Design of Near-Optimal Fractional-Order Adaptive FIR Filter

This paper presents the design and implementation of a near-optimal fractional order least mean squares (FO-LMS) adaptive finite impulse response (FIR) filter, leveraging fractional calculus (FC) to exploit its memory-preserving property for enhanced convergence, steady-state accuracy, and noise robustness. Three fractional derivative formulations Riemann Liouville (RL), Grünwald Letnikov (GL), and Caputo are employed to introduce fractionality into the LMS framework, providing additional degrees of freedom for improved adaptability in scenarios involving temporal and spatial dependencies. The proposed algorithm is evaluated on sinusoidal, chirp, and amplitude-modulated (AM) signals corrupted with additive white gaussian noise (AWGN), demonstrating its capability to suppress noise while preserving signal characteristics, including AM envelope integrity. Sensitivity analysis on the fractional order (FO) α and learning rates (μ1,μ2) is performed to guide parameter tuning and robustness assessment. Comparative experiments against the conventional integer order LMS (IO-LMS) show that all FO-LMS variants yield superior performance in terms of accuracy (ACC), standard deviation (STD), correlation coefficient (CORR), mean square error (MSE), and computational efficiency. Notably, the caputo-based FO-LMS algorithm consistently outperformed its counterparts across all three signal types, delivering higher accuracy, reduced steady-state error, and shorter computation time compared to the IO-LMS, thereby underscoring its efficacy for real-time signal estimation, denoising, and system identification.