QLMS-QHM: Accelerated Quaternion LMS Algorithm With Quasi-Hyperbolic Momentum

Quaternions adaptive filtering has become a defacto standard in multidimensional signal processing, however, challenges remain regarding in the balance between convergence speed and stability. To this end, we propose a novel quaternion least mean square algorithm with quasi-hyperbolic momentum (QLMS-QHM), which incorporates a quasi-hyperbolic momentum mechanism to adaptively integrate historical gradient information, thereby accelerating convergence and improving robustness in noisy environments. Theoretical analysis establishes step size bounds for asymptotic convergence, along with comprehensive evaluations of both mean and mean-square behaviors. Simulations demonstrate that the QLMS-QHM algorithm outperforms the classical QLMS, particularly in scenarios requiring rapid adaptation and interference suppression. Practical applications in quaternion-valued adaptive beamforming validate the effectiveness of the QLMS-QHM algorithm under dynamic interference conditions. Overall, the QLMS-QHM algorithm provides a unified framework for incorporating momentum mechanisms into quaternion adaptive filtering, enhancing the design and application of filters in real-world scenarios.