Nonlinear Thf‐Fxlms Algorithm For Active Noise Control With Loudspeaker Nonlinearity

Abstract

Adaptive algorithms are prevalently applied in the design of nonlinear active noise control (ANC) systems. The most important nonlinearity in ANC is the saturation effect produced by the electro‐acoustical sensors and transducers. The dominant saturation nonlinearity is in the transducers, which can be represented by a Wiener model. An effective solution to mitigate such nonlinear distortion is to employ the Nonlinear Filtered‐X Least Mean Square (NLFXLMS) algorithm. The controller compensates the nonlinearity using a model of the saturation effect represented by the scaled error function (SEF). However, the NLFXLMS is limited by two practical issues such that the degree of nonlinearity has to be known in advance and the SEF cannot be evaluated in real time. In this work, the NLFXLMS algorithm is modified by incorporating the tangential hyperbolic function (THF) to model the saturation effect of the loudspeaker. The proposed THF‐NLFXLMS algorithm, models the nonlinear secondary path and applies the estimated degree of nonlinearity in the control algorithm design. The results show that the Wiener secondary path, with saturation nonlinearity represented by SEF, can be modelled by THF with a certain degree of accuracy and can yield a reasonable estimate of the degree of nonlinearity. The performance of the proposed algorithm is comparable with the benchmark NLFXLMS and is superior to the conventional FXLMS as well as the second order Volterra algorithm of similar computational complexity with the proposed algorithm.

NLFXLMS and THF‐NLFXLMS Algorithms for Wiener‐Hammerstein Nonlinear Active Noise Control

“…Even though the values of the estimated secondary path may not necessarily converge to their true values, the controller will compensate this effect by adjusting its gain during adaptation. This gain swapping effect has been observed in the previous works .…”

“…It was shown that the degree of nonlinearity in can be related to the degree of nonlinearity in by : These relations are applicable to the Hammerstein and Wiener THF‐NLFXLMS algorithms, and are also applicable to the Wiener‐Hammerstein THF‐NLFXLMS algorithm.…”

“…In practical applications this prior information is not available and must be approximated. In the recent works by , the saturation nonlinearity represented by SEF was modeled using THF. The THF function is given by where α and β are two scaling parameters that determine the strength of the saturation nonlinearity.…”

“…From , in the case of perfect modeling, i.e. e s ( n )=0, the following relationship is satisfied: such that Similar to the Hammerstein and the Wiener NLFXLMS, the linear paths , , and the scaling parameters and do not necessarily have to converge to their true values, as long as (30) is satisfied. The proposed THF‐NLFXLMS algorithm is summarized in Table .…”

“…Since the SEF was used instead of THF to represent the true secondary path, the modeling is expected to be imperfect because THF approximate SEF up to certain degree of accuracy. Unlike in , S and cannot be compared in terms of error, because does not necessarily converge to S . Instead, the error between S and and K and must be calculated to evaluate modeling accuracy.…”