Fractional-Order Subband p-Norm Adaptive Filter via Transformation Nearest Kronecker Product Decomposition for Active Noise Control

Abstract: The conventional normalized subband p-norm (NSPN) algorithm achieves robustness in $α$-stable noise ($1<α\leq 2$) by utilizing low-order error moments. However, its performance degrades significantly under three scenarios: (1) non-Gaussian inputs, (2) $α$-stable noise with $0<α\leq 1$, and (3) sparse system identification. To address these limitations, this paper proposes a fractional-order NSPN algorithm based on the nearest Kronecker product (NKP) decomposition and fractional-order stochastic gradient descent, termed NKP-FoNSPN. Theoretical bounds for the fractional-order parameter $β$ are also derived. Notably, when $β=1$, the NKP-FoNSPN reduces to a new NKP-NSPN algorithm, while its non-NKP decomposition variant becomes the fractional-order NSPN (FoNSPN) algorithm. Furthermore, a novel transformation-based NKP (TNKP) decomposition technique is designed, which exhibits lower computational complexity than conventional NKP for specific filter structures. The resulting TNKP-based FoNSPN (TNKP-FoNSPN) achieves lower steady-state misadjustment and multiplication cost compared with the NKP-FoNSPN algorithm. Additionally, complete computational complexity analyses are provided. For active noise control (ANC) scenarios, we develop filtered-x variants: NKP-FxFoNSPN and TNKP-FxFoNSPN. From the former, two additional variants are derived: NKP-FxNSPN and FxFoNSPN. Simulations using diverse noise sources (pink, helicopter, gunshot, pile driver, and traction substation noise) demonstrate the superiority of the proposed algorithms. Finally, we validate their noise reduction performance in a real constructed single-channel duct ANC and a simulated multi-channel ANC systems.