Semi-widely linear estimation of CηCη-proper quaternion random signal vectors under Gaussian and stationary conditions

•Algorithms for estimating CηCη-proper quaternions under stationarity and Gaussianity conditions are provided.•The solutions incorporate the information supplied by the square quaternion observations.•The suggested estimators outperform (in MSE sense) their counterpart that ignore such information.

Recursive algorithms for estimating quaternion signal vectors are provided. Assuming stationarity, Gaussianity and CηCη-properness, two estimation problems are treated: the prediction and fixed-point smoothing problems. The problems are formulated in a very general way and solved following a semi-widely linear processing. The suggested solutions are derived taking additionally into account the information supplied by a square version of the quaternion observations. This extra information makes it possible to give improved estimators which outperform the corresponding full-widely linear estimators that ignore such information. A numerical example pertaining to the nonlinear prediction and fixed-point smoothing problems illustrates the application of the algorithms.