P-norm based Fractional-Order Robust Subband Adaptive Filtering Algorithm for Impulsive Noise and Noisy Input

TL;DR

The paper addresses robust subband adaptive filtering in impulsive, α-stable environments where existing NSPN methods falter for $0<α≤1$. It introduces FoNSPN, integrating fractional-order stochastic gradient descent into the mean $p$-power error framework to form a $β$-order gradient, with an update that reduces to NSPN when $β=1$. The authors derive stability conditions for the step-size $μ$ and fractional order $β$, along with a steady-state mean-square deviation model, and validate the approach through simulations under Gaussian, Cauchy, and α-stable noise, showing superior performance to several state-of-the-art fractional-order algorithms. The work provides practical guidelines for parameter selection and demonstrates FoNSPN’s potential for robust, fast-converging subband adaptive filtering in highly impulsive noise settings.

Abstract

Building upon the mean p-power error (MPE) criterion, the normalized subband p-norm (NSPN) algorithm demonstrates superior robustness in $α$-stable noise environments ($1 < α\leq 2$) through effective utilization of low-order moment hidden in robust loss functions. Nevertheless, its performance degrades significantly when processing noise input or additive noise characterized by $α$-stable processes ($0 < α\leq 1$). To overcome these limitations, we propose a novel fractional-order NSPN (FoNSPN) algorithm that incorporates the fractional-order stochastic gradient descent (FoSGD) method into the MPE framework. Additionally, this paper also analyzes the convergence range of its step-size, the theoretical domain of values for the fractional-order $β$, and establishes the theoretical steady-state mean square deviation (MSD) model. Simulations conducted in diverse impulsive noise environments confirm the superiority of the proposed FoNSPN algorithm against existing state-of-the-art algorithms.

Figures (1)

• Figure 1: Performance verification of the algorithms. (a) Steady-state MSDs versus the step-size when $\sigma_{n_k}^2=0.001$; (b) Theoretical and simulated MSDs versus; (c) $p=0.7$, $\gamma$ denotes the scaling factor, $\rho$ is the order of the fractional derivative, $\tau$ is the versoria parameter, and $\alpha^{'}$ is the fractional-order error parameter; (d) $p=0.7$.