Semi-Widely Simulation and Estimation of Continuous-Time $C^{\eta}$ -Proper Quaternion Random Signals

The repercussions of quaternion C^η-properness on the continuous-time simulation and estimation problems are studied. As a first step, a semi-widely linear quaternion Karhunen-Loève expansion is derived. Then this series representation is used in both problems to give the solutions. The simulation technique proposed is useful for transformations of nonstationary Gaussian quaternion signals. The estimation problem is also analyzed in a continuous-discrete setting and by considering two different formulations: the Cambanis formulation and the signal plus noise model. Initially, the estimators are developed on the basis of a semi-widely linear processing. Afterwards, the solutions are improved by incorporating the information supplied by the square quaternion observations and by using a semi-widely nonlinear processing. In this last scenario, we assume Gaussian observations and unknown probability distributions of the signal of interest.