Statistical analysis of narrowband active noise control using a simplified variable step-size FXLMS algorithm

Active noise control (ANC) is fundamental in a range of practical applications, including robotics, rotating machines, power transformers, just to name a few. ANC systems can be broadly classified into two: broadband ANC systems (BANCs) for handling broadband noise and narrowband ANC systems (NANCs) for mitigating narrowband noises generated by rotating machine components. It is important to improve the performance of these systems to enhance their ability to control noise. For decades, the most popular adaptive algorithms are those based on filtered-x LMS (FXMLS), RLS (FXRLS) and Kalman filtering techniques. Despite the effectiveness of the FXRLS and the Kalman filtering techniques, they incur higher implementation costs than FXMLS that is increasingly being used in most ANC systems. However, a common limitation for most existing algorithm schemes is that they can easily lose their efficiency due to poor tracking capabilities in the presence of nonstationary noise.

Variable step-size (VSS) FXMLS schemes have been developed to improve the tracking performance of both BANC and NANC systems. These schemes are based on VSS-LMS algorithms that are rooted in advanced adaptive filtering theory and are generally more effective in improving the steady-state performance and tracking capabilities than their counterparts with constant step sizes. Notably, improving the tracking capabilities of VSS-FXMLS requires the proper selection of VSS user parameters to avoid potential undesirable behaviors of the NANC systems. However, proper parameter selection has remained a challenge. Lately, extensive research has been conducted to address these problems associated with VSS-FXMLS schemes to improve the overall performance of NANC systems. Specifically, previous findings revealed that statistical analysis of VSS-FXMLS based NANC systems would provide a thorough understanding of the systems for designing high-performance NANC systems.

On this account, Dr. Yaping Ma from Jiangnan University, Professor Yegui Xiao from the Prefectural University of Hiroshima in collaboration with Dr. Liying Ma and Professor Khashayar Khorasani from Concordia University conducted an in-depth statistical analysis of a typical NANC system based on a simplified VSS-FXLMS algorithm. The authors started their research work by reviewing the SVSS-FXMLS scheme. They derived difference equations describing the NANC system convergent behaviors. Next, closed-form steady-state expressions were derived for mean, mean squared VSSs and mean residual noise power. Finally, extensive numerical simulations were carried out to illustrate and confirm the validity of the analytical findings. Their research work is currently published in the research journal, Signal Processing.

The researchers reported the successful derivation of difference equations and steady-state explicit expressions that are vital in describing the convergent behavior, dynamic properties and overall steady-state performance of the NANC system. Simulation results revealed that the theoretical curves obtained for the difference equations and explicit expressions excellently fitted the simulated points. Furthermore, results confirmed that the statistical analysis provided an in-depth understanding of NANC systems based on the SVSS-FXLMS algorithm. For example, it provided guidelines for proper selection of the user parameters and estimation of approximate noise reduction performance of NANC systems to be used in real-life applications.

In summary, the research team conducted a rigorous statistical analysis of an NANC system using a simplified variable SVSS-FXLMS algorithm. From the study results, SVSS-FXLMS scheme is superior to other schemes like FXLMS algorithm in terms of noise reduction performance. The proposed SVSS-FXLMS strategy enhanced the compression of the dynamic and steady-state properties of NANC systems. In a statement to Advances in Engineering, the authors believe their study findings advance our understanding to develop low-cost VSS-FXLMS algorithms for high-performance narrowband ANC systems with a wide range of real-life applications.