A frequency interval criterion for modeling secondary path in narrowband active noise control

The performance of narrowband active noise control (NBANC) systems is critically dependent on accurate secondary path modeling. As the core algorithm, narrowband filtered-x LMS (NBFxLMS) ensures convergence only if the phase deviation of the secondary path frequency response at the noise frequency remains below 90° Therefore, existing approaches estimate the target frequency response using a linear interpolation of online modeling results from two single-frequency auxiliary tones near the noise frequency. However, the interpolation interval must strike a balance: it needs to be sufficiently narrow to confine the phase error within the ±90° stability bound, yet wide enough to minimize mutual interference between the modeling process and the NBANC control loop, thereby avoiding performance degradation. Current methods lack theoretical guidance for interval selection, necessitating iterative tuning and exhibiting poor adaptability to dynamic secondary path variations, thereby significantly increasing implementation complexity. This paper introduces a frequency interval criterion derived from room acoustic modal theory, which correlates directly with acoustic reverberation time (T₆₀) and demonstrates robustness under multimodal coupling and in practical acoustic environments. Theoretical and simulation results confirm that this criterion guarantees algorithm convergence while maintaining perceptual inaudibility (i.e., ensuring the auxiliary tones remain inaudible) during secondary path modeling, providing a rigorous theoretical foundation for robust NBANC system design.