Real-time estimation of quaternion impropriety

An algorithm for tracking the degree of quaternion impropriety in real-time is proposed. This is achieved by exploiting the i-, j-, and k-pseudo-covariances, which makes it possible to introduce an impropriety measure as the minimum mean square error (MMSE) solution for estimating the quaternion involutions along the i, j, and k axes from the quaternion random variable itself. For rigour, convergence conditions for both the mean and mean square behavior of such a real-time impropriety tracker are established, and the effect of the degree of impropriety on the steady-state performance of the algorithm is analyzed. The concept is supported by simulations on both synthetic data and on real-world wind data recordings.