Quaternion Hyperbolic Tangent Geman-McClure for Adaptive Filtering

Quaternion Adaptive Filters play a crucial role in the analysis and processing of 3D and 4D signals. However, existing algorithms for quaternion adaptive filtering experience performance degradation when the output signal is disturbed by impulsive noises. To address this issue, we propose a novel and robust adaptive filtering algorithm for the Quaternion Widely Linear Model (QWL), named widely linear Quaternion Hyperbolic Tangent Geman-McClure (WL-QHTGM), specifically designed to counteract impulsive noise. The incorporation of the Hyperbolic Tangent Geman-McClure function in the WL-QHTGM algorithm enhances robustness and smoothness, effectively managing outliers and diminishing the effects of impulsive noise. This approach ensures the preservation of signal integrity and reliability. Furthermore, we conduct a steady-state analysis of the proposed WL-QHTGM algorithm utilizing quaternion theories and Taylor expansion. Subsequent simulations confirm the accuracy of the theoretical analysis and demonstrate the effectiveness and superiority of the WL-QHTGM algorithm in handling impulsive noise, compared to existing quaternion adaptive filtering algorithms.