Active Noise Control System Based on the Improved Equation Error Model

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) +µx0(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 processes 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)

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^ ^ ^ ' ' 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. Meanwhile, the optimal C(z) and A(z) must satisfy A(z) P(z) = (5) 1− C(z) S(z)

to minimize the error signal e(n) . Acoustics 2021, 3 356 3. The EE Adaptive IIR-Filter-Based ANC Algorithm

In order to solvec( nthe+ 1 )pole = c(n )problem + µyˆ0(n − 1caused)e(n) by S(z) in the(4) system shown in Figure ˆ0( ) = ( ) ∗ ( − ) 0( ) = ( ) ∗ ( ) ( ) = [ ( ) ( ) ]T where,2, the inestimatedy n sˆ nvaluey n of1 thex ndesiredsˆ n signalx n , a nd(n) a 0 isn synthesizeda1 n ... aNa−1 ,by means of digital sig- T x(n) = [x(n)x(n − 1) ... x(n − Na + 1)] , c(n) = [c1(n)c2(n) ... cNc (n)], sˆ(n) represents thenal impulse processing. response As of the shown secondary in Fi pathgureSˆ(z) 2,, and theµ isimpulse the step size. response Meanwhile, s( then) of the secondary path optimal C(z) and A(z=) must satisfy is S(z) . s(n) [s0s1...sL−1] , and the output signal can be expressed as follows: A(z) P(z) = T T (5) 1 −yC((nz)) =Sa(z) (n)x(n) + c (n)d(n −1) (6) ( ) to minimize the error− signal= e n .− − − T where d (n 1) [d (n 1)d (n 2)...d (n N c )] . The adaptations of the EE ANC sys- 3. The EE Adaptive IIR-Filter-Based ANC Algorithm temIn [21] order ca ton solve be derived the pole problem as follows: caused by S(z) in the system shown in Figure2, the estimated value of the desired signal d(n) is synthesized by means of digital signal ^ processing. As shown in Figure2, the impulse response s(n) of the secondary path is S(z). a(n +1) = a(n) + μe(n)(s(n)∗ x(n)) (7) s(n) = [s0s1 ... sL−1], and the output signal can be expressed as follows: and y(n) = aT(n)x(n) + cT(n)d(n − 1) (6)

T where d(n − 1) = [d(n − 1)d(n − 2) ... d(n − Nc)] . The adaptations of the EE^ ANC sys- ' tem [21] can be derived as follows: c(n +1) = c(n) + μe(n) d (n) (8) a(n + ) = a(n) + e(n)(sˆ(n) ∗ x(n)) 1 ^µ ^ (7) = + ∗ and Practically, we synthesize d(n) e(n) y(n) s(n) instead it. Therefore, the out- ˆ0 put signal can be rewrittenc(n + 1 )as = follows:c(n) + µe(n )d (n) (8) Practically, we synthesize dˆ(n) = e(n) + y(n) ∗ sˆ(n) instead it. Therefore, the output ^ signal can be rewritten as follows: T T y(n) = a (n)x(n) + c (n)d(n) (9) y(n) = aT(n)x(n) + cT(n)dˆ(n) (9)

FigureFigure 2. ANC 2. ANC system system based on based an equation on erroran equation (EE) adaptive error inﬁnite-impulse-response (EE) adaptive infinite-impulse-response (IIR) ﬁlter. (IIR)

filter.The complete ANC system block diagram based on the EE model is shown in Figure3 . As the EE model is composed of non-recursive terms, instability would be avoided. How- ever, thereThe are complete still problems ANC such system as a slow convergenceblock diagram speed andbased poor on noise the reduction EE model is shown in Figure performance. 3. As the EE model is composed of non-recursive terms, instability would be avoided.

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Acoustics 2021, 3 However, there are still problems such as a slow convergence357 speed and poor noise reduction performance.

FigureFigure 3. Complete 3. Complete block diagram block of diagram the ANC system of the when ANC applying system an EE adaptivewhen applying IIR filter. an EE adaptive IIR filter. 4. The EE Adaptive IIR-Filter-Based ANC Algorithm 4. TheIn order EE to Adaptive improve the IIR-Filter-Based convergence speed and ANC noise reduction Algorithm performance of the EE-based ANC system, an improved ANC system based on EE is proposed [18], where only twoIn adaptive order filters,to improveA(z) and theC(z) ,convergence are employed. Here, speed we adoptand annoise extra reduction filter performance of the namedEE-basedB(z). As ANC shown system, in Figure4, anA(z )improved, B(z), and C( zANC) are the syst transversalem based adaptive on filters EE is proposed [18], where for the EE-based model. The dotted square denotes the proposed secondary path modeling. only two adaptive filters, A(0 z) and2 C(z) , are employed. Here, we adopt an extra filter The squared error signal expressed as εEE = e (n), and the gradient of error surface is denoted as follows [18]: named B(z) . As shown0 in Figure 4, A(z) , B(z) , and C(z) are the transversal adap- ∇εEE = −2e(n)(s(n) ∗ u(n)) (10) wheretive sfilters(n) is the for impulse the response EE-based of the secondarymodel. pathTheS (doz) andtted the square input of the denotes adaptive the proposed secondary x(n) ' 2 filterpath is umodeling.(n) = The squared. The estimated error secondary signal pathexpressed is added, soas the ε equation= e (n) , and the gradient of d(n − 1) EE becomeserror surface is denoted as follows [18]: 0 ∇εEE = −2e(n)(sˆ(n) ∗ u(n)) (11) ∇ε ' = − ∗ Adaptations of the improved EE ANC systemEE can be derived2e(n)( ass follows:(n) u(n)) (10) a(n + 1) = a(n) + µe(n)(sˆ(n) ∗ x(n)) (12) where s(n) is the impulse response of the secondary path S(z) and the input of the b(n + 1) = bx((nn) +) µe(n)(y(n) ∗ sˆ(n)) (13) u(nc)(n=+1) = c(n) + µe(n)dˆ0(n) (14) adaptive filter is  −  . The estimated secondary path is added, so the equa- Finally, the output signal of the improvedd(n 1 EE) ANC system can be rewritten as follows:

tion becomes y(n) = aT(n)x(n) + cT(n)(e(n) + bT(n)yˆ(n)) (15)

A(z) ^ ∇ε=' P(z=) − ∗ (16) (11) 1 − C(z)B(z) EE 2e(n)(s(n) u(n)) It can be seen from Equation (15) that the improved EE model can avoid instability and improveAdaptations the accuracy of of the synthesized improvedd(n )EEsignal, ANC as it sy is astem nonrecursive can be system. derived as follows:

^ a(n +1) = a(n) + μe(n)(s(n)∗ x(n)) (12)

^ b(n +1) = b(n) + μe(n)(y(n)∗ s(n)) (13)

^ ' c(n +1) = c(n) + μe(n) d (n) (14)

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FigureFigure 4. The 4. improvedThe improved EE adaptive EE IIR-ﬁlter-based adaptive ANC IIR-filter-based algorithm. ANC algorithm. 4.1. The Step-Size Constraint WeFinally, assume that the reference output signal signalx(n) is a whiteof the noise improved signal with a meanEE ANC value of system zero can be rewritten as fol- and a variance of 1. The step size limits µA, µB, and µC of A(z), B(z), and C(z), are set as thelows: following Equation (16), respectively:

3 ^ 0 < µA < T T (17)T y(n) =(Naa +(3∆neq))xPx(0n) + c (n)(e(n) + b (n) y(n)) (15) 3 0 < µB < (18) (Nb + 3∆eq)Pyˆ A(z) 3 = P(z) (16) 0 < µC < − (19) (Nc + 3∆eq1)Pdˆ0C(z)B(z) 0 0 ˆ0 where Px , Pyˆ, and Pdˆ0 denote the power of x (n), yˆ(n), and d (n), respectively, and the is the equivalentIt can delay be ofseen the secondary from Equation path Equation (15) (16). ∆thateq can the be expressed improved as follows: EE model can avoid instability

and improve the accuracy of theL−1 synthesized d(n) signal, as it is a nonrecursive system. 2 ∑ lsl = ∆ = l 1 (20) eq L−1 2 4.1. The Step-Size Constraint ∑ sl l=0 We assume that reference signal x(n) is a white noise signal with a mean value of The maximum overall step-size µ of the ANC system should be the minimum of µA, µ , and µ , and the signal powers of P 0 , P , and P 0 in Equations (16)–(18) are set to be 1. zeroB andC a variance of 1. xTheyˆ stepdˆ size limits μ , μ , and μ of A(z) , B(z) , and Hence, the maximum step size and ﬁlter length are only affected by theA secondaryB path C delayC(z∆)eq,. are set as the following Equation (16), respectively: 4.2. Global Minimum Solutions 3 The mean square error (MSE) function based0 on< theμ improved< EE-based ANC system is expressed as follows: A + Δ (17) (Na 3 eq )P ' 0 2 T x ξEE(n) = E[d (n)] + a (n)Rx0x0 a(n) T T +b (n)Ryˆyˆb(n) + c (n)Rdˆ0dˆ0 c(n) T T 0 +2[a (n)Rx0dˆ0 c(n) + a (n)Rx yˆb(n) (21) T T T 3 +b (n)R c(n) − a (n)p 0 − b (n)p yˆdˆ0 0 < dxμ < dyˆ T B −c (n)p ˆ0 ] + Δ (18) dd (Nb 3 eq )P^ y

3 0 < μ < C + Δ (19) (Nc 3 eq )P^ d '

^ ^ ' P ' P^ P^ x'(n) y(n) d (n) where x , , and denote the power of , , and , respectively, y d ' Δ and the is the equivalent delay of the secondary path Equation (16). eq can be expressed as follows:

L−1 2 lsl Δ = l=1 eq L−1 (20) 2 sl l=0

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0 0 0 0 0 0 T T where Rx x = E[x (n)x (n)], Ryˆyˆ = E[yˆ(n)yˆ(n)], Rdˆ0dˆ0 = E[x (n)x (n)], 0 0 ˆ0T 0 ˆ0 Rx0dˆ0 = E[x (n)d (n)] and pdx = E[d(n)x (n)] pddˆ0 = E[d(n)d (n)]. It can be seen that the ANC system based on the improved EE model owns a global minimum. By calculating the gradient function of Equation (21), we have

0 0 ( ) 0 ( ) 0 ( ) T ∂ξEE = [ ∂ξEE n ∂ξEE n ··· ∂ξEE n ] ∂a(n) ∂a (n) ∂a (n) ∂a (n) 0 1 Na−1 (22) 0 0 0 0 = 2Rx x a(n) + 2Rx0dˆ0 c(n) + Rx yˆb(n) − 2pdx

0 ( ) 0 ( ) 0 ( ) 0 ( ) T ∂ξEE n = [ ∂ξEE n ∂ξEE n ··· ∂ξEE n ] ∂b(n) ∂b (n) ∂b (n) ∂b (n) 0 1 Na−1 (23) 0 = 2Ryˆyˆb(n) + 2Rx yˆa(n) + 2Ryˆdˆ0 c(n) − 2pdyˆ

0 ( ) 0 ( ) 0 ( ) 0 ( ) T ∂ξEE n = [ ∂ξEE n ∂ξEE n ··· ∂ξEE n ] ∂c(n) ∂c (n) ∂c (n) ∂c (n) 0 1 Na−1 (24) T T = 2Rdˆ0dˆ0 c(n) + 2Rx0dˆ0 a (n) + 2b (n)Ryˆdˆ0 − 2pddˆ0 0 0 0 Therefore, the optimal weight vectors, aEE(n) bEE(n), and cEE(n) can be derived by supposing that the gradient functions are equal to zero.

5. Computer Simulation 5.1. Test Environment and Noise Characteristics The noise of the compressor (product model VW-11/4) in a gas station is x(n), which possesses a strong periodicity and high noise input level. As shown in Figure5a, a dual- Acoustics 2021, 3 FOR PEER REVIEW channel audio signal analyzer (AWA6290M+) is used to transmit the audio signals collected7 by two signal sensors to the MATLAB2019a software on a personal computer for algorithm veriﬁcation processing. The distance from sensor1 to the computer is the primary path (1.2 m), and the distance from sensor to the computer is the secondary path (6.7 m). collected by two signal sensors to the0 MATLAB2019a software on a personal computer for The time–frequency diagram of the tested noise is obtained by the computer mea- algorithmsurement systemverification shown processing. in Figure The5b, indicatingdistance from that thesensor sound1 to pressure the computer level in is thethe lowpri-

maryfrequency path range(1.2 m), of and 0–1 the kHz distance is much from higher sensor than0 the to one the incomputer the high is frequency the secondary range. path To (6.7compare m). our algorithm with the ones in [1,5,18] under similiar conditions, the secondary path of this system is set with a delay of 24 samples and ﬁlter length of 64.

FigureFigure 5. 5. ((aa)) Test Test environment; environment; ( b) time–frequency graph of the collected noisenoise inin aa certaincertain periodperiod ofof time.time. 5.2. Simulation Results The time–frequency diagram of the tested noise is obtained by the computer meas- urementFirstly, system by inputtingshown in theFigure data 5b, into indicating Equations that (12)–(14) the sound of pressu the improvedre level in EE-based the low frequencyANC system range model, of 0–1 the kHz learning is much process higher of than the weightthe one coefficientsin the high canfrequency be calculated range. To as 0( ) = compareshown in our Figure algorithm6a–c. Moreover,with the ones we in can [1,5,18] compute under the similiar optimal conditions, weights athe0 n secondary0.683, 0( ) = 0( ) = pathb0 n of this0.579 system, and cis0 setn with1.196 a delay(the firstof 24 column samples vector and filter of the length optimal of 64. weight coefficient

5.2. Simulation Results Firstly, by inputting the data into Equations (12)–(14) of the improved EE-based ANC system model, the learning process of the weight coefficients can be calculated as shown 0 = in Figure 6a–c. Moreover, we can compute the optimal weights a0 (n) 0.683 , 0 = 0 = b0 (n) 0.579 , and c0 (n) 1.196 (the first column vector of the optimal weight coefficient matrix). Secondly, according to Equations (17)–(20), the step size bounds can be μ < μ < μ < computed as A 0.139 , B 0.346, and C 0.21.

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Acoustics 2021, 3 FOR PEER REVIEW 8 matrix). Secondly, according to Equations (17)–(20), the step size bounds can be computed as µA < 0.139, µB < 0.346, and µC < 0.21.

Figure 6. ConvergenceConvergence process process of ofweight weight coefficients coefﬁcients (a–c (a) –andc) and ANC ANC system system based based on improv on improveded EE in different EE in different step factors step factors(d). (d).

Therefore, the step size bound is µ <μ0.139< , i.e., µmax =μ 0.139= . The learning curves Therefore, the step size bound is 0.139 , i.e., max 0.139 . The learning with different step sizes are shown in Figure6d, and is achieved by applying the following curves with different step sizes are shown in Figure 6d, and is achieved by applying the Equation (22): following equation (22): ∂(n) = γ∂(n − 1) + (1 − γ)e2(n) (25) 2 ∂(n) = γ∂(n −1) + (1−γ )e (n) 2 (25) where γ = 0.97 is a forgetting factor. By taking n = 0, we have ∂ (0) = e (0). The improved EE-basedγ = ANC system converges when the step size= is less than 0.4∂µmax.= 2 whereWe compare0.97 the is a performanceforgetting factor. of the By conventional taking n FIR0 , we method have (FxLMS),(0) e the(0) OE-based. The im- μ provedmodel, theEE-based EE-based ANC ANC system systems converges in [12,17 when,18], andthe step the improved size is less EE-based than 0.4 ANCmax systems. withWe the measurementcompare the performance noise. Figure of7a the reveals conventional the average FIR sound method pressure (FxLMS), level the of OE-based the noise. model,The step the sizes EE-based are 0.018, ANC 0.015, systems 0.0016, in and [12,17 0.0007,18], for and the the FxLMS, improved OE, EE, EE-based and improved ANC sys- EE temsschemes, with respectively. the measurement The FxLMS-based noise. Figure algorithm7a reveals filterthe average length issound 200; thepressure filter lengthslevel of theof A noise.(z) and TheC( stepz) are sizes 42 for are both 0.018, the 0.015, OE-based 0.0016, and and EE-based 0.0007 for ANC the systems; FxLMS, andOE, theEE, filter and improvedlengths of EEA( zschemes,), B(z), and respectively.C(z) are 42 The for FxLM the improvedS-based algorithm EE-based system.filter length Obviously, is 200; the algorithms compared in this paper can effectively converge and achieve a certain noise filter lengths of A(z) and C(z) are 42 for both the OE-based and EE-based ANC sys- reduction effect. Although, for the noise of a compressor, the improved EE-based ANC tems;system and shows the filter better lengths convergence of A(z) speed,, B(z as) , and shown C( inz) Figure are 427 b.for Furthermore, the improved as EE-based shown system.in Figure Obviously,7a, the improved the algorithms ANC system compared can reducein this paper the noise can ofeffectively the compressor converge up and to achieve28 dBA ata certain frequency noise below reduction 2.5 kHz, effect. and Alth theough, amplitude for the is noise expressed of a compressor, by the A-weighted the im- provedsound pressure EE-based level. ANC In system addition, shows the time–frequency better convergence diagram speed, of the as ANCshown system in Figure through 7b. Furthermore,various algorithms as shown is shown in Figure in Figure 7a, the7c. improved ANC system can reduce the noise of the compressor up to 28 dBA at frequency below 2.5 kHz, and the amplitude is expressed by the A-weighted sound pressure level. In addition, the time–frequency diagram of the ANC system through various algorithms is shown in Figure 7c.

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FigureFigure 7.7.( a(a)) Indicates Indicates the the noise noise reduction reduction performance performance comparison comparison of different of different ANC systems,ANC systems, the ordinate the ordinate is the A-weighted is the A- soundweighted pressure sound level, pressure and level, the abscissa and the is abscissa the frequency. is the frequency. (b) The convergence (b) The convergence process ofprocess the mean of the square mean error square of eacherror algorithm.of each algorithm. (c) Time–frequency (c) Time–frequency diagram diagram of different of different algorithms algorithms after noise after reduction. noise reduction. 5.3. Computational Complexity 5.3. Computational Complexity We compare the calculation complexity of the FIR-based FXLMS algorithm, the IIR- We compare the calculation complexity of the FIR-based FXLMS algorithm, the IIR- based OE, the EE-based ANC system, and the improved EE-based ANC system in Table1 . based OE, the EE-based ANC system, and the improved EE-based ANC system in Table The filter lengths of W(z), A(z), B(z), and C(z) are N, N , N , and N , respectively. Gen- W (z) A(z) B(z) C(z) a b N N c N N erally1. The speaking,filter lengths in contrast of with, the FIR-based, , and method, theare IIR-based , a , method b , and can utilizec , re- fewerspectively. filter Generally lengths; in speaking, comparison in contrast with the with OE-based the FIR-based method, method, the EE-based the IIR-based method requiresmethod can an additionalutilize fewer filtering filter lengths; of Sˆ(z) into comparison obtain the outputwith the signal; OE-based in contrast method, with the theEE- EE-based method, the improved EE-based method^ adds an additional adaptive filter C(z) S(z) inbased order method to make requires the ANC an additional system more filtering accurate of and faster. to obtain the output signal; in contrast with the EE-based method, the improved EE-based method adds an additional adaptive filter C(z) in order to make the ANC system more accurate and faster.

Table 1. Computational complexity.

Algorithm Multplications Additions FxLMS 2N + L + 2 2N + L −1

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Table 1. Computational complexity.

Algorithm Multplications Additions FxLMS 2N + L + 2 2N + L − 1 OE ANC 2Na + 2Nc + 2L + 2 2Na + 2Nc + 2L − 2 EE ANC 2Na + 2Nc + 3L + 2 2Na + 2Nc + 3L − 2 Proposed EE ANC 2Na + 2Nb + 2Nc + 4L + 2 2Na + 2Nb + 2Nc + 4L − 2

Table1 displays the comparison of the computational complexity of the various algorithms. If we substitute the previous data into the table using the FxLMS algorithm for 450 multiplications and 447 additions, the OE-based method requires 298 multiplications and 294 additions, the EE-based method needs 362 multiplications and 358 additions, and the improved EE-based method requires 424 multiplications and 420 additions. Obviously, the IIR-based models cost less in computational complexity.

6. Conclusions This paper introduces an adaptive IIR ﬁlter based on an improved EE model employing an ofﬂine secondary path modeling method that is superior in convergence speed and noise reduction performance. The compressor noise is taken as the input noise data to simulate the performance of the system. The results show that it has a good noise reduction effect from 0 Hz to 2.5 kHz. However, the system will increase a certain amount computational complexity. The model proposed in this paper is suitable for processing noise signals in a low frequency range, which is typical among industrial equipment, such as engines and compressors with loud noise signals.

Author Contributions: Conceptualization, J.Y. and J.R.; methodology, J.Y.; software, J.L.; validation, A.Z., X.Z., and J.R.; formal analysis, J.L.; investigation, J.Y.; resources, J.R.; data curation, A.Z.; writing—original draft preparation, J.Y.; writing—review and editing, J.R.; visualization, J.L.; super- vision, X.Z.; project administration, J.R.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by the Research Funds of the Maritime Defense Technologies Innovation (YT19201705) and the work was supported by the Chongqing Major Project of Integrated Circuit Industry (cstc2018jszx-cyztzx0217). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The Maritime Defense Technologies Innovation Institute and Chongqing Uni- versity of Posts and Telecommunications are greatly acknowledged for their financial support in making this research possible. Conflicts of Interest: The authors declare that the essay serves no personal or organizational interest. The authors declare no conflict of interest.

References 1. Lasota, A.; Meller, M. Iterative Learning Approach to Active Noise Control of Highly Autocorrelated Signals with Applications to Machinery Noise. IET Signal Process. 2020, 14, 560–568. [CrossRef] 2. Nelson, P.A.; Elliott, S.J.; John, E.W.F. Active control of sound basics of interferometry. Phys. Today 1993, 46, 75. [CrossRef] 3. Fredianelli, L.; Del Pizzo, A.; Licitra, G. Recent developments in sonic crystals as barriers for road trafﬁc noise mitigation. Environments 2019, 6, 14. [CrossRef] 4. Koussa, F.; Defrance, J.; Jean, P.; Blanc-Benon, P. Acoustical efﬁciency of a sonic crystal assisted noise barrier. Acta Acust. United Acust. 2013, 99, 399–409. [CrossRef] 5. Kuo, S.M.; Morgan, D.R. Active Noise Control Systems; Wiley: New York, NY, USA, 1996; Volume 4. 6. Aggogeri, F.; Al-Bender, F.; Brunner, B.; Elsaid, M.; Mazzola, M.; Merlo, A.; Ricciardi, D.; De La ORodriguez, M.; Salvi, E. Design of piezo-based avc system for machine tool applications. Mech. Syst. Signal Process. 2013, 36, 53–65. [CrossRef] Acoustics 2021, 3 363

7. Guo, X.; Jiang, J.; Chen, J.; Du, S.; Tan, L. Bibo-stable implementation of adaptive function expansion bilinear filter for nonlinear active noise control. Appl. Acoust. 2020, 168, 107407. [CrossRef] 8. Lam, B.; Shi, C.; Shi, D.; Gan, W.S. Active control of sound through full-sized open windows. Build. Environ. 2018, 141, 16–27. [CrossRef] 9. Murao, T.; Shi, C.; Gan, W.S.; Nishimura, M. Mixed-error approach for multi-channel active noise control of open windows. Appl. Acoust. 2017, 127, 305–315. [CrossRef] 10. Shi, C.; Jia, Z.; Xie, R.; Li, H. An active noise control casing using the multi-channel feedforward control system and the relative path based vir-tual sensing method. Mech. Syst. Signal Process. 2020, 144, 106878. [CrossRef] 11. Carini, A.; Mathews, V.J.; Sicuranza, G.L. Sufficient stability bounds for slowly varying direct-form recursive linear filters and their applications in adaptive iir filters. IEEE Trans. Signal Process. 1999, 47, 2561–2567. [CrossRef] 12. Feng, D.Z.; Zheng, W.X. Fast rls-type algorithm for unbiased equation-error adaptive iir filtering based on approximate inverse- power iteration. IEEE Trans. Signal Process. 2005, 53, 4169–4185. [CrossRef] 13. Netto, S.L.; Diniz, P.S.; Agathoklis, P. Adaptive iir filtering algorithms for system identification: A general framework. IEEE Trans. Educ. 1995, 38, 54–66. [CrossRef] 14. Regalia, P. Adaptive IIR Filtering in Signal Processing and Control; Rout-ledge: London, UK, 2018. 15. Mandelbrot, B.B. The variation of certain speculative prices. In Fractals and Scaling in Finance; Springer: Berlin, Germany, 1997; pp. 371–418. 16. Bouvet, M.; Schwartz, S.C. Comparison of adaptive and robust receivers for signal detection in ambient underwater noise. IEEE Trans. Acoust. Speech Signal Process. 1989, 37, 621–626. [CrossRef] 17. Elliott, S.; Nelson, P. The active control of sound. Electron. Commun. Eng. J. 1990, 2, 127–136. [CrossRef] 18. Ho, C.Y.; Shyu, K.K.; Chang, C.Y.; Kuo, S.M. Development of equation-error adaptive iir-filter-based active noise control system. Appl. Acoust. 2020, 163, 107226. [CrossRef] 19. Zhao, T.; Liang, J.; Zou, L.; Zhang, L. A new fxlms algorithm with offline and online secondary-path modeling scheme for active noise control of power transformers. IEEE Trans. Ind. Electron. 2017, 64, 6432–6442. [CrossRef] 20. Kim, I.S.; Na, H.S.; Kim, K.J.; Park, Y. Constraint filtered-x and filtered-u least-mean-square algorithms for the active control of noise in ducts. J. Acoust. Soc. Am. 1994, 95, 3379–3389. [CrossRef] 21. Fraanje, R.; Verhaegen, M.; Doelman, N. Convergence analysis of the filtered-u lms algorithm for active noise control in case perfect cancellation is not possible. Signal Process. 2003, 83, 1239–1254. [CrossRef]