Adaptive convex combination approach for the identification of improper quaternion processes

Data-adaptive optimal modeling and identification of real-world vector sensor data is provided by combining the fractional tap-length (FT) approach with model order selection in the quaternion domain. To account rigorously for the generality of such processes, both second-order circular (proper) and noncircular (improper), the proposed approach in this paper combines the FT length optimization with both the strictly linear quaternion least mean square (QLMS) and widely linear QLMS (WL-QLMS). A collaborative approach based on QLMS and WL-QLMS is shown to both identify the type of processes (proper or improper) and to track their optimal parameters in real time. Analysis shows that monitoring the evolution of the convex mixing parameter within the collaborative approach allows us to track the improperness in real time. Further insight into the properties of those algorithms is provided by establishing a relationship between the steady-state error and optimal model order. The approach is supported by simulations on model order selection and identification of both strictly linear and widely linear quaternion-valued systems, such as those routinely used in renewable energy (wind) and human-centered computing (biomechanics).