Widely Linear Adaptive Filtering Based on Clifford Geometric Algebra: A unified framework [Hypercomplex Signal and Image Processing]

In this article, we present a powerful unifying framework for widely linear (WL) adaptive filters building on the concept of geometric algebra (GA), including recently proposed complex-valued (CV), quaternion-valued, and GA WL adaptive filters (WLAFs). We also consider and review WL adaptive filtering methods that feature robustness against impulsive noise, noisy input measurements, partial coefficient updates, subband structures, censoring, and composite structures under the unified framework. Furthermore, we propose innovative WL adaptive filtering algorithms for functional link polynomial (FLP) nonlinear filters, infinite-impulse response (IIR) systems, and kernel-based nonlinear system identification, showcasing the advantages of the unified framework. The article also investigates the relationship among WLAFs, graph filters, and Cayley–Dickson (CD)-valued adaptive filters, offering new insights into how the unified framework can be extended to graph signals and CD numbers. Finally, the article motivates future work on WL adaptive filtering based on GA and its special cases.