Selective Partial-Update Widely Linear Geometric-Algebra NLMS

Adaptive filters based on geometric algebra have demonstrated remarkable success in various applications, such as computer vision, data fusion, and linear prediction, particularly when dealing with high-dimensional multivectors as unknown parameters. However, the extensively used geometric-algebra adaptive filters, especially those applied to widely linear signals, often present significant computational complexity challenges. To address this, the partial-update widely linear geometric algebra least mean squares algorithm was introduced, but it unfortunately exhibits an underwhelming convergence performance. In this brief we develop a new selective partial update geometric-algebra normalized least mean square (NLMS) algorithm, along with its set-membership extension, to improve the convergence performance of partial-update algorithms. Simulation studies conducted to examine the performance of these proposed algorithms, specifically for multivector and quaternion-valued inputs, corroborate their effectiveness and improved performance.