The widely linear quaternion recursive least squares filter

A quaternion valued recursive least squares algorithm for the processing of the generality of quaternion valued random processes (both circular and noncircular) is introduced. This is achieved by extending the widely linear model from the complex domain, and accounting for the specific properties of quaternion algebra. Firstly, the widely linear quaternionic Wiener solution is introduced which uses the ‘augmented’ input and weight vectors and thus makes full use of the available second order information. Next, the widely linear quaternion recursive least squares (WL-QRLS) algorithm is derived and is shown to exhibit enhanced transient and steady state properties as compared to the standard widely linear quaternion least mean square (WL-QLMS). Simulations on real world 3D wind signal support the approach.