This letter addresses the problem of estimating wide-sense Markov signals within the$\beta$-quaternion domain, under conditions of first-order properness. This property offers a significant advantage that cannot be achieved in the real fields: a reduction in the dimensionality of the state-space model, resulting in lower computational load when running estimation algorithms. Once the equation of the wide-sense Markov signal model has been obtained, and assuming that the observations used to estimate the signal are affected by packet dropouts, a filtering algorithm based on first-order$\beta$-quaternion processing is devised for the optimal estimator. The effectiveness of the estimators proposed is illustrated through numerical comparison with those obtained by using traditional quaternion processing.

The filtering estimation problem under uncertainty conditions is addressed for a class of improper quaternion signals, called widely factorizable, characterized because their augmented correlation function is a factorizable kernel. From the knowledge of the correlation functions involved, a recursive algorithm is designed for the computation of the widely linear (WL) filtering estimate and its associated mean squared error. The main advantage of the proposed solution is that it can be applied in situations where a state-space model is not readily at hand. The benefits of the proposed WL filtering algorithm is analyzed through a simulation example where WL filtering errors are compared with respect to the strictly linear (SL) counterparts, showing the superior behavior of the former over the latter.

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.

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.

A numerical approach to the vector signal detection problem is proposed by using an approximate series representation for vector complex-valued random processes. This technique allows us to provide computationally feasible suboptimum receivers for the problem of detecting an improper or proper complex signal in additive white noise. Its implementation requires the knowledge of the correlation matrix of the vector process involved. The performance of the technique devised is assessed by some communication examples.

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.

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.

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.

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-Loeve 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.

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.

A new solution is given to the continuous-discrete linear estimation problem of a signal process under the assumption that the autocorrelation function of the signal is known. This approach is based on approximate series expansions of a stochastic process and it is valid for the entire class of measurable, smooth signals defined on half-open intervals of the real line. It includes as particular cases all earlier solutions based on the approximate Karhunen-Loeve expansions. The main advantage of the solution obtained is that it can be derived through an efficient algorithm similar to the Kalman filter.

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.

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.

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.