Juan Carlos Ruiz-Molina

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 solution to linear estimation problems using approximate Karhunen-Loeve expansions

An explicit and efficiently calculable solution is presented to the problem of linear least-mean-squared-error estimation of a signal process based upon noisy observations that is valid for finite intervals. This approach is based on approximate Karhunen-Loeve expansions of a stochastic process and can be extended to estimate a linear operation, in the sense of the quadratic mean, of the signal process.

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.

Differentiation of the modified approximative Karhunen-Loeve expansion of a stochastic process

A modification of the approximative Karhunen-Loeve expansion of a stochastic process is proposed and the convergence of both the modified expansion and its mth derivative are studied. Likewise, an expansion for the covariance function is provided.

Signal detection using approximate Karhunen-Loeve expansions

A new approach to the signal detection problem in continuous time is presented on the basis of approximate Karhunen-Loeve (K-L) expansions. This methodology gives approximate solutions to the problem of detecting either deterministic or Gaussian signals in Gaussian noise. Furthermore, for this last problem an approximate estimator-correlator representation is provided which approaches the optimum detection statistic.

ON THE DERIVATION OF A SUBOPTIMAL FILTER FOR SIGNAL ESTIMATION

A suboptimal filter to estimate a signal corrupted by a white noise, derived from the approximative Karhunen-Loeve expansion of the signal, is given. The convergence of the suboptimal filter is showed and a bound on the truncation error is found.

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.