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.