# Active Control for Marine Engine Room Noise Using an FXLMS Algorithm

Authors: Qianqian Yu; Erming Cao
Journal: Scientific Programming
 (2022)

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## Abstract

The noise in the marine engine room has always been a major cause of disturbance and damage to sailors’ physical and mental health; however, the antinoise countermeasures were being ignored by the industry. To further reduce the noise in the marine engine room, the system structure of active noise control (ANC) and the theoretical basis of the least mean square (LMS) and filteredX least mean square (FXLMS) algorithm are studied. Based on the FXLMS algorithm ANC system, the simulation study of active noise control in the marine engine room of a multigas carrier is carried out. According to the simulation results by the diagrams plotted on the time domain and spectrum, the ANC systems with the FXLMS algorithm can effectively reduce the noise in the marine engine room by about 20 dB, especially for the peak low-frequency range of 20–500 Hz with a large antinoise effect.


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## Body

## 1. Introduction

To deal with the global energy crisis and fulfill the demand for “low-carbon,” the global power engine has set off a wave of electrification. This electric engine is not only a little more efficient than the traditional internal combustion engine, but also has the advantages of zero pollutant emissions and low noise. But ocean-going ships, which are the main vehicle for global trade, are still far away from real electrification. It means that the internal combustion engine will remain the main driving engine or generator for ships. Although the internal combustion engine has the advantages of high efficiency and high reliability, its noise has been troubling the physical and mental health of the crew, reducing the safety and comfort of the ship. Some studies have pointed out that a crew exposed to high noise for a long time is more likely to feel tired, also can easily lead to inattention and reduce work efficiency even lead to safety accidents [1, 2]. What is more serious, it may be deleterious to cardiovascular, endocrine, and nervous systems, and it is associated with neuropsychiatric disorders [3, 4]. To keep the crew away from heavy noise on board, the 90th meeting of the International Maritime Safety Committee (MSC) in May 2012 approved the draft revision of the Noise Level Rules, which set higher requirements for the noise reduction performance of ships [5]. The shipbuilder and class should take reasonable means to control the cabin noise of ships.Traditional noise control solutions are focused on passive noise reduction methods. Zhu indicates that semiactive mufflers can control exhaust noise effectively [6]. Liu studied the flow and noise control performance of a compressor silencer [7]. Liang concluded the floating floor of the vessel’s engine room can reduce the vibration and noise [8]. Some researchers studied the fuel injection and control strategy from the perspective of the combustion side to reduce the combustion noise of engines [9, 10] while the speed and frequency variations challenge to passive noise reduction. Researchers pay more attention to controllable magneto-rheological damping with linear or nonlinear system control methods for better vibration and noise control [11–13].Because of the huge cost and less controllable flexibility of passive vibration and noise control methods, more and more industries are starting to use active noise control methods in specific regions [14–16]; those applications indicate the good potential of the active noise control method. The active noise controller recognized the reference noise frequency and then output the opposite amplitude to achieve the effect of reducing the noise. For different kinds of noise resources and applications, lots of different algorithms and solutions of ANC systems have been studied and applied during the past decades [17, 18]. Zeb simulates the improvement of active noise control for the vehicle by the filtered-input recursive least squares (FxRLS) algorithm [18]. Jiang evaluated the performance of the modified hybrid active noise control system (HANC) which combines the strengths of narrowband ANC and broadband ANC systems on vehicle noise reduction [19]. Even though many ANC solutions have been adopted for small space room applications, the studies of active noise control on large vessel engine rooms have almost not been reported before. Due to the good convergence and stabilities of the FXLMS algorithm, it has been adopted in this article to design the active noise controller for marine engine room noise control. Hence, the noise in the marine engine room has been identified, and the preliminary results of ANC control were reported in this article. The research in this article will provide some beneficial foreshadowing for further ANC applications applied on the vessel.


## 2. Materials and Methods

Active noise control is a noise control method that adds noise reduction by a control signal which is released by a secondary sound source during the initial noise signal propagation route. The interference theory of sound waves shows that two sound waves with the same frequency and opposite phase overlap each other in the propagation path to produce the interference phenomenon so that the acoustic amplitude of the original noise signal cancels each other. The active noise control achieves the purpose of controlling the noise by achieving the attenuation of the sound energy by interfering with the acoustic wave. In the interference phenomenon of sound waves, the phase and amplitude of two columns of interfering sound waves are two key factors that determine whether the sound energy can decay. In practical application, the reverse phase offset acoustic wave can be auxiliary generated by matching the adaptive digital processor and the electroacoustic device, and the effective suppression of primary noise can be realized. Figure1 shows the schematic diagram of the noise signal superposition principle.Figure 1
Schematic diagram of noise signal superposition principle.Primary noise is the original noise signal, and secondary noise is the noise generated by the speaker which is used to offset the original noise. When the system works normally, the output signal of the speaker interferes with the original noise signal in a specific space. To form a stable interference, the two-column waves need to meet the following three conditions: (1) the propagation direction is the same, (2) the phase difference remains constant, and (3) the vibration frequency is the same.Analysis from the perspective of mass point displacement: Let the noise signal be represented by equation (1), the cancellation signal be represented by equation (2), and the signal superimposed in the specified noise reduction area be represented by equation (3).(1)E1=A1sinω1t+φ1,(2)E2=A2sinω2t+φ2,(3)E3=E1+E2=A1sinω1t+φ1+A2sinω2t+φ2.The superimposed waveform stability is poor when the frequencyω1 is not equal to ω2. When the difference between the amplitude of A1 and A2 is large, the superimposed waveform amplitude cannot reduce the final noise compared with the original noise signal. When ω1 is equal to ω2,(4)E3=A∗2+2cosφ2−φ1∗sinω1t+φ1.For the positive noise reduction effect, the phase difference ofφ1 and φ2 should be in the order of 2nπ+3/3π,2nπ+4/3π, or the interference phase length phenomenon will occur; therefore, the noise reduction system cannot play a noise reduction effect in this condition. The adaptive process of the ANC noise reduction system is to adjust the initial phase of the cancellation signal to meet the phase requirements of the system. In the active control of noise, due to the time-change of the noise signal and environmental factors, an adaptive controller is needed to effectively control the accurate tracking of noise signal under the premise of system stability. The filter in the adaptive control system realizes the real-time signal processing function by adapting to the environmental changes of the adaptive algorithm. The adaptive controller consists of a digital filter and a corresponding adaptive algorithm for adjusting the filter parameters, which are the main focus of the controller design work. The objective of the adaptive algorithm is to set an objective function, which is to make the result approach a target value by constant iterative computation.


### 2.1. Active Noise Control System

The active noise control system usually consists of sensors, controllers, and speakers. The sensor can be divided into a reference signal sensor and an error sensor. The acoustic sensor, speed sensor, or acceleration sensor can be used as input signals. The controller mainly includes the signal processing hardware and the control function of the software programs.The active noise control system is divided into a feedback system and a feedforward system. The feedforward system usually selects the noise signal to collect the noise source in the reference signal selection, and the controller sends out the control signal after processing the input reference signal. The feedback system is often used when the control system cannot collect the initial reference signal.


#### 2.1.1. Feedforward Active Control System

The system equivalent diagram is shown in Figure2. In the figure, pn is the noise signal, Wz is the controller, H1z is the primary path transmission function, and H2z is the secondary path transmission function.Figure 2
Feedforward active control system equivalent diagram.The reference signal sensor picked up the signalxn which was released by the noise source, and then the speaker played the control signal yn which was processed by the controller algorithm. The error sensor picks up the error signal en after interfering with the noise source noise and the control signal emitted by the secondary sound source interferes and transmits it to the controller. The controller adjusted the weight coefficient by using the design algorithm to reduce the error signal until the system has become stable.


#### 2.1.2. Feedback Active Control System

The controller responds to the signal feedback by the error sensor when the feedback system processes the noise signal. Because the signal is picked up by the feedback control system through the error sensor and then used as the controller input, it is easy to be disturbed by other noise and reduces the stability of the system, which has a certain impact on the system noise reduction effect. In the feedback active noise control system, the signal processing takes an error signal as its input signal. The equivalent control system diagram is displayed in Figure3. Compared with the feedforward active noise system, its noise reduction performance is poor and rarely used.Figure 3
Feedback active control system equivalent diagram.The symbol definitions in Figure3 are omitted here as the same symbols were used in Figure 2.


### 2.2. LMS and FXLMS Algorithms for ANC System

Because of the time-varying characteristics of the noise signal, it is difficult to predict in advance and makes the noise difficult to be tracked in real time. ANC technology requires tracking the time-varying noise signal by adjusting the controller to make the generated secondary noise signal minimize the original noise signal. An adaptive filter can track the time-varying signals well and continuously by adjusting the required control signals to produce them through some optimization error criterion. This optimization error criterion belongs to the ANC algorithm. The adaptive LMS algorithm and FXLMS algorithm are widely used.


#### 2.2.1. LMS Algorithm

The equivalent diagram of the LMS algorithm is shown in Figure4.Figure 4
Block diagram of noise LMS algorithm active control system.The noise signalxn is processed by the primary path transmission function Pz and then becomes as the primary noise signal dn. The noise signal xn and the error signal en are transmitted to the adaptive filter to update the weight coefficient, and the adaptive filter outputs yn making the error signal en gradually reduce.Symbol definition:xn: noise signal; Pz: primary sound source transmission path; dn: initial noise signal of the error sensor; en: error signal; wn: the LMS adaptive filter; and yn: output signal of the LMS adaptive filter.The input signal at timen gets a response by a filter of order L:(5)Xn=xn,xn−1,……,xn−l+1T.The vector of the filtered weight coefficient at timen is expressed as(6)Wn=w1n,w2n,……wLnT.The filter output signalyn can be expressed as(7)yn=XTnWn=WTnXn=∑i=1Lwinxn−l+1.The error signal at timen is(8)en=dn−yn=dn−wTnxn.Using the minimum mean-variance error optimization criterion,(9)Jn=Ee2n=Edn−wTnxn2=Ed2n−2WTnEdnXn+WTnEXnXTnWn,delimit R and P as the equations below.(10)R=EXnXTn,P=EdnXn.R is the autocorrelation matrix of the reference signal and P is the intercorrelation matrix between the expected signal and the reference signal.Then formula (9) is reduced to (11)Jn=Ed2n−2WTnP+WTnRWn.Jn is the quadratic function about W, and to realize the minimum error signal, the gradient obtained by equation (11) is indicated by(12)∇n=∂Jn∂W|W=Wn=∂Jn∂W1,∂Jn∂W2,……,∂Jn∂WL=2RWn−2P.The above equation is made equal to 0, and then the filtering coefficientW0 for the minimum error signal is obtained, namely, the Wiener solution.(13)W0=R−1P.Returning to the formula (11),(14)Jmin=Ed2n−2PTR−1P+R−1PTRR−1P=Ed2n−PTW0,Jn=Jmin+Wn−W0TRWn−W0.When the autocorrelation matricesR and P are known, the adaptive filter algorithm can calculate the W0, that is, it only needs to know the autocorrelation matrix R and the intercorrelation matrix P to obtain the best filter weight coefficient. In practice, P and R are not necessarily certain; however, the minimum mean square algorithm does not require an autocorrelation matrix, it still can obtain the best-filtered weight coefficient through the steepest descent method.From the steepest descent theory, the filter weight vector at the next moment can be expressed aswn+1.(15)Wn+1=Wn−μ∇n.In the equation (15), μ is the convergence step and ∇n is the gradient of the n-th iteration.Taking the gradient in the LMS algorithm from the square root of the error signal as an unbiased estimate of∇n,(16)∇^=∂e2n∂W=−2enXn,(17)Wn+1=Wn+2μenXn.


#### 2.2.2. FXLMS Algorithm

The FXLMS equivalent diagram of the active noise control system is shown in Figure5.Figure 5
Block diagram of noise FXLMS algorithm active control system.In practice, the LMS algorithm ignores the existence of the secondary pathCz, that is, the control signal yk from the speaker can bring the error signal close to zero, but the signal yk after passing through the secondary path Cz does not necessarily make the error signal close to zero. Because the electrical signal emitted by the speaker transmits through a series of digital-analog conversions to the microphone, there is a certain time difference with the reference signal, which can easily lead to the instability of the system. Based on this, Morgan [20] proposes to place the same filter in the reference signal path to realize the weight update of the LMS algorithm, thus realizing FXLMS’s algorithm. C′z can estimate the statute of Cz precisely which is required in the algorithm, and the two are approximately equal. Adding the estimation C′z of the secondary path transmission function improves the LMS algorithm and indeed becomes the essential difference between FXLMS and LMS algorithms [21–30].Symbols definition:xk: noise signal; Pz: primary sound source transmission path; Pk: primary path transmission function; dk: initial noise signal of the error sensor; Cz: secondary sound source transmission path; Ck: secondary path transmission function; ek: error signal; wz: self-adapting filter; yk: output signal of the adaptive filter; sk: secondary noise signal passing through the secondary path; C′z: estimated secondary path; and Fk: noise signal generated through the estimated secondary path.The desired signal is transmitted through the primary path and can be represented as a convolution of the reference signal with the primary path transmission function.(18)dk=xk∗Pk.Similarly, the secondary noise signal emitted by the speaker may be expressed as(19)sk=yk∗Ck.The filter weight coefficient and the algorithm input signal vector are(20)Wk=w1k,w1k,……,wLkT,Fk=fk,fk−1,……,fk−l+1T=XkCk.The output of the filter ofL order at time k in the FXLMS algorithm is(21)yk=XTkWk=WTkFk=∑l=1Lwlkfk−l+1.To delimitrk as the filtered signal, which is the convolution of the reference signal and the secondary path transmission function can be expressed as(22)rk=Xk∗Ck.The derived error signal is(23)ek=dk+sk=dk+rTkWk.The relation of the filter weight can be derived by equation (17).(24)Wk+1=Wk−2μekrk.R is defined as the autocorrelation matrix of the reference signal, by which its orthogonal matrix can be expressed as Q.(25)R=QΛQT.Λ is the diagonal matrix composed of the eigenvalues of the R,(26)Λ=diagλ1,λ2,…,λL,Wn+1=Ι−2μQΛQTWn+2μP,EWn+1=QΙ−2μΛn+1Q−1EW0+2μP.In the above formula,W0 is the initial value of the filter weight; thus, the convergence conditions of the above formula are(27)1−2μλmax<1.Equation (27) can be transformed as(28)0<μ<1/λmax.λmax is the maximum eigenvalue of the positive definite matrix R, and the adaptive filtering weights converge when the convergence step size factor μ satisfies equation (28). Usually, the autocorrelation matrix R is unknown but the reference signal is available, and the mean square value of the reference signal and the autocorrelation matrix R satisfy the following relationship:(29)TrR=∑i=0Nλi=∑n=1LEx2n,λmax<TrR.Therefore, the convergence step lengthμ satisfies the following relationship:(30)0<μ<1∑n=1LEx2n.The sum mean square of the reference signal is known, and the value range of the convergence steps can be prejudged by the reference signal.


## 2.1. Active Noise Control System

The active noise control system usually consists of sensors, controllers, and speakers. The sensor can be divided into a reference signal sensor and an error sensor. The acoustic sensor, speed sensor, or acceleration sensor can be used as input signals. The controller mainly includes the signal processing hardware and the control function of the software programs.The active noise control system is divided into a feedback system and a feedforward system. The feedforward system usually selects the noise signal to collect the noise source in the reference signal selection, and the controller sends out the control signal after processing the input reference signal. The feedback system is often used when the control system cannot collect the initial reference signal.


### 2.1.1. Feedforward Active Control System

The system equivalent diagram is shown in Figure2. In the figure, pn is the noise signal, Wz is the controller, H1z is the primary path transmission function, and H2z is the secondary path transmission function.Figure 2
Feedforward active control system equivalent diagram.The reference signal sensor picked up the signalxn which was released by the noise source, and then the speaker played the control signal yn which was processed by the controller algorithm. The error sensor picks up the error signal en after interfering with the noise source noise and the control signal emitted by the secondary sound source interferes and transmits it to the controller. The controller adjusted the weight coefficient by using the design algorithm to reduce the error signal until the system has become stable.


### 2.1.2. Feedback Active Control System

The controller responds to the signal feedback by the error sensor when the feedback system processes the noise signal. Because the signal is picked up by the feedback control system through the error sensor and then used as the controller input, it is easy to be disturbed by other noise and reduces the stability of the system, which has a certain impact on the system noise reduction effect. In the feedback active noise control system, the signal processing takes an error signal as its input signal. The equivalent control system diagram is displayed in Figure3. Compared with the feedforward active noise system, its noise reduction performance is poor and rarely used.Figure 3
Feedback active control system equivalent diagram.The symbol definitions in Figure3 are omitted here as the same symbols were used in Figure 2.


## 2.1.1. Feedforward Active Control System

The system equivalent diagram is shown in Figure2. In the figure, pn is the noise signal, Wz is the controller, H1z is the primary path transmission function, and H2z is the secondary path transmission function.Figure 2
Feedforward active control system equivalent diagram.The reference signal sensor picked up the signalxn which was released by the noise source, and then the speaker played the control signal yn which was processed by the controller algorithm. The error sensor picks up the error signal en after interfering with the noise source noise and the control signal emitted by the secondary sound source interferes and transmits it to the controller. The controller adjusted the weight coefficient by using the design algorithm to reduce the error signal until the system has become stable.


## 2.1.2. Feedback Active Control System

The controller responds to the signal feedback by the error sensor when the feedback system processes the noise signal. Because the signal is picked up by the feedback control system through the error sensor and then used as the controller input, it is easy to be disturbed by other noise and reduces the stability of the system, which has a certain impact on the system noise reduction effect. In the feedback active noise control system, the signal processing takes an error signal as its input signal. The equivalent control system diagram is displayed in Figure3. Compared with the feedforward active noise system, its noise reduction performance is poor and rarely used.Figure 3
Feedback active control system equivalent diagram.The symbol definitions in Figure3 are omitted here as the same symbols were used in Figure 2.


## 2.2. LMS and FXLMS Algorithms for ANC System

Because of the time-varying characteristics of the noise signal, it is difficult to predict in advance and makes the noise difficult to be tracked in real time. ANC technology requires tracking the time-varying noise signal by adjusting the controller to make the generated secondary noise signal minimize the original noise signal. An adaptive filter can track the time-varying signals well and continuously by adjusting the required control signals to produce them through some optimization error criterion. This optimization error criterion belongs to the ANC algorithm. The adaptive LMS algorithm and FXLMS algorithm are widely used.


### 2.2.1. LMS Algorithm

The equivalent diagram of the LMS algorithm is shown in Figure4.Figure 4
Block diagram of noise LMS algorithm active control system.The noise signalxn is processed by the primary path transmission function Pz and then becomes as the primary noise signal dn. The noise signal xn and the error signal en are transmitted to the adaptive filter to update the weight coefficient, and the adaptive filter outputs yn making the error signal en gradually reduce.Symbol definition:xn: noise signal; Pz: primary sound source transmission path; dn: initial noise signal of the error sensor; en: error signal; wn: the LMS adaptive filter; and yn: output signal of the LMS adaptive filter.The input signal at timen gets a response by a filter of order L:(5)Xn=xn,xn−1,……,xn−l+1T.The vector of the filtered weight coefficient at timen is expressed as(6)Wn=w1n,w2n,……wLnT.The filter output signalyn can be expressed as(7)yn=XTnWn=WTnXn=∑i=1Lwinxn−l+1.The error signal at timen is(8)en=dn−yn=dn−wTnxn.Using the minimum mean-variance error optimization criterion,(9)Jn=Ee2n=Edn−wTnxn2=Ed2n−2WTnEdnXn+WTnEXnXTnWn,delimit R and P as the equations below.(10)R=EXnXTn,P=EdnXn.R is the autocorrelation matrix of the reference signal and P is the intercorrelation matrix between the expected signal and the reference signal.Then formula (9) is reduced to (11)Jn=Ed2n−2WTnP+WTnRWn.Jn is the quadratic function about W, and to realize the minimum error signal, the gradient obtained by equation (11) is indicated by(12)∇n=∂Jn∂W|W=Wn=∂Jn∂W1,∂Jn∂W2,……,∂Jn∂WL=2RWn−2P.The above equation is made equal to 0, and then the filtering coefficientW0 for the minimum error signal is obtained, namely, the Wiener solution.(13)W0=R−1P.Returning to the formula (11),(14)Jmin=Ed2n−2PTR−1P+R−1PTRR−1P=Ed2n−PTW0,Jn=Jmin+Wn−W0TRWn−W0.When the autocorrelation matricesR and P are known, the adaptive filter algorithm can calculate the W0, that is, it only needs to know the autocorrelation matrix R and the intercorrelation matrix P to obtain the best filter weight coefficient. In practice, P and R are not necessarily certain; however, the minimum mean square algorithm does not require an autocorrelation matrix, it still can obtain the best-filtered weight coefficient through the steepest descent method.From the steepest descent theory, the filter weight vector at the next moment can be expressed aswn+1.(15)Wn+1=Wn−μ∇n.In the equation (15), μ is the convergence step and ∇n is the gradient of the n-th iteration.Taking the gradient in the LMS algorithm from the square root of the error signal as an unbiased estimate of∇n,(16)∇^=∂e2n∂W=−2enXn,(17)Wn+1=Wn+2μenXn.


### 2.2.2. FXLMS Algorithm

The FXLMS equivalent diagram of the active noise control system is shown in Figure5.Figure 5
Block diagram of noise FXLMS algorithm active control system.In practice, the LMS algorithm ignores the existence of the secondary pathCz, that is, the control signal yk from the speaker can bring the error signal close to zero, but the signal yk after passing through the secondary path Cz does not necessarily make the error signal close to zero. Because the electrical signal emitted by the speaker transmits through a series of digital-analog conversions to the microphone, there is a certain time difference with the reference signal, which can easily lead to the instability of the system. Based on this, Morgan [20] proposes to place the same filter in the reference signal path to realize the weight update of the LMS algorithm, thus realizing FXLMS’s algorithm. C′z can estimate the statute of Cz precisely which is required in the algorithm, and the two are approximately equal. Adding the estimation C′z of the secondary path transmission function improves the LMS algorithm and indeed becomes the essential difference between FXLMS and LMS algorithms [21–30].Symbols definition:xk: noise signal; Pz: primary sound source transmission path; Pk: primary path transmission function; dk: initial noise signal of the error sensor; Cz: secondary sound source transmission path; Ck: secondary path transmission function; ek: error signal; wz: self-adapting filter; yk: output signal of the adaptive filter; sk: secondary noise signal passing through the secondary path; C′z: estimated secondary path; and Fk: noise signal generated through the estimated secondary path.The desired signal is transmitted through the primary path and can be represented as a convolution of the reference signal with the primary path transmission function.(18)dk=xk∗Pk.Similarly, the secondary noise signal emitted by the speaker may be expressed as(19)sk=yk∗Ck.The filter weight coefficient and the algorithm input signal vector are(20)Wk=w1k,w1k,……,wLkT,Fk=fk,fk−1,……,fk−l+1T=XkCk.The output of the filter ofL order at time k in the FXLMS algorithm is(21)yk=XTkWk=WTkFk=∑l=1Lwlkfk−l+1.To delimitrk as the filtered signal, which is the convolution of the reference signal and the secondary path transmission function can be expressed as(22)rk=Xk∗Ck.The derived error signal is(23)ek=dk+sk=dk+rTkWk.The relation of the filter weight can be derived by equation (17).(24)Wk+1=Wk−2μekrk.R is defined as the autocorrelation matrix of the reference signal, by which its orthogonal matrix can be expressed as Q.(25)R=QΛQT.Λ is the diagonal matrix composed of the eigenvalues of the R,(26)Λ=diagλ1,λ2,…,λL,Wn+1=Ι−2μQΛQTWn+2μP,EWn+1=QΙ−2μΛn+1Q−1EW0+2μP.In the above formula,W0 is the initial value of the filter weight; thus, the convergence conditions of the above formula are(27)1−2μλmax<1.Equation (27) can be transformed as(28)0<μ<1/λmax.λmax is the maximum eigenvalue of the positive definite matrix R, and the adaptive filtering weights converge when the convergence step size factor μ satisfies equation (28). Usually, the autocorrelation matrix R is unknown but the reference signal is available, and the mean square value of the reference signal and the autocorrelation matrix R satisfy the following relationship:(29)TrR=∑i=0Nλi=∑n=1LEx2n,λmax<TrR.Therefore, the convergence step lengthμ satisfies the following relationship:(30)0<μ<1∑n=1LEx2n.The sum mean square of the reference signal is known, and the value range of the convergence steps can be prejudged by the reference signal.


## 2.2.1. LMS Algorithm

The equivalent diagram of the LMS algorithm is shown in Figure4.Figure 4
Block diagram of noise LMS algorithm active control system.The noise signalxn is processed by the primary path transmission function Pz and then becomes as the primary noise signal dn. The noise signal xn and the error signal en are transmitted to the adaptive filter to update the weight coefficient, and the adaptive filter outputs yn making the error signal en gradually reduce.Symbol definition:xn: noise signal; Pz: primary sound source transmission path; dn: initial noise signal of the error sensor; en: error signal; wn: the LMS adaptive filter; and yn: output signal of the LMS adaptive filter.The input signal at timen gets a response by a filter of order L:(5)Xn=xn,xn−1,……,xn−l+1T.The vector of the filtered weight coefficient at timen is expressed as(6)Wn=w1n,w2n,……wLnT.The filter output signalyn can be expressed as(7)yn=XTnWn=WTnXn=∑i=1Lwinxn−l+1.The error signal at timen is(8)en=dn−yn=dn−wTnxn.Using the minimum mean-variance error optimization criterion,(9)Jn=Ee2n=Edn−wTnxn2=Ed2n−2WTnEdnXn+WTnEXnXTnWn,delimit R and P as the equations below.(10)R=EXnXTn,P=EdnXn.R is the autocorrelation matrix of the reference signal and P is the intercorrelation matrix between the expected signal and the reference signal.Then formula (9) is reduced to (11)Jn=Ed2n−2WTnP+WTnRWn.Jn is the quadratic function about W, and to realize the minimum error signal, the gradient obtained by equation (11) is indicated by(12)∇n=∂Jn∂W|W=Wn=∂Jn∂W1,∂Jn∂W2,……,∂Jn∂WL=2RWn−2P.The above equation is made equal to 0, and then the filtering coefficientW0 for the minimum error signal is obtained, namely, the Wiener solution.(13)W0=R−1P.Returning to the formula (11),(14)Jmin=Ed2n−2PTR−1P+R−1PTRR−1P=Ed2n−PTW0,Jn=Jmin+Wn−W0TRWn−W0.When the autocorrelation matricesR and P are known, the adaptive filter algorithm can calculate the W0, that is, it only needs to know the autocorrelation matrix R and the intercorrelation matrix P to obtain the best filter weight coefficient. In practice, P and R are not necessarily certain; however, the minimum mean square algorithm does not require an autocorrelation matrix, it still can obtain the best-filtered weight coefficient through the steepest descent method.From the steepest descent theory, the filter weight vector at the next moment can be expressed aswn+1.(15)Wn+1=Wn−μ∇n.In the equation (15), μ is the convergence step and ∇n is the gradient of the n-th iteration.Taking the gradient in the LMS algorithm from the square root of the error signal as an unbiased estimate of∇n,(16)∇^=∂e2n∂W=−2enXn,(17)Wn+1=Wn+2μenXn.


## 2.2.2. FXLMS Algorithm

The FXLMS equivalent diagram of the active noise control system is shown in Figure5.Figure 5
Block diagram of noise FXLMS algorithm active control system.In practice, the LMS algorithm ignores the existence of the secondary pathCz, that is, the control signal yk from the speaker can bring the error signal close to zero, but the signal yk after passing through the secondary path Cz does not necessarily make the error signal close to zero. Because the electrical signal emitted by the speaker transmits through a series of digital-analog conversions to the microphone, there is a certain time difference with the reference signal, which can easily lead to the instability of the system. Based on this, Morgan [20] proposes to place the same filter in the reference signal path to realize the weight update of the LMS algorithm, thus realizing FXLMS’s algorithm. C′z can estimate the statute of Cz precisely which is required in the algorithm, and the two are approximately equal. Adding the estimation C′z of the secondary path transmission function improves the LMS algorithm and indeed becomes the essential difference between FXLMS and LMS algorithms [21–30].Symbols definition:xk: noise signal; Pz: primary sound source transmission path; Pk: primary path transmission function; dk: initial noise signal of the error sensor; Cz: secondary sound source transmission path; Ck: secondary path transmission function; ek: error signal; wz: self-adapting filter; yk: output signal of the adaptive filter; sk: secondary noise signal passing through the secondary path; C′z: estimated secondary path; and Fk: noise signal generated through the estimated secondary path.The desired signal is transmitted through the primary path and can be represented as a convolution of the reference signal with the primary path transmission function.(18)dk=xk∗Pk.Similarly, the secondary noise signal emitted by the speaker may be expressed as(19)sk=yk∗Ck.The filter weight coefficient and the algorithm input signal vector are(20)Wk=w1k,w1k,……,wLkT,Fk=fk,fk−1,……,fk−l+1T=XkCk.The output of the filter ofL order at time k in the FXLMS algorithm is(21)yk=XTkWk=WTkFk=∑l=1Lwlkfk−l+1.To delimitrk as the filtered signal, which is the convolution of the reference signal and the secondary path transmission function can be expressed as(22)rk=Xk∗Ck.The derived error signal is(23)ek=dk+sk=dk+rTkWk.The relation of the filter weight can be derived by equation (17).(24)Wk+1=Wk−2μekrk.R is defined as the autocorrelation matrix of the reference signal, by which its orthogonal matrix can be expressed as Q.(25)R=QΛQT.Λ is the diagonal matrix composed of the eigenvalues of the R,(26)Λ=diagλ1,λ2,…,λL,Wn+1=Ι−2μQΛQTWn+2μP,EWn+1=QΙ−2μΛn+1Q−1EW0+2μP.In the above formula,W0 is the initial value of the filter weight; thus, the convergence conditions of the above formula are(27)1−2μλmax<1.Equation (27) can be transformed as(28)0<μ<1/λmax.λmax is the maximum eigenvalue of the positive definite matrix R, and the adaptive filtering weights converge when the convergence step size factor μ satisfies equation (28). Usually, the autocorrelation matrix R is unknown but the reference signal is available, and the mean square value of the reference signal and the autocorrelation matrix R satisfy the following relationship:(29)TrR=∑i=0Nλi=∑n=1LEx2n,λmax<TrR.Therefore, the convergence step lengthμ satisfies the following relationship:(30)0<μ<1∑n=1LEx2n.The sum mean square of the reference signal is known, and the value range of the convergence steps can be prejudged by the reference signal.


## 3. Results

### 3.1. Noise Sampling

This article collects the cabin noise as a reference noise signal on a multigas carrier. The noise source of the marine engine room is mainly derived from the noise of a 6-cylinder two-stroke low-speed engine, and its main parameters are shown in Table1.Table 1
Engine parameter for sampling noise signal in the marine engine room.
ItemsUnitValueBoremm400Strokemm1770Speedrpm125Cylinder number—6PowerkW4950Firing order—1-6-2-4-3-5Usually, for a certain type of engine, the engine room noise is greatly affected by the engine speed and power. The higher the speed, the greater the power, and the more obvious the corresponding noise. The position of the microphone of the noise reference signal is located on the top of a 1 m distance of the exhaust manifolder between cylinder 3 and cylinder 4. The sampling frequency of the reference signal collection is 8 kHz, under the condition that the engine speed of 125 rpm, the power is 3715 kW, which corresponds to a 75% load operating condition. All the generators are stopped, and the electric power consumption on the vessel is provided by the main engine shaft belt generator. Figure6 shows the actual drawing of the low-speed engine in the engine room. The main noise in the engine room comes from the friction, vibration, air transmission, and combustion noise of the two-stroke engine.Figure 6
Schematic diagram of reference noise signal picking up position in the marine engine room.


### 3.2. Signal Processing and Modeling

Figure7 is the characteristics of the noise signal in the marine engine room. Figure 7(a) shows the time domain map of the reference signal after being analyzed. The engine running noise in the engine room shows obvious periodicity, so the acquisition signal within 50 ms is selected as an input, and the analytical spectrum map shows that the frequency at the peak noise is concentrated between 80 and 400 Hz, which is displayed on Figure 7(b). These low-frequency noises are difficult to eliminate by the passive noise reduction method.Figure 7
Characteristics of the noise signal in the marine engine room. (a) The time domain diagram of reference noise signal; (b) the spectrum diagram of reference noise signal by fast Fourier transform (FFT).
(a)(b)The active noise reduction calculation model of engine cabin noise with the FXLMS algorithm has been established on the MATLAB platform. White noise is used as an excitation to identify the secondary path transmission functionC′z offline. It is assumed that the characteristics of C′z are variable and unknown at the first. The offline estimation method can be used to identify the secondary path transmission function, and the identification coefficient of C′z is used as the fixed coefficient of the FXLMS algorithm. The steps of offline identification are shown in Figure 8.Figure 8
The block diagram of offline identification for secondary transmission path function.A white noise signal in then moment is υn, the secondary path output value is dn, yυn is the value of the white noise through the secondary path identification, and e0n is the error value between the secondary path output value and secondary path identification value. It can be considered that the secondary path identification function C′^z is closer to the secondary path function C′z when the error value e0n is close to zero through the LMS algorithm’s constantly iterative update.Offline identification is a system identification independently separated from the ANC system, which will not increase the operation burden of the ANC system, and also will not damage the robustness of the ANC system. If the secondary path transmission function of the ANC system remains unchanged, offline identification has certain advantages.During the simulation tests, the primary and secondary transmission functions are set as equations (31) and (32).(31)pz=z−1−0.07z−3+0.006z−4,(32)Cz=1z+0.1.The filter order of the FXLMS algorithmN was set as 16, the convergence step factor μ was 0.1, the sampling frequency was 8000 Hz, the total simulation discrete sampling data was 8000, and the whole simulation time was set within one second.


### 3.3. Disclosed Results

Figure9 shows offline identification results for the secondary path transmission function and output control signal. The offline identification error and the iteration coefficient of the secondary path transmission function are shown in Figure 9(a). After the two-order iteration and the simulation discrete step length of 600, the offline system identification error tends to be zero. It shows that the LMS algorithm is accurate and efficient for the offline identification of the secondary path transmission function. As shown in the simulation calculation results in Figure 9(b), the control signal of the adaptive output of the speaker agrees well with the reference noise signal after 4000 iteration steps.Figure 9
Offline identification results for secondary path transmission function and output control signal. (a) The simulation results of offline identification by LMS algorithm; (b) the output control signal of FXLMS ANC systems.
(a)(b)The residual noise continuously drops under the action of the active noise reduction system quickly, and gradually stabilizes to a low error value of 0.01 after 5000 steps, and no divergence phenomenon occurs. Comparing the signal amplitude and sound pressure level on the time domain diagram, the blue curve in Figure10 is the original noise reference signal, and the red curve is the error signal processed by the FXLMS algorithm ANC noise reduction system, indicating that the ANC system significantly reduces the active noise reduction of marine engine room noise. From the perspective of SPL degrees, it shows that the maximum SPL is reduced by about 20 dB at the end of the simulation.Figure 10
The time domain collation map diagram of the amplitude and SPL between reference noise signal and residual noise signal by FXLMS ANC system. (a) Amplitude comparison. (b) Reference noisy signal SPL (dB) 1 ms.
(a)(b)The noise reduction effect was evaluated from the perspective of power, and Spectrum Analyzer was used to analyze the input noise signal and the output error signal by frequency spectrum. The analysis results are shown in Figure11. As can be seen from Figure 11, the error signal which is colored in blue is generally lower than the input noise signal which is colored in yellow in the full frequency range, and the noise reduction effect at the peak frequency of 125 Hz and 300 Hz is about 20 dB. It is obviously noticed that the ANC system has a better noise reduction effect in the low-frequency range of 20-500 Hz. The ANC system based on the FXLMS algorithm has a good active control effect on the cabin noise of the periodic, low-frequency, and high-intensity noise signal.Figure 11
The spectrum analyzer between reference noise signal and residual noise signal by FXLMS ANC system.When considering the impacts of the different convergence step factorsμ, three sets of different μ have been studied. The noise reduction performance is shown in Figure 12 for μ = 0.1, 0.5, and 0.9, respectively.Figure 12
Noise reduction performance for different convergence step factorsμ. (a) The time domain diagram for u = 0.9; (b) the spectrum analyzer diagram for u = 0.9; (c) the time domain diagram for u = 0.5; (d) spectrum analyzer diagram for u = 0.5; (e) the time domain diagram for u = 0.1; (f) spectrum analyzer diagram for u = 0.1.
(a)(b)(c)(d)(e)(f)The results show that the noise reduction performance is linearly improving with the decreasing with the convergence step factorμ. In Figures 12(a) and 12(b), when μ=0.9, the maximum SPL decrease is about 10 dB, not also a large fluctuation for the final residual noise signal but also a weak or even bad noise reduction performance for the frequency range of 1.5-4 kHz are observed. As a comparison for μ=0.5, the results in Figures 12(c) and 12(d) show a better noise reduction performance, the maximum SPL decrease is about 15 dB; for μ=0.1, the results in Figures 12(e) and 12(f) show the maximum SPL decrease is about 20 dB. The results show that the convergence step factor has great importance for the ANC workings.


## 3.1. Noise Sampling

This article collects the cabin noise as a reference noise signal on a multigas carrier. The noise source of the marine engine room is mainly derived from the noise of a 6-cylinder two-stroke low-speed engine, and its main parameters are shown in Table1.Table 1
Engine parameter for sampling noise signal in the marine engine room.
ItemsUnitValueBoremm400Strokemm1770Speedrpm125Cylinder number—6PowerkW4950Firing order—1-6-2-4-3-5Usually, for a certain type of engine, the engine room noise is greatly affected by the engine speed and power. The higher the speed, the greater the power, and the more obvious the corresponding noise. The position of the microphone of the noise reference signal is located on the top of a 1 m distance of the exhaust manifolder between cylinder 3 and cylinder 4. The sampling frequency of the reference signal collection is 8 kHz, under the condition that the engine speed of 125 rpm, the power is 3715 kW, which corresponds to a 75% load operating condition. All the generators are stopped, and the electric power consumption on the vessel is provided by the main engine shaft belt generator. Figure6 shows the actual drawing of the low-speed engine in the engine room. The main noise in the engine room comes from the friction, vibration, air transmission, and combustion noise of the two-stroke engine.Figure 6
Schematic diagram of reference noise signal picking up position in the marine engine room.


## 3.2. Signal Processing and Modeling

Figure7 is the characteristics of the noise signal in the marine engine room. Figure 7(a) shows the time domain map of the reference signal after being analyzed. The engine running noise in the engine room shows obvious periodicity, so the acquisition signal within 50 ms is selected as an input, and the analytical spectrum map shows that the frequency at the peak noise is concentrated between 80 and 400 Hz, which is displayed on Figure 7(b). These low-frequency noises are difficult to eliminate by the passive noise reduction method.Figure 7
Characteristics of the noise signal in the marine engine room. (a) The time domain diagram of reference noise signal; (b) the spectrum diagram of reference noise signal by fast Fourier transform (FFT).
(a)(b)The active noise reduction calculation model of engine cabin noise with the FXLMS algorithm has been established on the MATLAB platform. White noise is used as an excitation to identify the secondary path transmission functionC′z offline. It is assumed that the characteristics of C′z are variable and unknown at the first. The offline estimation method can be used to identify the secondary path transmission function, and the identification coefficient of C′z is used as the fixed coefficient of the FXLMS algorithm. The steps of offline identification are shown in Figure 8.Figure 8
The block diagram of offline identification for secondary transmission path function.A white noise signal in then moment is υn, the secondary path output value is dn, yυn is the value of the white noise through the secondary path identification, and e0n is the error value between the secondary path output value and secondary path identification value. It can be considered that the secondary path identification function C′^z is closer to the secondary path function C′z when the error value e0n is close to zero through the LMS algorithm’s constantly iterative update.Offline identification is a system identification independently separated from the ANC system, which will not increase the operation burden of the ANC system, and also will not damage the robustness of the ANC system. If the secondary path transmission function of the ANC system remains unchanged, offline identification has certain advantages.During the simulation tests, the primary and secondary transmission functions are set as equations (31) and (32).(31)pz=z−1−0.07z−3+0.006z−4,(32)Cz=1z+0.1.The filter order of the FXLMS algorithmN was set as 16, the convergence step factor μ was 0.1, the sampling frequency was 8000 Hz, the total simulation discrete sampling data was 8000, and the whole simulation time was set within one second.


## 3.3. Disclosed Results

Figure9 shows offline identification results for the secondary path transmission function and output control signal. The offline identification error and the iteration coefficient of the secondary path transmission function are shown in Figure 9(a). After the two-order iteration and the simulation discrete step length of 600, the offline system identification error tends to be zero. It shows that the LMS algorithm is accurate and efficient for the offline identification of the secondary path transmission function. As shown in the simulation calculation results in Figure 9(b), the control signal of the adaptive output of the speaker agrees well with the reference noise signal after 4000 iteration steps.Figure 9
Offline identification results for secondary path transmission function and output control signal. (a) The simulation results of offline identification by LMS algorithm; (b) the output control signal of FXLMS ANC systems.
(a)(b)The residual noise continuously drops under the action of the active noise reduction system quickly, and gradually stabilizes to a low error value of 0.01 after 5000 steps, and no divergence phenomenon occurs. Comparing the signal amplitude and sound pressure level on the time domain diagram, the blue curve in Figure10 is the original noise reference signal, and the red curve is the error signal processed by the FXLMS algorithm ANC noise reduction system, indicating that the ANC system significantly reduces the active noise reduction of marine engine room noise. From the perspective of SPL degrees, it shows that the maximum SPL is reduced by about 20 dB at the end of the simulation.Figure 10
The time domain collation map diagram of the amplitude and SPL between reference noise signal and residual noise signal by FXLMS ANC system. (a) Amplitude comparison. (b) Reference noisy signal SPL (dB) 1 ms.
(a)(b)The noise reduction effect was evaluated from the perspective of power, and Spectrum Analyzer was used to analyze the input noise signal and the output error signal by frequency spectrum. The analysis results are shown in Figure11. As can be seen from Figure 11, the error signal which is colored in blue is generally lower than the input noise signal which is colored in yellow in the full frequency range, and the noise reduction effect at the peak frequency of 125 Hz and 300 Hz is about 20 dB. It is obviously noticed that the ANC system has a better noise reduction effect in the low-frequency range of 20-500 Hz. The ANC system based on the FXLMS algorithm has a good active control effect on the cabin noise of the periodic, low-frequency, and high-intensity noise signal.Figure 11
The spectrum analyzer between reference noise signal and residual noise signal by FXLMS ANC system.When considering the impacts of the different convergence step factorsμ, three sets of different μ have been studied. The noise reduction performance is shown in Figure 12 for μ = 0.1, 0.5, and 0.9, respectively.Figure 12
Noise reduction performance for different convergence step factorsμ. (a) The time domain diagram for u = 0.9; (b) the spectrum analyzer diagram for u = 0.9; (c) the time domain diagram for u = 0.5; (d) spectrum analyzer diagram for u = 0.5; (e) the time domain diagram for u = 0.1; (f) spectrum analyzer diagram for u = 0.1.
(a)(b)(c)(d)(e)(f)The results show that the noise reduction performance is linearly improving with the decreasing with the convergence step factorμ. In Figures 12(a) and 12(b), when μ=0.9, the maximum SPL decrease is about 10 dB, not also a large fluctuation for the final residual noise signal but also a weak or even bad noise reduction performance for the frequency range of 1.5-4 kHz are observed. As a comparison for μ=0.5, the results in Figures 12(c) and 12(d) show a better noise reduction performance, the maximum SPL decrease is about 15 dB; for μ=0.1, the results in Figures 12(e) and 12(f) show the maximum SPL decrease is about 20 dB. The results show that the convergence step factor has great importance for the ANC workings.


## 4. Discussion

This article studied the significant noise reduction performance in the marine engine room for ANC systems with the FXLMS algorithm. The first part of the article explained the composition of active noise reduction systems and the modeling theory basis of LMS and FXLMS algorithms. The latter part of the article introduced the simulation conditions, ANC modeling, as well as the disclosed results. In detail, the reference marine engine room noise signal from a gas carrier has been imported and analyzed, which indicated that periodic and peak low frequency are the main characteristics of the noise signal. An ANC system with an FXLMS algorithm was designed to control the sample noise. The model calculation shows that the ANC system based on the FXLMS algorithm significantly suppressed the marine engine room noise. During the entire one-second simulation time, the residual marine engine room noise signal after active control is significantly lower than the input noise signal, as well as the SPL drops by about 20 dB. The controlled marine engine room noise is generally improved over the input noise over the full frequency range, especially a drop of SPL by about 20 dB for the peak low frequency at 125 Hz and 300 Hz which can be concluded on the spectrum analyzer diagram. While the noise reduction performance for ANC system with FXLMS algorithm has important variations on the convergence step factor. The residual signal error increases linearly with the convergence step factor when compared to the results for different convergence step factors 0.1, 0.5, and 0.9. It shows some limitations in the performance of ANC systems with FXLMS algorithms. When considered, the engine room noise is always stable and can even continue throughout the ship’s life cycle; the limitations can be minimized by the initial optimum for the convergence step factor. Further research will focus on multipath ANC systems design and variable convergence step factor optimum for ANC systems [31].



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*Source: 8372929-2022-08-18.xml*