of the proposed approach over its counterpart in the Hamilton quaternion domain.

A new approach to numerically evaluate an inner product or a norm in an arbitrary reproducing kernel Hilbert space (RKHS) is considered. The proposed methodology enables us to approximate the RKHS inner product with the desired accuracy avoiding analytical expressions. Furthermore, its implementation is illustrated by means of some classic examples and compared with the standard iterative method provided by Weiner for this purpose. Finally, applications in both the problem of representing approximately second-order stochastic processes by means of series expansions and in the problem of signal detection are studied.