[Paper Review] Active Noise Control with Sampled-Data Filtered-x Adaptive Algorithm

Masaaki Nagahara, Kenichi Hamaguchi+1 more|arXiv (Cornell University)|Aug 15, 2013

Advanced Adaptive Filtering Techniques17 references3 citations

TL;DR

This paper proposes a sampled-data filtered-x adaptive algorithm for active noise control that models the secondary path as a continuous-time system using lifting techniques from sampled-data control theory. By optimizing a continuous-time cost function and deriving an LMS-type algorithm with integral approximation, the method achieves improved stability and performance over conventional discrete-time approaches, with simulations showing a 1.8-fold wider stable step-size range.

ABSTRACT

Analysis and design of filtered-x adaptive algorithms are conventionally done by assuming that the transfer function in the secondary path is a discrete-time system. However, in real systems such as active noise control, the secondary path is a continuous-time system. Therefore, such a system should be analyzed and designed as a hybrid system including discrete- and continuous- time systems and AD/DA devices. In this article, we propose a hybrid design taking account of continuous-time behavior of the secondary path via lifting (continuous-time polyphase decomposition) technique in sampled-data control theory.

Motivation & Objective

• To address the limitation of conventional filtered-x algorithms that assume discrete-time secondary paths, which does not reflect real-world continuous-time behavior in active noise control systems.
• To design an adaptive algorithm that optimizes the continuous-time error signal rather than its sampled version, leading to better noise cancellation performance.
• To develop a computationally feasible implementation of the continuous-time optimization via lifting-based approximation of integral computations in the LMS algorithm.
• To ensure stability and convergence of the proposed algorithm under realistic system conditions, including slowly varying secondary paths.

Proposed method

• Uses lifting (continuous-time polyphase decomposition) to transform continuous-time signals into function-valued discrete-time signals, enabling exact hybrid system analysis.
• Defines the active noise control problem as minimizing a continuous-time cost function, leading to a Wiener solution in the lifted domain.
• Derives a steepest descent algorithm based on the Wiener solution, which involves integrals over finite intervals in continuous time.
• Applies a lifting-based approximation to convert the integral computations into finite-dimensional digital filter operations for real-time DSP implementation.
• Proposes an LMS-type algorithm with a step-size parameter, ensuring causality and stability through convergence analysis.
• Employs spectral analysis and Parseval’s identity to bound the eigenvalues of the Hessian matrix, enabling stability proofs under slowly varying conditions.

Experimental results

Research questions

• RQ1 Can a hybrid sampled-data control approach improve active noise control performance by modeling the secondary path as a continuous-time system?
• RQ2 How can a continuous-time cost function be optimized in a digital adaptive filtering framework?
• RQ3 What approximation method enables efficient and stable implementation of integral-based LMS updates in real-time DSP systems?
• RQ4 How does the stability region of the proposed algorithm compare to conventional filtered-x LMS in terms of step-size range?

Key findings

• The proposed method achieves a 1.8-fold wider stable step-size range compared to the conventional filtered-x LMS algorithm, with the stable interval extending to μ ∈ (0, 0.7257) versus (0, 0.4051).
• Simulation results show that the L² norm of the error signal is equal to or smaller for the proposed method across all tested μ values, indicating superior noise attenuation.
• The proposed algorithm maintains stability and performance over a significantly wider range of step-size parameters, demonstrating enhanced robustness.
• The lifting-based approximation enables accurate computation of continuous-time integrals using finite-dimensional digital filters, making the algorithm implementable on standard DSP hardware.
• Convergence theorems are established under assumptions of uniformly bounded and slowly varying system matrices, ensuring theoretical reliability.