Convex combination of two geometric-algebra least mean square algorithms and its performance analysis

Abstract

Recently, the geometric-algebra theory and the geometric-algebra based adaptive filters have been applied to numerous applications, such as 3D wind speed, computer vision and fusion prediction of dynamic pressure. However, similar to the real-valued adaptive filter, the geometric-algebra based adaptive filters also have the tradeoff problem between the low steady state error and the fast convergence speed. To overcome this shortcoming, this paper proposes a novel geometric algebra adaptive algorithm by convexly combining two geometric algebra least mean square algorithms with two different step sizes. Afterwards, this paper gives a detail steady state performance analysis of the CGA-LMS algorithm by using the geometric algebra theory. Moreover, to address the phenomenon that the slow filter may lag considerably behind the fast filter, which slows down the overall convergence of the combined geometric-algebra filter, we proposed a novel instantaneous transfer strategy, further leading to the CGA-LMS algorithm with transfer strategy (CGA-LMS-TS). To process the noncircular 3D and 4D signals, we have proposed the convex combination of widely linear GA-LMS (CWL-GA-LMS) algorithm. The CWL-GA-LMS with transfer strategy (CWL-GA-LMS) is also investigated. Simulation results for multivector-valued input are presented to verify the performance of the proposed algorithms and the correctness of the performance analysis.