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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. Index Terms--Estimation in colored noise, Hilbert space theory, improper complex-valued random signals.

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. Index Terms--Linear operations in weak sense, series representations of stochastic processes, simulation.

On line algorithms are designed for the linear least-squares estimates of a signal observed in the presence of white noise correlated with the signal. Recursive computational formulas for the estimates and the associated error covariances of different prediction, filtering, and smoothing problems, for both continuous as well as discrete cases, are obtained.

The linear least mean-square (LLMS) error estimation problem of a nonstationary signal corrupted by additive white noise is studied. The formulation of the problem is very general, in the sense that it deals with different estimation problems (smoothing, filtering, and prediction) involving correlation between the signal and the white noise and the possibility of estimating a linear operation (in quadratic mean) of the signal. The obtained solution is in the form of a suboptimum estimate and is derived by using the approximate series expansions for stochastic processes with the aim of solving the Wiener-Hopf equation in the general (nonstationary) case. The main characteristic of this new solution is that it can be computed efficiently using a recursive algorithm similar to the Kalman filter without requiring the signal to obey a state-space model. Index Terms--Approximate series representations of stochastic processes, linear least mean-square error estimation problems.

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. Index Terms--Linear mean-square estimation, series representations of stochastic processes, signal detection.

We define the approximate simultaneous orthogonal (ASO) expansions of two second-order stochastic processes from the Rayleigh-Ritz eigenfunctions and prove its convergence. We consider an example that illustrates the implementation of the proposed method and that allows us to assess the accuracy of the approximations achieved with such finite expansions. Finally, we give two specific applications: in the problem of estimating a Gaussian signal in noise and in the Gaussian signal detection problem. Index Terms--Approximate simultaneous orthogonal (ASO) expansions, mean-square estimation, signal detection.

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. Index Terms--Approximate Karhunen-Loeve expansions, linear least mean-squared-error estimation problems.

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 ex- ample and a performance analysis is presented. [ABSTRACT FROM AUTHOR]

The approximate Karhunen-Loève (KL) expansions are used as a tool for systems identification providing an approximate state-model for the signal of interest. The identified model can then be used to derive sequential algorithms for the nonlinear estimation problem by using standard estimation techniques. The proposed solution is specially indicated in those situations where a state equation is not at hand and the only information available is the correlation matrix of the signal. [ABSTRACT FROM AUTHOR]

In this letter, a recently presented solution to the problem of estimating a signal uncorrelated with observation noise is extended to include correlation between the signal and the noise. We propose a suboptimum estimator that estimates efficiently the signal through a recursive algorithm. For that, and firstly, by using the finite Karhunen-Loève expansion, we derive a new design procedure of the optimum estimator of the signal alternative to the classic form obtained through the direct solving of the Wiener-Hopf equation. Then, on the basis of this new approach, we devise the proposed suboptimum estimator. [ABSTRACT FROM AUTHOR]

The problem of estimating the intensity process of a doubly stochastic Poisson process is analyzed. Using the knowledge of the first and second-order moments of the intensity process, a recursive linear minimum mean-square error estimate is designed. Moreover, an efficient procedure for the computation of its associated error covariance is shown. The proposed solution becomes an alternative approach to the Kalman filter which is applicable under the only structural assumption that the intensity process to be estimated has a finite-dimensional covariance function. [ABSTRACT FROM AUTHOR]