# A Study of Engine Intake Noise Control Based on the Improved Filtered-x Least Mean Square Algorithm

Authors: Chaofeng Lan; Lei Zhang; Shou Lv; Pengwei Jiang
Journal: Shock and Vibration
 (2022)

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## Abstract

To reduce the intake noise of automobile engines, an active control system model of engine intake noise is established with the Filtered-x least mean square (FxLMS) algorithm. The offline identification method is adopted to identify the secondary path. The engine speed signal is used to construct the reference signal of a sound source to avoid interference of a secondary sound source to the reference signal. A variable-step algorithm is proposed, in which parameters are added to the normalized algorithm instead of the sinusoidal variable-step algorithm to adjust the amplitude range. This algorithm not only has the advantages of fast convergence speed and small steady-state error but also adapts to the characteristics of a time-varying reference signal and easy selection of parameters. In this paper, the noise of automobile engines under the New European Driving Cycle (NEDC) is studied and the proposed algorithm has faster convergence speed compared with the normalization algorithm, better adaptability to the change of the reference signal, and better stability compared with the sinusoidal variable step-size algorithm. The results show that the algorithm proposed can effectively reduce the intake noise of the engine at each speed and the noise reduction effect can reach 23.11 dB at a certain frequency. Meanwhile, the stability of the system is improved.


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## Body

## 1. Introduction

The noise of automobiles greatly influences users’ comfort and satisfaction. Cars bring convenience to people; at the same time, it also causes problems such as sub-health and environmental noise pollution. Therefore, reducing the noise of cars has attracted much attention [1, 2]. Traditional noise reduction methods, such as sound insulation, vibration isolation, and muffler, have little effect on controlling low-frequency noise [3]. With the development of electronic technology in recent years, the active noise control method has been noticed and has been widely discussed by researchers due to its effective low-frequency noise reduction and its goal orientation and target orientation.In the 1990s, the application of active noise control technology in the cabin received widespread attention [4]. Lockheed Company of the United States and P. A. Nelson of the University of Southampton in the United Kingdom, respectively, tested the active control of aircraft cabin noise and achieved good reduction effects [5]. It proves the broad prospects of active noise control technology in cabin applications such as aircraft, ships, and automobiles. The British Lotus Motor Company applied the noise active control technology to the air vehicle and also significantly reduced the low-frequency harmonic noise of the engine inside the car. The in-house roar sound corresponding to the ignition frequency of the engine ignition frequency can be reduced by about 10 dB [6]. Nissan Corporation of Japan is equipped with an active noise control system on a car with Blue Bird, which can reduce the noise inside the car by about 5-6 dB [7]. Burgess applied adaptive filtering theory to active noise control technology for the first time, made computer simulation research on the system's composition and algorithm, and proposed the Filtered-x least mean square (FxLMS) algorithm [8]. This algorithm is simple and effective, and it is still the most popular algorithm. Hu Xiao et al. put forward a vehicle indoor active noise control scheme based on multireference input and multierror output FxLMS algorithm and gave the computer simulation results [9]. Yuankai et al. proposed a feedforward active noise reduction method based on the normalized leakage FXLMS algorithm, which can significantly reduce the sound pressure in the control area under random excitation [10]. Yu proposed a sinusoidal variable step-size algorithm for active noise control, which overcomes the shortcomings of the fixed step-size FXLMS algorithm [11]. Zhang et al. proposed a variable step-size active noise equalization (ANE) algorithm based on the sinusoidal variable step-size FXLMS algorithm, which has faster convergence speed and smaller steady-state error [12]. Kukde et al. proposed some distributed active noise control systems, which have good noise cancellation effects under different noise conditions and acoustic environments [13–15].However, the performance of the sinusoidal variable step-size algorithm and ANE algorithm are easily affected by the reference signal; this paper proposed an improved variable step-size algorithm based on the advantages and disadvantages of the sinusoidal variable step-size algorithm and the normalization algorithm.In the past, scholars used the methods of active control to study the noise inside the vehicle and used the acoustic sensor to collect the primary sound source noise signal as the reference signal. This method can easily generate acoustic feedback during the control process and affects the stability of the control system [16–20]. In this paper, the intake noise reference signal is constructed by the engine speed signal and the improved variable step-size algorithm is used to control the noise. The advantages of the method and the traditional method are compared and analysed in terms of system stability, convergence speed, and noise reduction effect. Studies have shown that the system is simple and easy to implement and can significantly reduce the noise in the car caused by the engine intake system.In this paper, Section2 introduces the basic principles of active noise control; Section 3 analyses the intake noise characteristics of the car engine and proposes the use of engine speed to construct a reference signal; Section 4 analyses the performance of the fixed step-size, sinusoidal variable step-size, and normalization algorithms and proposes an improved variable step-size algorithm; Section 5 uses the improved algorithm to simulate under different engine speeds and NEDC conditions, and the advantages of the method and the traditional method are compared and analysed in terms of system stability, convergence speed, and noise reduction effect.


## 2. Basic Principles of Active Noise Control

The idea of active noise control is based on the principle of wave interference in physics that amplitudes of the secondary sound source and the noise signal are equal while their phases are opposite to eliminate noise. The adaptive algorithm is the core of the control system. Most active noise control systems, feedforward types or feedback types, use least mean square (LMS) algorithm and its improved algorithm to adjust the controller weight coefficient. Standard LMS algorithm makes the reference signal and error signal unequal in time due to the existence of a secondary sound path, which makes the active noise control system unstable. A new algorithm can be put forward by adding the weight estimation of the secondary sound path transfer function between the reference signal and the error signal to effectively improve the stability of the system [21]. This new algorithm filters the reference signal with the sound path model and updates the filter coefficients with the filtered reference signal and the system’s output error to modify the controller. Thus, the estimated sound path takes the place of the sound path in actual transmission [22]. This paper makes a study on the convergence speed, stability, and noise reduction effect of the system by using the feedforward active control system, whose structure is shown in Figure 1.Figure 1
Active noise control system model of FXLMS algorithm.In Figure1, k is the sampling time; n is the engine rotational speed; Pk is the primary noise source; dk is the primary path output; xk is the reference signal; yk is the driving signal of the secondary sound source; sk is the output signal of the secondary sound source at the error sensor; ek is the error signal; H1z and H2z are primary and secondary path transfer functions, respectively; and H∧2z is the estimate of the secondary path transfer function.The sound transfer function of the single-frequency periodic signal can be obtained by the actual measurement method in Figure1. However, the engine rotational speed varies continuously during the driving process, which leads to the change of the frequency of the single-frequency periodic signal. Therefore, it is difficult to obtain the transfer function H2z in Figure 1. This paper uses the transverse FIR filter of the LMS algorithm as the identification filter [23], and the offline identification method is used to estimate H2z in Figure 1 in this case, as shown in Figure 2.Figure 2
Schematic diagram of offline identification structure.In Figure2, the secondary path Hsz is served as the system under test or the unknown system, whose output is denoted as dk, that is, the expected signal recognized by the system. White noise xk is the excitation signal. yk is the output of the secondary path estimate H∧sz. The identification error ek calculated by dk and yk is offered to the LMS algorithm, to adjust and update the H∧sz coefficient. If the system converges, when the system runs long enough to the steady state, H∧sz can be provided to the FxLMS algorithm by the estimation of the secondary path Hsz.


## 3. The Characteristic of Engine Intake Noise

The noise of the engine intake system is the main noise source inside and outside the car, which has a significant effect on the noise inside the car [24]. Intake noise is generated by the periodic opening and closing of the intake valve. When the intake valve opens, a pressure pulse is formed in the intake pipe. As the piston moves, this pressure pulse is quickly damped and disappears. The same situation happens when the intake valve closes. In other words, there are two pressure pulses in one working cycle. The two pressure pulses occur periodically and form periodic noises whose frequencies are affected by the engine speed and can be expressed as follows:(1)fi=n⋅z⋅m60τHz,where m is the harmonic order: 1, 2, 3, ..., n is the engine rotational speed, z is the number of cylinders, and τ is the stroke coefficient, where four-stroke τ  = 2 and two-stroke τ  = 1.According to (1), the engine speed not only affects the noise frequency but also has a large impact on the noise magnitude. For the same engine, the noise changes with the rotational speed and is usually linearly distributed. Every time the speed doubles, the noise level increases by about 10 dB. When the engine speed is 1000 RPM, the intake noise level is 73 dB. Therefore, the relationship between noise level and rotational speed is expressed as follows:(2)SPL=n100+63.According to the definition of sound pressure level (SPL), the following can be deduced:(3)SPL=20lgpepref,where pe is the sound pressure effective value, pref is the reference sound pressure, and the value in the air is 2×10−5Pa.By combining (2) and (3), there is the following relationship between pe and engine rotational speed n:(4)pe=2×10n/2000−1.85.Since (1) and (4) show the quantitative relationship between engine rotational speed, engine intake noise frequency, and sound pressure, this paper simulates the effect of noise control and the stability of the system by using engine rotational speed signal to construct the reference signal of the sound source.


## 4. Variable Step-Size Algorithm Research

The simulation is based on an automobile engine with an in-line four-cylinder (z=4) four-stroke (τ=2) and a rotational speed of 3000 RPM[25]. Generally, the higher harmonic noise in the engine intake noise; that is, if over 3rd order is lower, it can be ignored [26]. Therefore, this paper selects the first 3 harmonics as the reference signal. According to (1), the frequencies of the fundamental, second, and third harmonics are 100Hz, 200Hz, and 300Hz, respectively. And, the first 3 sound pressure levels are 90dB, 76dB, and 70dB, respectively. (3) can be used to calculate the effective amplitude ratio of sound pressure [27], which is 1:0.159:0.1. Based on this amplitude ratio, the sound wave from the loudspeaker excited by the secondary sound source signal going through power amplifies at the same level with the sound wave calculated by (4), so the engine intake noise can be cancelled out. The reference signal can be set as follows:(5)xk=sin2πf1t+0.159sin2πf2t+0.1sin2πf3t,where f1=100Hz, f2=200Hz, and f3=300Hz.During the simulation process, the reference signal and the random noise, whose mean value and variance are 0 and 0.5, respectively, are superimposed to serve as the noise signalPk in Figure 1.(6)Pk=xk+0.5randn1,N,where N is the length of random noise.The primary and secondary sound path transfer functions [12] in Figure 1 are set as follows:(7)H1z=0.001+0.005z−1−0.005z−2+0.75z−3+0.5z−4−0.35z−5−0.4z−6+0.2z−7+0.4z−8−0.1z−9,H2z=0.01+0.01z−1+0.9z−2−0.01z−3−0.75z−4.The influences of step size on system performance and noise reduction are the most obvious in the FXLMS algorithm. The following research focuses on the effects of fixed step-size algorithm, sinusoidal variable step-size algorithm, and normalization algorithm on noise control.


### 4.1. Sinusoidal Step-Size Algorithm

Fixed step-size algorithm, as it suggests, has a fixed valueμ. According to the LMS algorithm, the larger the step size μ, the faster the convergence speed t, but the larger the steady-state error emin. Otherwise, a smaller step size results in slower convergence speed and smaller steady-state error. Therefore, the convergence speed and steady-state error in the fixed step-size algorithm are a pair of contradictions. In order to analyse the influence of step size on convergence time and steady-state error, the filter order L is set to be 20. Figure 3 shows the curve of how convergence time and steady-state error change with step μ, in which t is the convergence time and emin is the steady-state output error.Figure 3
Curve of convergence time and steady-state error changing with.μ.Figure3 shows that when 0<μ<0.02, the system can converge. The steady-state error increases with the increase in the step size and the convergence time becomes shorter. Therefore, when there is a higher requirement for noise reduction, the step size should be smaller, and when the requirement of convergence speed is higher, the larger step size should be chosen. If μ>0.02, the system will not converge and will fail to achieve noise reduction.In order to overcome the shortcomings of fixed step-size algorithms, many scholars have proposed improved algorithms. The most important one is variable step-size algorithm, using time-varying step size to replace fixed step size. There are many variable step-size algorithms, such as sigmoid function variable step-size algorithm, variable step-size algorithm of Lorentzian function, and sinusoidal variable step-size algorithm. This paper adopts the function relationship between step size and error signal in sinusoidal variable step-size algorithm [10].(8)μk=βsinαek−π2+1,where μk is the step size and α and β are the parameters. The parameter α controls the changing speed of step size. The parameter β controls the range of step size. The following simulation analyses the influence of error signal on step size when α is fixed while β equals to different values and when β is fixed while α equals to different values. The simulation parameters are as follows: aα=2,β=0.01,0.02,0.04bβ=0.01,andα=0.5,1,2. The function relationship between step size μk and the error signal ek is shown in Figure 4.Figure 4
Function relationship betweenμk and ek. (a) α  = 2 and β  = 0.01, 0.02, 0.04. (b) β  = 0.01 and α  = 0.5, 1, 2.
(a)(b)It can be seen from Figure4 that when the value of ek is large, so is the corresponding μk value. And, the convergence speed of the algorithm is fast. When the algorithm is in the convergence state, ek reaches its minimum, so does the corresponding μk. Besides, the larger the value of β, the larger the value range of the step size. The parameter α can control the changing speed of step size.In order to compare and analyse the influence of fixed step-size algorithm and sinusoidal variable step-size algorithm on system performance, there is a simulation and analysis of the relationship curve between output error signal changes and time by using fixed step-size algorithm. The simulation results are shown in Figure5(a). The simulation parameters are as follows: L=20,μ=0.01,0.02. Figure 5(b) is the results of the simulation and analysis of the relationship curve between output amplitude and time by using a sinusoidal variable step-size algorithm. The simulation parameters are α=2 and β=0.01 as in Figure 5(b).Figure 5
Curves of error signal time. (a) Fixed step-size algorithm. (b) Sinusoidal variable step-size algorithm.
(a)(b)It can be seen from Figure5 that when the steady-state error calculated by the fixed step-size algorithm is as small as that calculated by the sinusoidal variable step-size algorithm, its convergence speed is slower than that of the sinusoidal variable step-size algorithm. If the fixed step-size algorithm achieves a faster convergence speed, the steady-state error will be larger than that of the sinusoidal variable step-size algorithm. Therefore, the sinusoidal variable step-size algorithm alleviates the contradiction between steady-state error and convergence speed. In other words, it achieves smaller steady-state error and meanwhile maintains a faster convergence speed. Therefore, compared with fixed step-size algorithm, sinusoidal variable step-size algorithm can avoid the inherent defects of fixed step-size algorithm and is regarded as an effective improved variable step-size algorithm.


### 4.2. Normalization Variable Step-Size Algorithm

As is known to all, it is impossible for the car to maintain a constant speed in the driving process, so will be the engine rotational speed of the automobile. Therefore, the reference signal of the active noise control system is unstable and time-varying. Sinusoidal variable step-size algorithm only thinks about the function relationship between step size and error but fails to take into account the influence of reference signal’s amplitude variation on system performance. In an active noise control system using a sinusoidal variable step-size algorithm, if the reference signal’s amplitude is changeable, the error signal may not converge. Therefore, it is necessary to find an algorithm that can adjust the step size by the reference signal.The normalization algorithm is one of the variable step-size algorithms that adjusts the step size of the algorithm based on filter input. As the input increases, the steady-state error of the filter also grows. Therefore, it is necessary to reduce the steady-state error of the filter by adjusting the step size. The step size of the “normalizes” squared Euclidean norm is chosen to determine the reconstructed signal. The relationship between the step size and the reference signal is expressed as follows:(9)μk=μ1xk2<μ1<1.(9) shows that when the parameter μ selects a suitable value, the step size μk will adjust with the change amplitude of the reference signal xk. xk increases while μk decreases. Or xk decreases while μk increases. Thus, the system can adjust the change of xk by using this algorithm and improve the stability of the system.In order to verify that the normalization algorithm can enhance the stability of the system, the reference signalxk is doubled at 0.25 seconds to discuss how the error signal changes with time. The simulation parameters of normalization algorithms are μ1=0.01. And, the simulation results are shown in Figure 6(a). The simulation parameters of sinusoidal variable step-size algorithms are β=0.01andα=2. And, the simulation results are shown in Figure 6(b). Figures 6(a) and 6(b) represent the curve that the error signal changes with time when the reference signal is invariant and the curve that the error signal changes with time when the amplitude of the reference signal is doubled in 0.25 seconds, respectively.Figure 6
Curves of reference signal changing with time. (a) Normalization algorithm. (b) Sinusoidal variable step-size algorithm.
(a)(b)It can be seen from Figures6(a) and 6(b) that sinusoidal variable step size can converge before 0.25 seconds. After the reference signal xk is doubled at 0.25 seconds, the error signal diverges and does not converge. If it wants to achieve further convergence, parameters α and β need to be readjusted. The reference signal constructed in the active noise control system of the automobile varies with time, leading to the necessity to constantly adjust the parameters in order to achieve a better effect of noise reduction. Therefore, the sinusoidal variable step-size algorithm cannot be directly applied to the active noise reduction system inside the automobile. When the normalization algorithm doubles the reference signal xk at 0.25 seconds, the error signal can converge again. However, different from the situation in the sinusoidal variable step-size algorithm, the step size in the normalization algorithm cannot adjust itself with the error signal. When the amplitude of xk is constant, the value of μk is fixed. It is the same with the situation in fixed step-size algorithm that the contradiction between the convergence speed of the system and the steady-state error cannot be alleviated. Besides, when xk2 is very small, the step size will be too large and make the system unstable. Therefore, this paper proposes an improved variable step-size algorithm to improve the noise reduction effect inside the automobile.


### 4.3. An Improved Variable Step-Size Algorithm

Based on the advantages and disadvantages of the sinusoidal variable step-size algorithm and normalization algorithm, the amplitude range of step size can be adjusted by adding a parameterγ to the normalization algorithm and using it to replace parameter β in the sinusoidal variable step-size algorithm. The relationship among step size, reference signal, and the error signal is as follows:(10)uk=μ1γ+xk2sinaek−π2+10≤γ≤1.In equation (10), γ is set to avoid xk2 being too small and cause a too large step size. After the parameters μ1 and γ are selected, the step size can be adjusted in real time according to xk. When ek is large, μk increases. When ek is small, μk decreases. The parameter α can adjust the trend of step size changes with an error signal. Therefore, the improved algorithm possesses the advantages of both the sinusoidal step-size algorithm and the normalization algorithm.In order to analyse the convergence and stability of the improved variable step-size algorithm system, the reference signalxk is doubled and quadrupled at 0.25 seconds, respectively, and there is a simulation and analysis of the curve that the error signal changes with time. Simulation parameters are γ=1,μ1=0.01,andα=2. The simulation results are shown in Figure 7.Figure 7
The curve of error signal changing with time in the improved algorithm. (a) Reference signal is doubled at 0.25 seconds. (b) Reference signal is quadrupled at 0.25 seconds.
(a)(b)It can be seen from Figure7(a) that although the error signal increases after the reference signal xk is doubled, it can converge quickly again and it converges faster than the normalization algorithm, and the steady-state error is small. Therefore, the improved algorithm not only has the characteristics that the error signal can converge again after the amplitude of the xk signal changes by the normalization algorithm but also absorbs such advantages of sinusoidal variable step-size algorithm as fast convergence and small steady-state error, which can be well applied to the active noise control system of an automobile. It can be seen from Figure 7(b) that if the amplitude of the reference signal changes greatly and if the simulation parameters are not changed, the system may not converge and the improved algorithm cannot obtain a good noise reduction effect.In order to evaluate the noise reduction effect of the improved algorithm, there is a simulation and analysis of the amplitude-frequency curve of the engine noise signal and the error signal. The results are shown in Figure8.Figure 8
Amplitude-frequency curves of noise and error signals.In Figure8, the amplitude of the noise signal at 100 Hz is denoted as P1 and the amplitude of error signal at 100Hz is denoted as P2. Then, the equation for calculating the amount of noise change is as follows:(11)ΔE=20lgP1P2.The simulation results in Figure8 show that the amplitudes of the noise signal at 100Hz, 200Hz, and 300Hz are 1.4930, 0.2467, and 0.1669, respectively. The amplitudes of the error signal at 100Hz, 200Hz, and 300Hz are 0.1461, 0.0362, and 0.0273, respectively. According to (11) the noise reduction values at 100Hz, 200Hz, and 300Hz are 20.19dB,6.67dB,and5.73dB, respectively, indicating that this method has a satisfying noise reduction effect on the engine.


## 4.1. Sinusoidal Step-Size Algorithm

Fixed step-size algorithm, as it suggests, has a fixed valueμ. According to the LMS algorithm, the larger the step size μ, the faster the convergence speed t, but the larger the steady-state error emin. Otherwise, a smaller step size results in slower convergence speed and smaller steady-state error. Therefore, the convergence speed and steady-state error in the fixed step-size algorithm are a pair of contradictions. In order to analyse the influence of step size on convergence time and steady-state error, the filter order L is set to be 20. Figure 3 shows the curve of how convergence time and steady-state error change with step μ, in which t is the convergence time and emin is the steady-state output error.Figure 3
Curve of convergence time and steady-state error changing with.μ.Figure3 shows that when 0<μ<0.02, the system can converge. The steady-state error increases with the increase in the step size and the convergence time becomes shorter. Therefore, when there is a higher requirement for noise reduction, the step size should be smaller, and when the requirement of convergence speed is higher, the larger step size should be chosen. If μ>0.02, the system will not converge and will fail to achieve noise reduction.In order to overcome the shortcomings of fixed step-size algorithms, many scholars have proposed improved algorithms. The most important one is variable step-size algorithm, using time-varying step size to replace fixed step size. There are many variable step-size algorithms, such as sigmoid function variable step-size algorithm, variable step-size algorithm of Lorentzian function, and sinusoidal variable step-size algorithm. This paper adopts the function relationship between step size and error signal in sinusoidal variable step-size algorithm [10].(8)μk=βsinαek−π2+1,where μk is the step size and α and β are the parameters. The parameter α controls the changing speed of step size. The parameter β controls the range of step size. The following simulation analyses the influence of error signal on step size when α is fixed while β equals to different values and when β is fixed while α equals to different values. The simulation parameters are as follows: aα=2,β=0.01,0.02,0.04bβ=0.01,andα=0.5,1,2. The function relationship between step size μk and the error signal ek is shown in Figure 4.Figure 4
Function relationship betweenμk and ek. (a) α  = 2 and β  = 0.01, 0.02, 0.04. (b) β  = 0.01 and α  = 0.5, 1, 2.
(a)(b)It can be seen from Figure4 that when the value of ek is large, so is the corresponding μk value. And, the convergence speed of the algorithm is fast. When the algorithm is in the convergence state, ek reaches its minimum, so does the corresponding μk. Besides, the larger the value of β, the larger the value range of the step size. The parameter α can control the changing speed of step size.In order to compare and analyse the influence of fixed step-size algorithm and sinusoidal variable step-size algorithm on system performance, there is a simulation and analysis of the relationship curve between output error signal changes and time by using fixed step-size algorithm. The simulation results are shown in Figure5(a). The simulation parameters are as follows: L=20,μ=0.01,0.02. Figure 5(b) is the results of the simulation and analysis of the relationship curve between output amplitude and time by using a sinusoidal variable step-size algorithm. The simulation parameters are α=2 and β=0.01 as in Figure 5(b).Figure 5
Curves of error signal time. (a) Fixed step-size algorithm. (b) Sinusoidal variable step-size algorithm.
(a)(b)It can be seen from Figure5 that when the steady-state error calculated by the fixed step-size algorithm is as small as that calculated by the sinusoidal variable step-size algorithm, its convergence speed is slower than that of the sinusoidal variable step-size algorithm. If the fixed step-size algorithm achieves a faster convergence speed, the steady-state error will be larger than that of the sinusoidal variable step-size algorithm. Therefore, the sinusoidal variable step-size algorithm alleviates the contradiction between steady-state error and convergence speed. In other words, it achieves smaller steady-state error and meanwhile maintains a faster convergence speed. Therefore, compared with fixed step-size algorithm, sinusoidal variable step-size algorithm can avoid the inherent defects of fixed step-size algorithm and is regarded as an effective improved variable step-size algorithm.


## 4.2. Normalization Variable Step-Size Algorithm

As is known to all, it is impossible for the car to maintain a constant speed in the driving process, so will be the engine rotational speed of the automobile. Therefore, the reference signal of the active noise control system is unstable and time-varying. Sinusoidal variable step-size algorithm only thinks about the function relationship between step size and error but fails to take into account the influence of reference signal’s amplitude variation on system performance. In an active noise control system using a sinusoidal variable step-size algorithm, if the reference signal’s amplitude is changeable, the error signal may not converge. Therefore, it is necessary to find an algorithm that can adjust the step size by the reference signal.The normalization algorithm is one of the variable step-size algorithms that adjusts the step size of the algorithm based on filter input. As the input increases, the steady-state error of the filter also grows. Therefore, it is necessary to reduce the steady-state error of the filter by adjusting the step size. The step size of the “normalizes” squared Euclidean norm is chosen to determine the reconstructed signal. The relationship between the step size and the reference signal is expressed as follows:(9)μk=μ1xk2<μ1<1.(9) shows that when the parameter μ selects a suitable value, the step size μk will adjust with the change amplitude of the reference signal xk. xk increases while μk decreases. Or xk decreases while μk increases. Thus, the system can adjust the change of xk by using this algorithm and improve the stability of the system.In order to verify that the normalization algorithm can enhance the stability of the system, the reference signalxk is doubled at 0.25 seconds to discuss how the error signal changes with time. The simulation parameters of normalization algorithms are μ1=0.01. And, the simulation results are shown in Figure 6(a). The simulation parameters of sinusoidal variable step-size algorithms are β=0.01andα=2. And, the simulation results are shown in Figure 6(b). Figures 6(a) and 6(b) represent the curve that the error signal changes with time when the reference signal is invariant and the curve that the error signal changes with time when the amplitude of the reference signal is doubled in 0.25 seconds, respectively.Figure 6
Curves of reference signal changing with time. (a) Normalization algorithm. (b) Sinusoidal variable step-size algorithm.
(a)(b)It can be seen from Figures6(a) and 6(b) that sinusoidal variable step size can converge before 0.25 seconds. After the reference signal xk is doubled at 0.25 seconds, the error signal diverges and does not converge. If it wants to achieve further convergence, parameters α and β need to be readjusted. The reference signal constructed in the active noise control system of the automobile varies with time, leading to the necessity to constantly adjust the parameters in order to achieve a better effect of noise reduction. Therefore, the sinusoidal variable step-size algorithm cannot be directly applied to the active noise reduction system inside the automobile. When the normalization algorithm doubles the reference signal xk at 0.25 seconds, the error signal can converge again. However, different from the situation in the sinusoidal variable step-size algorithm, the step size in the normalization algorithm cannot adjust itself with the error signal. When the amplitude of xk is constant, the value of μk is fixed. It is the same with the situation in fixed step-size algorithm that the contradiction between the convergence speed of the system and the steady-state error cannot be alleviated. Besides, when xk2 is very small, the step size will be too large and make the system unstable. Therefore, this paper proposes an improved variable step-size algorithm to improve the noise reduction effect inside the automobile.


## 4.3. An Improved Variable Step-Size Algorithm

Based on the advantages and disadvantages of the sinusoidal variable step-size algorithm and normalization algorithm, the amplitude range of step size can be adjusted by adding a parameterγ to the normalization algorithm and using it to replace parameter β in the sinusoidal variable step-size algorithm. The relationship among step size, reference signal, and the error signal is as follows:(10)uk=μ1γ+xk2sinaek−π2+10≤γ≤1.In equation (10), γ is set to avoid xk2 being too small and cause a too large step size. After the parameters μ1 and γ are selected, the step size can be adjusted in real time according to xk. When ek is large, μk increases. When ek is small, μk decreases. The parameter α can adjust the trend of step size changes with an error signal. Therefore, the improved algorithm possesses the advantages of both the sinusoidal step-size algorithm and the normalization algorithm.In order to analyse the convergence and stability of the improved variable step-size algorithm system, the reference signalxk is doubled and quadrupled at 0.25 seconds, respectively, and there is a simulation and analysis of the curve that the error signal changes with time. Simulation parameters are γ=1,μ1=0.01,andα=2. The simulation results are shown in Figure 7.Figure 7
The curve of error signal changing with time in the improved algorithm. (a) Reference signal is doubled at 0.25 seconds. (b) Reference signal is quadrupled at 0.25 seconds.
(a)(b)It can be seen from Figure7(a) that although the error signal increases after the reference signal xk is doubled, it can converge quickly again and it converges faster than the normalization algorithm, and the steady-state error is small. Therefore, the improved algorithm not only has the characteristics that the error signal can converge again after the amplitude of the xk signal changes by the normalization algorithm but also absorbs such advantages of sinusoidal variable step-size algorithm as fast convergence and small steady-state error, which can be well applied to the active noise control system of an automobile. It can be seen from Figure 7(b) that if the amplitude of the reference signal changes greatly and if the simulation parameters are not changed, the system may not converge and the improved algorithm cannot obtain a good noise reduction effect.In order to evaluate the noise reduction effect of the improved algorithm, there is a simulation and analysis of the amplitude-frequency curve of the engine noise signal and the error signal. The results are shown in Figure8.Figure 8
Amplitude-frequency curves of noise and error signals.In Figure8, the amplitude of the noise signal at 100 Hz is denoted as P1 and the amplitude of error signal at 100Hz is denoted as P2. Then, the equation for calculating the amount of noise change is as follows:(11)ΔE=20lgP1P2.The simulation results in Figure8 show that the amplitudes of the noise signal at 100Hz, 200Hz, and 300Hz are 1.4930, 0.2467, and 0.1669, respectively. The amplitudes of the error signal at 100Hz, 200Hz, and 300Hz are 0.1461, 0.0362, and 0.0273, respectively. According to (11) the noise reduction values at 100Hz, 200Hz, and 300Hz are 20.19dB,6.67dB,and5.73dB, respectively, indicating that this method has a satisfying noise reduction effect on the engine.


## 5. Results and Discussion

There are different noises characteristics for vehicle engines under different working conditions. To verify that the improved algorithm can be applied to an automobile active noise control system, it is necessary to analyse the noise control effect on the engine under different working conditions. The tire size of this car is 195/55R15, and the final gear ratio is 3.722. The gear ratio in different gears is shown in Table1.Table 1
The gear ratio in each gear.
Gear1ST2ND3RD4TH5THGear ratio3.5452.1581.4781.1290.886For a specific vehicle, the tire size and final transmission ratio are fixed parameters, and for a certain gear, the gear ratio is also fixed. Therefore, as long as the gear and engine speed are known, the corresponding vehicle speed can be calculated. Under the five-gear condition, the relationship between engine rotational speed and vehicle speed of a domestic vehicle is shown in Table2.Table 2
Corresponding relationship of rotational speed and vehicle speed.
Engine speed (RPM)18002400300036004200Vehicle speed (km/h)61.381.7102.1122.5142.9Noise fundamental frequency (Hz)6080100120140In order to verify the noise reduction effect of the improved algorithm at various engine rotational speeds, the reference signals are constructed at several speeds in Table2. The simulation parameters of each rotational speed are the same: γ=1,μ1=0.01,andα=2. The result at 1800 RPM is shown in Figure 9.Figure 9
Amplitude-frequency curves of noise and error signals at 1800RPM.In Figure9, the amplitude of the noise signal at 60Hz is denoted as P1 and the amplitude of error signal at 60Hz is denoted as P2. According to (11), the amount of noise reduction at 60Hz can be calculated to be 16.31dB.There is a simulation and analysis of the amplitude-frequency curve of the noise signal and the error signal at different speeds, and the amplitude of the noise signal and the error signal is obtained; the noise reduction under different rotational speeds can be calculated with Eq. (11) and is shown in Table 3.Table 3
Noise reductions at different rotational speeds.
RPMNoise frequencyOrderNoise amplitudeError amplitudeReduction (dB)18006010.18900.019419.7712020.12230.014918.2618030.03390.007413.2524008010.37650.037120.1216020.09410.018014.3324030.21070.029517.06300010011.00000.069923.1120020.15900.019418.2630030.10000.014816.58360012011.23880.094422.3624020.69370.064220.6736030.78040.071720.73420014013.83970.299422.1628020.34560.106210.2542031.22860.148218.37In order to further verify that the improved algorithm is suitable for the engine noise control system, the improved algorithm is used to simulate the vehicle under NEDC operating conditions, which is the European endurance test standard. It is mainly used in Europe, China, Australia, and other countries. NEDC cycle conditions include urban operation cycle and suburban operation cycle. This paper selects a part of the suburban operation cycle in the NEDC operating condition. This part contains the conditions of the car's uniform speed, acceleration, deceleration, and gear shifting. The reference signals can be constructed based on the data in Table1. The reference signal is shown in Figure 10. The convergence effect of the error signal is simulated by using the improved algorithm and the reference signal in Figure 10.Figure 10
The curve of the simulated reference signal.The simulation parameter of normalization algorithm isμ1=0.01, and the simulation results are shown in Figure 11(a). The simulation parameters of sinusoidal variable step-size algorithms are β=0.01andα=2, and the simulation results are shown in Figure 11(b). The simulation parameters are γ=1,μ1=0.01,andα=2, and the results of the improved algorithm are shown in Figure 11(c).Figure 11
The curve of the error signal changing with time. (a) Normalization algorithm. (b) Sinusoidal variable step-size algorithm. (c) The improved algorithm.
(a)(b)(c)It can be seen from Figure11 that after the error signal of the normalization algorithm converges, the error signal can maintain convergence regardless of the change of the reference signal, but the convergence speed is relatively slow. The convergence speed of the sinusoidal variable step-size algorithm is relatively fast, but when the reference signal amplitude is large, the error signal is large. So, the normalization algorithm and the sinusoidal variable step algorithm cannot be directly applied to the active noise control of the automobile engine. The improved algorithm has a better control effect on this kind of signal. Once the error signal converges, the error signal can stay in a stable state and needs no time for adjustment to converge even if the reference signal changes. After the parameters γ, μ1, and α of the improved algorithm are determined, the controller can automatically adjust the step-size parameters according to the input of xk and ek without any further changes. There will be no case that the system cannot converge due to the acceleration, deceleration, and gear shifting of the vehicle. Besides, it can achieve a better noise reduction effect. Therefore, the improved algorithm proposed in this paper is suitable for the application of an active noise control system inside the automobile.The improved algorithm combines the advantages of both the sinusoidal variable step-size algorithm and the normalization algorithm. Compared with the normalized algorithm, the improved algorithm has a faster convergence speed. Compared with the sine variable step-size algorithm, the improved algorithm can better adapt to the change of the reference signal and has a better stability. Meanwhile, the improved algorithm has a fast convergence speed and small steady-state error which is suitable for time-varying reference signals. And, the parameters are easy to select.The noise reduction can reach more than10dB at each frequency when the improved algorithm is applied to the FXLMS algorithm. Besides, the active noise control system has an obvious effect on the noise reduction of the vehicle under NEDC conditions.



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*Source: 8011171-2022-05-27.xml*