Transient Performance Analysis of Geometric Algebra Least Mean Square Adaptive Filter

Recently, a new class of multivector-valued adaptive filters, called geometric-algebra adaptive filters (GA-AFs), have been proposed and applied to 3D registration of point clouds, rotation estimation in computer vision, and so on. To offer a complete theoretical foundation for GA-AFs, we present a transient behavior evaluation of the geometric algebra least mean square (GA-LMS) algorithm. Specifically, the transient mean square deviation (MSD) is obtained by using the results of the mean square theoretical behavior of the GA-LMS. Furthermore, the transient excess mean square error (EMSE) is also given by using the results of MSD. The analytical results rely on the independence theory which has been commonly used in the convergence analysis of real-valued adaptive filter. Considering the non-commutative geometric algebra, the conventional method of analysis is not suitable for the GA-LMS algorithm. Therefore, we propose a novel method to analyze the GA-LMS algorithm under white noise assumption by separating the weight-error array. The obtained theoretical results can be used to accurately predict the transient performance of the GA-LMS. Moreover, the stability condition and steady-state performance are also analyzed. Our proposed steady-state model is more accurate than the previous model. Finally, numerical experiments are presented to confirm the accuracy of the theoretical analysis results.