Power Normalized and Fractional Power Normalized Least Mean Square Adaptive Beamforming Algorithm

With the rapid deployment of high-speed maglev transportation systems worldwide, the operational velocity, electromagnetic complexity, and channel dynamics have far exceeded those of conventional rail systems, imposing more stringent requirements on real-time capability, reliability, and interference robustness in wireless communication. In maglev environments exceeding 600 km/h, the channel becomes predominantly line-of-sight with sparse scatterers, exhibiting strong Doppler shifts, rapidly varying spatial characteristics, and severe interference, all of which significantly degrade the stability and convergence performance of traditional beamforming algorithms. Adaptive smart antenna technology has therefore become essential in high-mobility communication and sensing systems, as it enables real-time spatial filtering, interference suppression, and beam tracking through continuous weight updates. To address the challenges of slow convergence and high steady-state error in rapidly varying maglev channels, this work proposes a new Fractional Proportionate Normalized Least Mean Square (FPNLMS) adaptive beamforming algorithm. The contributions of this study are twofold. (1) A novel FPNLMS algorithm is developed by embedding a fractional-order gradient correction into the power-normalized and proportionate gain framework of PNLMS, forming a unified LMS-type update mechanism that enhances error tracking flexibility while maintaining O(L) computational complexity. This integrated design enables the proposed method to achieve faster convergence, improved robustness, and reduced steady-state error in highly dynamic channel conditions. (2) A unified convergence analysis framework is established for the proposed algorithm. Mean convergence conditions and practical step-size bounds are derived, explicitly incorporating the fractional-order term and generalizing classical LMS/PNLMS convergence theory, thereby providing theoretical guarantees for stable deployment in high-speed maglev beamforming. Simulation results verify that the proposed FPNLMS algorithm achieves significantly faster convergence, lower mean square error, and superior interference suppression compared with LMS, NLMS, FLMS, and PNLMS, demonstrating its strong applicability to beamforming in highly dynamic next-generation maglev communication systems.