comparative study of frequency domain filtered-x lms algorithms applied to vehicle powertrain noise...

36 Int. J. Vehicle Noise and Vibration, Vol. 5, Nos. 1/2, 2009

Copyright 2009 Inderscience Enterprises Ltd.

Comparative study of frequency domain filtered-x LMS algorithms applied to vehicle powertrain noise control

Jie Duan, Mingfeng Li and Teik C. Lim* Vibro-Acoustics and Sound Quality Research Laboratory, Department of Mechanical Engineering, University of Cincinnati, 598 Rhodes Hall, P.O. Box 210072, Cincinnati, OH 45221 0072, USA E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author

Ming-Ran Lee, F. Wayne Vanhaaften, Ming-Te Cheng and Takeshi Abe Powertrain NVH R&D Department, Research and Advanced Engineering Center, Ford Motor Company, Dearborn, MI 48124, USA E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

Abstract: Currently, passive noise control treatment is widely applied to treat vehicle powertrain noise. However, passive noise control technology is often not effective in the low frequency range where the response is typically the most dominant component. With the rapid development of digital signal processing, active noise control (ANC) can be a feasible alternative. In this study, an enhanced frequency domain filtered-x least mean square (FXLMS) algorithm is proposed as the basis of an active control system for treating powertrain interior noise. Compared to the time domain algorithms, the approach can save computing time especially for long controllers filter length. Furthermore, unlike traditional ANC techniques for suppressing response, the proposed frequency domain FXLMS algorithm is targeted at tuning vehicle interior response in order to achieve a desirable sound quality. Several frequency domain algorithms are studied numerically by applying the analysis to treat vehicle interior noise recorded from an actual vehicle.

Keywords: frequency domain; filtered-x; LMS algorithm; noise control; powertrain.

Comparative study of frequency domain filtered-x LMS algorithms 37

Reference to this paper should be made as follows: Duan, J., Li, M., Lim, T.C., Lee, M-R., Vanhaaften, F.W., Cheng, M-T. and Abe, T. (2009) Comparative study of frequency domain filtered-x LMS algorithms applied to vehicle powertrain noise control, Int. J. Vehicle Noise and Vibration, Vol. 5, Nos. 1/2, pp.3652.

Biographical notes: Jie Duan received a BSc in Electronic Science and Engineering from Nanjing University, China, in 2006. He is currently pursuing a PhD in Mechanical Engineering at the University of Cincinnati. He is a Student Associate of the Institute of Noise Control Engineering. His research interests include active noise control and adaptive signal processing.

Mingfeng Li is a Research Associate in the Mechanical Engineering Department at the University of Cincinnati, and a member of ASME and SAE. His research interest includes acoustics, vibrations and active noise control. He received a BSc (1994) and MSc (1999) in Acoustics from Nanjing University and the Institute of Acoustic at Chinese Academy of Sciences, respectively. He earned an MSc (2002) and a PhD (2005) in Mechanical Engineering from the University of Alabama and the University of Cincinnati, respectively.

Teik C. Lim is a Professor and Department Head of Mechanical Engineering at the University of Cincinnati, a member of the Editorial Board for the Int. J. Vehicle Noise and Vibration and a Fellow of the ASME and SAE. His research interest is in power transmissions, structural dynamics, vibro-acoustics, sound quality and active noise control. He has published over 130 technical papers and advised 25 graduate students. He received his BSc (1985), MSc (1986) and PhD (1989) in Mechanical Engineering from the Michigan Technological University, University of Missouri-Rolla and Ohio State University, respectively.

Ming-Ran Lee is a Technical Expert in Powertrain noise, vibration and harshness (NVH) at Ford Motor Company. His expertise is in the areas of powertrain sound quality, intake/exhaust NVH, engine impulsive noise and active noise control. He has been working for Ford since 1993 in Product Development, Research and Advanced Engineering. He has a BSc (1984) in Mechanical Engineering from the National Taiwan University, MSc (1989) in Mechanical Engineering from the Pennsylvania State University and PhD (1993) in Mechanical Engineering from the Ohio State University.

F. Wayne VanHaaften is a Noise and Vibrations Engineer in Powertrain Development at Ford Motor Company. He has been with Ford since 1996 in research and advanced engineering, product development and test validation and development. His primary focus, while at Ford, has been on transmission- and engine-related NVH development along with active noise control. He received his BSc (1995) and MSc (2000) in Mechanical Engineering from the New Mexico State University and Purdue University, respectively.

Ming-Te Cheng is the Advanced Powertrain and Powertrain System NVH Technical Leader at Ford Motor Company. He graduated from the National Cheng-Kung University with a BSc in Mechanical Engineering (1985). He received his MSc and PhD specialising in dynamics and acoustics from the Iowa State University (1993). Since then, he has been working in the powertrain NVH area and joined Ford Motor Company at 1998. His primary area of focus has been the development of advanced powertrain NVH technologies in hardware, methods, facilities and analysis.

Takeshi Abe is a Henry Ford Technical Fellow in NVH at Ford Motor Company. His expertise includes Powertrain (i.e. engine, transmission, driveline and mounting system) NVH-related test/development, CAE/hybrid engineering methods, innovative test facility/advanced technology development and pass-by noise (test/simulation methods). He has a BSc and MSc in

38 J. Duan et al.

Mechanical Engineering from Keio University, an MSc in Sound and Vibration from ISVR, University of Southampton, and a PhD in Mechanical Engineering specialising in Psychoacoustics from Osaka University.

1 Introduction

Vehicle NVH (noise, vibration and harshness) characteristics have always been an important consideration in automotive design and manufacturing (Harrison, 2004). With the development of vehicle design, simply reducing the vehicle interior noise is not enough. Therefore, to satisfy the increasing customer demand for better NVH performance, automotive engineers are interested in designing vehicles with more pleasing sound quality. However, the traditional passive noise control technique is generally not effective for treating low-frequency response and also difficult to perform structural-acoustic tuning to meet certain sound quality criteria. To address these needs, an active noise control (ANC) system for tuning sound quality is developed and studied (Duan et al., 2009; Kuo and Ji, 1995; Kuo and Morgan, 1996; Kuo et al., 1997; Sorosiak et al., 2008).

In 1997, Kuo proposed a novel ANC system based on the use of frequency domain algorithm (Kuo et al., 1997). Compared to the conventional time domain technique, this frequency domain algorithm possesses several advantages. One of advantages is the significant saving in computational cost because it allows dynamic signal to be processed block by block, which enables most convolutions and correlations to be performed in frequency domain via the Fast Fourier Transform (FFT). However, this also introduces block delay into the system due to the inherent data processing technique. Fortunately, although block delay in general can be a problem, for harmonic control such as powertrain noise, it is less of a concern as explained later. Furthermore, faster convergence can be achieved by the frequency domain algorithms as reported in References (Kuo et al., 2007; Ogue et al., 1983).

In our previous study, the fast least mean square (FLMS) algorithm was successfully applied to control the powertrain noise (Duan et al., 2009). Compared to the basic version of the frequency domain LMS (FDLMS) algorithm, the FLMS algorithm is suitable for a broad range of frequencies (Ferrara, 1985). However, FLMS is much more complex and consume more computational cost. Mansour and Gray (1982) proposed an unconstrained frequency domain least mean square (UFLMS) algorithm that is capable of achieving further computational saving and faster convergence speed. In this article, the performances of the FLMS and UFLMS algorithms applied to powertrain noise are studied and their results are compared. In addition, the advantage of frequency domain algorithm, which is the ability to tune the step size for each frequency bin independently, is also investigated.

This article is organised as follows. First, the fundamental configuration of the proposed active powertrain noise control system is presented in Section 2. Second, in Section 3, the block delay caused by buffer and un-buffer is analysed, and how computational saving can be achieved is discussed. Finally, in Section 4, the two frequency domain algorithms are implemented numerically to tune the response of selected orders of powertrain noise and to match the corresponding sound spectrum with a predefined set of desired amplitudes. The performances of the different frequency domain algorithms applied in this study are also compared Section 4.


2 Frequency domain ANC controllers

2.1 Reference signal

Vehicle powertrain noise is usually dominated by harmonics that are related to the engine orders. These orders can be derived using the engine crankshaft speed data estimated from the measured raw tachometer signals. Also, based on the engine speed result, a sine wave generator can be employed to create the appropriate reference signal expressed generally as,

1

2( ) sin

Ni

isi

nfx n a

fS

(1)

where ia is the amplitude of the ith order, if is the frequency of the ith order, and sf is the sampling rate. With the information contains in the above equation, it is then possible to target the specific harmonics for either reduction or enhancement.

It is well known that the convergent rate is determined by the power response of the filtered signal that is generally frequency dependent even if the original reference signal prior to being filtered by the secondary path model is uniform. This is because the responses of the secondary path at different frequencies are usually quite different. For a specific harmonic, the convergent rate depends on the step size and the secondary path response amplitude at that frequency. To improve the performance of an ANC system, the convergence has to be nearly the same rate over a broad frequency range. In order to achieve this condition using the time domain algorithm, the amplitudes of reference sinusoids are modified according to the following equation (Kuo et al., 1999; Li et al., 2004)

^1

i

i

ah

(2)

where ^

ih is the magnitude response of ( )h z

at frequency if . However, in some applications, modifying the amplitudes of reference sinusoids is not practical. Thus, the frequency domain algorithm that allows the tuning of step size independently at each frequency bin can be employed as the alternative. This advantage will be discussed later.

2.2 Frequency domain control system configuration

The structure of simple FDLMS algorithm is illustrated in Reference (Kuo et al., 1997) and the FLMS algorithm is illustrated in Ferrara (1985). Although the configuration using FLMS algorithms shows a compromised control effect, the payoff in applying it is a simplified control structure as well as significant computational saving. Mansour and Gray (1982) proposed a new frequency domain algorithm named the UFLMS algorithm that removes two FFT operations per iteration. Since the amount of computations is limited by the digital signal processing (DSP) hardware, it is important to study the possibility of the simplified control structure. The main difference between the FLMS and the UFLMS algorithms is that the UFLMS one removes the gradient constraint procedure within the dashed rectangle as shown in Figure 1. The theoretical equations are derived in the ensuing discussion.

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Figure 1 Basic configuration of the proposed frequency domain active noise control systems

To tune the sound spectrum, the pseudo-error signal defined as the difference between the primary disturbance ( )PT n at the error microphone location and the output of secondary sound ( )y n from the secondary path can be expressed as

'( ) ( ) ( )e n e n d n (3)

where ( )e n is the residual error signal sensed by the error microphone sensor and ( )d n is the desired sound pressure that is synthesised according to a certain pre-determined vehicle interior sound quality criteria.

The reference signal has to be filtered by the estimated model of the secondary path to avoid signal path distortion. This is basically the well-known FXLMS algorithm. In this study, another filter used to compensate the block delay is added to the signal path. Thus, the filtered reference signal is computed as

' ( ) ( )* ( )* ( )x n x n h z s z

(4)

where ( )h z

is the estimated model of the secondary path, ( )s z is the block delay compensation filter and * is the linear convolution operation. For the 50% overlap


method where the trailing half of one block of filtered reference signal data overlaps with a previous block, a new block of pseudo-error signal data padded with N zeros are accumulated in the buffer separately to yield two vectors with 2N-point signal data, namely ' ( )x k and '( )e k

' ' ' ' '( ) [ ( ) ( 1) ( ) ( 1)]x k x kN N x kN x kN x kN N (5) ' ''( ) [0 0 0 ( ) ( 1)]e k e kN e kN N (6)

where k is the block index. The above two vectors, ( )x k and '( ),e k are then transformed once every N samples by a 2N-point FFT to produce a pair of frequency domain vectors described by

' ' '( ) { ( )}, '( ) { ( )}X k FFT x k E k FFT e k (7)

Hence, the update strategy for the filter weights is

' '( 1) ( ) ( ) ( )W k W k X k E kP (8)

Only the last N terms of the 2N-point '( )y k is used to drive the secondary source as given by

'( ) [ ( ) ( 1)] last terms of ( )Ty k y kN y kN N N y k (9)

where ' ( )y k is the inverse FFT operation on the output of the frequency domain adaptive filter that is expressed as

''( ) IFFT{ ( )} IFFT{ ( ) ( )}y k Y k X k W k (10)

Since ( )y k is calculated every N samples, where k is the block index, it is important to note that an un-buffer device is needed to generate the serial signal ( )y n before driving the secondary control speaker.

2.3 Variable step size

Usually, the step size is adjusted based on the corresponding power response of the filtered-x reference signal as (Wu et al., 2008)

2'm

mX

PP (11)

where P is the normalised step size, < indicates the Euclidean norm operator of the

vector, m is the frequency bin index and 'X is the filtered reference signal. However, the drawback of this method is that mP has to be calculated in every iteration. To simplify this process, the step size is modified in the following equation

2^m

mh

PP

(12)

42 J. Duan et al.

The basic idea is similar to equation (2) that normalises the amplitudes of the reference sinusoids. Accordingly, the equation (8) can be expressed as

' '( 1) ( ) ( ) ( )m m mW k W k X k E kP (13)

3 Block delay and complexity analysis

3.1 Block delay analysis

There is a tendency for frequency domain algorithms to add an unwanted delay into signal path (Morgan and Thi, 1995). That is because the algorithm processes the naturally a serial data block-by-block, which requires a buffer to structure the data block before the actual FFT operation. Similarly, the IFFT calculation has to be followed by an un-buffer process to translate the data block back into a serial one. In doing so, one block delay will be caused by either the buffer or un-buffer processes. Thus, the filter ( )D z shown in Figure 2 is added to the weights update path to compensate for the inherent block delay. The above Equation (8) that is used to update the adaptive filter weights can be rewritten as

( 1) ( ) ( )* ( ( ), ( ))W k W k h k f x k e kP

(14)

Figure 2 Comparison of the convergent rates of different frequency domain active control algorithms

Keys: solid line , simple frequency domain least mean square (FDLMS) algorithm; dashed line , FLMS algorithm; and dotted line , UFLMS algorithm.


where k is block index, is the step size, ( )h k

is the estimated impulse response of the

secondary path filter ( ),H z

f function expresses the total process based on the fast LMS algorithm by treating ( )x k and ( )e k as variables, and the operator * again represents the convolution. If the block delay is taken into consideration, the update equation becomes

( 1) ( ) ( )* ( ( 1), ( 1))W k W k h k f x k e kP

(15)

Furthermore, the above equation (11) can be re-expressed as

1 2( 1) ( ) ( )* ( ( )* ( ), ( )* ( ))W k W k h k f s k x k s k e kP

(16)

Here, 1( )s k is the estimated impulse response of the block delay caused by the buffer process in the error signal path, and 2 ( )s k is the estimated impulse response of the block delay caused by the un-buffer in the output signal path as illustrated in Figure 2. However, both delayed blocks are equal to one block delay in this analysis. Since the delay is independent of the FFT or linear convolution for harmonic signals, Equation (16) can be simplified as

( 1) ( ) '( )* ( ( ), ( ))W k W k h k f x k e kP

(17)

where 1 2'( ) ( )* ( )* ( ).h k h k s k s k

From Equation (17), it is seen that the block delays can be compensated by modifying the estimated secondary path filter. In this analysis, only two block delays are taken into account. Thus, the estimated secondary path transfer

function ( )h z

can be changed into ( ) ( ),h z s z

where ( )s z is impulse response of the 2Nsampling time delay and the symbol * once more refers to the linear convolution.

3.2 Complexity analysis

This section studied the computational cost of the FLMS and UFLMS algorithms as well as the traditional time domain algorithm using a number of real-valued multiplications. In our analysis, the frequency domain algorithm shows significant computational saving compared to the time domain algorithm. The Nth order adaptive filter, Mth order secondary path filter and N-block size are used in this analysis. For the traditional time domain algorithm, the total computations of adaptive filter output and weights update require 2N multiplications. In addition, calculation of the filtered reference signal is M multiplication. Therefore, to produce N output samples, total number of multiplications are N(2N+M). In the case of the FLMS algorithm, the adaptive filter requires four 2N-point FFT and two 2N-point IFFT. Also, the adaptive filter weights work in complex number form. Thus, the total number of multiplications per block signals for the FLMS is 2(12log 8 )N N M (Duan et al., 2009). The UFLMS algorithm saves additional two FFT operations by removing the gradient constraint.

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Table 1 Multiplication ratios of the FLMS and UFLMS relative to the conventional time domain algorithm

Multiplication ratio Block size FLMS UFLMS

32 1.0110 0.9560 64 0.8879 0.8318

128 0.7050 0.6547 256 0.4975 0.4581 512 0.3142 0.2870

Therefore, 2(8log 8 )N N M real-valued multiplications are required for N output signals, respectively. The ratios of real-valued multiplications required by the FLMS and UFLMS algorithms relative to the traditional time domain algorithm are, respectively, given by

212log 82

N MN M

(18)

28log 82

N MN M

(19)

To further examine the computational costs of the frequency and time domain algorithms, the following case study is considered. The length of secondary path filter M is 300 in the proposed system. From the results listed in Table 1, which are computed from the above two equations, it is clear that the proposed FLMS algorithm will cost less computation time than the traditional time domain algorithm when the block size is larger than 32. This is shown by those ratios that are less than the value of 1. When the block size is equal to 32, the computational costs of these two algorithms are nearly the same. In addition, the UFLMS algorithm can achieve more computational saving than the FLMS algorithm because two FFT operations can be further eliminated per iteration.

4 Computer simulation

4.1 Convergent rate

For the time domain algorithm, the step size has to be fixed for all frequencies. Hence, the selected step size may be optimised for only one harmonic, but not for other harmonics. In fact, some of the other harmonics may become divergent if the step size is not suitably selected. This is already shown in our previous study (Duan et al., 2009), and therefore will not discuss again in this article. Here, the convergence rates of several different frequency domain algorithms, that are FDLMS, FLMS and UFLMS, will be analysed below.

The UFLMS algorithm is expected to achieve faster convergence as compared to the FLMS one. This is because the UFLMS algorithm has smaller eigenvalue spread in the input autocorrelation matrix. To verify the performance of the UFLMS algorithm, numerical simulations are performed. Figure 2 shows the comparison of convergent rate of different frequency domain algorithms, namely FLMS, UFLMS and FDLMS. The


block size is chosen to be 128 for all three simulations. The primary noise consists of a single harmonic at 150 Hz mixed in with some white noise. The reference signal is the same sinusoid of unit amplitude.

As shown in Figure 2, the FDLMS algorithm converges faster as compared to the others, but its mean square error (MSE) is unacceptable. The reason is because the FDLMS possesses a circular convolution distortion as discussed above. From these three curves, the UFLMS algorithm has the greatest potential for use in active control of powertrain noise. It converges faster than the FLMS one but has less MSE compared to the FDLMS algorithm. Similar to earlier observation, the first 700 samples do not converge in all simulations due to the problem of block delay. This appears to imply that block delay affects the convergence of all frequency domain algorithms.

4.2 Vehicle powertrain noise control

The performance of the proposed active control systems using different frequency domain algorithms are studied and compared numerically using Matlab/Simulink (MATLAB/Simulink R2007b). The primary powertrain disturbances along with tachometer signal were recorded on an actual vehicle. Two cases are studies in this article. One is a steady-state case when engine speed is set to 3,500 rpm (revolution per minute). The other one involves the engine speed running up from 1,000 to 3,500 rpm. A transfer function of the secondary path, which is from the driving signal of the secondary control speaker to the error microphone location, is synthesised from a fast numerical model for vehicle interior acoustics (Sorosiak et al., 2008). The secondary path

was modelled using a 256-tap finite impulse response filter ( )h z

for all simulations as shown in Figure 3. The lengths of frequency domain adaptive filter are N = 128. Thus, 256-point FFT is applied in the corresponding simulations.

Figure 3 Frequency response function (dB reference to 1.0 3Pascal Sec / m ) of the secondary path

46 J. Duan et al.

In the first simulation, the performance difference of the two algorithms (FLMS and UFLMS) is examined with powertrain noise when the engine is running at constant speed. For demonstration purpose, the desired signal is designed to reduce the response of the third order and to enhance the response of fourth order to meet the desired amplitude. Hanning window is used in the analysis. In these simulations, the last 8,192 samples of pseudo-error signal after reaching convergence is taken as the steady-state response. The desired value of the 4th order is labelled by the asterisk symbol (*). The solid curve is the original response of the powertrain noise when the control is off. The pseudo-error signals when the control is on using the FLMS and UFLMS algorithms are shown in Figures 4(a) and (b), respectively. As shown in the plot 4(a), the reduction at the 3rd order is very obvious, that is about 18 dB. Also, the resultant response at the 4th order is enhanced to the desired level. From these results, it is concluded that the FLMS algorithm is quite promising for both response attenuation and enhancement. However, in Figure 4(b), there are some undesired harmonics when the control is turned on. Based on further investigation, these harmonics came from the output of the adaptive controller, as shown in Figure 4(c), even although the reference signal is clean. This is because the UFLMS algorithm removes the gradient constrain, which causes a larger MSE than the FLMS algorithm. If the control target is attenuation only, the undesired harmonics will be at the background noise level. In other words, there are no visible unintended harmonics when the control is on as shown in Figure 4(d).

Figure 4 Active noise control simulation results for a steady-state case using the FLMS and UFLMS algorithms; (a) The objective is to reduce the 3rd order response and enhance the 4th order response using FLMS; (b) The objective is to reduce the 3rd order response and enhance the 4th order response using UFLMS; (c) The objective is to reduce the 3rd order response and enhance the 4th order response using FLMS and UFLMS; and (d) The objective is to reduce the 3rd order response only using UFLM


Figure 4 Active noise control simulation results for a steady-state case using the FLMS and UFLMS algorithms; (a) The objective is to reduce the 3rd order response and enhance the 4th order response using FLMS; (b) The objective is to reduce the 3rd order response and enhance the 4th order response using UFLMS; (c) The objective is to reduce the 3rd order response and enhance the 4th order response using FLMS and UFLMS; and (d) The objective is to reduce the 3rd order response only using UFLM (continued)

48 J. Duan et al.

Figure 4 Active noise control simulation results for a steady-state case using the FLMS and UFLMS algorithms; (a) The objective is to reduce the 3rd order response and enhance the 4th order response using FLMS; (b) The objective is to reduce the 3rd order response and enhance the 4th order response using UFLMS; (c) The objective is to reduce the 3rd order response and enhance the 4th order response using FLMS and UFLMS; and (d) The objective is to reduce the 3rd order response only using UFLM (continued)

Since the UFLMS algorithm has larger MSE as shown in Figure 3, the poor performance of UFLMS is expected. However, if it is targeted to attenuate the primary noise only, the UFLMS algorithm is expected to have comparable performance with the FLMS algorithm.

In the traditional time domain algorithm, the amplitude of the reference signal of each order is adjusted separately to achieve similar convergent rate for different frequencies. This method is clearly not as convenient. However, as noted above, the tuning of the variable step size in the frequency domain algorithm provides an alternative for the proposed active control system. Furthermore, tuning step size for each frequency bin according to the magnitude of secondary path estimate can lead to faster overall convergence rate. In this simulation, the responses of the 3rd and 3.5th orders are chosen for reduction as much as possible. Since the magnitudes of secondary path function at the corresponding frequencies are 8 dB difference from each other, the optimal step size at these two frequencies are expected to be different. In our analysis, the block size for the frequency domain algorithm is 128 with sampling rate of 4,096. The resultant convergence time is 2 sec. The frequency domain spectrum of the last 4,096 samples of error signal is shown in Figures 5 and 6. The step size of fixed step size simulation and the normalised step size of variable step size simulation are determined experimentally for each algorithm by finding the largest stable value to achieve largest reduction for the 3rd order. Furthermore, the step size for each frequency bin of the variable step size simulation is calculated by Equation (12). For the implementation shown in Figure 5,


which is applied employing the FLMS algorithm, it can be seen that variable step size can achieve more reduction at the 3.5th order compared to the fixed step size. Since the convergence time is the same, the results imply that the convergent rate at the frequency of 3.5th order is improved. As a result, the algorithm can achieve nearly the same convergent rate at the frequencies corresponding to the 3rd and 3.5th orders. In Figure 6, the UFLMS algorithm shows similar level of performance.

The last simulation involves the engine crankshaft speed running up. The time duration for the entire process is 30 sec, during which the speed increases from 1,000 to 3,500 rpm. The goal is to reduce the response of the 3rd order. Thus, the control frequency range is from 50 to 175 Hz. Figure 7 shows the simulation results using the FLMS algorithm, while Figure 8 shows the results using the UFLMS algorithm. The black curves in Figures 7 and 8 are the original response of the measured powertrain noise. The dashed curves and dotted are the sound pressure response after the controller is switched on using fixed and variable step sizes, respectively. It is noted that the variable step size has better performance at the lower engine speed around 1,700 rpm and higher engine speed around 3,400 rpm. The corresponding frequencies are 85 and 170 Hz, respectively. That is because the magnitudes of secondary path at these frequencies, as shown in Figure 2, are smaller than the responses at the frequencies of the middle engine speeds around 2,500 rpm, that corresponds to around 125 Hz. Thus, different step sizes are need for these various frequency ranges to achieve faster convergence. This simulation also verifies that the FLMS and UFLMS algorithms have comparable performance ability to achieve the desired attenuation.

Figure 5 Active noise control simulation results for a steady-state case using the FLMS algorithm. The objective is to reduce the 3rd and 3.5th order responses

Keys: solid line , baseline noise response; dashed line , fixed step size; and dotted line , variable step size.

50 J. Duan et al.

Figure 6 Active noise control simulation results for a steady-state case using the UFLMS algorithm. The objective is to reduce the 3rd and the 3.5th order responses


Figure 7 Active control result for engine speed run-up case using FLMS algorithm. The objective is to reduce the 3rd order response



Figure 8 Active control result for engine speed run-up case using UFLMS algorithm. The objective is to reduce the 3rd order response


5 Conclusion

The active control systems equipped with two different frequency domain algorithms, namely the FLMS and UFLMS algorithms, for tuning the vehicle powertrain noise spectrum have been implemented, analysed and compared. Numerical case studies were conducted for both constant and run-up engine speed conditions. Calculation results show that the proposed active control system can provide faster overall convergent rate by tuning the step sizes of the algorithm, and also save computation cost by adopting the FFT operation. Simulation results show that when we attempt to achieve both attenuation and enhancement operations simultaneously, the FLMS algorithm performs better than the UFLMS one. However, if the objective is only to attenuate the primary noise response, both algorithms can accomplish comparable level of noise reduction. Finally, it may also be noted that the computational savings achieved by the UFLMS algorithm is more than that of the FLMS one.

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