acoustics Article Active Noise Control System Based on the Improved Equation Error Model Jun Yuan 1, Jun Li 1, Anfu Zhang 2, Xiangdong Zhang 2,* and Jia Ran 3,* 1 Department of Microelectronics Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China; [email protected] (J.Y.); [email protected] (J.L.) 2 Wuhan Second Ship Design and Research Institute, Wuhan 430064, China; [email protected] 3 Department of Electromagnetic Field and Wireless, Chongqing University of Posts and Telecommunications, Chongqing 400065, China * Correspondence: [email protected] (X.Z.); [email protected] (J.R.) Abstract: This paper presents an algorithm structure for an active noise control (ANC) system based on an improved equation error (EE) model that employs the offline secondary path modeling method. The noise of a compressor in a gas station is taken as an example to verify the performance of the proposed ANC system. The results show that the proposed ANC system improves the noise reduction performance and convergence speed compared with other typical ANC systems. In particular, it achieves 28 dBA noise attenuation at a frequency of about 250 Hz and a mean square error (MSE) of about −20 dB. Keywords: active noise control; improved equation error; mean square error 1. Introduction Citation: Yuan, J.; Li, J.; Zhang, A.; Zhang, X.; Ran, J. Active Noise In order to reduce acoustic noise in workplaces and living environments, passive Control System Based on the and active approaches [1] have been employed. For passive approaches, the energy of the Improved Equation Error Model. noise is absorbed by dampers, barriers, mufflers, or sonic crystals [1–4]. However, at low Acoustics 2021, 3, 354–363. https:// frequencies, the attractiveness of the passive approach decreases because of the increase doi.org/10.3390/acoustics3020024 in volume, mass, and cost of passive attenuators [1,5]. In this case, active approaches like active noise control (ANC) technology is proposed because of its low cost, effectiveness Academic Editor: Claudio Guarnaccia at a low frequency, and simple implementation [6,7]. In the ANC system, there is a signal with the same frequency and amplitude, while a phase difference of 180 degrees with Received: 7 December 2020 the original noise signal is generated to eliminate the reference noise. Currently, ANC Accepted: 25 May 2021 technology has achieved great success in noise-canceling headphones. It will likely be Published: 31 May 2021 applied in noise reducing senarios in smart cities, buildings, and manufacturing [8–10]. ANC systems can generally be classified into feedback systems and feedforward Publisher’s Note: MDPI stays neutral systems. The former only apply residue signals at the point of interests, which limits with regard to jurisdictional claims in their applicability to narrowband cases only. The latter employ both the reference signal published maps and institutional affil- generated by the source of the noise and the residue signal, making it effective at attenuating iations. both wideband and narrowband disturbances [1,5]. Furthermore, the combination of feedback and feedforward approaches would be employed in hybrid systems. Currently, the adaptive filter-based ANC feedforward system is widely applied, in which coefficients of its filters adjust with changing the statistics of the signals to be filtered [11]. It can Copyright: © 2021 by the authors. be implemented with the following two structures: adaptive finite impulse-response Licensee MDPI, Basel, Switzerland. (FIR) filters and adaptive infinite-impulse-response (IIR) filters. By employing both poles This article is an open access article and zeros, one can model the desired filter with fewer parameters through adaptive IIR distributed under the terms and filters, leading to less computational complexity than the all-zero-based adaptive FIR conditions of the Creative Commons filters [5,12]. Conventionally, there are two kinds of adaptive IIR filters, namely, the output- Attribution (CC BY) license (https:// error (OE) [12–16] and the equation-error (EE) models [12,17]. The OE method updates creativecommons.org/licenses/by/ the filter coefficients directly in a pole-zero form, which might lead to an unstable pole or 4.0/). Acoustics 2021, 3, 354–363. https://doi.org/10.3390/acoustics3020024 https://www.mdpi.com/journal/acoustics Acoustics 2021, 3 FOR PEER REVIEW 2 coefficients directly in a pole-zero form, which might lead to an unstable pole or local minimum. On the other hand, the EE-based IIR filter has only zeros but not a pole. Hence, the EE method for adaptive IIR filtering can operate in a stable manner when the step size is properly selected [12,18]. Moreover, the mean square error (MSE) function of the EE- based method is a quadratic function, and thus the global minimum can be achieved. In [18], an equation-error adaptive IIR-filter-based active noise control system is developed, Acoustics 2021, 3 where the convergence condition is addressed to assure stability,355 and the optimal solution to achieve the global minimum of the MSE is also derived. Adaptive ANC algorithms localbased minimum. on offline On the othersecondary hand, the EE-based path modeling IIR filter has only can zeros ensure but not system a pole. convergence and maintain- Hence,ability, the EEmeanwhile method for adaptive reducing IIR filtering transformer can operate in anoise stable manner[19]. whenIn this the paper, the off-line secondary step size is properly selected [12,18]. Moreover, the mean square error (MSE) function ofpath the EE-based modeling method algorithms is a quadratic function,are applied and thus in the EE global IIR minimum filter-based can be noise control (ANC) systems achieved.and the In noise [18], an equation-errorreducing near adaptive a compressor IIR-filter-based active in a noise gas control station system is taken as an example to utilize is developed, where the convergence condition is addressed to assure stability, and the optimalthis ANC solution system. to achieve In the contrast global minimum with of the the MSE FIR-based, is also derived. OE-based, Adaptive and EE-based ANC systems ANCproposed algorithms in based [18], on offline the secondaryimproved path modelingEE IIR can filter-based ensure system convergence ANC system with off-line secondary and maintainability, meanwhile reducing transformer noise [19]. In this paper, the off-line secondarypath modeling path modeling algorithms algorithms are appliednot only in EE enhances IIR filter-based the noise noise control (ANC)reducing performance greatly, but systemsalso asks and the for noise less reducing computation near a compressor complexity. in a gas station Mo is takenreover, as an examplethe improved system can still main- to utilize this ANC system. In contrast with the FIR-based, OE-based, and EE-based ANC systemstain stability proposed inwhen [18], the inputting improved EE high IIR filter-based noise level ANC systemsignals. with off-line secondary path modeling algorithms not only enhances the noise reducing performance greatly, but also asks for less computation complexity. Moreover, the improved system can still2. The maintain OE-Model-Based stability when inputting ANC high noise System level signals. 2. The OE-Model-BasedIn order to solve ANC System the problem of acoustic feedback, a feedback structure is added to theIn secondary order to solve thechannel problem ofbased acoustic on feedback, the aFxLMS feedback structure algorthm. is added Figure to 1 reveals the OE-model- the secondary channel based on the FxLMS algorthm. Figure1 reveals the OE-model-based ANCbased system ANC [20,21 system]. [20,21]. FigureFigure 1. 1.The The output-error output-error (OE)-model-based (OE)-model-based active noise control active (ANC) noise system control using the (ANC) system using the FURLMS algorithm. FURLMS algorithm. where P(z) and S(z) are the primary and secondary paths, respectively. The error signal is given by the following: where P(z) and S(z) are the primary and secondary paths, respectively. The error e(n) = d(n) − s(n) ∗ y(n) (1) signal is given by the following: where s(n) is the impulse response of S(z), d(n) is the primary noise signal, and the 0∗0 denotes the convolution operation. y(n) is expressed as follows: ∗ (1) Na−1 Nc−1 where s(n) isy(n )the = ∑impulseai(n)x(n − responsei) + ∑ cj(n )ofy(n −Sj)(z) , d(n) is(2) the primary noise signal, and i=0 j=0 the '∗' denotes the convolution operation. y(n) is expressed as follows: where, in ai(n)(i = 0, 1, ... , Na−1) and cj(n)(j = 0, 1, ... , Nc−1), Na and Nc are the filter lengths. According to the OE-model-based ANC system, the weight updating processes − − can be expressed as follows: Nac1 N 1 y(n) = a (n)x(n − i) + c (n)y(n − j) a(n + 1) = a(n) +mx0(n)ie(n) j (3) (2) i=0 j=0 = = where, in ai (n)(i 0,1,..., Na−1) and c j (n)( j 0,1,..., N c−1 ) , N a and Nc are the filter lengths. According to the OE-model-based ANC system, the weight updating pro- cesses can be expressed as follows: ' a(n +1) = a(n) + μx (n)e(n) (3) ^ ' c(n +1) = c(n) + μ y (n −1)e(n) (4) Acoustics 2021, 3 FOR PEER REVIEW 3 ^ ^ ^ ' ' where, in y (n) = s(n)∗ y(n −1) x (n) = s(n)∗ x(n) , a(n) = [a (n)a (n)...a ]T 0 1 N a−1 , ^ x(n) = [x(n)x(n −1)...x(n − N +1)]T , c(n) = [c (n)c (n)...c (n)] , s(n) represents a 1 2 N c ^ the impulse response of the secondary path S(z) , and μ is the step size.