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Recursive algorithms are designed for the computation of the optimal linear filter and all types of predictors and smoothers of a signal vector corrupted by a white noise correlated with the signal. These algorithms are derived under both continuous and discrete time formulation of the problem. The only hypothesis imposed is that the correlation functions(More)
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(More)
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(More)
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(More)
Recursive estimation algorithms for discrete complex-valued second-order stationary signals are derived following a widely linear processing approach. The formulation is very general in that it allows for a variety of estimation problems. The results are applied on a simulation example and a performance analysis is presented.
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(More)
This paper deals with the nonlinear minimum mean-squared error estimation problem by using a quaternion widely linear model. On the basis of the information supplied by a Gaussian signal and its square, a quaternion observation process is defined and then, by applying a widely linear processing, an optimal estimator for the continuous-time setting is(More)