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.