Rosa M. Fernández-Alcalá

Widely Linear Estimation Algorithms for Second-Order Stationary Signals

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.

A Quaternion Widely Linear Series Expansion and Its Applications

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.

Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory

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.

A Quaternion Widely Linear Model for Nonlinear Gaussian Estimation

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 provided. The special structure of the estimator proves its superiority over the complex-valued widely linear solution. The continuous-discrete version of the problem is also studied where the solution takes the form of a suboptimal estimator useful in practical applications. In addition, the particular case of signal plus noise is considered in which the suboptimal solution and its associated error can be implemented through an iterative algorithm. Two numerical simulation examples are presented showing the advantages of the proposed approach.

Prediction of wide-sense stationary quaternion random signals

An efficient widely linear prediction algorithm is introduced for the class of wide-sense stationary quaternion signals. Specifically, using second order statistics information in the quaternion domain, a multivariate Durbin-Levison-like algorithm is derived. The proposed solution can be applied under a very general formulation of the problem, allowing for the estimation of a function of the quaternion signal which is observed through a system with both additive/multiplicative noises.

Linear and nonlinear filters based on the improper Karhunen-Loève expansion

Suboptimal linear and nonlinear continuous-discrete filters for improper complex valued signals are given. The estimators are derived from a generalized improper Karhunen-Loeve expansion of the signal involved and take the form of recursive algorithms which can easily be implemented in practice. Two examples show that the technique is feasible.

Linear least-square estimation algorithms involving correlated signal and noise

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 involved are factorizable kernels. The main contribution of this work with respect to previous studies lies in allowing correlation between the signal and the observation noise, which is useful in applications to feedback control and feedback communications. Moreover, recursive computational formulas are obtained for the error covariances associated with the above estimates.

Widely linear prediction for transfer function models based on the infinite past

The problem of widely linear (WL) prediction for both WL ARMA models and WL transfer function models on the basis of infinite past information is studied. A recursive algorithm to obtain a suboptimum predictor for WL ARMA systems is first given. Then this algorithm is used to develop another recursive algorithm which performs WL prediction for transfer function models. The suggested solutions become an alternative to the WL prediction based on a finite number of observations provided the size of the time series is sufficiently large.

Approximate series representations of linear operations on second-order stochastic processes: application to Simulation

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.

Approximate series representations of second-order stochastic processes: applications to signal detection and estimation

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.