Active Control of Vehicle Powertrain and Road

A dissertation submitted to the

Graduate School of the University of Cincinnati

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

in the School of Dynamic Systems, Mechanical Engineering

of the College of Engineering and Applied Science

June 2011

By

Jie Duan

B.S., Electronic Science and Engineering, Nanjing University, P.R. China, 2006

M.S., Mechanical Engineering, University of Cincinnati, USA, 2009

Committee chair: Dr. Teik C. Lim

Members: Dr. Ronald L. Huston

Dr. Jay H. Kim

Dr. Manish Kumar

Dr. David F. Thompson

ABSTRACT

Noise, , and harshness (NVH) has been an important factor in the development of modern motor vehicles since the 1980s. One of the challenges is the control of low- powertrain and road noise inside passenger cabin. Traditional passive control approach uses heavier and/or thicker materials for low-frequency noise reduction, which worsens the fuel efficiency of the vehicle due to the added weight. To satisfy the increasing demand for both fuel efficiency and better NVH performance, active noise control (ANC) that works better at low- frequency noise attenuation with slight increase in weight, can be a promising solution. The most common ANC system uses feedforward control approach formulated with filtered-x least mean square (FXLMS) algorithm. However, the conventional method experiences some difficulties when applying to vehicle low-frequency acoustic noise control. The focus of this dissertation is to develop a feasible ANC system with advanced control algorithms for use inside the passenger compartment of motor vehicles.

Powertrain noise that is dominated by a large number of is most perceivable when vehicle is at idle or changing speed conditions. Because of the tonal nature, it can negatively impact quality inside the passenger cabin. The slow convergence behavior of the conventional FXLMS algorithm is one of the factors that degrade the overall performance of powertrain noise control. In this dissertation, virtual secondary path algorithm is proposed to improve the convergence of the adaptive algorithm. Another challenge is to control multiple orders of powertrain response simultaneously. When the conventional FXLMS algorithm is applied, interference may occur that often results in overshoot at some adjacent orders.

Twin-FXLMS algorithm is proposed to address this problem, by splitting the adaptive filter into two sets, such that the adjacent sinusoids are spaced out farther apart. In addition, traditional

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ANC system is aimed to reduce the sound pressure level as much as possible. However, powertrain response carries some useful information about the speed and power. To achieve a better vehicle interior sound quality, active powertrain response tuning system is presented to either enhance or attenuate the powertrain order selectively.

Road noise is the dominant source when the vehicle is driving at middle or high speed. In contrast to powertrain noise, road noise is more fatiguing and irritating than having benefit. Thus, road noise must be well treated. In practice, it is difficult to obtain reference that are well correlated with the targeted noise in a broad frequency range. A combined feedforward-feedback control approach is proposed to solve this problem, which is uniquely formulated with subband

FXLMS algorithm. In addition, the computational complexity is another important consideration of the control algorithm. However, the conventional FXLMS algorithm can requite huge computational burden, especially for the multi-reference multi-channel control system. Here, time-frequency-domain FXLMS algorithm is utilized to significantly reduce the computational complexity. Furthermore, a novel channel equalization concept is proposed to overcome the channel dependent convergence behavior of the multichannel FXLMS algorithm.

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ACKNOWLEDGMENTS

I would like to thank my research advisor, Dr. Teik C. Lim, for giving me the opportunity to be introduced to the field of active control. I thank him for his helpful guidance, support, advice, encouragement, and patience during my doctoral studies. Also, my gratitude is expressed to Dr. Jay H. Kim, Dr. David F. Thompson, Dr. Manish Kumar, and Dr. Ronald L. Houston for serving as my doctoral supervisory committee members.

Among the many colleagues in the Vibro- and Sound Quality Research

Laboratory here at University of Cincinnati, I would like to express my appreciation to Dr.

Mingfeng Li for helping me with the fundamental knowledge on active noise control, and reading and revising this dissertation. Working with him has been a truly exciting experience. I also wish to thank Dr. Pravin Sondkar, Dr. Brent Budd, Dr. Tao Peng, Mr. Junyi Yang, Mr.

Guohua Sun and many others for their help in this research and most important their friendship.

This research has been partially supported by Ford Motor Company. I wish to thank Dr.

Ming-Ran Lee, Dr. Ming-Te Cheng, Dr. Takeshi Abe, and Mr. Wayne Vanhaaften for their insightful suggestions and help in various experiments. I would also like to acknowledge that financial support was provided by the School of Dynamic Systems at University of Cincinnati and Ford Motor Company.

Finally, I would like to thank my parents for their love and encouragement.

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TABLE OF CONTENTS

ABSTRACT ...... iii

ACKNOWLEDGMENTS ...... vi

TABLE OF CONTENTS ...... vii

LIST OF FIGURES ...... x

LIST OF SYMBOLS ...... xvii

Chapter 1. Introduction ...... 1

1.1 Background ...... 1

1.2 Active Noise Control ...... 3

1.3 Organization of the Dissertation ...... 5

Chapter 2. Literature Review...... 9

2.1 Journal and Conference Papers Review ...... 10

2.2 Patents Review ...... 19

2.3 Summary ...... 29

Chapter 3. Virtual Secondary Path Algorithm for Multichannel Active Control of Powertrain

Noise ...... 31

3.1 Introduction ...... 31

3.2 Convergence Analysis of MIMO FXLMS Algorithm ...... 34

3.3 Channel Equalization Algorithm for Powertrain Noise ...... 41

3.4 Numerical Simulation ...... 49

3.5 Conclusions ...... 54

Chapter 4. Twin-FXLMS Algorithm for Active Control of Transient Powertrain Noise ...... 55

4.1 Introduction ...... 55

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4.2 Basic Configuration of Twin-FXLMS Algorithm ...... 57

4.3 Nonlinearity of FXLMS Algorithm ...... 60

4.4 Numerical Simulation ...... 61

4.5 Conclusions ...... 74

Chapter 5. An Active Sound Tuning System using Computational-Efficient Algorithm for

Powertrain Response ...... 75

5.1 Introduction ...... 75

5.2 Time-Frequency-Domain Active Sound Tuning System Applied to Powertrain Response77

5.2.1 Window-Function Implementation ...... 80

5.2.2 Overlap-Save Implementation ...... 83

5.3 Computational Complexity Analysis ...... 84

5.4 Numerical Simulation ...... 87

5.5 Conclusions ...... 95

Chapter 6. A Combined Feedforward-Feedback Active Control of Road Noise ...... 97

6.1 Introduction ...... 97

6.2 Selection of Reference ...... 99

6.3 Control System Design ...... 104

6.3.1 Combined Feedforward-Feedback Controller ...... 104

6.3.2 Feedforward Control Algorithm ...... 107

6.3.3 Feedback Control Algorithm ...... 112

6.4 Numerical Simulation ...... 113

6.5 Conclusions ...... 120

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Chapter 7. A Computational-Efficient Algorithm for Multichannel Active Control of Road

Noise ...... 122

7.1 Introduction ...... 123

7.2 Multi-Reference Multi-Channel ANC System by using FXLMS Algorithm ...... 124

7.3 Multi-Reference Multi-Channel ANC System by using TF-FXLMS Algorithm ...... 128

7.4 Computational Complexity Analysis ...... 131

7.5 Numerical Simulation ...... 137

7.6 Conclusions ...... 146

Chapter 8. Channel Equalization Algorithm for Multichannel Active Control of Road Noise

...... 147

8.1 Introduction ...... 147

8.2 Multichannel ANC System for Road Noise...... 149

8.3 Channel Equalization Algorithm for Road Noise ...... 153

8.4 Numerical Simulation ...... 161

8.5 Conclusions ...... 168

Chapter 9. Conclusions and Recommendations ...... 169

9.1 Conclusions ...... 169

9.2 Recommendations for Future Studies ...... 175

BIBLIOGRAPHY ...... 177

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LIST OF FIGURES

Figure 1-1. Active noise control based on the principle of superposition. The left column plots

the primary and secondary wave, and the right column plots the resultant sound by

adding the two waves: (a) ap=1, a p =∆= 1,φ 0 ; (b) ap=1, a p =∆= 1,φ 0.5 π ; (c)

ap=1, a p =∆= 1,φ 0.9 π ; (d) ap=1, a p = 0.8, ∆=φ π ; (e) ap=1, a p =∆= 1, φ π . (Keys:

solid red line , resultant wave; dashed black line , primary wave; and

dotted blue line , secondary wave)...... 4

Figure 3-1. Basic configuration of the proposed MIMO ANC systems with virtual secondary path

algorithm for treating vehicle interi or powertrain noise...... 34

Figure 3-2. Magnitude and phase responses of estimated and EE virtual secondary path: (a)

Magnitude response ; (b) Phase response. (Keys: solid black line , estimated

secondary path; and dotted blue line , EE virtual secondary path)...... 41

Figure 3-3. Proposed 2I2O ANC system...... 46 ) Figure 3-4. Magnitude responses of four estimated secondary path transfer functions: (a) S11 and

) ) ) S21 ; (b) S12 and S22 ...... 47

Figure 3-5. Magnitude responses of four EE -CE virtual secondary path transfer functions: (a)

e e e e S11 and S21 ; (b) S21 and S22 ...... 49

Figure 3-6 Comparison of active noise control results between EE virtual secondary path

algorithm and the EE-CE virtual secondary path algorithm: (a) Error 1; (b) Error 2.

(Keys: solid black line , baseline noise response; dashed blue line , EE

virtual secondary path algorithm; and dotted red line , EE-CE virtual secondary

path)...... 52

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Figure 3-7. Comparison of active noise control results between the EE -CE virtual secondary

path algorithm with a nd without turning point: (a) Error 1; (b) Error 2. (Keys: solid

black line , baseline noise response; dashed blue line , without turning

point; and dotted red line , with turning point)...... 54

Figure 4-1. Basic configuration of the proposed active noise control system based on the twin -

FXLMS algorithm...... 57

Figure 4-2. Estimated engine speed from tachometer for the ramp -up engine speed case.

...... 62

Figure 4-3. Frequency response function of the secondary path dynamics...... 63

Figure 4-4. Comparison of control results between conventional FXLMS and twin -FXLMS

algorithms: (a) 1.5 th order; (b) 2 nd order; (c) 2.5 th order; (d) 3 rd order; (e) 3.5 th order.

(Keys: solid black line , baseline nois e response; dashed blue line ,

conventional FXLMS algorithm; and dotted red line , proposed twin -FXLMS

algorithm)...... 67

Figure 4-5. Effects of adaptive filter length on nonlinearity phenomenon by using the

conventional FXLMS algorithm: (a) 1.5 th order; (b) 2 nd order; (c) 2.5 th order; (d) 3 rd

order; (e) 3.5 th order. (Keys: solid black line , baseline noise response; dashed

blue line , L=128; and dotted red line , L=256)...... 70

Figure 4-6. Effects of adaptive filter length on nonlinearity phenomenon by using the twin -

FXLMS algorithm: (a) 1.5 th order; (b) 2 nd order; (c) 2.5 th order; (d) 3 rd order; (e) 3.5 th

order. (Keys: solid black line , baseline noi se response; dashed blue line ,

L=64; and dotted red line , L=128)...... 74

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Figure 5-1. Block diagram of the proposed AST system for vehicle powertrain response with

window-function implementation of TF -FXLMS algorithm...... 78

Figure 5-2. Block diagram of the proposed AST system for vehicle powertrain response with

overlap-save implementation of TF -FXLMS algorithm...... 79

Figure 5-3. Normalized computa tional complexity based on the number of computations, I=256.

(Keys: up-triangle marker , real multiplications; circle marker , real

additions; dashed blue line , window-function implementation; and solid black line

, overlap-save implementation)...... 86

Figure 5-4. Magnitude and phase responses of the estimated secondary path transfer function. 88

Figure 5-5. Active sound tuning simulation results for a constant engine speed of 3500 rpm case.

(Keys: solid black line , baseline noise response; dashed blue line ,

window-function implementation of TF -FXLMS; dotted red line , overlap-save

implementation of TF-FXLMS; and asterisk *, desired value)...... 89

Figure 5-6. Active sound tuning simulation results for an engine speed ramp -up case: (a) 3 rd

order response reduction. (b) 4 th order response enhancement. (Keys: solid black line

, baseline noise response; dashed blue line , window -function

implementation of TF -FXLMS algorithm; dotted red line , overlap-save

implementation of TF-FXLMS algorithm; and asterisk *, desired value)...... 92

Figure 5-7. Comparison of the performance of TF -FXLMS algorithms with or without gradient

estimate adjust factor: (a) 3 rd order response reduction; (b) 4 th order response

enhancement. (Keys: solid black line , baseline noise re sponse; dotted red line

, without gradient estimate adjust factor ; dashed-dotted green line , with

gradient estimate adjust factor; and a sterisk *, desired value)...... 95

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Figure 6-1. Principle component analysis of twenty -one signals...... 102

Figure 6-2. Multiple coherence function and potential maximum noise reduction in decibels of

best set of six accelerometers, along with sound pressure level of typical road noise.

(Keys: solid blue line , sound pressure level of typical road noise, labeled as the

left y-axis; dashed green line , multiple reference function, labeled as the right y -

axis; dotted red line , potential maximum noise reduction; and shadow area

, frequency range that has high SPL of road noise but low multiple coherence

value)...... 103

Figure 6-3. Block diagram of the proposed combined feedforward -feedback active road noise

control system...... 105

Figure 6-4. Feedforward control part of the proposed active road noise control system based on

subband FXLMS algorithm...... 108

Figure 6-5. Feedback control part of the proposed active road noise control system based on

IMC architecture with FXLMS algorithm...... 112

Figure 6-6. IRF and FRF of the measured secondary path: (a) IRF; (b) FRF...... 115

Figure 6-7. Comparison of feedforward active noise control results between the subband FXLMS

algorithm and the conventional time -domain FXLMS algorithm. (Keys: solid black line

, baseline road noise response; dashe d blue line , subband FXLMS

algorithm; and dotted red line , conventional time-domain FXLMS algorithm). 117

Figure 6-8. Comparison between feedforward road noise control result by using subband

FXLMS algorithm and potential maximum noise reduction. (Keys: solid black line ,

baseline road noise response; dashed blue line , subband FXLMS algorithm; and

solid gray line , po tential maximum noise reduction)...... 118

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Figure 6-9. Feedback active noise control result based on IMC architecture with FXLMS

algorithm. (Keys: solid black line , baseline road noise response; and dotted red

line , feedback-only control based on IMC architecture with FXLMS algorithm).

...... 119

Figure 6-10. Comparison of active noise control results between the proposed combined

feedforward-feedback control system and the feedforward only control system with

subband FXLMS algorithm. (Keys: solid black line , baseline road noise response;

dashed blue line , combined feedforward -feedback control system; and dotted red

line , feedforward -only control system with subband FXLMS algorithm)...... 120

Figure 7-1. Block diagram of the multi -reference multi-channel active road noise control system

with conventional FXLMS algorithm...... 126

Figure 7-2. Block diagram of the proposed multi -reference multi-channel active road noise

control system with TF-FXLMS algorithm...... 129

Figure 7-3. Normalized computational complexities with M= [1:8], K=[1:8], J=6, L=256, I=256:

(a) Real multiplications; (b) Real additions...... 135

Figure 7-4. Normalized computational complexities with M=2 , K=2, J=[1:10], L=256, I=256 .

(Keys: solid line with up -triangle marker , real multiplications; and solid line with

circle marker , real additions)...... 136

Figure 7-5. Normalized computational complexitie s with M=2 , K=2, J=6, L=[32, 64, 128, 256,

512, 1024, 2048], I=256 . (Keys: solid line with up-triangle marker , real

multiplications; and solid line with circle marker , real additions)...... 136

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Figure 7-6. Normalized computational complexities with M=2 , K=2, J=6, L=256 , I=[32, 64, 128,

256, 512, 1024, 2048] . (Keys: solid line with up-triangle marker , real

multiplications; and solid line with circle marker , real additions)...... 137 ) Figure 7-7. Magnitude and phase responses of the sec ondary path transfer functions: (a) S11

) ) ) ( ) and S21 ( ); (b) S22 ( ) and S12 ( )...... 139

Figure 7-8. Multiple coherence function between a set of reference signals and targeted road

noise: (a) Error 1; (b) Error 2...... 143

Figure 7-9. Comparison of active noise control results between using conventional FXLMS

algorithm and proposed TF -FXLMS algorithm with six reference acceleromet ers: (a)

Error 1; (b) Error 2. (Keys: solid black line , baseline road noise response;

dashed blue line , conventional FXLMS algorithm; and dotted red line ,

proposed TF-FXLMS algorithm)...... 144

Figure 7-10. Comparison of active control results between the conventional FXLMS algorithm

with two reference accelerometers and the TF -FXLMS algorithm with six reference

accelerometers: (a) Error 1. (b) Error (2). (Keys: solid black line , baseline road

noise response; dashed blue line , conventional FXLMS algorithm; and dotted red

line , proposed TF -FXLMS algorithm)...... 146

Figure 8-1. Block diagram of the multichannel ANC system using conventional FXLMS

algorithm for treating road noise...... 150

Figure 8-2. Block diagram of the proposed multichannel ANC system with channel equalization

algorithm for treating road noise...... 155

Figure 8-3. Magnitude spectrums of original and equalized reference signals: (a) Original

reference signals; (b) Equalized reference signals...... 160

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) Figure 8-4. Magnitude responses of four estimated secondary path transfer functions: (a) S11

) ) ) ( ) and S21 ( ); (b) S22 ( ) and S12 ( )...... 161

Figure 8-5. Magnitude responses of virtual secondary path transfer functions by using channel

) ) equalization algorithm. (Keys: S ( ) and S ( ))...... 161 22 12

Figure 8-6. Multiple coherence functions between a set of reference signals and targeted road

. (Keys: solid black line , error 1; and dotted blue line , error 2). . 163

Figure 8-7. Comparison of active noise control results between using conventional FXLMS

algorithm and proposed channel equalization algorithm: (a) Error 1; (b) Error 2. (Keys:

solid black line , baseline road noise response; dashed blue line ,

conventional FXLMS algorithm; and dotted red line , proposed channel

equalization algorithm)...... 165

Figure 8-8. Magnitude spectrums of assumed reference signals...... 166

Figure 8-9. Comparison of active noise control results between using conventional FXLMS

algorithm and proposed channel equalization algorithm by using assumed reference

signals: (a) Error 1; (b) Error 2. (Keys: solid black line , baseline road noise

response; dashe d blue line , conventional FXLMS algorithm; and dotted red line

, proposed channel equalization algorithm)...... 168

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LIST OF SYMBOLS

frequency-domain pseudo-error signal vector at index frequency-domain the k-th error signal vector at index frequency-domain power estimate vector of the jkm -th filtered reference signal index power spectrum matrix of filtered reference signal frequency response of secondary path matrix frequency response of estimated secondary path matrix virtual secondary path matrix feedforward control filter frequency-domain reference signal matrix frequency-domain filtered reference signal matrix frequency-domain reference signal vector at index frequency-domain the j-th filtered reference signal vector at index frequency-domain the jkm -th filtered reference signal vector at index gradient estimate matrix derivative operation with respect to the adaptive filter gradient estimate of TF-FXLMS algorithm at index gradient estimate of the adaptive filter of TF-FXLMS algorithm at index signal vector of the estimated primary disturbance error signal picked by error microphones

xvii

gain vector for adjusting reference signals primary powertrain noise at error microphones impulse response of secondary path matrix impulse response of estimated secondary path matrix. a small amount of white noise for MIMO system identification coefficient vector of the first adaptive filter of the twin-FXLMS algorithm

coefficient vector of the second adaptive filter of the twin-FXLMS algorithm

coefficient vector/matrix of adaptive filter reference signal equalized reference signal the first reference signal vector of the twin-FXLMS algorithm the second reference signal vector of the twin-FXLMS algorithm filtered reference signal the first filtered reference signal vector of the twin-FXLMS algorithm the first filtered reference signal vector of the twin-FXLMS algorithm the km -th filtered reference signal vector filtered by the km -th estimated secondary path for single reference control system ̂ the jkm -th filtered reference signal vector that is obtained by filtering by the estimated secondary path for multiple references ̂ control system

adjust factor of gradient estimate xviii

adjust factor of gradient estimate for adaptive filter at index decimation factor Z-transform of the error signal statistical expectation operator ∙ spectrum density of the output signal cross spectrum density of input signal and output signal cross spectrum density function of input signals and extraneous noise output spectrum predicted linear output spectrum frequency response function of the transfer path from the input signal to the output signal m-th filter bank order of estimated secondary path filter number of reference sensors number of error microphones number of secondary speakers number of subband of SFXLMS algorithm block size maximum potential noise reduction of sound pressure level at frequency

the order of the adaptive filter the order of the subband filter 1

xix

the order of the filter banks autocorrelation matrix of the filtered reference signal frequency-domain value of estimated secondary path in the -th frequency bin

frequency response of the estimated secondary path from the m-th secondary speaker to the k-th error microphone

frequency response of the secondary path from the m-th secondary speaker to the k-th error microphone

Z-transform of the secondary path Z-transform of the estimated secondary path main path of the i-th column of eigenvalue-equalized virtual secondary path of period undesired time-varying components the first adaptive filter of the twin-FXLMS algorithm the second adaptive filter of the twin-FXLMS algorithm feedback control filter frequency-domain value of the reference signal in the -th frequency bin Z-transform of the output signal amplitude of the i-th order of powertrain noise amplitude of the j-th order of powertrain noise amplitude of primary wave

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amplitude of secondary wave desired signal amplitude of the i-th order of the desired signal primary road noise estimated primary disturbance primary road noise at the k-th error microphone primary road noise at the k-th error microphone that is caused by the p-th noise source

error signal picked by the error microphones error signal picked by the i-th error microphone pseudo-error signal fundamental frequency of engine rotation in Hz frequency of the i-th order powertrain response sampling rate impulse response of the m-th filter bank ℎ mean magnitude value ∙ time index primary powertrain noise engine speed in rpm a small amount of white noise for SISO system identification the first reference signal of the twin-FXLMS algorithm that contains all integer orders of powertrain noise

the second reference signal of the twin-FXLMS algorithm contains all xxi

half orders of powertrain noise

reference signal picked by the j-th reference sensor the km -th filtered reference signal filtered by the km -th estimated secondary path for single reference control system ̂ the filtered reference signal that is obtained by filtering the -th reference signal by the estimated secondary path for multiple references ̂ control system

output signal of the control filter output signal of the feedback control filter output signal of the feedforward control filter output signal of the control filter for the m-th secondary speaker cancellation sound at the k-th error microphone impulse response of the secondary path impulse response of the estimated secondary path ̂ impulse response of the estimated secondary path from the m-th ̂ secondary source to the k-th error microphone

main path of the m-th column of ̂ reference value of the virtual secondary path reference value of the reference signal a small positive constant circular frequency of the i-th order powertrain response circular frequency of the j-th order powertrain response

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eigenvalue spread of the autocorrelation matrix the largest eigenvalues of the autocorrelation matrix the smallest eigenvalues of the autocorrelation matrix forgetting factor multiple coherence function : step size (convergence factor) block index phase difference between primary and secondary waves ∆ instantaneous squared magnitude of error signal the derivative of with respect to the adaptive filter at time n the derivative of with respect to the adaptive filter at time n gradient estimate complex conjugate operation ∙ Euclidean norm operator ‖∙‖ dot product operation .× convolution operation ∗

Superscripts

Hermitian transpose transpose operation

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Chapter 1. Introduction

1.1 Background

Vehicle noise, vibration, and harshness (NVH) issues are increasing important factors in the development of modern vehicle, as can be noticed in various manufacturers in which quiet vehicles are claimed. Since the sound quality of the vehicle is directly related to the product quality of the vehicle from customer point of view, vehicle manufacturers have devoted significant time and resource into developing more pleasant and comfortable sound environment of the vehicle cabin. Vehicle cabin noise is caused by several sources, including powertrain, road, wind and a variety of electromechanical accessories. Amongst these sources, powertrain and road noise are the most difficult noises to control. Since most powertrain and road noise are transmitted by the structure-borne transfer path, this kind of the noise falls into the low frequency range. Traditionally, noise control is achieved through passive control method. Normally, the passive control approach is more effective at middle and high frequency ranges, but is less effective for low frequency range. To reduce low frequency noise, passive control method often requires very heavy or thick materials, which significantly increases the weight of the vehicle or takes up space. In recently years, the commercial demand of lower fuel consumption and CO 2 emission has led vehicle manufacturers to make a lighter-weight, smaller-size vehicle. However, the lighter vehicle structure often transmits more low frequency noise to the vehicle cabin than the heaver structure does, which exacerbates the noise problem. To solve this conflict, active noise control (ANC) approach is considered to be a promising method, which works better at low frequency noise with slight, if any, increase in the weight of the vehicle.

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It is well known that the perceived vehicle interior sound quality is not only related to sound pressure level, but also to the temporal and spectral characteristic of the sound. Hence, the effective way to achieve better sound quality of the vehicle cabin is to tune the spectrum of the sound response rather than simply suppress the sound. However, the widely used passive control approach is not able to achieve this goal satisfactorily, even though it can obtain some degree of noise attenuation. In contrast, active control approach has a potential to reshape the sound spectrum to meet a certain vehicle interior sound quality criteria. One of the examples is the active powertrain response tuning system. The powertrain response is typically dominated by a large number of harmonics, and the amplitude and frequency of each harmonic are functionally related to the rotational speed of the engine. These characteristics of the harmonics are associated with the sound quality. The idea of powertrain response tuning is to reduce some orders of the powertrain response, while increase the other orders, such that the perceived powertrain response becomes less annoying, but the driver can still hear some audible information about engine speed to be able to control the vehicle safely. It also can be used to achieve more powerful engine feeling by increasing some associated orders of the powertrain response.

Another advantage of active control approach is that most of components required for

ANC system, such as , microphone, amplifier, and digital signal processor are already presented in the audio and communication systems of the modern vehicle. Tachometer sensor is the standard unit of the vehicle as well. These allow the ANC system to be applied with limited increase in cost. However, the calculation power of the digital signal processor is the major limitation of the application, especially for multi-reference multi-channel ANC system.

Thus, it is important to find a computational-efficient adaptive algorithm without sacrificing system performance. This is one of the subjects that will be addressed in this dissertation.

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1.2 Active Noise Control

Active noise control is to use additional secondary sources that generate secondary sound to cancel the primary disturbance. It is based on the principle of superposition. The ANC concept can be extended to active sound tuning, which cancels, attenuates, or enhances the primary disturbance based on certain sound quality requirement. The examples of two sine waves with different amplitudes and phases are shown in Figure 1-1. In this figure, the dashed black line is the primary wave, the dotted blue line is the secondary wave generated by secondary sources, and the solid red line is the resultant wave. denotes the amplitude of the primary wave, denotes the amplitude of the secondary wave, and represents the relative phase of the two ∆ waves. Note in Figure 1-1(a) that the two lines representing the primary and secondary waves are directly on top of each other. It is seen that the amplitude of the resultant sound depends on both amplitudes and relative phases of the two waves. However, only when the secondary sound has the same amplitude and out of phase as the primary disturbance, the resultant sound can be cancelled out. This same principle holds for more complicated waveforms and sound fields.

2 2

0 0

Amplitude -2 Amplitude -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time Time

(a) Enhancement, ap = 1, as =1, ∆φ = 0

2 2

0 0

Amplitude -2 Amplitude -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time Time

3

(b) Enhancement, ap = 1, as =1, ∆φ = 0.5 π

2 2

0 0

Amplitude -2 Amplitude -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time Time

(c) Attenuation, ap = 1, as =1, ∆φ = 0.9 π

2 2

0 0

Amplitude -2 Amplitude -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time Time

(d) Attenuation, ap = 1, as = 0.8 , ∆φ = π

2 2

0 0

Amplitude -2 Amplitude -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time Time

(e) Cancellation , ap = 1, as =1, ∆φ = π

Figure 1-1. Active noise control based on the principle of superposition. The left column plots the primary and secondary wave, and the right column plots the resultant sound by a dding the

two waves: (a) ap=1, a p =∆= 1,φ 0 ; (b) ap=1, a p =∆= 1,φ 0.5 π ; (c) ap=1, a p =∆= 1,φ 0.9 π ;

(d) ap=1, a p = 0.8, ∆=φ π ; (e) ap=1, a p =∆= 1, φ π . (Keys: solid red line , resultant wave; dashed black line , primary wave; and dotted blue line , secondary wave ).

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1.3 Organization of the Dissertation

After introduction in Chapter 1, the following chapters of this dissertation are organized as:

Chapter 2 provides a literature review as well as a discussion regarding limitations of previous research work related to the active control of vehicle interior noise. This chapter reviews published scientific papers, conference articles, and recent patents.

In Chapter 3, the convergence property of multichannel filtered-x least mean square

(FXLMS) algorithm is analyzed in frequency domain in order to understand the physical meaning of the convergence behavior of the FXLMS algorithm. The analysis shows that the convergence of the FXLMS algorithm is relative slow, due to the widely eigenvalue spread of the autocorrelation matrix of the filtered reference signal. The slow convergence of the FXLMS algorithm is an important factor that limits the system performance. Virtual secondary path algorithm is then proposed to overcome the frequency convergence dependent behavior of the conventional FXLMS algorithm and to reduce the variation of convergent speed inherent in multichannel cases, in order to improve the overall convergence of the control system. Hence, the overall reduction at multiple error sensors can be improved eventually.

Chapter 4 presents active powertrain noise control system uniquely formulated with a twin-FXLMS algorithm to prevent harmonic interference that often resulted in overshoot at some adjacent orders, especially at low engine speed range where the sinusoids are close together. The basic design of the twin-FXLMS algorithm is to split the adaptive filter into two sets, such that the reference sinusoidal can be spaced out farther apart. This algorithm allows controlling multiple number orders of the engine noise response simultaneously. Nonlinearity analysis of the

FXLMS algorithm is also presented in this chapter.

5

Chapter 5 proposes an active powertrain response tuning system. Traditional active noise control system is intended to reduce the sound pressure level as much as possible. However, the sound quality of the vehicle cabin has been largely ignored. The proposed active sound tuning system is designed to either enhance or attenuate the powertrain response selectively in order to achieve a better sound quality criterion. In fact, the heavy computational burden of the conventional FXLMS algorithm is one of the critical problems that plagued the practical use of this . To reduce the computational requirement of the control system, the control algorithm is updated in frequency domain. The computational complexity analysis shows that the proposed algorithm significantly reduces computational complexity in comparison with the conventional time-domain FXLMS algorithm.

Chapter 6 proposes a combined feedforward-feedback active control system for road noise reduction. Based on literature research, pure feedforward control system with conventional

FXLMS algorithm has been widely used. The feedforward control approach requires the reference signal that is well correlated with the targeted noise. However, in practice, it is difficult to obtain a good coherence value between the reference signal and targeted noise in a broad frequency range, which results in poor noise reduction at the where the coherence value is low. In this chapter, we propose to add a feedback loop to handle the noise that cannot be attenuated by feedforward control alone. In addition, subband FXLMS algorithm is proposed to replace the conventional FXLMS algorithm, in order to achieve more attenuation in a broader frequency range.

In Chapter 7, active road noise control system based on computational-efficient multi- reference multi-channel algorithm is proposed. Previous studies show that the performance of the road noise control system is limited by calculation power of the hardware. This is due to the

6 heavy computational burden of conventional FXLMS algorithm, especially for multi-reference multi-channel control system, such as road noise ANC application. The time-frequency-domain

FXLMS algorithm is proposed to address the problem. In the proposed algorithm, both control filtering and filter coefficients updating are implemented in time domain to minimize delay, however, the gradient estimate used to update the filter coefficients is calculated in frequency domain to reduce computational loads. As shown in example system, the computational load of the proposed algorithm can be reduced to 24% of that of the conventional FXLMS algorithm for real multiplications and 27% for real additions.

Chapter 8 presents the channel equalization algorithm that is used to attenuate road noise.

In practice, because of the arrangement and sensitivities of the reference sensors, secondary sources and error transducers, the magnitude level of the reference signals and the secondary paths can be very different. These differences result in minimal attenuation at some error sensor locations, which significantly degrades the overall performance of the multichannel ANC system.

The idea of channel equalization algorithm is to equalize the magnitude level of the reference signals and main secondary paths, in order to overcome the channel dependent behavior of the convergence of the FXLMS algorithm. By applying the proposed channel equalization algorithm, the attenuation at multiple error sensors can be balanced, therefore, the overall noise reduction of the control system is improved.

In Chapter 9, the general conclusions and achievements of this dissertation are summarized and also some recommendations for future research are made.

The performance of the proposed control algorithms and systems are evaluated through computer simulation by using actual data from test vehicle, and the simulation results are presented and discussed in the following chapters. Most of the control algorithms and systems

7 are validated by live vehicle experiment. However, the experimental results are required to be confidential and not shown in this dissertation.

8

Chapter 2. Literature Review

Lueg (1936) was the first one who described the active noise control system in 1936 in his U.S. patent. He gave some illustrations that are related to one-dimensional sound propagation in a duct. This patent is wildly cited in most literatures. Actually, in 1933, Lueg (1937) filed his first patent application on this topic in German (Heering, 1993). In that patent, Lueg outlined the principle: “The unwanted sound is picked up by one or more microphones, their electrical signals feed, after amplification, to one or more such that the sound wave produced is in phase opposition to the primary unwanted sound and cancels it.” (Guicking, 1990). As mentioned by Guicking in 1990 (Guicking, 1990), Lueg also filed the patent application in

France, Italy and Austria.

Although the concept of the active noise control was released in the 1930’s, the idea did not attract the attention of many researchers until decades later when rapid development of computing power, digital and hardware manufacturing made practical applications more feasible. Some of the earlier applications of ANC are controlled by transformers (Conover, 1956; Ross, 1978; Wheeler et al., 1978). Also, examples of ANC applications in enclosed spaces such as airplane and vehicle cabins are also seen. The first study on active interior noise control can be traced back to 1950s (Olson & May, 1953; Olson, 1956).

In that study, an electronic sound absorber was proposed in the effort to achieve both global and local sound attenuation. After that early work, very limited studies on the active attenuation of enclosed sound fields were reported in the next couple of decades. In a rare instance, Parkin and

Morgan (1965) worked on assisted resonance in concert halls whereby active noise control is applied to modify the reverberation characteristics of the enclosed space. Widespread attention to active control applications for enclosed sound fields did not increase drastically until 1980s,

9 which coincided with the start of rapid development of fast digital signal processor technology that made the implementation of such systems much more feasible. Many efforts were made at the treatment of interior noise in vehicles, because recent advancement in materials and the demands to increase fuel economy resulted in lighter-weight structures with the propensity for noisier or poorer sound quality interior. Passive noise control treatments are not ideal because they almost always add more weight into the vehicle design. Hence, active noise control appears to be a more feasible option, especially at low to middle frequencies, because it does not usually lead to substantial increase in weight. Today, the industrial applications of active noise control technology are limited to control the plane wave sound, such as gas turbine exhausts and engine exhausts, steady-state harmonic sound such as severs, or a small control area such as headphones.

The comprehensive theory of active control techniques can be found in some textbooks by Nelson and Elliott (1992), Hansen and Snyder (1997), and Fuller, et al. (1996). In addition, the signal process technique of active control can be found in textbooks Elliott (2001) and Kuo and Morgan (1996). In this chapter, an attempt is made to summarize the relevant literatures to the application of vehicle low-frequency noise problems.

2.1 Journal and Conference Papers Review

In 1984, Oswald (1984) introduced the first active noise control system for vehicle application, which is a single-input single-output (SISO) system for engine noise reduction.

Tachometer signal, which can precisely estimate the engine speed, was used to generate the reference signal, such that the reference signal and the engine disturbance were highly correlated.

This ANC system was reported to obtain significant noise reduction below 200 Hz.

In 1987, Nelson et al. (1987), Bullmore et al. (1987), and Elliott et al. (1987) proposed an active approach to achieve global reductions in a harmonically excited enclosed sound field. The

10 three publications include theory, computer simulation and experimental verification. However, these papers were intended to deal with steady-state sound field. In 1988, Elliott et al. (1988) applied the above theory to engine noise control. In their system, four error microphones and two control speakers were used to achieve global noise attenuation for a single engine .

About 10 to 15 dB reduction was obtained for a range of engine speeds from 3500 to 6000 revolution per minute (rpm), since the higher engine speed produce larger noise in sound pressure level. Unfortunately, the noise reduction at lower engine speed was not gained.

There has been a lot of research work on active control applied to vehicle road noise in the early 1990s, which was done jointly by Lotus Engineering and the Institute of Sound and

Vibration Research (ISVR) at the University of Southampton (Sutton et al., 1990; Sutton et al.,

1994). In their publications, the effect of delay to noise attenuation has been discussed in details.

When the delay in the controller is relatively short, the controller is able to generate the cancellation sound by using a causal filter. However, if the delay is too long, the controller is unable to predict the primary noise, which significantly degrades the performance of the control system. In the reference (Sutton et al. 1990), multiple coherence function was used to predict the performance of the active control of road noise. The reference (Sutton et al. 1990) showed that the active control of road noise was feasible, and an overall reduction of approximately 5 dB could be expected by using six accelerometers attached to the underside of vehicle body. In 1994,

Sutton and Elliott et al. (1994) proposed an active control system with six accelerometers based on their previous studies. This paper presented a design guideline for the selection of reference signals, including detection of road noise source and selection of least number of reference accelerometers. Multiple coherence function and principle component analysis were used. The selection of secondary speakers was also reported in this paper. The control system was designed

11 by using six reference accelerometers, two loudspeakers, and two microphones, where two control loudspeakers were placed in the front right and left doors, and the two microphones were positioned on the outer ear positions. The control filters used FIR filter with 128 taps that was adapted by the conventional filtered-x least mean square (FXLMS) algorithm. When the test vehicle was driven at constant speed 60 km/h over a coarse road surface, around 7 dB of A- weighting sound pressure level reduction could be achieved in the frequency range 100-200 Hz.

When the vehicle speed was increased to 80 km/h, 5-7 dB of A-weighting sound pressure level reduction was perceived in the frequency range 100-200 Hz. If both the secondary loudspeakers and microphones were installed in the passenger seat headrest, an additional 2-3 dB reduction could be measured in the higher frequency range 300-400 Hz, because the arrangement of loudspeaker and microphone had less delay than the previous one. Pure feedforward control approach was adopted in the above studies.

In 1994, Heatwole and Bernhard (1994) from Herrick Laboratories proposed a method to predict the noise reduction in sound pressure level of structure-bone road noise by using adaptive controller. This method used the frequency response functions associated with the structural acoustic system and the control filters to predict the controllable sound pressure level. By using this method, the number of reference sensors and the number of adaptive filter coefficients could be determined prior to designing the control system. One year later, Heatwole and Bernhard

(1995) investigated the relationship between the convergence rate of the FXLMS algorithm and the selection of reference sensors. Based on their analysis, the convergent speed of the FXLMS algorithm is directly related to the independence of the reference signals. In the same year,

Bernhard (1995) conducted a study on active control of road noise, which was originally intended to investigate the feasibility of active road noise control. The feasibility study showed

12 that the interior noise was highly coherent with the suspension vibration and that the prediction of interior noise by using the suspension vibration was causal. In addition, global control of road noise was feasible for the low-frequency range below 200 Hz. After the feasibility study, an active noise control system was designed to reduce the road noise. It is also showed that a large number of the adaptive filter coefficients were required to achieve a good noise reduction.

However, this paper failed to show any actual road noise control result.

In 1998, to reduce the computational burden of active road noise control, orthogonal virtual references method was proposed by Dehandschutter et al. (1998). In 1999, similar method was proposed by Akiho et al. (1999). In their methods, singular value decomposition (SVD) of the power spectrum matrix of the reference signals was used to find the number of independent noise sources, such that the comparable control performance could be achieved by using a reduced set of the virtual references instead of the full set of measured references. By using these methods, the computational complexity was reduced and the convergence rate of the FXLMS algorithm was enhanced. Simulation showed that the reduced set of the virtual references could yield the comparable reductions as the original set of references.

In 2001, Sano et al. (2001) was the first one who developed an active noise control system that is combined with an audio system. This implementation significantly reduced the cost of vehicle application of ANC system. In addition, active road noise control system often requires many accelerometers to detect the road noise, which dramatically increase the cost. To further reduce the cost of the control system, Sano et al. utilized feedback control approach, which did not require any reference sensor comparing to feedforward control approach.

Furthermore, a fixed feedback controller was designed by using the control method and was implemented by analogue circuits, to control the noise in the front seat area. However, when the

13 drumming road noise of 40 Hz was reduced in the front seat, the noise at the same frequency in the rear seat area was amplified by 3dB. To resolve this problem, a feedforward control based on

FXLMS algorithm was designed to suppress the amplified noise. The reference signal of the feedforward control was the output signal from the error microphone used in feedback control.

Experiment was conducted when the test vehicle was driven on a rough road surface at a constant speed 50 km/h, which resulted in a 10 dB reduction of drumming noise in the front seat without a noise increase in the rear seat. However, the targeted drumming noise is a very narrow band noise from 35 to 45 Hz.

In 2002, Oh et al. (2002) developed a feedforward active control system to reduce the road booming noise that has strong nonlinear characteristics. The selection of reference accelerometer was also introduced in this paper. A leaky constraint filtered-x LMS algorithm with FIR-based filter was used in the control system, which was claimed to have a faster convergent speed compared to the FXLMS algorithm. Oh et al. found that the number of FIR filter coefficients could be limited by the computational power of the hardware system. To address this problem, IIR-based filter was proposed to reduce the complexity of the control algorithm. An on-road test was conducted on the rough asphalt and turtle back road at a constant speed of 60 km/h. In the control system, six accelerometers were selected by using the multiple coherence function, two error microphones were positioned at the outer ears of two front seats, and two loudspeakers were located behind the front seat. By comparing the experimental result using different control algorithms, IIR filter showed better performance than the FIR filter did, which obtained an additional 1-2 dB reduction. However, even with using FIR filer, a reduction around 5 dB of A-weighting sound pressure level was only achieved at the booming noise from

240 to 260 Hz.

14

In 1999, Couche and Fuller (1999) proposed an active noise control system for powertrain and road noise control with the most common FXLMS algorithm. As discussion in this paper, finding proper locations for the reference sensors was a key factor to control road noise. Two methods for reference sensor placement optimization were proposed. A principle component analysis was performed to determine the number of independent noise sources and multiple coherence function between candidate reference signals and the cabin noise was used to estimate the achievable maximum noise reduction by using a given set of reference sensors. Two sets of control actuators were also investigated: the conventional loudspeakers and the piezoelectric speakers. Experimental results showed that the control of powertrain noise was feasible by using the piezoelectric speakers above 150 Hz. However, it was not possible to control the engine firing order below 150 Hz, due to the low output of these control sources at frequencies below 150 Hz. When the piezoelectric speakers were used to control road noise, no noise reduction was obtained at the error sensors below 150 Hz. In contrast, the conventional loudspeakers had better performance than the piezoelectric speakers.

In 2002, Park et al. (2002) presented a real-time active noise control system for road

noise reduction. The work was continued on the research by Couche (1999). In this study, a

survey of the interior acoustic response to acoustic excitation was conducted to provide a guide

of actuators and sensors placement. In addition, LMS based system identification was used to

evaluate the causality from reference sensors to the cabin noise that was targeted to control. The

test vehicle was parked in the laboratory and excited by using two external shakers. The

excitation signals were used as the reference signals of the control system, such that the reference

signals were well correlated with the cabin noise. Four microphones and four loudspeakers were

used to form the control system. It is seen that 3-4 dB reduction could be achieved, which was

15 much less than the maximum potential reduction around 10 dB, which was estimated by the coherence function. A continued work was reported by Park et al. (2004) in 2004, which was uniquely designed to combine two independent control systems. One control system was designed to focus on reducing low frequencies noise from 30 to 300 Hz, while the other control system with embedded actuators focused on attenuating the noise having frequencies from 250 to

500 Hz. Several on-road tests were conducted on a rough road surface. An overall reduction around 3.6 dB of A-weighting sound pressure level was reported in this paper.

As discussed above, all of these studies attempted to reduce noise level. However, reducing noise level does not always improve the perceived sound quality inside the vehicle cabin. Instead, reshaping of the time-frequency characteristics of the primary noise may yield a better solution. This is an area that has not received much attention in the past and studies have been fairly sparse.

In 1993, Kuo and Ji et al. (1993, 1993, 1994) proposed an active noise equalizer (ANE) based on adaptive noise filter to tune the spectrum of a single pure tone signal. This system could be run in four different operation modes: cancellation, attenuation, pass and enhancement. In order to use LMS adaptive algorithm, pseudo error signal was used to replace the residual acoustic noise. In 1996, this ANE system was extended to control the broadband noise based on the FIR adaptive filter (Kuo & Yang, 1996). Unlike the ANE system for controlling harmonically noise, which used a desired signal, a shaping filter was used to shape the residual acoustic noise for controlling broadband noise. In 2004, a multichannel ANE system was proposed by Diego et al. (Diego et al., 2004; Gonzalez et al., 2006). In 2006, Rees and Elliott

(2006) developed a phase scheduled command filtered-x least mean square (FXLMS) approach to improve the stability of the algorithm. This algorithm was designed to treat pure tone noise

16 only. Later on, a research team led by Kuo (Kuo et al., 2007, 2008) used filtered-error least mean square and several frequency-domain algorithms to reshape the noise spectrum. However, these studies are limited to very simple cases using only idealized response, signals and secondary path representation. Also, the desired spectrum shape is only hypothetical and actually not related to any sound quality metric. Since the actual dynamics are more complex, more detailed studies are needed to quantify the performance under practical condition.

In early 2000s, the active sound quality control started to appear in vehicle application. In

2002, Scheuren et al. (2002) proposed an active sound design system for improving engine sound quality. The general concept was to alter the sound spectrum rather than simply suppress it. In their studies, active sound design system was aimed at reducing the predominant engine orders to reduce the booming noise, while at the same time increasing higher orders at some rpm to deliver a higher sound quality of engine. One of the examples involved producing more sports feeling for a 4 cylinders car, such as generating virtual sporty sound as 6 cylinders, 8 cylinders or 12 cylinders . However, the global control for higher frequencies that required a higher number of secondary speakers might not be practical at real cars.

In 2006, Carme et al. (2006) built an active noise profiling system to fulfill the pre- defined sound quality requirements for vehicle cabin. This system consisted of two loudspeakers placed on the roof-liner, two error microphone sensors placed on the seat-headrest and a specific hardware electronic. This control system, which based on command FXLMS algorithm, was intended to either enhance or reduce the engine orders in the vehicle cabin according to predefined sound pressure level for each order from 800 rpm to 6000 rpm. In the lab experiment, six engine orders were designed to control simultaneously. However, when the engine speed was

17 lower than 1600 rpm, some engine orders could not be controlled effectively. Furthermore, the acceleration of the test engine was not reported in this paper.

In 2008, Kobayashi et al. (2008) proposed an active sound control system based adaptive notch filter with FXLMS algorithm. When the engine speed was smaller than 2500 rpm, the control system was designed to reduce the engine booming noise as the conventional active noise control system, while the engine speed was higher than 2500 rpm, the control system was designed to adjust the acceleration sound. To achieve both quietness while cruising and good acceleration sound while accelerating simultaneously, the sound pressure level must be adjusted in accordance with pedal operation. The gain setting process for each order was also discussed in this paper. As shown in the example results, noise reduction of approximately 10 dB was achieved at the engine booming noise. In addition, when the vehicle was accelerating with full open throttle, the active sound system could either increase the sound pressure level or effectively reduce peaks and dips of the engine order response, which allowed hearing linear acceleration sound at the passenger ear positions.

In 2009, Sorosiak et al. built a numerical acoustic cavity model for vehicle cabin. The model incorporated multiple acoustic cavities joined by flexible panels to represent adjacent vehicle compartment, which was validated by comparing to the experimental results. This model could be used to find the optimal locations of the secondary speakers and error microphones that maximize the effectiveness of the ANC system. Based on this model, Li et al. (2009) proposed an active sound quality control system based on FXLMS algorithm to tune the powertrain response rather than simply suppress it. The control system had been successfully applied to tuning both steady-state and transient powertrain response. In 2009, Duan et al. (2009; 2009) improved the above active sound quality control system by using a frequency-domain FXLMS

18 algorithm, such that the computational complexity could be significantly reduced. However, the control systems were only demonstrated to control one reduced and one enhanced orders.

Further information can be found in three review papers on active control of automotive applications (Sano et al., 2002; Mackay et al., 2004; Elliott, 2008).

2.2 Patents Review

Besides the above-mentioned research studies that have been reported publicly, numerous patents related to this field have been granted as well in the past decades. Part of this review work has been published on Journal of Recent Patents on Mechanical Engineering.

In 1997, a patent was granted to Sano et al. in 1997 (Sano et al., 1997). The claim of this patent was motivated by the following reason. In the conventional active noise control system, the adaptive filter is used not only for the feedforward controller but also for the feedback one.

Accordingly, the performance of the conventional active noise control system suffers from processing an extremely large amount of digital operations required to perform the calculation of the adaptive algorithm. This obviously results in very large processing time. In the invention by

Sano et al., the principle objective was to provide an active noise control method and a corresponding system that could be operated with less processing time. This invention decreased the computational load on a signal processing unit, and made it possible to perform sufficient noise suppressive control. Their invention was based on an internal model controller (IMC) concept. However, instead of using an adaptive filter as the IMC controller with the least mean square (LMS) algorithm, their proposed system constructed the IMC controller from the plant model with a certain degree of uncertainty in the plant representation. The plant model represented the transfer function from the input of the control speaker to the output of the error sensor. Thus, the amount of calculation needed was greatly reduced. Furthermore, the

19 development of this invention accounted for potential variations in the transfer function of the plant as factors within the environment changes, such as increase or decrease in the number of passengers, states of opening or closing of windows, and changes in performances of the microphone and speaker. The degree of variation of the plant transfer function was approximated and set by previously estimating the additive perturbation over a range of operating frequency.

The application also assumed the estimation was performed before final packaging was done.

In 1998, Tsuji et al. (1998), proposed an active vibration or noise control system for applications. The invention was reported to be capable of quickly responding to changes in the traveling conditions, such as vehicle speed, road surface and riding state, of the vehicle by resetting the designed adaptive filter. Also, there were no strict requirements on the performance and size of the identification system. The identification system referred to the adaptive system for designing the adaptive filter. Hence, the overall cost could be reduced by using the identification system with lower performance and smaller in size. In one example of this implementation, two sets of memory chips were employed. The first memory chip stored the initial coefficient values of the adaptive filter, and the second memory chip stored the coefficient values modified by a filter coefficient-updating algorithm. Both cases corresponded to the predetermined traveling conditions. When a change occurred in the traveling condition was detected, the control system would read the initial filter coefficient values corresponding to the traveling condition detected immediately before and after the occurrence of the change from the first and second memory chips. Then, the controller would utilize those two different sets of coefficient values to generate the control signal needed to suppress the targeted vibration or noise response. The specific set of filter coefficient values that resulted in a smaller residual noise would be selected and applied to the adaptive filter. Furthermore, the coefficient values of the

20 adaptive filter could be updated by a suitable algorithm such as the least mean square (LMS) algorithm.

In early 2002, a patent was granted to Billoud (2002) that aimed to reduce the noise response within a closed space caused by a vibratory disturbance source. The proposed active noise control system was stated to be highly efficient, which comprised of four major components; a reference sensor for deriving the reference signal that was correlated to the disturbance source, an error sensor for sensing the residual sound pressure level, a speaker for generating the cancelling sound wave, and a controller for providing the control signal to the control speaker. The proposed ANC invention could be used in a vehicle passenger compartment to reduce the unwanted interior noise due to engine excitation. In this application, the speaker assemblies were mounted to a trim that can be located under the seats, on the window platform, or in the front of the rear seats. In this design, as claimed by the patent, the speaker was inversely mounted in an enclosure to provide a more efficient noise cancellation effect. Each enclosure assembly and installation was intentionally designed to produce a planar wave guide and at the same time constrain the canceling wave form. Hence, the generated control sound would come from the enclosure in a direction that was mostly parallel to the trim’s surface.

In 2005, Astorino (2005) proposed an active noise control system with a horn sound feature. Basically, this invention was an active noise control system that also provided a horn sound function even when the vehicle was not running. The invention took advantage of basic capabilities of the ANC system and provided additional features or functions for the consumer, such as horn sound, in order to minimize the overall cost of vehicle components. One example system designed according to this invention included a speaker producing horn sound with and without the vehicle running. On one hand, the controller operated in a normal mode while the

21 vehicle was running, and would drive the speaker to achieve the desired noise cancellation.

When the horn switch activation was detected, the controller interrupted the active noise cancellation temporarily such that the speaker can produce the intended horn sound. On the other hand, the controller operated in a power saving mode when the vehicle is not running. However, the controller could be woken up from the power saving mode through a horn switch activation device. The controller then drove the speaker to emit a horn sound and returns to the power saving mode after the horn switch activation ceases.

In the same year later in 2005, a relevant active control patent was granted to Vaishya

(2005). This patent was an active engine noise control system that relies upon a single sensor, which was rather simple and inexpensive. The control system was claimed to serve as an after- market product that can be easily installed on a vehicle. In their one example system shown, a single pressure sensor was used to estimate both the engine speed and throttle open position that were needed to compute the precise phase information of the primary engine noise excitation.

Specifically, the pressure sensor signal had a frequency component that can be used to estimate the engine speed in RPM and a static or DC component to estimate the throttle valve position.

The controller then used the estimated engine speed and throttle position to generate the appropriate control signal for suppressing engine noise.

In 2006, Nakamura et al., (2006) proposed an active noise control system for reducing low frequency vehicle noise without producing an abnormal or distortional noise from the speaker. The proposed active noise control system included four components, namely a microphone to sense the unwanted noise, a signal generator for producing a controlling signal, a limiting amplifier and a speaker for generating the cancellation sound. Obviously, the proposed system was a feedback control system. Hence, according to the classic control theory, the

22 resultant noise level at the microphone location would depend on the open loop transfer function of the system. The signal generator was used to adjust the open-loop transfer function within the frequency range that encompasses the low frequency noise response of interest. The function of the limiting amplifier was used to enhance the output from the signal generator dynamically in accordance with the amplitude of the signal fed into the speaker. Furthermore, the limiting amplifier had a predetermined threshold or maximum value associated with its output signals.

Hence, the amplitude of the output signal from the limiting amplifier would not exceed the predetermined threshold level when in use. In this way, the system was capable of controlling the noise inside the cabin without causing distortional sound from the speaker.

Also in 2006, a patent was issued to Daly (2006). In this patent, Daly proposed an active control system for controlling and modifying the acoustic noise response created by a vehicle door closing. The system used a two-stage switch corresponding to two different positions of the door. Each switch would deliver a signal to the processor when the door is at the corresponding position. By measuring the elapse time between the two measured signals from the switches, the proposed processor could determine the door velocity as it closes. The door velocity would also be linked to a sound amplitude control unit such that the system can send a control signal through a speaker near the closing door event. In one example, it was proposed that each door on the vehicle has its own switch. This would enable the processor to select a proper door speaker to output the controlling sound. The spectrum of the control signal was determined based on the desired characteristics of the door closing sound. For example, the frequency spectrum of the control signal might be derived to treat higher frequency content of the door closing response and at the same time treat the reverberation sound generated when the vehicle door closes. The resulting response should then be a more pleasing sound that has a lower frequency spectrum

23 with no reverberation. In another reported demonstration, the invented system worked in the feedback way. In that implementation, the resulting acoustic noise was compared with a template reflecting the desired response. Based on that comparison, the proposed system adjusted the control signal to minimize any deviation between the resulting acoustic noise and the template to achieve the desired response.

To accomplish a similar function implemented in the earlier mentioned patent by

Nakamura (2006), Onishi et al. (2008) in 2008, proposed the active noise control system that is capable of avoiding abnormal acoustic noise resulting from the divergence of the adaptive controller. Furthermore, the patent claimed that the system could also stop the potential abnormal acoustic noise when it detected an abnormality in the output signal from the error microphone.

Their control system also included a monitor for tracking the signals used as the indication of abnormal level. For example, when the signals sensed by the microphone and supplied to the adaptive controller had the same positive or negative sign for a predetermined duration, the monitor would halt the controlling signal being output from the adaptive controller. One possible sign of abnormality was the ratio between the duration of the positive sign of output signals from the microphone and that of the negative sign being greater than or equal to a predetermined value.

This feature allowed the system to sense an abnormal level that was indicative of the output signal from the microphone having a mean value or DC offset, and thereby stopped the cancellation acoustic noise from being generated. According to the patent, there were also some other means for measuring the abnormal level, including, monitoring the abnormal increase in the temperature of speaker voice coil , the change in the speaker magnetic flux density , the level of noise canceling signal delivered to the amplifier, and the vibration of the speaker diaphragm.

24

Also, in 2008, another ANC related patent was granted to Inoue et al. (2008). Having acknowledging the fact that the convolution operation in most active control algorithms was time consuming, the inventors proposed a novel approach for vehicle noise control with the capability of reducing the calculation requirement greatly. The proposed system used the adaptive notch filter with a simple way to produce the filtered reference signals. The filter coefficients of the adaptive notch filters were updated to minimize the error signal from an error sensor placed inside the vehicle compartment using the least mean square (LMS) algorithm. The novelty of this invention was the means to calculate the filtered reference signal that was required for the filter coefficients updating task. In the traditional way, the filtered reference signal was obtained through the convolution of the reference signal with the finite impulse response (FIR) of the secondary path (Kuo & Morgan, 1996), which is the transfer function between the speaker and error sensor. This procedure was computationally demanding and time consuming. When there are many reference signals or the order of the estimated model of the signal transfer characteristic (i.e. estimated secondary path filter) is very high, the calculation burden increases dramatically. Here, the inventors proposed a very simple way to calculate the filtered reference signals. The first filtered reference signal was produced by subtracting the product of a sine corrective value and the reference signal from the product of a cosine corrective value and the reference cosine wave signal. The sine and the cosine corrective values were based on the sine and cosine values of the phase characteristics of the secondary path with respect to the frequencies of the reference signals. Similarly, the second filtered reference signal was produced by adding the product of a sine corrective value and the reference cosine wave signal and the product of the cosine corrective value and the reference sine wave signal. Hence, the computational process only required four multiplications and two additions for generating the

25 two filtered reference signals each time the filter coefficients of the two adaptive notch filters were updated. Obviously, the amount of calculations for obtaining the two filtered reference signals is much smaller than the case where the actual FIR filters are used. This feature allows the production control system to be relatively inexpensive. Furthermore, according to the claims in the patent, since the two filtered reference signals were calculated as optimally corrected signals directly from the reference signals, the contours of constant square error curves became concentric circles. This makes the system capable of controlling the unwanted noise much quicker.

Additionally, in the invented system discussed above, to determine the sine and cosine corrective values, the amplitude response and phase characteristics of a predetermined frequency in the secondary path was measured. However, the measured amplitude value had to be adjusted to obtain a corrected gain through a series of calculations with no change in the phase.

Subsequently, the cosine corrective value and the sine corrective value would be determined based on the corrected gain and measured phase information. The obtained cosine and sine corrective values were then stored in a storage device along with the frequencies of the reference signals for use in calculating the filtered reference signals later on. Since the cosine and sine corrective values were based on the measured characteristics of the secondary path, the filtered reference signals or the filter coefficients of the adaptive notch filters could be calculated with a high degree of accuracy. Therefore, the proposed system was expected to yield a much better performance. Furthermore, to increase the convergent speed of the algorithm and in turn result in better responsiveness of the system, the step size of the updating algorithm of the adaptive notch filters could be optimally adjusted.

26

In 2010, two more patents were granted to same group of researchers led by Sakamoto and Inoue (Sakamoto et al., 2010; Inoue et al., 2010). These two patents were related to that patent in 2008 discussed above. They both used adaptive notch filter to control engine noise.

However, one patent (Sakamoto et al., 2010) focused on the generation of reference signal. A series of sinusoidal waveform data were stored in the memory and output at different sampling time sequentially to create reference signal with the same frequency as the engine noise for the adaptive control algorithm. Conventionally, the system outputted the waveform data sequentially at increment 1. To make the frequency of reference signal the same as the engine noise, a variable sampling rate was used. The instantaneous sampling rate was calculated based on the engine speed and the number of the waveform data (N). In that design, the division number, defined as the number of sampling point within one period of sine wave, was the same as N which has to be a positive integer number. Hence, as, the control range (the range between maximum and minimum frequencies/periods of reference signal) would be limited by the processing capacity (CPU speed and calculation burden) and noise canceling capability that was characterized by the longest sampling period (or the lowest sampling rate). The relation between the period of reference sine wave signal and the period of the required sampling rate could be characterized by a straight line. In the invented system, the division number was a real number

(less than N). Hence, the system outputted the waveform data with an increment larger than 1.

Furthermore, the system used multiple division numbers for different range. In this way, the control range could be widened, and less CPU processing burden is required. As a result, the cost of system would be reduced.

The second related patent (Inoue et al., 2010) focused on preventing the ANC system from generating abnormal sound when the error microphone is covered unintentional or

27 malfunction. Unlike traditional adaptive control system, two filter coefficient updating algorithms were used. These two algorithms worked under different operating conditions. A switch decision unit chose corresponding updating algorithm according to the comparison result between the filter coefficient values and the predetermined threshold values. The threshold values might vary with the frequency, and therefore a table of the candidate values was used.

When the system sensed normal condition, the first updating algorithm would be used to adapt the filter coefficients as usual. Otherwise, the second updating algorithm, that was actually a forgetting process (current filter coefficient value will set to the multiplication between the old value and a forgetting factor), would be chosen. In this way, the potential abnormal sound due to the malfunction or unintentional cover of the microphone would be avoided.

Also, in 2010, a patent was issued to McCain et al. (2010) that was applying ANC to de-activation engine system. Cylinder de-activation engine system was designed to use engine cylinders selectively, such that it can improve fuel efficiency. For instance, when power requirements are small, only part of engine cylinders are active. Unfortunately, when fewer cylinders are utilized, the engine even generates more noise, because engine mounts, which is designed based on the full-cylinder operating mode, are less effective in the partial-cylinder operating mode. Thus, the ANC system can play a greater role. When the engine operating mode changes, for example, from a three cylinders utilization mode to a six cylinders one, there may be a ‘pop’ noise that occurs during the transient period. This patent focused on cancelling the unwanted ‘pop’ noise. One of the examples was the operating mode change from three cylinders to six cylinders. The targeted engine order to be controlled was changed from 1.5th order to 3rd order, which were the firing orders that dominate the engine noise. In the past, the ANC system was either turned completely off resulting in no cancellation signal or changed immediately to a

28

3rd order cancellation signal from a 1.5th order one yielding higher net amplitude than the primary engine noise. To overcome these effects, the authors uniquely proposed two steps to smoothen the transition process, namely ‘delay’ and ‘wait’. When the transition occurred, the

1.5th order cancellation signal was extended for a short period of time (delay time) to reduce the

‘pop’ noise, and then reduced smoothly for another short period of time (waiting time) to prevent generating undesired high amplitude of cancellation signal, before a change to 3rd order cancellation signal was imposed. Furthermore, the author suggested the preferred delay times and waiting time for six different cylinder mode transitions.

2.3 Summary

In spite of extensive active noise control studies on vehicle application in the last three decades, it practical application is still not widespread. One of the reasons is due to most studies as discussed above used the conventional FXLMS algorithm. In fact, the conventional FXLMS algorithm has many disadvantages. Firstly, the FXLMS algorithm suffers from slow convergence in broad frequency range, especially when the length of adaptive filter is long. Hence, it works only for the steady-state or slow varying narrowband response. Secondly, the inherent nonlinearity of the FXLMS algorithm, which results in a certain distortion of the filter output signal and significantly reduces the overall performance of the system. It also prevents the ANC system in use of controlling multiple orders of powertrain response simultaneously, when the engine speed changes fast. Thirdly, the computational load required by the FXLMS algorithm is very high, which plagues the use of inexpensive applications. This problem is more severe when the control system requires multiple reference sensor or/and multiple channels. Last but not least, the channel dependent convergence behavior of the multichannel FXLMS algorithm may cause minimal attenuation at some of the error sensors, which can lead to a significant degradation in

29 the overall performance of the control system. These issues will be addressed in the following chapters.

Besides the control algorithm, most previous research used pure feedforward or feedback control systems to road noise control. The feedforward control approach requires the reference signal is well correlated with the targeted noise. However, in practice, it is difficult to obtain a good reference signal in the broad frequency range of interest. In addition, as seen in the reference (Sano et al., 2001), feedback control approach only works for attenuating a very narrow band noise, because the frequency range of attenuation is limited by the delay of the secondary path. In this dissertation, a combined feedforward-feedback ANC system will be proposed to address this problem.

30

Chapter 3. Virtual Secondary Path Algorithm for Multichannel Active

Control of Powertrain Noise

In this chapter, an enhanced multiple-input multiple-output (MIMO) FXLMS algorithm using improved virtual secondary path is proposed as the basis for an ANC system for treating vehicle powertrain noise. This new algorithm is developed to overcome the limitation caused by the frequency dependent property of the standard FXLMS algorithm, and to reduce the variation of convergent speed inherent in multichannel cases, in order to improve the overall performance of the control system. In this study, the convergence property of the proposed algorithm is analyzed in frequency domain in order to yield a better understanding of the physical meaning of the virtual secondary path. In practice, because of the arrangement and sensitivities of the actuators (speakers), transducers (microphones) and physical environment, the magnitude response of the main secondary paths can be very different from each other. This difference will cause difficulty in the overall convergence of the algorithm, which will result in minimal attenuation at some of the channels. The proposed channel equalized (CE) virtual secondary path algorithm is designed to tackle this difficulty by equalizing the mean magnitude level of the main secondary paths and by adjusting other secondary paths correspondingly to keep the coupling effects amongst the control channels unchanged. The performance of the proposed algorithm is validated by analyzing a two-input two-output active powertrain noise control system.

3.1 Introduction

Active noise control system is one of the countermeasures to tackle powertrain noise, which has also been described in References (Carme et al., 2006; Couche & Fuller, 1999; Duan,

2009, 2011; Elliott et al., 1987; Hasegawa et al., 1992; Inoue et al., 2004; Mackay & Kenchingto,

31

2004; Oliveira et al., 2009; Oswald, 1984; Scheuren et al., 2002; Sorosiak et al., 2009). There are two primary characteristics of vehicle powertrain noise that are critical to the current analysis: (i) powertrain noise is typically dominated by a large number of harmonics; and (ii) the amplitude and frequency of each harmonic are functionally related to the rotational speed of the engine.

Therefore, the frequency range of interests can span a broad range. It has been found that the properties of the convergent speed are affected by the eigenvalue spread of the autocorrelation matrix of the filtered reference signals (Elliott, 2001; Kuo, 1996). In general, the eigenvalues of the autocorrelation matrix of the filtered reference signals are varying throughout the entire frequency range, which leads to frequency dependent behavior of the convergent speed. Hence, each frequency will have its own optimal step size (convergent factor). To maintain the stability, the step size of the control algorithm has to be chosen based on the frequency that has the smallest optimal step size. Otherwise, the system will become unstable at that frequency. Also, this tends to slow down the convergent speed for other frequencies and degrade the overall performance of the ANC system, because the step size chosen in that way is only optimal for that particular frequency and not optimal for other frequency components.

Several variations of the FXLMS algorithm have been developed to overcome the performance degradation. A precondition least mean square (LMS) algorithm was developed by

Elliott et al. (2000, 2000, 2001), which adds an additional filter to flatten the magnitude response of secondary path. To improve the convergent speed of the standard FXLMS algorithm, Chen et al. (1993, 1994, 1995, 1996, 2004) proposed that the determinant of autocorrelation matrix should be as flat as possible. They also developed three approaches to achieve that, including adjusting the positions of secondary sources and error sensors, increasing the number of secondary sources, and adding an inverse filter of the secondary path. In addition, Kuo et al.

32

(1999) suggested that the amplitude of the reference signals should be chosen to be inversely proportional to the magnitude response of the secondary path at the corresponding frequency.

Duan et al. (2009) further treated the problem of slow convergent speed in powertrain noise control using a frequency-domain fast least mean square (FLMS) algorithm. In spite of numerous studies done in the past as described above, most of the approaches either increase the computational load or increase the complexity of the control algorithm. Recently, a new approach that is eigenvalue equalization filterer-x least mean square (EE-FXLMS) algorithm has been developed by Sommerfeldt et al. (2008), Thomas et al. (2008, 2008) and Lovstedt et al.

(2008). The technique is designed to overcome the frequency dependent performance problem.

However, this method may experience difficulties when applied to multichannel active noise control system, especially when there are imbalances in the secondary path responses.

In this study, the convergence property of the MIMO FXLMS algorithm for multichannel

ANC system is analyzed in frequency domain. This allows the convergence characteristic to be evaluated in each frequency component, which enables a better understanding of how the convergence property is affected by the step size, power spectra of the filtered reference signal and multiple secondary paths. Furthermore, based on frequency analysis, the current study proposes an improved virtual secondary path algorithm to normalize the convergent speed amongst the control channels such that the overall performance of the ANC system can be improved. Finally, the proposed approach is verified numerically by employing a 2×2 (two error sensors and two secondary sources) ANC system to control actual vehicle powertrain noise.

This chapter is organized as follows. Firstly, the MIMO FXLMS algorithm is analyzed in frequency domain in the next section. Section 3.2 also presents the concept of virtual secondary path. Secondly, the virtual secondary path algorithm with channel equalization for application to

33 multichannel ANC control system is proposed in Section 3.3. Then in the subsequent section, the proposed new implementation is applied numerically to control the vehicle powertrain noise.

3.2 Convergence Analysis of MIMO FXLMS Algorithm

Multiple noise Tachometer Signal

S

Speed Calculator x(n) y(n) K & Sine Wave M Generator

MK channels

v(n) - + Noise Generator LMS 2 MK K LMS 1 e(n)

Equalization

Figure 3-1. Basic configuration of the proposed MIMO ANC systems with virtual secondary path

algorithm for treating vehicle interior powertrain noise.

A flowchart of the proposed MIMO ANC system for treating vehicle powertrain noise is

illustrated in Figure 3-1. For this setup, a single reference signal is used for all the (assuming is the number of secondary speakers) adaptive filters. The frequencies of the 34 reference signal can be calculated by using the engine speed data estimated from the measured raw tachometer signals. Then, assuming is a superposition of a series of pure sine waves, it can be expressed as

(3.1) = sin 2 ⁄ where is the amplitude of the i-th engine order, is the frequency of the i-th engine order, and is the sampling rate. With the information contains in the above equation, it is then possible to target the specific harmonics for attenuation.

The objective of controller design is to minimize the sum of the squares of residual noises

measured by error microphones. Accordingly, the adaptive filter coefficients updating process is given by

(3.2) + 1 = + where , is the number of adaptive filter; each ≡ ⋯ ≡ ( ) represents an adaptive filter with , , ⋯ , = 1, 2, ⋯ , coefficients; is time index; is step size that controls the convergent speed of the algorithm; is the residual noises at error sensors; and finally is the = ⋯ filtered reference signal matrix expressed as

⋯ (3.3) ⋯ = ⋮ ⋯ × Each element represents the filtered ≡ , − 1 ⋯ − + 1 reference signal vector. The filtered reference signal is defined as the convolution of the input reference signal , and represents the impulse response of the secondary path from

35 the m-th secondary source to the k-th error sensor. In practice, is unknown. However, it can be identified by injecting a small amount of white noise through the control input speakers and measuring the responses at the error microphones. Then, the secondary paths are modeled

by a series of I-th order finite impulse response (FIR) filters, whose coefficients are updated using the LMS algorithm. This is fundamentally an off-line system identification process. The estimated secondary path is denoted by . Thus, the filtered reference signal is calculated from ̂ , where denotes the convolution process, = ∗ ̂ ∗

̂ ̂ ⋯ ̂ (3.4) ̂ ̂ ⋯ ̂ = ∗ = ∗ ⋮ ̂ ̂ ⋯ ̂ × where is the secondary path transfer function matrix. Furthermore, in Figure 3-1 represents the primary powertrain noise. The convergent speed of the algorithm is determined by the eigenvalue spread of the autocorrelation matrix of the filtered reference signal and can be written as [3]

(3.5) = where represents the statistical expectation operator. Accordingly, the eigenvalue spread, ∙ denoted by , is defined as the ratio between the largest and smallest eigenvalues of the autocorrelation matrix , (3.6) = The convergent speed can also be analyzed in frequency domain. The frequency-domain filtered reference signal matrix can be defined by

36

, , ⋯ , (3.7) , , ⋯ , , = ⋮ , , ⋯ , × where . Here, is the reference signal of in frequency , = , , domain, and is the frequency response of the estimated secondary path transfer function from the m-th secondary source to the k-th error sensor. Equation (3.7) can be expanded as

follows:

⋯ (3.8) , = ⋯ ⋮ ⋯ ×

⋯ 0 ⋯ 0 (3.9) 0 ⋯ 0 , = ⋯ = ⋮ ⋮ ⋯ × 0 0 ⋯ × It may be noted that has been omitted from Equations (3.8) and (3.9) for brevity and , therefore is implied. Equation (3.5) can also be written in frequency domain as follows:

(3.10) = , , = where the operation indicates the Hermitian transpose. The power spectrum matrix is ∙ a matrix at each frequency . Therefore, the convergence behavior at the frequency × can be clearly observed. The convergence properties are still determined by the eigenvalue spread of at the corresponding frequencies. Accordingly, Equation (3.6) can be expressed in the frequency domain as

(3.11) =

37 where and are the largest and smallest eigenvalues at the frequency , respectively. It is well known that a smaller can achieve a faster convergent speed. Observing the convergent speed over the whole frequency range, the eigenvalue spread can be revised into the following form:

(3.12) = where and represent the maximum and minimum eigenvalues over the entire frequency range. Hence, to study the convergence property of the overall control

algorithm, we have to calculate all eigenvalues of autocorrelation matrix , which can be computationally demanding especially when the length of the control filter is long. Therefore, instead of calculating the eigenvalues of the matrix , the convergent speed can be evaluated approximately by computing the determinant spread of over the frequency range using the following equation (Chen et al., 1995):

(3.13) || = || where is the determinant of matrix , which can be expressed as || (3.14) || = || ∙ Since the reference signal in this study is generated by a sine wave generator, the reference signal can be set with equal amplitude at all harmonics over the entire frequency range. Hence, the power spectrum of the reference signal at each frequency will be the same. Thus, Equation (3.13) can be further derived as follows:

(3.15) =

38

To achieve faster convergence, should be as small as possible. This implies that should be as flat as possible over the entire frequency range. By examining the eigenvalue equalization filtered-x least mean square (EE-FXLMS) algorithm (Lovstedt et al., 2008;

Sommerfeldt et al., 2008; Thomas et al., 2008, 2008), it is easily seen that the EE-FXLMS algorithm is a way to flatten the magnitude response of for every secondary paths in the system by modifying the magnitude in frequency domain. To maintain the stability, the phase is kept unchanged.

The proposed procedure is outlined as follows (Lovstedt et al., 2008; Sommerfeldt et al.,

2008; Thomas et al., 2008, 2008): (i) obtain the estimated time-domain impulse response of the secondary path transfer function ; (ii) perform the Fast (FFT) process to ̂ obtain ; (iii) divide each value in the frequency domain by its magnitude, as , and then multiply the result by the mean value of the magnitude responses in the frequency range of interests that may not necessary be the whole frequency range available; and (iv) compute the modified secondary path impulse response function by using the inverse

FFT. The process is repeated for each estimated secondary path. The process of flattening the amplitude response of the secondary path employing the eigenvalue equalization (EE) process yield the eigenvalue-equalized virtual secondary path expressed as . Thus, the estimated secondary path matrix will be replaced by the virtual secondary path matrix that is expressed as in the implementation of FXLMS algorithm. The process to obtain virtual secondary path is illustrated in Figure 3-1 as the block on “Equalization”. If the virtual secondary path is modified by using the EE virtual secondary path algorithm only, it can be expressed as

39

⋯ (3.16) = ⋯ ⋮ ⋯ × An example of EE virtual secondary path is shown in Figure 3-2. In this plot, the solid black line is the estimated secondary path while the dotted blue line is the EE virtual secondary path. It is seen that the magnitude response is relatively flat over the entire frequency range after applying eigenvalue equalization.

30 10dB 20

10

0 Magnitude (dB) Magnitude

-10

-20 0 200 400 600 800 1000 Frequency (Hz)

(a) Magnitude response

40

4

3

2

1

0

-1 Phase(rad)

-2

-3

-4 0 200 400 600 800 1000 Frequency (Hz)

(b) Phase response

Figure 3-2. Magnitude and phase responses of estimated and EE virtual secondary path: (a)

Magnitude response ; (b) Phase response. (Keys: solid black line , estimated secondary

path; and dotted blue line , EE virtual secondary path).

3.3 Channel Equalization Algorithm for Powertrain Noise

When the virtual secondary path is applied in the FXLMS algorithm, in Equation (3.15) is replaced by . In the single-input, single-output (SISO) ANC system, the determinant of becomes the power of the virtual secondary path . Since possesses a flat magnitude in each frequency bin, the overall convergent speed | | in a ll frequency bins is naturally improved. How the specific magnitude of chosen is not critical for the SISO system, because the magnitude difference can be compensated by tuning the value of the step size parameter (Snyder & Hansen, 1990). However, thi s is not the case in

41 multiple-input, multiple-output (MIMO) system in which the selection of the specific magnitude of the virtual secondary path is important. In fact, the selection of the magnitude of the secondary path (the estimated secondary path or the virtual secondary path ) is critical to the convergent speed for each channel. Moreover, it is assumed that the multichannel ANC

control system has identical number of control speakers and error sensors, that is M=K . Thus, in

the conventional FXLMS algorithm, the updating equation of the i-th adaptive filter can be written as

+ 1 = + ̂ ∗ + ⋯ (3.17)

+ ̂ ∗ + ⋯ + ̂ ∗ where . One can notice that the adaptive algorithm utilizes all error signals to = 1, 2, ⋯ , adjust the controller of each secondary source. Also, each error signal is weighted by the

corresponding estimated secondary path. The estimated secondary path transfer function matrix

in frequency domain can be expressed as

⋯ (3.18) = ⋯ ⋮ ⋯ × It is assumed that the estimated secondary path is the main path of the i-th column of such that the main paths of each column vectors are the diagonal elements of the secondary path matrix . The main path of the i-th column is defined as the path from the i-th control speaker, which generates the most dominant sound amongst all the error sensors, to the i-th error sensor.

In fact, the i-th column of are the weights used to update the i-th adaptive filter as described in Equation (3.17) Hence, the main path is the largest weighting value for updating . Mathematically, the mean magnitude value of is larger than the mean magnitude value of

42

, where . Here, note that is the cross coupling term in the estimated , = 1, 2, ⋯ , , secondary path for the i-th channel. In practice, the magnitude of each main path is typically not

identical to each other. Without losing generality, the mean magnitude value of is assumed larger than the mean magnitude value of , expressed as . The extreme > case of this phenomenon is studied as follows. If , because the coupling ≫ secondary paths are even smaller than the main path, the adaptive filter will converge much slower than the adaptive filter when applying Equation (3.17). To improve the overall convergent speed of the MIMO ANC system, channel

equalization (CE) is proposed. The idea is to normalize the mean magnitude value of the main

secondary paths of each column of the secondary path matrix and keep the coupling ratios between mean magnitude of coupling secondary paths and that of the corresponding main path

unchanged. The normalization process is implemented as

⋯ (3.19) ⋯ = ⋮ ⋯ × Again, is the virtual secondary path. Here, the eigenvalue equalization technique is not used to modify the secondary path, but instead the proposed channel equalization technique is applied.

More specifically, the virtual secondary path adopting channel equalization (CE) technique alone is named as channel-equalized virtual secondary path. In Equation (3.19), is a reference value that can be set to any desired value. Usually, is set to unity value of 1, or the mean magnitude of the main secondary path that possesses the maximum mean magnitude amongst all main paths

43 from columns 1 to . Also, in the above equation, ( ) is the mean = 1, 2, ⋯ , magnitude value of the secondary path in the frequency range of interests. For more general cases, the number of the control speakers may be different from the

number of error sensors. Hence, the estimated secondary path transfer function matrix in the

frequency domain as shown in Equation (3.18) can be expressed in the form of

⋯ (3.20) = ⋯ ⋮ ⋯ × To implement the proposed channel equalization, the first step is to find the main path of the i-th column of . It is assumed that is the main path of the i-th column and is a reference value. Then, the magnitude of the main path is equalized according to the reference value as

⋯ (3.21) ⋯ = ⋮ ⋯ × In this way, the mean magnitudes of the main secondary paths of secondary path matrix are all

equalized to the reference value . The mean magnitudes of other secondary paths in the column are adjusted correspondingly to keep the coupling effects unchanged. Furthermore, the channel equalization technique can be combined with eigenvalue

equalization technique to achieve better performance of MIMO ANC system in powertrain noise

application. In this combined approach, still represents the virtual secondary path, namely EE-CE virtual secondary path. The procedure is to equalize the magnitude among frequencies of

44 each secondary path by eigenvalue equalization technique and then equalize the mean magnitude amongst the control channels by the proposed channel equalization technique. Further demonstration is performed using a two-input, two-output (2I2O) ANC system as shown in

Figure 3-3. After applying the eigenvalue equalization, the virtual secondary path becomes

(3.22) = × It is assumed that and are the main paths of the first and second column of secondary path matrix , respectively, and the . Then, by further applying the > channel equalization technique, the main paths of virtual secondary path are normalized to the

mean magnitude value of , which means that the reference value is set as . = Thus, the virtual secondary path is given by

(3.23) = × Consider the example of the experimentally obtained estimated secondary path transfer functions for the 2I2O ANC system, that are given by , , , and . The measured transfer functions from an actual test vehicle are shown in Figure 3-4. The secondary path

transfer functions are modeled by using an off-line system identification method. All four

secondary path transfer functions have been modeled with a 256-tap finite impulse response (FIR)

filter. The application of the EE-CE virtual secondary path as dictated by Equations (3.22) and

(3.23) yields four virtual secondary paths that are represented in Figure 3-5. It is seen that the

magnitude of is equal to magnitude of , and the ratios of and ⁄

45

stay the same as the original ratios of and ⁄ ⁄ in the frequency range of interests of 50 Hz to 300 Hz. ⁄

S11

Speaker 1 S21 Microphone 1

S12

S22 Speaker 2 Microphone 2

Figure 3-3. Proposed 2I2O ANC system.

10dB Amplitude (dB) Amplitude

0 100 200 300 400 Engine speed (rpm)

(a) ( ) and ( )

46

10dB Amplitude (dB) Amplitude

0 100 200 300 400

Engine speed (rpm)

(b) ( ) and ( ) ) Figure 3-4. Magnitude responses of four estimated secondary path transfer functions: (a) S11 and

) ) ) S21 ; (b) S12 and S22 .

47

15

10

5

Amplitude (dB) Amplitude 0

-5 0 100 200 300 400

Engine speed (rpm)

(a) ( ) and ( ) 15

10

5

Magnitude (dB) Magnitude 0

-5 0 100 200 300 400

Frequency (Hz)

(b) ( ) and ( )

48

Figure 3-5. Magnitude responses of four EE-CE virtual secondary path transfer functions: (a)

e e e e S11 and S21 ; (b) S21 and S22 .

3.4 Numerical Simulation

The performance of proposed active noise control system can be examined by performing a series of numerical simulations using experimental data for powertrain noise response. The simulation model is constructed using Matlab/Simulink (The Mathworks, Inc.). The ANC system is designed to attenuate the powertrain noise around the driver’s and passenger’s head areas.

Two error microphones are positioned at the ceiling of the passenger compartment directly above the driver’s and passenger’s head positions. Also, two loudspeakers are placed at the lower areas of left and right front doors. The test vehicle that has a V-6 engine is driven on asphalt road surface with a constant acceleration of 0.2 g (gravitational acceleration). The time duration for the entire acceleration operation is about 12 seconds, during which the engine changes from about 1000 to 5500 rpm. Tachometers signal is recorded to generate the needed reference signal for the control system as discussed earlier in Section 3.2. The primary powertrain noises at two error microphones along with tachometer signal were recorded by a computer through a dSPACE (dSPACE, Inc.) system with a sampling frequency of 4096 Hz. The estimated secondary path transfer functions were measured experimentally by off-line system identification method described in Section 3.2. Both the conventional FXLMS algorithm using estimated secondary path transfer functions and proposed virtual secondary path algorithm using combined

EE-CE technique are applied. Note that the magnitudes of these transfer functions are shown in

49

Figures 3-4 and 3-5. For demonstration purpose, the aim of these simulations is designed to reduce the response of the 1.5 th , 2 nd , and 3 rd engine order as much as possible.

Figure 3-6 shows the simulation results for acoustic noise reduction at the 3 rd engine order, which is the primary firing order in this case. Here, we only show the control results of the

3rd engine order for simplification. However, the performances of the 1.5 th and 2 nd engine order responses are similar to the performance of the 3 rd engine order response. Figure 3-6(a) is the control result at the error microphone 1 (error microphone above the driver’s head), and Figure

3-6(b) is the control result at the error microphone 2 (error microphone above the passenger’s head). The step size for each implementation is set to the largest value while still maintaining stability. The solid black lines in Figures 3-6(a) and 3-6(b) are the primary response of the powertrain noise at error microphones 1 and 2, respectively. The dashed blue lines are the sound pressure response after the control is activated using the EE virtual secondary path algorithm.

The dotted red lines show the control result of the implementation of the EE-CE virtual secondary path algorithm. It can be seen that the controlled response at error microphone 1 applying the conventional EE virtual secondary path algorithm (dashed blue line in Figure 3-6(a)) does not show significant reduction. However, the reduction at error microphone 2 depicted by the comparison of the dashed blue and solid black lines in Figure 3-6(b) can be clearly seen. Also, from the figures, it is obvious that the one using EE virtual secondary path algorithm yields an unbalanced noise reduction result between the two error microphones 1 and 2. This is because the magnitude of is much higher than , and the primary noise response at microphone 2 is higher than the noise response at microphone 1. On the other hand, from the results represented by the dotted red line in Figure 3-6, it can be seen that the EE-CE virtual secondary path algorithm can yield a much more balanced noise reduction result between the two different error

50 microphones. Also, this improved virtual secondary path algorithm with CE applied produced a greater level of attenuation.

The performance can be further improved by adding a turning point where the step size changes. The turning point is determined experimentally based on the specific case under consideration. In this experimental case, 2400 rpm has been chosen as a turning point. Figures 3-

7(a) and 3-7(b) show the simulation results using the improved virtual secondary path with or without turning point for errors 1 and 2, respectively. In Figure 3-7, the step size of the dotted red line is 1.6 times larger when the speed is below than 2400 rpm and is the same as the dashed blue line one when the speed is higher than 2400 rpm. Also, it can be clearly seen that the dotted red line is able to achieve more reduction when the engine speed is lower than 2400 rpm by using a larger step size. Amplitude (dBA) Amplitude

10dB

1,000 2,000 3,000 4,000 5,000

Engine speed (rpm)

(a) Error 1

51

Amplitude (dBA) Amplitude

10dB

1,000 2,000 3,000 4,000 5,000

Engine speed (rpm)

(b) Error 2

Figure 3-6 Comparison of active noise control results between EE virtual secondary path

algorithm and the EE-CE virtual secondary path algorithm: (a) Error 1; (b) Error 2. ( Keys: solid black line , baseline noise response; dashed blue line , EE virtual secondary

path algorithm ; and dotted red line , EE-CE virtual secondary path) .

52

Amplitude (dBA) Amplitude

10dB

1,000 2,000 3,000 4,000 5,000 Engine speed (rpm)

(a) Error 1 Amplitude (dBA) Amplitude

10dB

1,000 2,000 3,000 4,000 5,000

Engine speed (rpm)

(b) Error 2

53

Figure 3-7. Comparison of active noise control results between the EE -CE virtual secondary

path algorithm with and without turning point: (a) Error 1; (b) Error 2. ( Keys: solid black line

, baseline noise response; dashed blue line , without turning point ; and dotted red

line , with turning point).

3.5 Conclusions

This chapter proposed a MIMO ANC system with virtual secondary path for applying active noise control of powertrain noise inside the vehicle cabin. The proposed channel equalization technique substantially improves the overall performance of the control sy stem by equalizing the amplitude levels of the estimated response of all primary channels in order to reduce the variation of convergent speed inherent in the multichannel setup . Numerical simulation studies are conducted for engine speed run -up condition. Simulation results show that the proposed advanced virtual secondary path algorithm with channel equalization demonstrates better performance for 2I2O ANC control system when the magnitude responses of physical secondary paths are quite different, as comp ared to the conventional system. Furthermore, the performance of the MIMO ANC system can be further refined by adding a turning point for the step size.

54

Chapter 4. Twin-FXLMS Algorithm for Active Control of Transient

Powertrain Noise

In this chapter, an ANC applying advanced FXLMS algorithms is employed to reduce the vehicle interior noise by targeting these harmonics. The proposed ANC system is designed to control multiple orders of the engine noise response simultaneously. It is also uniquely formulated with a twin-FXLMS algorithm to prevent harmonic interference that often resulted in overshoot at some adjacent orders, especially at low engine speed range where the reference sinusoids are close together. In fact, the interference issue is one of the critical problems that previously plagued the use of the conventional FXLMS algorithm. The basic design of the twin-

FXLMS algorithm splits the adaptive filter into two sets. This allows different sum of reference sinusoids to be fed into each adaptive filter in order to widen the frequency separation between two adjacent harmonics. Finally, the performances of proposed twin-FXLMS are validated by numerical simulations. Part of this work has been published on Journal of Dynamic Systems,

Measurement and Control.

4.1 Introduction

In our previous study, a modified narrowband active noise equalizer based on FXLMS was

successfully applied to tune powertrain noise response (Duan et al., 2009, 2009; Li et al., 2008,

2009). However, the application meant to control the peak response amplitudes at only two

engine orders; the 3 rd engine order was targeted for reduction while the 4 th engine order was

targeted for amplification. In general, powertrain noise contains a large number of sinusoidal

components. These frequency components appear not only at the odd and even harmonics of the

fundamental frequency of engine rotation, but also at the half order harmonics. Therefore, the

55 peak response tend to cluster closely together especially at low engine speed where the frequency separation between adjacent orders are small. When the powertrain noise is steady-state response, such as the engine is running at a constant speed, conventional ANC system is able to achieve more than 10dB reduction at each targeted harmonics (Li et al., 2008, 2009). However, when the powertrain noise is transient response as engine speed ramp-up case, due to the close proximity of the adjacent orders, applying the active noise control to multiple components simultaneously will tend to cause overshoot in some of the orders rendering the algorithm ineffectual. The overshoot usually occurs at half order components, because half order components have relative lower response compared to integer order components. To address this problem, a twin-FXLMS algorithm is proposed in this chapter.

The twin-FXLMS algorithm contains two adaptive filters. Each filter has an individual sinusoid generator to provide the reference signal. One generator is designed to synthesize sinusoidal components corresponding to even and odd harmonics, and the other one contains the frequencies of half order components. In this manner, the adjacent noise components are spaced out farther apart to prevent the interference between two closely spaced orders being controlled.

A similar idea that applies a direct/parallel configuration was also developed by Yuan et al.

(1994) to improve the convergence of the ANC system for automotive application. In that study,

Yuan et al. (1994) only controlled the half order components when the engine speed is higher than 2500 rpm where the frequency separation between two adjacent sinusoids is greater and less sinusoidal interference occurs. However, in order to have an effective ANC application inside the vehicle compartment that works over a broad engine speed range, the ability to control a large number of integer and half orders is needed. Moreover, although the parallel configuration can further widen the frequency separation, the computational cost will increase correspondingly,

56 which is undesired. These concerns are addressed here using the proposed twin-FXLMS algorithm for transient powertrain noise application.

This chapter is organized as follows. Firstly, the twin-FXLMS algorithm is proposed in the Section 4.2. Secondly, the nonlinearity of the FXLMS algorithm is analyzed in Section 4.3.

In Section 4.4, numerical simulation is conduced to validate the proposed twin-FXLMS algorithm by using actual transient powertrain noise. The simulation results are also discussed in this section.

4.2 Basic Configuration of Twin-FXLMS Algorithm

LMS 1

S$( z )

Sine Wave Powertrain y (n) Generator x (n) 1 Noise 1 W1(z) th th th 1.5 , 2.5 , 3.5 … P(n)

+ + Tachometer S(z) e(n) Signal + +

Sine Wave W2(z) Generator x2(n) y2(n) 2nd , 3rd …

S$( z )

LMS 2

Figure 4-1. Basic configuration of the proposed active noise control system based on the twin-

FXLMS algorithm.

57

The conventional FXLMS algorithm for active control of vehicle powertrain noise control can be found in Reference (Li et al., 2008, 2009). Here we only introduce the proposed twin-FXSLMS algorithm for powertrain noise control. Figure 4-1 shows the flowchart of the

ANC system of powertrain noise based on the proposed twin-FXLMS algorithm. In this algorithm, there are two individual reference signal generators along with two corresponding adaptive filters. To ensure that the frequencies of all sinusoidal components in each reference signal generator are not too close to each other, the reference signal is assigned all integer orders of powertrain noise, while the second reference signal contains only the half orders. The first reference signal is handled by controller 1 and can be expressed as (4.1) 2 ⁄ = cos , ℎ = 2, 4, ⋯ , 2 2 where is the fundamental frequency of engine rotation in Hz (revolution per second), and is the sampling rate. Here, is computed from the instantaneous engine crankshaft speed data measured using a tachometer, is an integer. Similarly, the second reference signal is the target for controller 2 and can be written as

(4.2) 2 ⁄ = cos , ℎ = 1, 3, ⋯ , 2 − 1 2 where corresponds to the index number for half orders (i.e. represents the 1.5 order). = 3 Thus, combining and yields the reference signal vector, (4.3) ≡

58 where and ≡ − 1 ⋯ − + 1 ≡ − 1 ⋯ − are the reference signal vectors at time for those two adaptive controllers, and is the + 1 order of the adaptive filters. The weight vector of adaptive filter is

(4.4) ≡ where and are the ≡ , ⋯ , ≡ , ⋯ , coefficient vectors corresponding to the adaptive controllers and at time . The output signal at time is the sum of the output from the two filters, which can be expressed as (4.5) = where denotes the transpose. The residual signal that is the measured response at the error ∙ microphone can be written as

(4.6) = + ∗ where is the impulse response of the secondary path, the operator represents the ∗ convolution and the is the primary disturbance. The secondary path is given by in Figure 4-1, which is the transfer function from the adaptive filter output to the error signal . The cost function of LMS adaptive algorithm is given by the instantaneous squared magnitude of error signal (Haykin, 1996; Widrow & Stearns, 1985),

(4.7) = || The algorithm minimizes the cost function by updating the controller coefficient vector in the negative gradient direction, which is given by

(4.8) + 1 = − ∇ 2

59

In the above Equation (4.8), is the gradient of the cost function at time n and can be ∇ expressed as (Kuo and Morgan, 1996)

(4.9) ∇ = 2 where is the filtered reference signal vector having similar form as the reference signal vector , (4.10) ≡ in which and ≡ − 1 ⋯ − + 1 ≡ − . In addition, the filtered reference signal can be calculated from 1 ⋯ − + 1 = , with , and is the estimated impulse response of the secondary path. ∗ ̂ = 1, 2 ̂ Substituting Equation (4.9) into the adaptive filter update Equation (4.8) essentially yields the

twin-FXLMS algorithm given by

(4.11) + 1 = − where is the step size. The Equation (4.11) can also be expressed in the following equivalent form,

(4.12) + 1 = − + 1

4.3 Nonlinearity of FXLMS Algorithm

The analysis of the LMS algorithm applied to sinusoidal interference was introduced in

Reference (Glover, 1977). A sum of reference sinusoids is expressed as (4.13) = cos , ℎ = 1, 2, ⋯ ,

60 where is sampling rate and . By processing the LMS algorithm, the output 1⁄ = = 2 signal in Z-transform can be obtained as follows (Glover, 1977),

= + 4

(4.14) − + , , 4 2

+ + , , 4 2 where represents the undesired time-varying components, is the adaptive filter length, is the step size, and represent the Z-transform of the error signal and output signal respectively, and are the amplitude of the i-th and j-th sinusoids, and are the circular frequencies of the i-th and j-th sinusoids, presents the filter coefficients = 1⁄ − 1 update strategy, and the function is defined as (4.15) sin , = sin Furthermore, Glover (1977) concluded that when the reference sinusoids are close together, a long filter is required to eliminate all of the unwanted components. Here, the overshoot at the response of half order is one of the phenomena caused by unwanted components.

4.4 Numerical Simulation

The performance of the proposed twin-FXLMS algorithm was evaluated numerically using Matlab/Simulink (The MathWorks, Inc.) and its results are discussed in this section. The primary powertrain disturbances along with tachometer signal are recorded on an on-road test

61 vehicle with a constant acceleration of 0.2 g (gravitational acceleration). The time duration for the entire process is about 12 seconds, during which the engine speed changes from about 1000 to 5500 rpm. Tachometer signal is a series of pulse trains. One pulse represents one revolution of engine crankshaft. Hence, from tachometer signal, one can obtain the engine crankshaft speed in revolution per minute (rpm) that is shown in Figure 4-2. Then, reference signals can be produced by sinusoidal generators based on the calculated shaft speed to ensure coherence between the reference and disturbance signals.

6000

5000

4000

3000

2000 Estimated speed (rpm) speed Estimated

1000

0 0 2 4 6 8 10 12 Time (Sec)

Figure 4-2. Estimated engine speed from tachometer signal for the ramp-up engine speed case.

The transfer function of the secondary path from the driving signal of the control speaker to the error microphone response in the passenger compartment is also measured inside the test vehicle. In practical, the secondary path is modeled off-line by injecting an uncorrelated white noise through the control speaker and identifying the system dynamics using adaptive FIR filter

62 and standard LMS algorithm, before the active control is implemented. The magnitude and phase functions of the secondary path are plotted in Figure 4-3. The secondary path is modeled using a

256-tap finite impulse response filter for all simulations. The length of the adaptive filters is ̂ 128 for both the conventional FXLMS algorithm and the proposed twin-FXLMS algorithm.

30

20

10

0

-10 Magnitude (dB) Magnitude -20 0 100 200 300 400 5

0

-5

Phase (rad) Phase 0 100 200 300 400 Frequency (Hz)

Figure 4-3. Frequency response function of the secondary path dynamics.

As demonstration, the objective of this analysis is to reduce the response of five strong

interior noise components as much as possible, which are the 1.5 th , 2 nd , 2.5 th , 3 rd , and 3.5 th orders.

Figure 4-4 shows the magnitude spectrum of the primary disturbance and comparison to the

control results. The Hanning window is used in the signal processing of the data. The solid black

line is the baseline response of the powertrain noise when the control is off. The dashed blue and

dotted red lines are the resultant responses when the control is on for the cases using

conventional FXLMS algorithm and twin-FXLMS algorithm, respectively. As shown in the plots

63

4(a), 4(c) and 4(e), which are the simulation results of controlling the half orders, the dashed blue line for conventional FXLMS algorithm is seen higher than the solid black line (baseline response) in the low engine speed range. This is the overshoot problem due to the sinusoids interference discussed earlier, which occurs at the half orders when the engine speed is relatively low.

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000 Engine speed (rpm)

(a) 1.5 th order

64

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000 Engine speed (rpm)

(b) 2 nd order

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000 Engine speed (rpm)

(c) 2.5 th order

65

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000 Engine speed (rpm)

(d) 3 rd order

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000 Engine speed (rpm)

(e) 3.5 th order

66

Figure 4-4. Comparison of control results between conventional FX LMS and twin -FXLMS

algorithms: (a) 1.5 th order; (b) 2 nd order; (c) 2.5 th order; (d) 3 rd order; (e) 3.5 th order. (Keys:

solid black line , baseline noise response; dashed blue line , conventional F XLMS

algorithm; and dotted red line , proposed twin-FXLMS algorithm).

It is interesting to note that the overshoot is not observed at all for the 2 nd and 3 rd orders in

Figures 4-4(b) and 4-4(d) even for the conventional FXLMS algorithm. This is because the responses of the integer orders at low engine speed range are much larger than the half orders, and are therefore less susceptible to interference from the adjacent half orders. In contras t, even small interference between two adjacent orders will tend to increase the responses at the half order frequencies.

When the twin-FXLMS algorithm is applied, the overshoot at half orders is significantly suppressed as clearly seen by comparing the d otted red (controlled response) and solid black lines (baseline response) in Figures 4-4(a), 4-4(c) and 4-4(e). In fact, by observing the dotted red lines in all the plots of Figure 4-4, the responses of all target orders are seen much lower than the basel ine response in the low and middle speed range (<3000 rpm). Furthermore, it may be pointed out that the proposed twin -FXLMS algorithm (dotted red line s) is able to not only suppress the overshoot at half orders but also yield greater reduction in integer o rders, compared to the conventional FXLMS algorithm (dashed blue lines), as shown in Fig ure s 4-4(b) and 4-

4(d).

67

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(a) 1.5 th order

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(b) 2 nd order

68

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(c) 2.5 th order

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(d) 3 rd order

69

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(e) 3.5 th order

Figure 4-5. Effects of adaptive filter length on nonlinearity phenomenon by using the

conventional FXLMS algorithm : (a) 1.5 th order; (b) 2 nd order; (c) 2.5 th order; (d) 3 rd order; (e)

3.5 th order. (Keys: solid black line , baseline noise response; dashed blue line ,

L=128 ; and dotted red line , L=256).

As Glover (1977) said when the reference sinusoids are close together, a long filter has potential to reduce the overshoot between each order. The second simulation is designed to test the effects of overshoot by in creasing the control filter order. Since the frequency separation is different using conventional FXLMS and twin -FXLMS algorithms, two different filter length ( ) sets are used to demonstrate the effect of filter length on overshoot. In the case of applyi ng conventional FXLMS, and are used for the comparison study, and in the case = 128 = 256 of applying twin-FXLMS and are selected instead. The selected filter lengths = 64 = 128

70 are different because the larger frequency separation between the adjacent sinusoids using twin-

FXLMS algorithm requires less adaptive filter taps.

The simulation studies of interest here are still designed to control the 5 major response components given by 1.5 th , 2 nd , 2.5 th , 3 rd , and 3.5 th orders. Figure 4-5 shows the control results by

using the conventional FXLMS algorithm, while Figure 4-6 shows the results by using the twin-

FXLMS one. In both Figures 4-5 and 4-6, it can be seen that increasing the filter length leads to

improved performance. In the case of Figure 4-5 showing results of the conventional FXLMS

algorithm, even though longer filter length provides some relief to the overshoot problem, the

overshoot is still seen at the lower speed range. On the other hand, in Figure 4-6, it is seen that

there is a small overshoot at 2.5 th and 3.5 th orders when it is using filter of 64 coefficients.

However, by increasing filter order to 256, the overshoot can be completely eliminated.

Moreover, as compared to the conventional algorithm, only 128 taps are needed to achieve satisfactory performance. The smaller tap size will reduce process cost and time.

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

71

(a) 1.5 th order

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(b) 2nd order

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(c) 2.5 th order

72

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(d) 3 rd order

10dB Amplitude (dBA) Amplitude

1000 2000 3000 4000 5000 6000

Engine speed (rpm)

(e) 3.5 th order

73

Figure 4-6. Effects of adaptive filter length on nonlinearity phenomenon by using the twin-

FXLMS algorithm: (a) 1.5 th order; (b) 2 nd order; (c) 2.5 th order; (d) 3 rd order; (e) 3.5 th order.

(Keys: solid black line , baseline noise response; dashed blue line , L=64; and

dotted red line , L=128).

4.5 Conclusions

The proposed twin-FXLMS has been implemented and analyzed numerically. The engine speed ramp up case has been studied successfully. Compared to the conventional FXLMS algorithm, the newly proposed active noise controller is able to suppress the unwanted ov ershoot and yield more reduction.

74

Chapter 5. An Active Sound Tuning System using Computational-

Efficient Algorithm for Powertrain Response

Conventional vehicle ANC methods aim to attenuate the vehicle cabin noise, which fail to take into account of sound quality effects. In this chapter, active sound tuning (AST) system is proposed to reshape the vehicle powertrain response based on the predetermined vehicle interior sound quality criteria. Since it is important to develop a computational-efficient system in real- world application, the time-frequency-domain filtered-x least mean square (TF-FXLMS) algorithm is utilized in the AST system. The proposed TF-FXLMS algorithm significantly reduces the computational complexity compared to the conventional time-domain FXLMS algorithm by calculating gradient estimate in frequency domain. Also, as the gradient estimate of

TF-FXLMS algorithm can be adjusted for each frequency bin, it is potential to improve the overall convergent speed and tracking ability of the algorithm. The proposed AST system applying to vehicle powertrain response is validated by tuning individual engine order response, which is targeted to either enhance or attenuate, under both steady-state and transient engine speed conditions.

5.1 Introduction

Traditional passive noise control treatment tends to add weight back to the vehicle

structure, which is undesirable in producing a fuel efficient vehicle. An alternative treatment for

low-frequency noise is active noise control. Both these two approaches are aimed to reduce the

noise level as much as possible. However, the perceived quality of sound is often the underlying

concern. To avoid bulky passive control approach and to tune the powertrain response perceived

at the vehicle occupant location rather than simply suppress it, active sound tuning (AST) is

75 presented in this chapter to either enhance or attenuate the powertrain response selectively in order to achieve a better sound quality criterion.

In addition, the performance of active control system is limited by the computational power of DSP. Therefore, a computationally efficient adaptive control algorithm plays a critical role to improve the ANC system performance by reducing the computational load. Most active control applications of vehicle interior noise utilized the time-domain FXLMS algorithm

(Couche & Fuller, 1999; Duan et al., 2011; Elliott et al., 1988; Elliott, 2008; Kuo & Morgan,

1996; Kobayashi et al., 2008; Li et al., 2009; Mackay & Kenchington, 2004; Oswald, 1984;

Scheuren et al., 2002; Sutton et al., 1994). Despite its popularity, the conventional time-domain

FXLMS algorithm suffers from heavy computational burden, especially for high-order adaptive filter implementation or multichannel control system. To address this drawback of the time- domain FXLMS algorithm, some efforts have been made, such as frequency-domain adaptive filter (Dentino et al., 1978; Duan, 2009; Ferrara, 1980; Kuo et al., 1997; Mansour & Gray, 1982;

Shynk, 1992), block processing adaptive filter (Clark et al., 1981; Shen & Spanias, 1996) and partial updating adaptive filter (Douglas, 1997). While useful, these methods often introduce an additional delay to the signal path which could result in poorer tracking ability. However, in the transient disturbance applications, such as engine ramp-up noise, the tracking ability of adaptive algorithm is one of the most important factors to achieve a better noise control performance. In this chapter, the AST system was designed based on the TF-FXLMS algorithm, whose both control filtering and filter coefficients updating are implemented in time-domain to minimize delay, however, the gradient estimate used to update the filter coefficients is calculated in frequency-domain to reduce computational load. Similar idea was described by Morgan and Thi

(1995) for delayless subband adaptive filtering. Due to the gradient is estimated in frequency-

76 domain, some efforts must be taken to prevent circular correlation effects. Two approaches are implemented and compared in this chapter, i) filtering both reference and error signals by window functions before the FFT process (Kasaka et al., 1997; Kuo et al., 2008), ii) overlap-save implementation (Shynk, 1992). Furthermore, a unique design of adjusting the gradient estimate for powertrain noise application is presented to further improve the tracking ability of the TF-

FXLMS algorithm.

This chapter is organized as follows. Firstly, the cost-efficient AST system for vehicle interior powertrain response based on TF-FXLMS algorithms is presented in Section 5.2.

Secondly, Section 5.3 analyzes the computational complexity of the TF-FXLMS algorithms and compares with the conventional time-domain FXLMS algorithm. Finally, in Section 5.4, the proposed AST system is demonstrated using steady-state and transient powertrain response examples.

5.2 Time-Frequency-Domain Active Sound Tuning System Applied to

Powertrain Response

The block diagram of the proposed AST system for powertrain response based on TF-

FXLMS algorithm is shown in Figures 5-1 and 5-2. In these implementations, feedforward control strategy is used which requires high coherence between the reference signal and primary disturbance response. Since the powertrain response contains a large number of harmonics that functionally related to the rotational speed of the engine, a tachometer sensor is used to estimate the engine crankshaft speed by calculating the time of one revolution in real time. Then, the reference signal generator synthesizes a set of sine waves having frequencies that are multiples of the engine rotational speed. The reference signal can be expressed as:

77

(5.1) = sin 2 ⁄ where is the engine order index, is the sampling rate and is the frequency of the i-th order. , where is the engine speed in rpm (revolution per minute). The reference signal = ⁄ 60 only contains the engine order frequencies that are targeted to control, which is selective control approach.

Tachometer Signal

Powertrain Reference Signal Response Generator

2N -Data Drop the last N 2N -Data Coefficients

Window function Window function IFFT

FFT LMS FFT

Figure 5-1. Block diagram of the proposed AST system for vehicle powertrain response with

window-function implementation of TF-FXLMS algorithm.

78

Tachometer Signal

Powertrain Reference Signal Response Generator N-order filter

N-Data Drop the last N N-Data Coefficients Padding previous N-Data Padding N-Zeros Previous N-Data N-Data N -Zeros N-Data IFFT

FFT LMS FFT

Figure 5-2. Block diagram of the proposed AST system for vehicle powertrain response with

overlap-save implementation of TF-FXLMS algorithm.

To achieve more pleasant sound quality at the vehicle occupant locations, the AST system is designed to tune the powertrain response rather than simply suppress it. Thus, the proposed AST system introduces a desired signal, namely . Unlike the traditional active noise control approach that intends to achieve zero response at the error microphone position, the proposed control system is aimed to let the resultant response picked by error microphone follows the predefined desired response. The desired signal is also generated by sine wave generator based on the current engine speed, but its amplitude is synthesized according to a

certain pre-determined vehicle interior sound quality criteria. is expressed as 79

(5.2) = sin 2 ⁄ where is the amplitude of the i-th order determined by vehicle interior sound quality criteria. If equal to zero, the response of the i-th order is targeted to reduce as much as possible. To be presented in the control algorithm, pseudo-error signal will be used to update the adaptive filter coefficients instead of the error signal in the traditional active noise attenuation approach, which is defined as

(5.3) = − where is the residual error signal sensed by the error microphone and is the difference between the primary disturbance (uncontrolled response) and the secondary sound generated by control speaker.

5.2.1 Window-Function Implementation

The window-function implementation of proposed AST system by TF-FXLMS is presented in Figure 5-1. In the conventional FXLMS algorithm, the gradient estimate, defined as

, is updated sample by sample. In contrast, the gradient estimate of TF-FXLMS ∇ = is updated in a lower rate, which is block by block. Thus, the adaptive filter coefficients are also updated in a lower rate, such that the computational complexity can be significantly reduced.

However, lower update rate will not lead to slower convergence rate, because the block estimate provides more accurate estimation of the real gradient, due to the gradient estimate are averaged in block. To further reduce the computational complexity, the block gradient estimate is calculated in frequency-domain by taking computational advantage of the FFT process. Since the

FFT process inherently perform circular correlation, while the adaptive filter update process

80 requires linear correlation, some efforts have to be made to eliminate wrap-around effects caused by undesired circular correlation. Then the block gradient estimate is transferred back to the time-domain and ready for updating the adaptive filter coefficient. We assume that the block size

is equal to the order of the adaptive filter . To obtain L-th order time-domain adaptive filter, the 2N samples of reference signal and the 2N samples of pseudo-error signal are accumulated into buffers separately to form two 2N -point data vectors, namely and given by

(5.4) = − 2 + 1 − 2 + 2 ⋯ (5.5) = − 2 + 1 − 2 + 2 ⋯ where denotes the transpose operation and is block index. To eliminate wrap-around ∙ effects, proper window functions are needed for the reference and error signals before the FFT processes. Such windows include rectangular window, rectangular window plus zero padding,

Hanning window, zero padding plus rectangular window. The detailed discussion of choosing window functions can be found in reference (Kosaka et al., 1997). The windowed two data vectors and are then transformed once every 2N samples by a 2N -point FFT to produce a pair of frequency-domain vectors, which are expressed as

(5.6) ≡ ⋯ = (5.7) ≡ ⋯ = To avoid distortion caused by secondary path, the reference signal has to be filtered by the secondary path transfer function that relates the speaker control signal to the sound pressure response at the error microphone in the passenger compartment. This is the well-known FXLMS algorithm. In practice, the secondary path transfer function is usually unknown. However, it can be identified by injecting a small amount of white noise through the control input speaker 81 and measuring the response at the error microphone. Then, the secondary path is modeled by I- th order finite impulse response (FIR) filter, whose coefficients are updated using the least mean square (LMS) algorithm. These procedures are called off-line system identification, which results in estimated secondary path . Here, to further reduce the computational complexity, ̂ the filtered reference signal is also calculated in frequency domain. To do so, the estimated secondary path transfer function is transformed into frequency domain transfer function ̂ by using FFT process, where . Then, the time-domain ≡ ⋯ gradient estimate is calculated as:

(5.8) = where denotes the complex conjugate operation and represents the adjust factor vector, ∙ such that the gradient estimate can be adjusted for each frequency bin. Because there are only N coefficients of adaptive filter in time domain, the last N components of the frequency-domain gradient estimate are dropped. Different from references (Kosaka et al., 1997; Kuo et al., 2008), the TF-FXLMS algorithm in this study only estimates the gradient estimate that is used to update adaptive filter coefficients in time domain, thus there is no need to calculate the frequency- domain adaptive filter coefficients. Therefore, the adaptive filter coefficients are updated in time domain as:

(5.9) + 1 = + where is the step size that control the convergent speed of the adaptive algorithm. So far, it can be seen that the adaptive filter updates in time domain very 2N samples. The N-th order time domain adaptive filter can be expressed as . Hence, the output of = ⋯

82 the adaptive filter , which is used to drive the secondary speaker, is given by = . 5.2.2 Overlap-Save Implementation

Overlap-save technique is a method to convert circular correlation to linear correlation

(Shynk, 1992). To generate N correct output signals, it will be necessary to use FFT of length

larger than . In this study, the length of FFT process is set to 2N , which is referred to 50% 2 − 1 overlap. This implementation is illustrated for the active powertrain response tuning system, as

shown in Figure 5-2. The differences between the overlap-save implementation and the window-

function implementation presented in the last sub-section are highlighted by using bold blocks

shown in Figure 5-2. To be specific, only N samples of reference and pseudo-error signals are accumulated in the buffers to form two N-point data vectors and , comparing to 2N - point in the window-function implementation.

(5.10) = − + 1 − + 2 ⋯ (5.11) = − + 1 − + 2 ⋯ However, on one hand, the overlap-save implementation keeps the previous block data of reference signal , and then pads it with the current block data to form 2N -point − 1 data vector, expressed as . On the other hand, it pads N-point zero data with − 1 the vector to form 2N -point vector, given as . This process is shown as 0 ⋯ 0 padding the bold dashed blocks to the bold solid blocks in Figure 5-2. Then the frequency- domain reference and pseudo-error signals are calculated by 2N -point FFT, expressed as

(5.12) ≡ ⋯ = − 1 (5.13) ≡ ⋯ = 0 ⋯ 0

83

Similarly, the gradient estimate is calculated as Equation (5.8) and the adaptive filter coefficients are update in time-domain as Equation (5.9). As noted, in the overlap-save implementation, the adaptive filter coefficients update every N samples, which is as twice frequently as the window-function implementation do. Due to this reason, the AST system with overlap-save implementation of TF-FXLMS algorithm is expected to have better tracking ability for transient powertrain response compared to the window-function one. However, it will slightly increase the computational complexity.

5.3 Computational Complexity Analysis

This section evaluates the computational complexity of the TF-FXLMS algorithm as well as the conventional time-domain FXLMS algorithm. The detailed derivation of the time-domain algorithm can be found in reference (Kuo & Morgan, 1996). This study is based on comparing the number of real multiplications and real additions used for producing 2N output samples of adaptive filter. The N-th order adaptive filter and I-th order estimated secondary path filter are

used in this analysis. For the time-domain algorithm, the computational requirement for the

adaptive filter output per sample is real multiplications and real additions, since the − 1 output calculation equation is . The computations required in calculating = filtered reference signal is real multiplications and real additions per sample, by using − 1 equation , where is the impulse response of the estimated secondary = ̂ ∗ ̂ path transfer function and denotes the convolution process. To produce the pseudo-error signal, ∗ 1 real addition is required. Furthermore, the adaptive filter update equation + 1 = + requires real multiplications and N real additions. Therefore, to produce 2N + 1

84 samples of output signal, the time-domain FXLMS algorithm requires real 22 + + 1 multiplications and real additions in total. 22 + − 1 The different computations required by TF-FXLMS algorithm include two 2N -point

FFTs and one 2N -point inverse FFT processes. Each 2N -point FFT process requires 4 ∙ real multiplications and real additions. Since the estimated secondary 2 4 ∙ 2 path transfer function has been transformed to frequency domain, there is no need to calculate the filtered reference signal separately as the time-domain FXLMS algorithm. It is noted that the transformation of the estimated secondary path transfer function from time domain to frequency domain is implemented off-line, thus, it is not counted into the online computational complexity analysis discussed here. The gradient estimate Equation (5.8) requires 20N real multiplications

and real additions (Noting that a complex multiplication requires four real multiplications 8 and two real additions and that a complex addition requires two real additions), while the

algorithm update Equation (5.9) requires N real multiplications and N real additions. Thus, to

produce 2N samples of output signal, the total computations required by window-function

implementation of the TF-FXLMS algorithm are real 2 ∙ + 12 2 + 21 multiplications and real additions. The overlap-save 2 − 1 + 12 2 + 9 implementation updates as twice frequently as the window-function implement. Thus, the total computations required by overlap-save implementation of the TF-FXLMS algorithm are

real multiplications and 2 ∙ + 24 2 + 42 2 − 1 + 24 2 + 18 real additions.

85

1

0.8

0.6

0.4

0.2

0 0 200 400 600 800 1000 1200

Normalized Computational Complexity Computational Normalized Filter length

Figure 5-3. Normalized computational complexity based on the number of computations, I=256.

(Keys: up-triangle marker , real multiplications; circle marker , real additions;

dashed blue line , window -function implementation; and solid black line , overlap-

save implementation).

How much computations can be saved by using TF -FXLMS depends on the choice of the orders of adaptive filter and the estimated secondary path filter . Here, some examples are shown in Figure 5-3. In Figure 5-3, the order of the estimated secondary path filter is set to be a fixed value of 256, and the order of the adaptive filter N is varied from 32, 64, 128, 256, 512 and 1024. The x-axis is the ada ptive filter length and the y -axis is the normalized computational complexity (NCC) based on the number of computations. Normalized computational complexity is defined as a ratio of the number of computations (real multiplications or real additions) of TF -

FXLMS algorithm over that of the conventional time -domain FXLMS algorithm. The solid black

86 line represents NCC of overlap-save implementation of TF-FXLMS, while the dashed blue line represents NCC of window-function implementation of TF-FXLMS. In addition, the line with up-triangle marker represents NCC of real multiplications, while the line with circle marker represents NCC of real additions. It can be seen from the Figure 5-3 that at least 50% computations can be saved by TF-FXLMS algorithm compared to the conventional time-domain

FXLMS algorithm. For the active powertrain response tuning simulations which will be presented in the next section, 128-th order of the adaptive filter and 256-th order of the estimated secondary path filter are selected. In this case, about 52% computations can be saved by overlap- save implementation of TF-FXLMS algorithm and 65% by window-function implementation of

TF-FXLMS. Even though the TF-FXLMS algorithm is demonstrated by single-input single- output (SISO) AST system in this study, it can be easily extended to multiple-input multiple- output (MIMO) system.

5.4 Numerical Simulation

The proposed cost-efficient AST systems of vehicle powertrain response with two implementations of TF-FXLMS algorithms are studied and compared numerically by using

Matlab/Simulink (The MathWorks, Inc.). The primary powertrain noises along with the tachometer signals are recorded experimentally on a test vehicle with a 6-cylinder engine. The tachometer signal is used to generate the reference and desired signals as discussed in section 5.2.

The secondary speaker is placed at headrest position of the driver, while the error microphone is located at the ceiling of the passenger compartment directly above the driver’s head position.

The secondary path transfer function is measured by using system identification approach with

LMS algorithm (Kuo and Morgan, 1996) and modeled as 256-th order FIR filter, whose

87 amplitude and phase responses are shown in Figure 5-4. In all simulations, the adaptive filter length is 128 and the sampling rate is 4096 Hz.

30

20

10

0

-10 Magnitude (dB) Magnitude -20 0 200 400 600 800 1000 5

0

-5

Phase (rad) Phase 0 200 400 600 800 1000 Frequency (Hz)

Figure 5-4. Magnitude and phase responses of the estimated secondary path transfer function.

Two specific cases are studied in this study. The first case is a steady-state example, while the second one involves run-up engine speed. For demonstration purpose, in both simulations, the goal is to reduce the response of the 3 rd engine order as much as possible and enhance the response of the 4 th engine order to a predetermined value, while keep the other orders unchanged. In the first case, the engine is running at constant speed 3500 rpm and the spectrum of the powertrain response is shown as the solid black line in Figure 5-5. The control period is 5 seconds, while the last 8192 samples of pseudo-error signal after reaching convergence is taken as the steady-state response. The simulation results are also shown in

Figure 5-5. The desired response level of the forth order is labeled by the asterisk symbol ( ). ∗

88

The dashed blue line and dotted red line are control results by using window -function and overlap-save implementations of TF -FXLMS algorithm, respectively. Their resultant responses are very c lose to each other. It is seen that the reduction of response of the 3rd engine order is very obvious, which is about 20 dB, while the resultant response of the 4th engine order is enhanced up to the desired level.

10 dB 4th

3rd Magnitude (dB) Magnitude

100 150 200 250 300 350 400 Frequency (Hz)

Figure 5-5. Active sound tuning simulation results for a constant engine speed of 3500 rpm case.

(Keys: solid black line , baseline noise response; dashed blue line , window-

function implementation of TF-FXLMS ; dotted red line , overlap-save implementation of

TF-FXLMS ; and asterisk *, desired value).

The second case involves the engine speed ramping up. This simulation is designed not only to test the active sound tuning performance for transient powertrain response, but also to analyze the tracking ability of the proposed TF -FXLMS algorithms. In this case, the amplitude

89 and frequency of the primary powertrain are time-varying, which related to the varying engine speed. Again, the goal is to reduce the response of the 3 rd order and enhance the response of the

4th order. One can expect that the algorithm which has better tracking ability will yield more reduction at the 3 rd order and more close to the desired value at the 4th order. The time duration

for the entire process is 30 seconds, during which the speed increases from 1000 rpm to 3600

rpm. Comparison control results for the two implementations of TF-FXLMS algorithm are

shown in Figures 5-6(a) and 5-6(b). The step size for each simulation is chosen to be the largest value while still ensuring that the adaptive algorithms remain stable. In Figure 5-6, the solid black line is baseline response, while the dashed blue line and dotted red line present the control results by using the window-function and overlap-save implementations of TF-FXLMS algorithm, respectively. It can be seen that the overlap-save implementation yields more reduction in Figure 5-6(a) and more close to the desired value in Figure 5-6(b) compared to the window-function implementation. Thus, the overlap-save implementation has better tracking ability than the window-function implementation as our expectation.

90

10 dB Amplitude (dBA) Amplitude

1000 1500 2000 2500 3000 3500 Engine speed (rpm)

(a) 3 rd order response reduction

10 dB Amplitude (dBA) Amplitude

1000 1500 2000 2500 3000 3500

Engine speed (rpm)

(b) 4 th order response enhancement

91

Figure 5-6. Active sound tuning simulation results for an engine speed ramp-up case : (a) 3 rd

order response reduction. (b) 4 th order response enhancement. (Keys: solid black line ,

baseline noise response; dashed blue line , window-function implementation of TF -

FXLMS algorithm; dotted red line , overlap-save implementation of TF-FXLMS algorithm;

and asterisk *, desired value).

From Figures 5-6(a) and 5-6(b), one can notice that none of the algorithms achieve any reduction for 3 rd order or close to the desired value for the 4 th order when the engine speed is lower than 1800 rpm. In fact, since the adaptive filter is updated in the time domain, the step size is fixed for all frequencies. That means the step size is large enou gh to achieve faster convergence in the high speed range, however, it will be relatively small in the low engine speed range, due to the lower power of the filtered reference signal for the low frequency range. If one further increase the step size for ach ieving better performance at low speed range will result in the system unstable when the engine speed reaches the high speed range. Fortunately, the TF -

FXLMS algorithm calculates the gradient estimate in frequency domain. Thus, we can adjust the gradient e stimate for each frequency bin by tuning in Equation (5. 8). Generally, the adjust factor of gradient estimate for each frequency bin can be normalized with respect to the inverse power of the filtered reference signal (Shynk, 1992), as

(5.14) 1 ̂ = where is the adjust factor for the m-th frequency bin, indicates the Euclidean norm ̂ ‖∙‖ operator of the vector, and represents the filtered reference signal in the corresponding frequency bin . However, in the active powertrain response tuning application, the reference

92 signal only contains sinusoidal signals that correlated with the targeted engine order, such that the frequency-domain reference signal may have zero power in some frequency bins. If the Equation (5.14) is used, the adjust factor for those frequency bins could be infinite. Even though one can add a small constant value to the estimated power, the algorithm is still easy to become unstable. A modification has been made to avoid the infinite inverse power in this study.

Since the reference signal is generated by a sine wave generator, the amplitude of the reference signal can be arbitrary set to unit value. To follow the same logic above, the adjust factor for each frequency bin in the frequency range of interest can be modified as

(5.15) ̂ = The simulations are conducted by using the same primary powertrain response as the second case. Figures 5-7(a) and 5-7(b) shows the comparison results of the overlap-save implementation of TF-FXLMS algorithm with and without the adjust factor for each frequency bin. It is note that the Equation (5.15) can be also used by the window-function implementation of TF-FXLMS algorithm. It is seen that the two implementations have similar performance when the engine speed is higher than 2500 rpm but show fairly large discrepancy at low engine speed range. The TF-FXLMS algorithm with adjust factor of gradient estimate (dash-dotted green line) shows much better performance than the one without adjust factor (dotted red line) when the engine speed is lower than 2500 rpm. The enhancements appear to improve the system performance when the amplitude response of secondary path varies significantly with frequency.

This is clearly shown in Figure 5-4 where the amplitude response of secondary path transfer function is more dynamic at low speed range (about 20 dB change between 50 Hz and 125 Hz

93 that corresponds to 1000 –2500 rpm range) than it at high speed range (about 5 dB change between 125 Hz and 180 Hz that corresponds to 2500 –3600 rpm range).

10 dB Amplitude (dBA) Amplitude

1000 1500 2000 2500 3000 3500 Engine speed (rpm)

(a) 3 rd order response reduction

94

10 dB Amplitude (dBA) Amplitude

1000 1500 2000 2500 3000 3500

Engine speed (rpm)

(b) 4 th order response enhancement.

Figure 5-7. Comparison of the performance of TF -FXLMS algorithms with or without gradient

estimate adjust factor: (a) 3 rd order response reduction ; (b) 4 th orde r response enhancement.

(Keys: solid black line , baseline noise response; dotted red line , without gradient

estimate adjust factor; dashed -dotted green line , with gradient estimate adjust factor ;

and asterisk *, desired value).

5.5 Conclusions

This chapter proposed a computational-efficient active sound tuning system for vehicle powertrain response, which allows reshape the powertrain response rather than simply suppresses it. Both constant engine speed and run -up engine speed cases are simulated. Even though only one enhanced orde r and one reduced order are exemplified, the proposed control

95 system are capable of tuning more complex cases based on different sound quality requirements.

Two implementations of TF-FXLMS algorithm are compared in their computational complexity and tracking ability. Based on the analysis, the overlap-save implementation shows better tracking ability for transient powertrain response, but requires slight more computations than the window-function implementation. However, compared to the standard time-domain FXLMS algorithm, either one significantly reduces computational complexity. Furthermore, the tracking ability of TF-FXLMS algorithm can be improved by adjusting the gradient estimate for each frequency bin.

96

Chapter 6. A Combined Feedforward-Feedback Active Control of Road

Noise

Conventional active control of road noise inside vehicle cabin uses pure feedforward control system with the standard FXLMS algorithm. It can give good reduction when the reference signal is well correlated with the targeted noise. However, in practical, it is not always possible to obtain a reference signal that has high coherence value with the target noise in a broad frequency range. In this chapter, an active noise control system with combined feedforward-feedback controller has been developed to improve performance of attenuating road noise. To take full advantage of feedforward control, subband FXLMS algorithm, which can achieve noise attenuation in a broader frequency range, is used to replace the conventional

FXLMS algorithm. A feedback controller based on internal model control architecture is also introduced to reduce the road noise components that has strong but poorly correlated with the reference signals. The proposed combined feedforward-feedback ANC system has been demonstrated by a control system with six reference accelerometers, two control loudspeakers and one error microphone using actual data measured from the test vehicle. The results show that the performance of proposed combined system is better than either feedforward or feedback controller alone, and is able to achieve 3.8 dB of A-weighted sound pressure level of overall noise reduction.

6.1 Introduction

ANC is widely studied on several applications, such as the engine and exhaust noise in the passenger vehicle (Carme et al., 2006; Couche & Fuller, 1999; Duan et al., 2009, 2011;

Elliott et al., 1988; Kobayashi et al., 2008; Kuo & Morgan, 1996; Li et al., 2009; Oswald, 1984;

97

Scheuren et al., 2002;) and aircraft (Smith et al., 1996), gearbox noise (Guan et al., 2005; Li et al.,

2005), fan noise (Gee & Sommerfeldt, 2004; Lauchle et al., 1997) and magnetic resonance imaging noise (Li et al., 2010; Milani et al., 2009). However, road noise reduction is one of the most difficult applications due to its random nature and multiple sources. In the past two decades, several studies were carried out to control the road noise by ANC approach (Bernhard, 1995;

Dehandschutter & Sas, 1998; Oh et al., 2002; Park et al., 2002, 2004; Sutton et al., 1994).

However, most of the studies used pure feedforward control system formulated with the conventional FXLMS algorithm. A feedforward control system can give good control performance only if the reference signal is well correlated to the targeted noise. However, in real case, there is no guarantee that a good reference signal can be obtained in the broad frequency range of interests. In addition, the conventional FXLMS algorithm also has some disadvantages when it is applied to road noise control. Road noise is a colored broadband noise and most of its energy is in the low frequency range, 80-400 Hz. Normally, broadband noise requires long order of the adaptive filter, which results in high computational burden and slow convergence, especially applying to control colored noise comparing to harmonic noise and white noise, because the specified step size of FXLMS is not optimal for all frequencies. To address these difficulties, a more advanced control system is proposed.

In this study, a combined feedforward-feedback active noise control system for vehicle road noise is proposed. The feedforward control part utilizes the subband adaptive FXLMS algorithm instead of the conventional FXLMS algorithm, which is able to significantly reduce the computational complexity, due to the adaptive filter update process is implemented in a lower sampling rate and increases the convergent speed, because the convergence factors can be adjusted for each frequency band. The location and number of reference sensors are selected by

98 evaluating the control performance based on the multiple coherence functions of the experimental reference signals. Furthermore, the feedback control part uses the internal model control (IMC) architecture (Morari & Hafiriou, 1989) that is formulated with the conventional

FXLMS algorithm. The feedback control is designed to suppress the dominant peak in the residual noise that is uncorrelated with reference signals, which results in significant improvement in the overall performance of the control system. The control performance of the proposed combined feedforward-feedback ANC system will be investigated through computer simulation using experimental data.

This chapter is organized as follows. Firstly, the selection method of the reference accelerometers is presented in Section 6.2. Secondly, Section 6.3 proposed the combined feedforward-feedback control system with the corresponding adaptive algorithms. Finally, simulation results of the proposed active road noise control system are discussed in Section 6.4.

6.2 Selection of Reference Accelerometers

The cancellation signal of feedforward ANC system is derived from the reference signal by passing through a set of linear filters. Hence, to yield more reduction, the reference signals have to be chosen that are well correlated with the targeted vehicle interior noise. Actually, the road noise is caused by the interaction between the vehicle tires and road surface, and transmitted to vehicle cabin mostly through structure-borne transfer paths. One possible means of detecting the road noise prior to it transmits into vehicle cabin is to attach accelerometers in the transfer paths, such as wheels, sub-frames, suspension or other body structures. Road noise is caused by multiple noise sources. At least, each tire can be considered as an independent noise source.

Because of the random nature of the road surface, the noise or vibration caused by each independent source is usually not related to each other. Therefore, a large number of

99 accelerometers are required to detect all the independent sources of the noise. The challenge is to determine how many accelerometers are necessary and where they should be located.

In this study, multiple coherence function and principle component analysis are used to determine the optimal locations and the least number of accelerometers. Multiple coherence function is the extension of concept of ordinary coherence function, which measures the linear relationship between the output signal and a set of input signals (Bendat & Piersol, 1980). Here, output signal is the vehicle interior noise to be controlled, expressed as , and the input signals are the set of reference signals picked by the accelerometers, expressed as , , where is the number of accelerometers. The spectrum density of the output signal = 1,2, ⋯ , can be considered as the sum of idea predicted linear output spectrum and the extraneous noise output spectrum , which can be represented as: (6.1) = + where is the spectrum density of the output signal. Then, the multiple coherence function can be defined as

(6.2) : = varies between 0 and 1 over the entire frequency range. In addition, can be : calculated using

(6.3) ∗ = where is the frequency response function of the transfer path from the input signal to the output signal and is the cross spectrum density function of input signal and . The can be obtained by solving the following set of Equations

100

(6.4) = where is the cross spectrum density of input signal and output signal . Once the multiple coherence function is obtained, the maximum potential noise reduction of sound pressure level (SPL) can be estimated as

(6.5) = −10 log 1 − : In general, larger multiple coherence value is, more reduction can be achieved. The multiple coherence function can be used to measure the quality of reference signal set, however it doesn’t directly determine the minimum number and optimal locations of accelerometers. To find the optimal set of accelerometers, the following experiment is performed. Seven possible candidates for accelerometer locations are selected before the experiment. Then twenty-one single-axial accelerometers are attached at the selected position. The twenty-one reference signals are recorded while the test vehicle is driving on a proven road surface at a constant speed of 30 mph. To estimate the minimum number of the accelerometers, principle component analysis is performed. Figure 6-1 shows the singular values between 60 and 400 Hz that is the frequency range of interests in this study, because the road noise has the most energy in the frequency range. Note that there should be twenty-one lines in total in Figure 6-1, however, only the largest ten singular values are shown for simplification. As discussed by Sutton et al. (1994), the number of significant singular values at each frequency can be considered as the number of significant noise sources at that frequency. Thus, it also predicts the minimum number of the accelerometers at that frequency. In the frequency range shown in Figure 6-1, the amplitudes of four singular values are larger than others, which imply that there are mainly four independent noise source. It is because each tire can be considered as an independent noise source. However,

101 at some frequency range, such as 100-170 Hz, the amplitudes of the fifth and sixth largest singular values are also large, and they cannot be simply ignored. To maximize the value of multiple coherence function, at least six accelerometers should be selected. The next step is to determine optimal locations of six accelerometers from the twenty-one candidate locations.

There are 54264 combinations if six accelerometers are selected out of twenty-one options. A

Matlab program is used to search for the best combination that provides the maximum sum of multiple coherence value in the frequency range 60–400 Hz.

-30

-35

-40

-45

-50

-55

Singular value Singular -60

-65

-70 0 50 100 150 200 250 300

Frequency (Hz)

Figure 6-1. Principle component analysis of twenty-one accelerometer signals.

The multiple coherence function of the best combination of six accelerometers is shown

in Figure 6-2 as the dashed green line. In Figure 6-2, the solid blue line is a typical road noise

picked by error microphone that is placed as driver’s overhead position. The dotted red line in

the bottom part of the figure shows the maximum potential reduction based on the multiple

102 coherence function at that frequency. It is interest ing to see that the coherence is very p oor in the frequency range 70-100 Hz, where the road noise has largest linear SPL. It is meant that the pure feedforward control is unable to get much reduction at the peak range of the road noise, where is the most important frequency range to be reduced. The bottom plot of the Figure 6-2 also shows that the potential reduction at frequency range 70 -100 is very small. To address this problem by using pure feedforward control approach , an additional feedback loop is proposed to handle the peak noise that cannot be attenuated by feedforward control alone. The combined feedforward - feedback controller for road noise attenuation is presented next.

1 10dB 0.8

0.6

0.4

0.2 Coherence Multiple Sound Pressure Level (dB) Level Pressure Sound 0 0 50 100 150 200 250 300 350 400 450 500500500 10

5

Reduction (dB) Reduction 0 0 50 100 150 200 250 300 350 400 450 500 Frequency (Hz) Figure 6-2. Multiple coherence function and potential maximum noise reduction in decibels of

best set of six accelerometers, along with sound pressure level of typical road noise. (Keys: solid

blue line , sound pressure level of typical road noise, labeled as the left y -axis; dashed

green line , multiple reference function, labeled as the right y-axis ; dotted red line ,

103

potential maximum noise reduction; and shadow area , frequency range that has high

SPL of road noise but low multiple coherence value ).

6.3 Control System Design

In this section, the configuration of the proposed combined feedforward -feedback controller, as well as the control algorithms are presented . For simplicity, multiple -reference single-input single-output (SISO) ANC system is described as an example. However, it can be easily extended to multiple-input multiple -output (MIMO) ANC system.

6.3.1 Combined Feedforward -Feedback Controller

Figure 6-3 shows the diagram of the proposed combined feedforward -feedback control system. The control system includes two controllers, which are feedforward controller and feedback controller . As discussed in the last section, accelerometers are attached to vehicle body to detect reference signals for the feedforward control. In the active road noise control application, the reference signals cannot be simply summed or summed by weighting factors to form a single reference signal, otherwise some useful information will be lost. Thus, each reference signal needs an individual control filter. Vector represents the feedforward adaptive filter coeffi cients associated with adaptive filters. The coefficients vector of adaptive filters are stacked up into a long vector, as , where ≡ ⋯ ≡ , with . is the coefficients vector of the j-th , , ⋯ , = 1, 2, ⋯ , adaptive filter, which is a -th order finite impulse response (FIR) filter. The output signal of 1

104 the feedforward controller is the weighted sum of all reference signals, , and can be expressed as , , ⋯ , (6.6) = where the whole reference signal vector is constructed by ≡ ⋯ stacking individual reference signal vector together to form a column vector with ∙ 1 × elements. . The j-th individual reference signal vector element is a column vector as, 1 ≡ with . − 1 ⋯ − 1 + 1 = 1, 2, ⋯ ,

Accelerometers

Reference signal + generator +

+ _

̂

Figure 6-3. Block diagram of the proposed combined feedforward-feedback active road noise

control system.

105

The feedback control shown in Figure 6-3 uses the internal model control architecture.

The plant relates the speaker control signal to the sound pressure response at the error microphone in the passenger compartment, called secondary path transfer function. In practice,

the secondary path transfer function is usually not known. However, it can be identified by using

off-line system identification method in the initial stage before control. In this study, the

secondary path transfer function is modeled by an I-th order FIR filter, defined as ̂ ≡ . is estimated impulse response of the secondary path. It is assumed ̂0 ̂1 ⋯ ̂ − 1 ̂ that is time-invariant during the control process. The idea of IMC architecture is to transform ̂ the feedback control problem to an equivalent feedforward one. The feedback controller contains an internal model , which is driven by the output signal of the controller. Then, the output signal ̂ of the internal model is subtracted from the observed error signal, , to obtain the estimated primary disturbance, which can be expressed as:

(6.7) = − ̂ − Then, is fed into the feedback controller . If the internal model perfectly represents the secondary path, such that , the feedback controller reduces to a pure feedforward ̂ = system. Therefore, the output signal of the feedback controller can be obtained by the following

equation:

(6.8) = where represents the coefficients vector of feedback ≡ , , ⋯ , control filter of order , and is the signal vector of 2 ≡ − 1 ⋯ − 2 + 1 the estimated primary disturbance. The adaptive filter update algorithms of both feedforward and feedback controller will be presented in the next two subsections. In the combined feedforward-

106 feedback control system as shown in Figure 6-3, the outputs of feedforward and feedback controller are summed up to drive the secondary control loudspeaker, which can be expressed as

(6.9) = +

6.3.2 Feedforward Control Algorithm

The conventional time-domain FXLMS algorithm is widely used in the feedforward

ANC system, including several road noise applications (Bernhard, 1995; Dehandschutter & Sas,

1998; Oh et al., 2002; Park et al., 2002, 2004; Sutton et al., 1994). However, for a broadband noise control application such as road noise, the adaptive filter often requires hundreds of filter taps. The long-length adaptive filter results in heavy computational burden, which precludes their use for many low-cost applications. In addition to computational complexity, the adaptive filter with many taps suffers from slow convergent speed, especially when the eigenvalues of the correlation matrix of the input signal are widely spread. Slow convergent speed can result in poor system performance. Recently, subband techniques have been developed to handle the above problems (Shynk, 1992; Vaidyanathan, 1990; Vetterli, 1987). The subband techniques allow us to treat the wideband signal in each small subband. Hence, the faster convergence is possible, because the eigenvalues of the correlation matrix of the input signal are much less spread in a smaller dynamic range and the step size can be adjusted in each subband. Furthermore, the computational complexity can be significantly reduced by decimating the number of adaptive filter taps and weight update rate in each subband. However, the conventional subband methods

(Shynk, 1992; Vaidyanathan, 1990; Vetterli, 1987) introduce a delay into the signal path by subband signal generation process. In ANC problem, the delay is a factor that significantly limits the performance. To avoid the signal path delay, while retaining the advantages of subband

107 processing, Morgan and Thi (1995) proposed a delayless subband adaptive filter. Later, Park et al. (2001) proposed a frequency-domain implementation of the delayless subband adaptive filter.

+ + + +

Weight transformation

DFT ̂ DFT

filter filter banks banks ̂

⋮ ⋮ ⋮ ⋮ ⋮ ↓ ↓ ̂

Figure 6-4. Feedforward control part of the proposed active road noise control system based on

subband FXLMS algorithm.

Here, the SISO subband filtered-x least mean square (SFXLMS) algorithm is developed

based on reference (Morgan & Thi, 1995) and (Park et al, 2001) and extended to the multiple-

reference application. The diagram of the proposed subband adaptive algorithm for updating the

feedforward controller is shown in Figure 6-4. In this study, the Discrete Fourier Transform

(DFT) filter banks (Lee et al., 2009) are used to decompose the reference signal and error

108 signal into sets of subband signals. The filter banks are derived from a prototype filter via modulation. The prototype filter is designed by using the Matlab , where 1 − 1, 1⁄ is the filter length of the prototype filter and is the number of the filter banks. Then the other filter banks ( ) are obtained by complex modulation. The − 1 , , ⋯ , modulation process in time domain can be expressed as

′ (6.10) / ℎ = ℎ where is the impulse response of the m-th filter bank , , and is the i- ℎ = 0, 1, ⋯ , − 1 th coefficient of , . The coefficients of and , for ℎ = 0, 1, ⋯ , ℎ ℎ = , are complex conjugate of each other. Hence, for real signals, only the first 1, 2, ⋯ , ⁄2 − 1 subbands need to be processed. Furthermore, the center frequencies of filter banks are ⁄2 + 1 uniformed spaced and all filters have equal bandwidth, because of the modulation, which is

called uniform filter banks. By decomposing the full band signal into subband signals by using

uniform filter banks, each subband signal contains only of the original frequency band. 1⁄ Thus, the subband signal can be maximum decimated by the factor , without losing any information. It is assumed that the decimation factor is . The decomposition process of reference signals and error signal can be expressed as

(6.11) , = ℎ −

(6.12) = ℎ − Where and are the j-th reference signal and error signal in the m-th subband, , respectively, and ; and is the subband index, = 1, 2, ⋯ , = 0, 1, ⋯ , − 1 = −

109

, with . The uniform DFT-modulated filter bank can be implemented 1/ = 0, 1, 2, ⋯ efficiently by polyphase FFT (Lee et al., 2009; Morgan & Thi, 1995). To further reduce the computational complexity, the estimated secondary path transfer function is decomposed into a ̂ set of subband functions, expressed as . Each impulse response of subband ̂, ̂, ⋯ , ̂ secondary path contains coefficients. Hence, the filtered reference signals are calculated in ⁄ each subband, as

(6.13) , = , ∗ ̂ where denotes the convolution process. ∗ Here, represents the subband adaptive filter of the j-th reference signal in the m-th , subband with length of . Each subband adaptive filter coefficients are updated individually 1⁄ by using complex least mean square (CLMS) algorithm, which can be expressed as

(6.14) , + 1 = , + ,, where is the signal vector of the j-th , ≡ , , − 1 ⋯ , − 1⁄ filtered reference signal in the m-th subband, , ≡ ,, ,, ⋯ ,,/ represents the subband filter at index , and denotes the complex conjugate. In addition, is ∙ a step size vector that can be adjusted for each subband. In practical, the step size is normalized with respect to the inverse filtered reference signal power in the corresponding subband, which can be expressed as

(6.15) = ′,, + where is the normalized step size, and is a small constant value to prevent infinite step size. As shown in Figure 6-4, the filtered reference signal vector and subband adaptive filter , can be stacked up into a long vector in each subband, which are expressed as , ≡ 110

and , where , , ⋯ , ≡ , , ⋯ , = . However, in the programming, only the first subbands need to be 1, 2, ⋯ , ⁄2 + 1 processed.

The next step is to transform a set of subband filter coefficients to an equivalent full band

one. Several weight transformation techniques have been developed, such as the FFT-stacking

method (Morgan & Thi, 1995), FFT-2 stacking method (Huo et al., 2001), DFT-FIR weight

transform (Huo et al., 2001), and linear weight transform (Larson et al., 2002). In this study, the

FFT-stacking method is adopted. Again, is the filter length of the full band adaptive filter and 1 is the number of subband filters. The subband filter coefficients is transformed into , frequency domain by -point FFT, expressed as . 1/ , ≡ ,0 ,1 ⋯ ,1/ Then all the frequency domain coefficients for the j-th reference signal, with , = , are properly stacked in a elements array . 0, 1, ⋯ , − 1 1 ≡ 0 1 ⋯ 1 − 1 The elements array is the frequency domain coefficient of the full band filter for the j-th 1 reference signal. The stacking rule can be expressed as

a) , for ′ ′ = , ⁄∙⁄ ∈ 0, 1⁄2 b) , for (6.16) = 0 = 1⁄ 2 c) = 1− where and are the -th frequency domain coefficient of the full band filter and m- , th subband filter, respectively. rounds the element to the nearest integer and /1 /1 stands for modulus . Finally the impulse response of the full band filter ∙⁄ 2 ∙ 1⁄ with coefficients is obtained by taking inverse FFT transform of the elements array as 1 1

111

, where . Then, the set of adaptive filters is stacked = ≡ , , ⋯ , up into a long vector, . ≡ ⋯ 6.3.3 Feedback Control Algorithm

+ + + +

_ +

̂

̂

Figure 6-5. Feedback control part of the proposed active road noise control system based on

IMC architecture with FXLMS algorithm.

As we discussed above, the internal model architecture of feedback control is to

transform the feedback problem into an equivalent feedforward one. As shown in Figure 6-5,

once the estimated the primary disturbance is calculated by Equation (6.6), will be fed into the feedback controller and the adaptive algorithm as a reference signal. In this study, the feedback controller is designed to reduce the peak response of the road noise at 70-100 Hz, where the feedforward control is less effective at. The conventional FXLMS algorithm is used to update the feedback control filter , because the LMS algorithm inherently utilize more

112 resources to attenuate the high level of the error signal. Thus, the feedback adaptive filter coefficients are updated as

(6.17) + 1 = + where is the step size of feedback control, is the error signal picked up by the error microphone inside the vehicle cabin, and is the ≡ , − 1, ⋯ , − 2 + 1 filtered reference signal vector. It can be obtained by convolution between the estimated primary

disturbance and estimated secondary path function, as . = ∗ ̂ 6.4 Numerical Simulation

In order to evaluate the effectiveness of the proposed combined feedforward-feedback active road noise control system, a simulation model is built using actual data from test vehicle.

For demonstration purpose, the control system is designed to include six accelerometers, two secondary speakers, and one error microphone as shown in Figure 6-3. Six single-axis accelerometers are mounted on the vehicle body where give the best multiple coherence between the reference signals from accelerometers and the targeted road noise. The locations of the six accelerometers are selected by the method discussed in Section 6.2 and their multiple coherence function is shown as the dashed green line in Figure 6-2. The ANC system is designed to attenuate the road noise around driver’s head area. Hence, the error microphone is placed at the ceiling of the passenger compartment directly above the driver’s head position. Two commercial loudspeakers are positioned at the left and right sides of driver’s headrest positions. The two control speakers are grouped as a single control speaker set. One microphone signal and six reference signals are filtered by low-pass filters with cut-off frequency of 500 Hz and then converted to digital signals with a sampling frequency of 2048 Hz. The primary road noise signal

113 when the active control is off along with six reference signals are recorded when the test vehicle is driven on a proven road surface at a constant speed 30 mile per hour (mph).

15

10

5

0

-5 Impulse response Impulse -10

-15 0 50 100 150 200 250

Tap number

(a) IRF

40

30

20

10

0 Magnitude (dB) Magnitude -10 0 100 200 300 400 500 200

0

-200 0 100 200 300 400 500 Phase (degree) Phase Frequency (Hz)

114

(b) FRF

Figure 6-6. IRF and FRF of the measured secondary path: (a) IRF; (b) FRF.

The transfer function of the secondary path from the driving signal of the control speaker

to the error microphone response in the passenger compartment is also measured inside the test

vehicle. In practical, the secondary path is modeled off-line by injecting a white noise through

the control speaker and identifying the system dynamics using adaptive 256-tap FIR filter

( ) and LMS algorithm, before the active control is implemented. The impulse response = 256 function (IRF) and frequency response function (FRF) of the measured secondary path are plotted in Figures 6-6(a) and 6-6(b).

A simulation model is built in Matlab/Simulink (The MathWorks, Inc.). The diagram of the simulation model is shown in Figure 6-3. Recorded reference signals from six accelerometers and primary road noise disturbance are imported into this simulation model. In all the () control simulations, 40,960 samples are used, while the last 8192 samples of error signal after

algorithm reach convergence is taken as the steady-state response. The secondary path is also

simulated by using measured secondary path transfer function. All full band feedforward control

filters and feedback control filter are designed to have 256 coefficients ( () () 1 = 256 and ). In the feedforward control, the filtered reference and error signals are 2 = 256 decomposed into 16 subband signals ( ) and the decimation factor is chosen as , = 16 = 8 which is 2x oversampling. The lowpass prototype filter for the DFT filter bank is designed to have 128 taps using Matlab command , hence, each subband adaptive filter has 1(127, 1/16) 32 taps ( ). 1⁄ = 32 The first simulation involves using feedforward controller only and temporarily let the

feedback controller ineffective. Figure 6-7 shows the comparison of control results by using the

115 proposed SFXLMS algorithm and the conventional FXLMS algorithm. The feedforward control filters of the conventional FXLMS algorithm have the same 256 coefficients as the SFXLMS one.

The solid black line is the baseline when the active control is off. Figure 6-7 shows that the

SFXLMS algorithm can achieve more noise reduction when the frequency is higher than 250 Hz compared to the FXLMS algorithm, which means the SFXLMS algorithm is able to attenuate road noise in a broader frequency range. Especially in the frequency range 330-360 Hz, the

SFXLMS algorithm can yield maximum 4 dB reduction, while the FXLMS algorithm cannot get any reduction at all. The reason is SFXLMS algorithm can adjust the step size in each subband with respect to the inverse power of the corresponding filtered reference signal, while the

FXLMS algorithm only have one step size in the full band frequency range. In fact, the full band signal is decomposed into 16 subbands, so that each subband spans 128 Hz with sampling frequency of 2048 Hz. Hence, 0-256 Hz is the first two subbands and 256-512 is the third and fourth subbands. In the control process, when the SFXLMS algorithm is used, the step sizes of the third and fourth subbands should be larger than the ones of the first two subbands. In contrast, the step size of the FXLMS algorithm is the same for all frequencies, which means the step size is large enough to achieve good attenuation for the low frequency range, however, it is small for the high frequency range. If one further increases the step size for achieving more attenuation in the high frequency range, it will result in the algorithm diverge in the low frequency range.

Furthermore, once the multiple coherence function between the reference signals and the targeted noise is obtained, the maximum potential reduction by using feedforward controller alone can be estimated by using Equation (6.5). The solid grey line in Figure 6-8 shows the attenuation limit based on multiple coherence analysis. It is seen that the performance of the

116 proposed feedforward control system with SFXLMS algorithm is very close to the theoretical limit.

5dB Sound Pressure Level (dBA) Level Pressure Sound

0 100 200 300 400 500

Frequency (Hz)

Figure 6-7. Comparison of feedforward active noise control results between the subband FXLMS

algorithm and the conventional time -domain FXLMS algorithm. (Keys: solid black line ,

baseline road noise response; dashed blue line , subband FXLMS algorithm ; and dotted

red line , conventional time-domain FXLMS algorithm).

The second simulation involves using feedback controller alone and the control results are shown in Figure 6-9. The solid black line is the baseline and the dotted red line is the resultant noise when the feedback control is on. Figure 6-9 shows that the reduction is nearly invisible, because the control speaker is 20 inches away from the error microphone, which cause a large delay of the secondary path. The delay is the mai n factor limiting the bandwidth of reduction that feedback control can achieve. Therefore, it is difficult to achieve a promising

117 reduction for a broadband noise. If the control speaker is close r to error microphone, the control performance of feedback con troller will be improved eventually. However, it is not practical to place the error microphone close to the secondary loudspeakers.

5dB Sound Pressure Level (dBA) Level Pressure Sound

0 100 200 300 400 500

Frequency (Hz)

Figure 6-8. Comparison between feedforward road noise control result by u sing subband

FXLMS algorithm and potential maximum noise reduction. (Keys: solid black line ,

baseline road noise response; dashed blue line , subband FXLMS algorithm ; and solid

gray line , potential maximum noise reduction).

By observing Figure 6-7, the residual disturbance for the feedforward control alone has a narrow peak at the frequencies from 70 to 100 Hz. Even though the feedback control is difficult to attenuate a broadband noise because of plant delay, it has potential to reduce the narrow peak response. The third simulation is designed to reduce the road noise by activating both the feedforward and feedback controllers. In Figure 6-10, the dashed blue line is the control result by

118 using proposed combined feedforward -feedback control, while the dotted red line is the control resulting by using feedforward control alone. By comparing the two resultant line s, it is seen that the feedback part of the combined controller is able to suppress the relatively narrow spectral peak in the residual spectrum caused by the feedforward controller by 4 dBA. The post -test calculation shows that the combined controller not only give a more uniform residual spectrum, but achieve a further 0.9 dBA of overall reduction. As a result, the p roposed combined feedforward-feedback control yield more than 6 dBA reduction in the frequency range 100 -170

Hz and 3.8 dBA of overall reduction.

5dB Sound Pressure Level (dBA) Level Pressure Sound

0 100 200 300 400 500

Frequency (Hz)

Figure 6-9. Feedback active noise control result based on IM C architecture with FXLMS

algorithm. (Keys: solid black line , baseline road noise response; and dotted red line

, feedback-only control based on IMC architecture with FXLMS algorithm ).

119

5dB Sound Pressure Level (dBA) Level Pressure Sound

0 100 200 300 400 500

Frequency (Hz)

Figure 6-10 . Comparison of active noise control results between the proposed combined

feedforward-feedback control system and the feedforward only control system with subband

FXLMS algorithm. (Keys: solid black line , baseline road noise response; dashed blue

line , combined feedforward -feedback control system; and dotted red line ,

feedforward-only control system with subband FXLMS algorithm).

6.5 Conclusions

Active control of road noise using combined feedforward and feedback controller has been investigated in this study . Six accelerometers are chosen from twenty -one candidate accelerometers to give the good multiple coherence value between the reference signals and targeted road noise with minimum number of accelerometers. SFXLMS algorithm is used to update the feedforward adaptive filter to achieve more noise attenuation in a broader frequency

120 range, compared to the conventional FXLMS algorithm. Feedback control based on IMC architecture successfully suppress the narrow peak spectrum of road noise where cannot be attenuated by the feedforward control alone. The performance of the proposed combined controller has been studied through computer simulation with using actual data measured in the test vehicle. The simulation results show that the performance of combined feedforward- feedback control system is better than the performance by using feedforward and feedback controller separately. It is seen that the SFXLMS algorithm pushes the feedforward control part closing to the theoretical limit of reduction. Thus, to improve the performance of active road noise control system, it is critical to find a set of accelerometers that provide even better multiple coherence function between the reference signals and the targeted road noise.

121

Chapter 7. A Computational-Efficient Algorithm for Multichannel

Active Control of Road Noise

The conventional time-domain FXLMS algorithm, which widely used in ANC applications, experiences a large computational burden. This problem is more serious when the application requires multiple references, multiple channels, or high-order adaptive filters, such as the active road noise control system. In practice, the limitation of the computational power of the control system limits the number of the reference sensors, the number of control channels, or the order of the adaptive filters, which significantly degrades the overall noise attenuation performance. To address this problem, a computational-efficient ANC system for reducing the road noise inside the vehicle cabin is proposed in this chapter. The proposed system employs time-frequency-domain FXLMS algorithm, which calculates the gradient estimate in frequency- domain to reduce the computational complexity. The computational complexity analysis shows that the proposed algorithm can significantly reduce the computation requirement compared to the conventional FXLMS algorithm. Furthermore, because the gradient estimate of the proposed algorithm is normalized in each frequency bin, the proposed control system has potential to achieve more noise reduction compared to the conventional one. The performance of the novel computational-efficient active road noise control system is validated by a demonstration system that is composed of multiple reference accelerometers, two control loudspeakers, and two error microphones using experimental data. The results show that the proposed system not only reduces the computational complexity but also improves the noise reduction.

122

7.1 Introduction

To control road noise by using ANC, most of the researchers utilized the conventional time-domain FXLMS algorithm (Sutton et al. (1994), Bernhard (1995), Dehandschutter et al.

(1998), J. Courche et al. (1999), Oh et al. (2002) and Park et al. (2002; 2004)), which has heavy computational burden, especially for multi-reference multi-channel ANC systems. As discussed in Reference (Couche & Fuller, 1999), because of the computational limitation, the length of the adaptive filter must be reduced if more references are used. This reduction of the length of the adaptive filter can lead to a significant degradation in the overall performance of the noise control system. Similarly, Oh et al. (2002) limited the number of reference signals in their study, because of the limitation of calculation power. Therefore, to take full advantage of ANC system and/or reduce its cost, it is important to develop a computational-efficient control algorithm of active road noise control system.

Frequency-domain least mean square (LMS) was developed to reduce computational complexity (Dentino et al., 1978; Duan et al., 2009; Ferrara, 1980; Kuo et al., 1997; Mansour &

Gray, 1982; Shynk, 1992). One major disadvantage of such frequency-domain algorithm is a delay introduced into the signal filtering path, because of buffering process from signal samples into signal vectors. In ANC application, the delay is a factor that seriously limits the control performance. To avoid the signal path delay, while retaining the advantage of computational efficiency, time-frequency-domain FXLMS (TF-FXLMS) algorithm is proposed. The idea is similar to the delayless subband filter architecture by Morgan and Thi (1995) and frequency- domain filter architecture by Elliott (2001). In the proposed algorithm, both control filtering and filter coefficients updating are implemented in time-domain to minimize delay, however, the gradient estimate used to update the filter coefficients is calculated in frequency-domain to

123 reduce computational loads. The TF-FXLMS algorithm is then extended to multi-reference multi-channel approach and applied to active control of road noise. The computational complexity of the multi-reference multi-channel TF-FXLMS algorithm is analyzed and compared to that of the conventional FXLMS algorithm. The sensitivities of the system parameters to the computational saving by using TF-FXLMS algorithm compared to FXLMS algorithm are also discussed. Furthermore, since the gradient estimate of TF-FXLMS algorithm is normalized in each frequency bin, it has potential to achieve more attenuation compared to the conventional FXLMS algorithm.

This chapter is organized as follows. Firstly, the multi-reference multi-channel active road noise control system based on the conventional time-domain FXLMS algorithm is presented in Section 7.2. Secondly, Section 7.3 proposed the computational-efficient active road noise control system with TF-FXLMS algorithm. In Section 7.4, the computational complexity of the proposed TF-FXLMS algorithm is discussed and compared to the conventional FXLMS algorithm. Finally, simulation results of the active road noise control system are presented in

Section 7.5.

7.2 Multi-Reference Multi-Channel ANC System by using FXLMS Algorithm

The general configuration of the multi-reference multi-channel active road noise control

system by using the conventional time-domain FXLMS algorithm is shown in Figure 7-1. accelerometers are attached to vehicle body to detect the reference signals for the feedforward control system. Each reference signal is amplified by a battery charged pre-amplifier and then converted to a digital signal by A/D convertor. denotes the j-th reference signal. In the () active road noise control application, the reference signals can’t be simply summed or summed by weighting factors to form a single reference signal, otherwise some useful information will be

124 lost. Thus, each reference signal is individually processed in a multichannel ANC system. It is assumed that the adaptive controller generates cancellation signals to drive the () secondary control loudspeakers. Therefore, the controller is represented by a matrix that × each element is an adaptive finite impulse response (FIR) filter with coefficients. The cancellation signal, driving the m-th control loudspeaker before D/A convertor, is made up of the contributions from reference signals, , and expressed as (), (), ⋯ , ()

(7.1) () = ()() where represents the coefficient vector of () ≡ [,() ,() ⋯ ,()()] the L-th order adaptive filter at time ; is the reference signal index and is the secondary loudspeakers index; is the j-th reference signal () ≡ [() ( − 1) ⋯ ( − + 1) ] vector. error microphones have been placed in the passenger compartment to pick up the error signals. As shown in Figure 7-1, relates the loudspeaker control signals to the sound pressure response at the error microphones, which is defined as

⋯ (7.2) () = ⋮ ⋱ ⋮ ⋯ where is the impulse response function from the m-th secondary loudspeaker to the k-th error microphone. is named as secondary path. Hence, the cancellation sound at the k-th error microphone location, composed of the contribution from secondary loudspeakers, which can be expressed as

(7.3) () = () ∗ ()

125 where denotes linear convolution and is the cancellation sound at the k-th error ∗ () microphone position. By substituting Equation (7.1) into Equation (7.3), we can obtain

(7.4) () = () ∗ ()( − )

Accelerometers

Multiple road noise sources

J Pre-amplifier S M K A/D D/A () () () A/D KM channels

()

()JKM K LMS ()

Figure 7-1. Block diagram of the multi-reference multi-channel active road noise control system

with conventional FXLMS algorithm.

The error signal measured by the k-th error microphone can be expressed as

(7.5) () = () − ()

126 where is the primary road noise at the k-th error microphone, which is caused by () independent road noise sources and can be expressed as . () = ∑ () () represents the primary road noise at the k-th error microphone that is caused by the p-th noise

source.

The cost function of least mean square (LMS) adaptive algorithm is given by the sum of

the instantaneous squared errors (Haykin, 1996; Widrow & Stearns 1985), which is defined as

(7.6) () = () The LMS algorithm minimizes the cost function by updating the adaptive filter coefficient vector

individually in the negative gradient direction, given by () (7.7) ( + 1) = () − () In the above equation, represents the gradient vector, which is the derivative of () () with respect to the adaptive filter at time n. By using the Equations (7.1), (7.3) and (7.5), () the gradient can be expressed as

(7.8) () = −2 ( () ∗ ())() In practice, the impulse response function of secondary path, , is usually not available. () However, it can be modeled by an I-th order FIR filter using off-line system identification

method (Kuo & Morgan, 1996). The estimated secondary path is represented as . Then, ̂ () the filtered reference signal is obtained by filtering by the estimated secondary path () , which can be expressed as . ̂ () () ≡ [ () ( − 1) ⋯ ( − + 1) ]

127

Substituting Equation (7.8) into the adaptive filter update Equation (7.7) essentially yields the multi-reference multi-channel time-domain FXLMS algorithm, which is given by

(7.9) ( + 1) = () + ()()

7.3 Multi-Reference Multi-Channel ANC System by using TF-FXLMS

Algorithm

In this section, we proposed a multi-reference multi-channel ANC system based on TF-

FXLMS algorithm that can significantly reduce the computational complexity compared to the conventional FXLMS algorithm presented in the Section 7.2 above. The block diagram of the proposed control system for road noise is shown in Figure 7-2. In contrast to the traditional frequency-domain adaptive filtering (Shynk, 1992), the proposed system performs the filtering processing in the time-domain to minimize the delay of the controller that caused by the buffering process. The cancellation signals driving secondary loudspeakers are calculated by using the Equation (7.1), which is same as the time-domain FXLMS algorithm. However, the gradients (correlation between the filtered reference signals and the error signals) involved in adaption of the filter coefficients are computed in frequency-domain by taking computational advantages of the FFT process. In addition, the gradients are calculated in a lower rate, which is block by block compared to sample by sample in the time-domain algorithm. Consequently, the adaptive filter coefficients are updated in a lower rate, such that the computational complexity can be further reduced.

128

Accelerometers

Multiple road noise sources

J Pre-amplifier S M

A/D D/A K () () () A/D

KM channels ()

N-Data N-Data Drop the last N coefficients N-Zero N-Data

FFT () IFFT FFT () JKM

K CLMS () () () ()

Figure 7-2. Block diagram of the proposed multi-reference multi-channel active road noise

control system with TF-FXLMS algorithm.

A block size of N points is assumed to be equal to the length of the adaptive filter, . = The adaptive filter will be updated after the acquisition of every new block ( N-point) of () reference signals , where . Because FFT inherently perform () () = () () ⋯ () circular correlation, while the adaptive filter update process requires linear correlation, unconstraint overlap-save technique with 50% overlap is used in this study to covert circular

129 correlation into linear correlation (Mansour & Gray, 1982; Shynk, 1992). Hence, 2N -point FFT is employed. In each adaption, the new block of reference signals padded with the previous block data and the new block of error signals padded with one block of zeros are accumulated in the buffer separately to yield two vectors with 2N -point signal data, namely

(7.10) () = [( − ) ⋯ ( − 1) ( ) ⋯ ( + − 1)] (7.11) () = [0 ⋯ 0 ( ) ⋯ ( + − 1)] where is the block index. Then, these two signal vectors are transformed into the frequency domain by applying the 2N -point FFT routine. The resultant frequency spectra are given by

(7.12) () = () = ,() ,() ⋯ ,() (7.13) () = () = ,() ,() ⋯ ,() Similar to the time-domain FXLMS algorithm, to avoid the distortion caused the secondary path,

filtered reference signal is required. In the approach, the filtered reference signal is calculated in frequency-domain to further reduce the computational complexity, which is represented as

(7.14) () = () where is the frequency response of the secondary path from the m-th secondary loudspeaker to the k-th microphone, which can be obtained by taking the FFT process of off-line. Hence, the gradient in Equation (7.8) can be estimated by using the frequency-domain filtered reference

and error signals, which can be expressed as

() (7.15) = ()()()

130

denotes the block gradient estimate of the adaptive filter , presents the complex (∙) conjugate operation, and is the adjust factor of gradient estimate. Because there are only N coefficients of adaptive filter in time domain, the last N components of the gradient estimate are dropped. Generally, the gradient adjust factor is normalized according to the filtered reference signal power in that frequency bin. Thus, the adjust factor can be calculated as () = , where is vector that represents an power estimate of the jkm -th 1⁄ [ () + ] () filtered reference signal in each frequency bin and is a small positive constant that prevents being zero . In this study, the estimation of the signal power is done by using () () an exponentially weighted average (Florian & Bershad, 1988; Shynk, 1992; Sommen et al., 2003)

for each filtered reference signal, expressed by

(7.16) () = () + (1 − ) () Since the gradient estimate is normalized according to the filtered reference signal power, it can be considered as the step size is optimized in each frequency bin of each channel. Thus, the performance of the normalized TF-FXLMS algorithm is expected to have better performance than that of the conventional time-domain FXLMS algorithm, especially for controlling colored noise. Once the gradient estimate is obtained, the mj -th adaptive filter coefficients can be updated by

(7.17) ( + 1) = () + () Again, is the block index. Therefore, the each adaptive filter is updated block by block. 7.4 Computational Complexity Analysis

This section evaluates the computational complexity of the proposed multi-reference multi-channel TF-FXLMS algorithm as well as the conventional time-domain FXLMS algorithm.

131

This study is based on comparing the number of real multiplications and real additions used for producing N samples of adaptive filter output signal. In this analysis, the number of secondary

loudspeakers is M, the number of error microphones is K, the number of reference

accelerometers is J, the length of the adaptive filters is L ( ), and the length of the = secondary path filter is I. The cancellation signal is generated by adaptive filter using Equation

(7.1). To drive all M secondary loudspeakers, the computational requirement for the adaptive

filter output per sample is real multiplications and real additions. To update ( − 1) adaptive filters, filtered reference signals are required, which is calculated by using equation . This filtering process requires real multiplications () = ̂ () ∗ () and real additions. Furthermore, the adaptive filter update Equation (7.9) for all ( − 1) adaptive filters requires real multiplication and real additions. Therefore, to ( + 1) produce samples of output signals, the multi-reference multi-channel time-domain FXLMS algorithm requires real multiplications and + +( +1) ( − 1) + real additions in total. ( − 1) + The multi-reference multi-channel TF-FXLMS algorithm utilizes the same Equation (7.1) as the time-domain algorithm to produce the cancellation signals. However, the adaptive filter update process from Equations (7.12) to (7.17) is different. Since N is the block size that equal to the adaptive filter length, 2N -point FFT is used in Equations (7.12) and (7.13). Each 2N -point

FFT process requires multiplications and addtions. Hence, the 4 ∙ (2) 4 ∙ (2) transformation from time-domain to frequency-domain for all reference signals requires real multiplications and real additions by using Equation ∙ 4 ∙ (2) ∙ 4 ∙ (2) (7.12). Similarly, to obtain frequency-domain error signals, Equation (7.13) requires ∙ 4 ∙ real multiplications and real additions. The filtered reference (2) ∙ 4 ∙ (2)

132 signals are also calculated in frequency-domain by Equation (7.14). Thus, to obtain filtered reference signals, real multiplications and real additions are required (one ∙ 8 ∙ 8 complex multiplication requires four real multiplications and two real additions). The Equation

(7.16) is repeated for filtered reference signal, which requires real multiplications ∙ 8 and real additions. For updating each adaptive filter coefficients , Equation (7.15) ∙ 4 requires real multiplications and ∙ 12 + 4 ∙ (2) ∙8+4(−1)+4∙ real additions, while Equations (7.17) requires real multiplications and real (2) additions. Equations (7.15) and (7.17) will repeated times for adaptive filters. Therefore, to produce samples of output signal, the multi-reference multi-channel TF-FXLMS algorithm requires real multiplications and ( + + ) ∙ 4 ∙ (2) + ∙ ( + 28 + ) real additions. ( + + ) ∙ 4 ∙ (2) + ∙ ( + 20 − 3) − Normalized computational complexities (NCC) based on the number of real

multiplications or real additions are presented in Figures 7-3 to 7-6. Normalized computational

complexity is defined as a ratio of the total number of real multiplications or real additions of the

proposed TF-FXLMS algorithm to that of the conventional time-domain FXLMS algorithm.

Thus, if NCC is less than 1, the TF-FXLMS algorithm is more computationally efficient than the

conventional FXLMS algorithm, and vice versa. Figures 7-3(a) and 7-3(b) show NCC of real

multiplications and real additions, by using , and . = 6, = 256, = 256 = [1: 8] = [1: 8] It is seen that the computational benefits of the proposed TF-FXLMS algorithm become more

significant as the number of error microphone ( ) increases and it is less sensitive to the number of secondary loudspeakers ( ). In Figures 7-4 to 7-6, the line with up-triangle marker represents the NCC of real multiplications and the line with circle marker represents the NCC of real additions. Figure 7-4 shows the NCC with respected to varied number of reference

133 accelerometers ( ) and constant ( ). It has been seen , , , = 2, = 2, = 256, = 256 that increasing cannot further reduce the NCC significantly. Figure 7-5 analyzes the relationship between NCC and the adaptive filter length ( ). It is shown that increasing the adaptive filter length will slightly increase NCC, however, when is larger than a particular value, the increment of NCC become very small. Figure 7-6 shows the NCC with varied

secondary path filter length ( ) and constant ( ). It has , , , = 2, =2, =6, =256 been seen that the NCC decreases as increases, because the proposed TF-FXLMS algorithm calculates the filtered reference signal in frequency-domain.

0.35

1 0.3 0.8

0.6 0.25

0.4

0.2 0.2

0 8 6 8 0.15

Normalized computational complexity computational Normalized 4 6 4 2 2 0 No. of Error Mics 0 No. of Speakers

(a) Real multiplications

134

0.45

0.4 1

0.8 0.35

0.6 0.3 0.4 0.25 0.2

0 0.2 8 6 8 0.15 Normalized computational complexity computational Normalized 4 6 4 2 2 0 No. of Error Mics 0 No. of Speakers

(b) Real additions

Figure 7-3. Normalized computational complexities with M= [1:8], K=[1:8], J=6, L=256, I=256:

(a) Real multiplications; (b) Real additions.

1

0.8

0.6

0.4

0.2

0 0 2 4 6 8 10

Normalized Computational Complexity Computational Normalized No. of Reference Accelerometers

135

Figure 7-4. Normalized computational complexities with M=2 , K=2, J=[1:10], L=256, I=256 .

(Keys: solid line with up-triangle marker , real multiplications; and solid line with circle

marker , real additions).

1

0.8

0.6

0.4

0.2

0 0 500 1000 1500 2000 2500

Normalized Computational Complexity Computational Normalized Adaptive Filter Length

Figure 7-5. Normalized computational complexities with M=2 , K=2, J=6, L=[32, 64, 128, 256,

512, 1024, 2048], I=256 . (Keys: solid line with up-triangle marker , real multiplications;

and solid line with circle marker , real additions).

136

1

0.8

0.6

0.4

0.2

0 0 500 1000 1500 2000 2500

Normalized Computational Complexity Computational Normalized Secondary Path Filter Length

Figure 7-6. Normalized computational complexities with M=2 , K=2, J=6, L=256 , I=[32, 64, 128,

256, 512, 1024, 2048] . (Keys: solid line with up-triangle marker , real multiplications;

and solid line with circle marker , real additions).

In any cases shown in Figures 7-3 to 7-6, the proposed multi-reference multi -channel TF-

FXLMS algorithm significantly reduces the computational complexity compared to the conventional FXLMS algorithm.

7.5 Numerical Simulation

The effect iveness of the proposed multi -reference multi-channel active road noise control system based on TF-FXLMS algorithm is validated numerically to control actual vehicle road noise . For demonstration purpose, the ANC system composed of two secondary loudspeakers and two error microphones is designed to attenuate road noise around driver’ s head area , on the

137 test car with the V-6 engine. Two error microphones are placed at positions that are two inches from driver’s left and right ears, while two commercial loudspeakers are positioned at the left and right sides of driver’s headrest positions. The secondary paths from the input signals of the control loudspeakers to the output responses of error microphones are modeled by using FIR filters with 256 taps ( ) experimentally, before the active control is implemented. The = 256 frequency response functions of four measured secondary paths are plotted in Figures 7-7(a) and

7-7(b). A number of accelerometers are mounted on vehicle body to detect the reference signals.

The reference signals are amplified by a set of amplifiers and filtered by low-pass filters with cut-off frequency of 500 Hz. Primary road noise from two error microphones along with a set of reference signals are acquired by dSPACE (dSPACE, Inc.) and recorded on computer with a sampling frequency of 2048 Hz. The test vehicle is driven on a glen-eagle road surface at constant speed 30 mile per hour (mph). The simulation model is built by Matlab/Simulink (The

MathWorks, Inc.) as the system configurations of Figures 7-1 and 7-2.

40

20

0 Magnitude (dB) Magnitude -20 0 100 200 300 400 500 200

0

-200 0 100 200 300 400 500 Phase (Degree) Phase Frequency (Hz)

138

(a) ( ) and ( ) 40

30

20

10

0 Magnitude (dB) Magnitude -10 0 100 200 300 400 500 200

0

-200 0 100 200 300 400 500 Phase (Degree) Phase Frequency (Hz)

(b) ( ) and ( ) ) Figure 7-7. Magnitude and phase response s of the secondary path transfer functions : (a) S11

) ) ) ( ) and S21 ( ); (b) S22 ( ) and S12 ( ).

The feedforward control system requires the reference signals are well correlated with the targeted noise. Therefore, to achieve good attenuation, the selection of reference accelerometers is very important. In this study, multiple coherence function (Bendat & Piersol,

1980) and principle component analysis (Sutton et al., 1994) are used to determine the locations and number of the reference accelerometers. Multiple coherence function measures the linear relationship between the targeted noise and a set of reference signa ls. It can be used to predict the maximum sound pressure level reduction that a particular set of reference accelerometers is capable of achieving. Once the multiple coherence function is obtained, the maximum potential

139 noise reduction in decibels can be estimated as , where () = −10log (1 − :()) :() is the multiple coherence function. In general, more reference accelerometers is used, larger multiple coherence function value can be achieved. However, it will raise the computational complexity of the control algorithm, and eventually increase the cost of the control system.

Therefore, there is a tradeoff between the cost and performance of the control system. Based on principle component analysis, at least six reference accelerometers should be used to achieve a good multiple coherence function. However, it does not tell us where to place these six reference signals. The next step is to determine the locations of the six reference accelerometers. Before the experiment, we selected twenty-one possible candidate locations for the accelerometers.

Then, we recorded the twenty-one reference signals while the test vehicle was driving on the glen-eagle road at constant speed of 30 mph. A program is running off-line to search the best combination of six accelerometers out of twenty-one candidates, which provides the maximum multiple coherence value in the frequency range of interest, 60 – 400 Hz. Figures 7-8 (a) and 7-

8(b) show the multiple coherence functions between the selected set of reference signals and targeted road noise at left (error 1) and right (error 2) error microphones, respectively.

First simulation is conducted to compare the performance of road noise control system between the conventional FXLMS and the proposed TF-FXLMS algorithm. For both control algorithms, and . In all the control simulations, 40,960 =2,=2,=6,=256, = 256 samples (20 seconds) are used, while the last 8192 samples of error signals after the algorithms reach convergence are taken as the steady-state response. Figures 7-9(a) and 7-9(b) show the simulation results at error 1 and error 2. The solid black line is the baseline response of road noise when the active control system is off. The dashed and dotted red lines are the resultant responses when the control is on for the cases using the conventional FXLMS algorithm and the

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TF-FXLMS algorithm, respectively. As shown in Figure 7-9(a), the two algorithms have competitively reduction at error 1. However, in Figure 7-9(b), the proposed TF-FXLMS algorithm shows more reduction at higher frequency range (150-350 Hz) compared to conventional FXLMS algorithm at error 2. The reason is the TF-FXLMS algorithm normalizes the gradient estimate with respected to the inverse power of filtered reference signal for each frequency bin in each channel. The most important improvement is that the computational load of the TF-FXLMS algorithm can be reduced to 24% of that of the time-domain algorithm for real multiplications and 27% for real additions.

In practice, the number of reference accelerometer is limited by the calculation capability of the control system. The second simulation is designed to investigate the advantage of the proposed TF-FXLMS algorithm by applying to active control system of road noise with limited calculation power. It is assumed that the maximum number of real multiplications of the control system is 7000 per sample time (1/2048 second). Based on the computational complexity analysis in section 7.4, the proposed TF-FXLMS algorithm with six reference accelerometers

( ) requires 4476 real multiplications per sample time, = 2, = 2, = 6, = 256, = 256 which is much lower than the system capability of computation. However, the control system cannot support the conventional FXLMS algorithm with six reference accelerometers ( = ), because it requires 18432 real multiplications per sample 2, = 2, = 6, = 256, = 256 time. There are two approaches to reduce the computations by using the conventional FXLMS

algorithm. One approach is to reduce the order of the adaptive filter. The highest order of the

adaptive filter that can be supported by the control system is 16 ( =2,=2,=6,= ), which requires 6912 real multiplications. However, road noise is a broadband 16, = 256 noise, whose ANC system requires much higher order filter than 16. Otherwise, it is very

141 difficult to achieve noise cancellation by using such small number of adaptive filter coefficients.

The other approach is to reduce the number of the reference accelerometers. In fact, the maximum number of reference accelerometers that can be afforded by the control system is 2

( ), which requires 6144 real multiplications. In the = 2, = 2, = 2, = 256, = 256 simulation, the set of two accelerometers is chosen, which provides the maximum multiple coherence function value among all possible combinations. As shown in Figures 7-10 (a) and 7-

10(b), since the TF-FXLMS algorithm has more reference accelerometers, it can achieve much more reduction compared to the conventional FXLMS algorithm at both error microphones.

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Figure 7-8. Multiple coherence function between a set of reference signals and targeted road

noise: (a) Error 1; (b) Error 2.

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Figure 7-9. Comparison of active noise control results between using conventional FXLMS algorithm and proposed TF-FXLMS algorithm with six reference accelerometers : (a) Error 1; (b)

Error 2. (Keys: solid black line , baseline road noise response; dashed blue line ,

conventional FXLMS algorithm; and dotted red line , proposed TF-FXLMS algorithm).

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Figure 7-10 . Comparison of active control results between the conventional FXLMS algorithm

with two reference accelerometers and the TF -FXLMS algorithm with six reference

accelerometers: (a) Error 1. (b) Error (2). (Keys: solid black line , baseline road noise

response; dashed blue line , conventi onal FXLMS algorithm; and dotted red line ,

proposed TF -FXLMS algorithm).

7.6 Conclusions

This chapter presents the development of a computational -efficient active noise control system for reducing low-frequency road noise inside the passenger compartment. Computational complexity analysis shows that the proposed TF -FXLMS algorithm can significantly reduce the computational load compared to the conventional FXLMS al gorithm, especially for multi - reference multi-channel ANC system. The performance of the proposed computational -efficient active road noise control system has been validated by computer simulatio n using actual vehicle data. The simulation results show that the TF -FXLMS algorithm not only reduce the computational load, but has potential to achieve more noise attenuation compared to the conventional FXLMS algorithm. Even though the computational -eff icient ANC system is demonstrated by vehicle interior road noise application, it can be extended to other applications that require low computational complexity of the control algorithm.

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Chapter 8. Channel Equalization Algorithm for Multichannel Active

Control of Road Noise

To control low-frequency road noise inside large-dimension vehicle cabin, multichannel

ANC system is often required. The most common control algorithm for the feedforward multichannel ANC system is multi-reference multi-channel FXLMS algorithm. The convergence performance of this algorithm is limited by the variance of reference signals and the secondary paths. In practice, because of the arrangement and sensitivities of the reference sensors, secondary sources, and error transducers, the magnitude level of the reference signals and the secondary paths can be very difference. This difference will result in minimal attenuation at some error sensor locations, which degrades the overall performance of the multichannel ANC system. To address this problem, channel equalization algorithm is proposed to equalize the magnitude level of the reference signals and that of the main secondary paths without changing the coupling effects amongst the control channels. The effectiveness of the proposed channel equalization algorithm will be demonstrated through an ANC system with six accelerometers, two secondary loudspeakers, and two error microphones, to control actual road noise.

8.1 Introduction

To achieve larger control area in the three-dimensional vehicle cabin, multichannel ANC system that involves the use of multiple secondary sources and multiple error sensors is often used. In addition, a large set of reference sensors are required to give a good coherence between the reference signals and the primary noise to be cancelled. The most common control algorithm for the multichannel ANC system is the multichannel filtered-x least mean square algorithm

147

(Elliott et al., 1987; Elliott & Nelson, 1993; Elliott, 2001), which is an extension of the FXLMS algorithm (Kuo & Morgan, 1996).

One of the limitations of the multichannel FXLMS algorithm with multiple references is the channel dependent convergence behavior. Actually, the convergence performance of each channel depends on the corresponding reference signal and secondary paths (Chen et al., 2008;

Elliott et al., 1992). Thus, each channel will have its own optimal step size. However, the step size of the system must be chosen to guarantee that the system is stable for the channel that has the smallest optimal step size. If a small step size is chosen, error signals at different error sensors will converge at different speed. After some iteration, error signals will result in a certain sound pressure spectrum, where there is less sound reduction at some error sensors with faster convergence than those with slower convergence. This leads a significant degradation in the overall performance of the control system.

To address the channel dependent convergence behavior of multi-reference multi-channel

FXLMS algorithm, channel equalization algorithm is proposed in this study. Channel equalization algorithm intents to equalize the magnitude levels of the multiple reference signals and the main secondary paths, such that each channel will have similar convergence. Eventually, the overall performance of the multichannel ANC system can be improved. This approach is relatively simple to implement, without increasing the computational burden of the algorithm.

The effectiveness of the proposed approach will be demonstrated by employing a multichannel

ANC system with six reference accelerometers, two loudspeakers and two error microphones to control actual vehicle road noise.

This chapter is organized as follows. Firstly, the multichannel ANC system for vehicle road noise application is presented in the next section. Secondly, the channel equalization

148 algorithm for improving the performance of the multichannel ANC system is proposed in

Section 8.3. Finally, in Section 8.4, the proposed new implementation of multichannel ANC system is applied numerically to control the vehicle interior road noise.

8.2 Multichannel ANC System for Road Noise

The block diagram of the proposed multichannel active road noise control system with multiple reference sensors is shown in Figure 8-1. It is assumed that accelerometers are attached to vehicle body to detect vibration induced by the wheels. These vibration signals are

then used as the reference signals for the feedforward ANC control system. To achieve better

performance, feedforward control system requires the reference signals to be chosen that are well

correlated with the targeted road noise inside vehicle cabin. One way to employ multiple

reference sensors by multichannel ANC system is to form a single reference signal by combining

reference signals. However, this method may lose some useful information of the reference signals. In the ANC of road noise application, simply summing a set of reference signals into one

single reference signal can significantly degrades the coherence value, which will result in poor

noise attenuation. Thus, in this study, a more general method that each reference signal is

individually processed in a multichannel ANC system is employed. It is assumed that the

multichannel ANC system in Figure 8-1 contains secondary speakers and error microphones. Thus, the adaptive controller is represented by a matrix that each element × is a FIR with coefficients, as

() () ⋯ () (8.1) () () ⋯ () () = ⋮ () () ⋯ ()×

149

Accelerometers Multiple road noise s ources

J S M

K () (+) () +

KM channels

- + () Noise LMS Generator 2 () JKM K LMS 1 ()

Figure 8-1. Block diagram of the multichannel ANC system using conventional FXLMS

algorithm for treating road noise.

The secondary signal that drives the m-th control speaker is made up of the contributions from reference signals and expressed as (8.2) () = ()() where represents the coefficient vector of () ≡ ,() ,() ⋯ ,()() the mj -th adaptive filter at time n; m is the secondary speaker index; j is the reference signal

150 index, and is the j-th reference signal vector. () ≡ [() ( − 1) ⋯ ( − + 1) ] The Equation (8.2) can be written in a matrix form as

(8.3) () = ()() where and . relates the () ≡ [() () ⋯ ()] () ≡ () () ⋯ () secondary speaker control signals to the sound pressure response at the error microphones, which contains impulse response function. It can be expressed as ×

⋯ (8.4) ⋯ = ⋮ ⋯ × where denotes the response from the m-th secondary speaker to the k-th error microphone. Hence, the cancellation at error microphone positions, composed of the contribution from secondary speakers, can be expressed in matrix form as (8.5) () = ∗ () where , is the cancellation sound at the k-th error () ≡ [() () ⋯ ()] () microphone, with ; and denotes linear convolution. = 1, 2, ⋯ , ∗ The error signal vector measured by error () ≡ [() () ⋯ ()] microphone can be expressed as

(8.6) () = () − () where is the primary road noise vector, caused by multiple () ≡ [() () ⋯ ()] road noise sources; denotes the primary noise at the k-th error microphone. The cost () function of the LMS algorithm is defined by the sum of the instantaneous squared error (Haykin,

1996; Widrow & Stearns, 1985), which is expressed as

151

(8.7) () = () The LMS algorithm minimizes the cost function in Equation (8.7) by updating the adaptive filter coefficients of in the negative gradient direction, which is given by (8.8) ( + 1) = () − () 2 where is the step size that control the convergent speed of the adaptive algorithm; and () represents the gradient matrix, which can be expressed as

⋯ (8.9) ⋯ () = ∙ () ⋮ ⋯ × where represents the derivative operation with respect to the adaptive filter . By using the Equations (8.3), (8.5), and (8.6), each element in can be calculated as () (8.10) () = −2 ∗ ()() In practice, is normally unknown. However, the updating Equations (8.8) and (8.10) require the knowledge of the secondary path. Assuming that the characteristics of the secondary path are time-invariant, off-line modeling can be used to estimate before the active control. The procedures of off-line secondary path modeling are as follows, i) generating random noise and

used as the input to the m-th secondary speaker; ii) measuring the sound pressure response at error microphones; iii) modeling the each secondary path by an I-th order FIR filter, whose coefficients are updated using LMS algorithm. The process is repeated for each secondary speaker. The estimated impulse response of secondary path matrix is expressed as

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̂ ̂ ⋯ ̂ (8.11) ̂ ̂ ⋯ ̂ = ⋮ ̂ ̂ ⋯ ̂ × Therefore, the updating equation for FXLMS algorithm is given by

(8.12) ( + 1) = () + ()() where is the filtered reference signal () ≡ [ () ( − 1) ⋯ ( − + 1) ] vector, which is obtained by filtering the reference signal by the corresponding estimated () secondary path . ̂ 8.3 Channel Equalization Algorithm for Road Noise

In general, the convergent speed of the FXLMS algorithm is determined by the value of the step size. A larger step size can result in a faster convergence of the adaptive algorithm, however, further increasing step size may cause the algorithm unstable. In practice, the maximum step size can be used in the FXLMS algorithm is inversely proportional to the filtered reference signal power (Elliott & Nelson, 1993), which means weaker filtered reference signals can use a larger step size and stronger filtered reference signal have to use a smaller step size. In general, in the multi-reference multi-channel FXLMS algorithm, the filtered reference signal

is different from each other, which result in the different convergence of each channel. This is channel dependent convergence behavior of the multi-reference multi-channel FXLMS algorithm. To ensure the stability of the control system, the step size must be chosen based on the filtered reference signal that has the largest power. In this way, the channel that has the largest filtered reference signal power implements on its fast convergent speed, however, other channels have to experience slow convergent speed. Small reduction is expected at the channels

153 whose convergent speed is slow, which significantly degrades the overall performance of the

ANC system.

Generally speaking, errors in the estimation of the secondary path have an influence on the stability of the system. However, it has been shown that the phase errors of less that approximately have little effect on the system performance (Boucher et al., 1991) and the 45° magnitude errors can be compensated by tuning the step size that have no effects on the system

stability (Snyder & Hansen, 1990, 1994). To address the channel dependent convergence

problem, channel equalization algorithm is proposed, which equalizes the convergent speed of

each channel by adjusting the magnitude level of the reference signals and the secondary paths.

The updating equation of the FXLMS can be expanded as

( + 1) = () + [()() + ()() + ⋯ (8.13)

+ ()()] The maximum step size can be used to update adaptive filter is determined by the filtered reference signal that has the largest power amongst . It is that assumed that , , ⋯ , is the filtered reference signal with the largest power and is ∈ [, , ⋯ , ] named as main filtered reference signal of the corresponding adaptive filter . If the × main filtered reference signals have the same power, it is possible to choose a step size that is optimal to update adaptive filters simultaneously. The idea of the proposed channel × equalization algorithm is to equalize the mean magnitude of in the frequency range of interests. Since the is obtained by filtering the j-th reference signal through the corresponding estimated secondary path , the reference signals and secondary paths can be ̂ equalized separately.

154

Accelerometers

Multiple road noise sources J

() S M K () (+) () +

KM channels

- + () Noise LMS Generator 2 () JKM K LMS 1 ()

Channel Equalization

Figure 8-2. Block diagram of the proposed multichannel ANC system with channel equalization

algorithm for treating road noise.

The first step is to equalize the mean magnitude of the reference signals. Firstly, we take a period of time samples of reference signals , where () ≡ [() () ⋯ ()] () denotes the signal from the j-th accelerometer. Secondly, the reference signals are transformed

155 into frequency domain by using FFT transformation, and expressed as , ≡ [ ⋯ ] where represents the frequency domain coefficients of the j-th reference signal. The mean magnitude value of can be calculated as , [] = [] [] ⋯ [] where denotes the mean magnitude value of in the frequency range of interests. As shown in Figure 8-2, a gain vector is used to adjust the magnitude of the reference signals. The equalized reference signals can be calculated as

(8.14) () = [.× ()] where is the equalized reference signals; is () ≡ [ () () ⋯ ()] ≡ 1./[] the adjust gain vector; denotes the dot product operation; and is a reference value that can .× be set to any desired value. Usually, is set to unit value 1, or the mean magnitude of the reference signal that has the maximum mean magnitude amongst all reference signals. The second step is to equalize the mean magnitude of the secondary paths. After equalizing the reference signals, the Equation (8.13) can be rewritten as

( + 1) = () + ̂ ∗ ()() + ̂ ∗ ()() (8.15)

+ ⋯ + ̂ ∗ ()() As above discussion, the maximum step size of updating is determined by () = ̂ ∗ , which has the largest signal power amongst . is the () (), (), ⋯ , () ̂ secondary path that has the largest mean magnitude among . In fact, the ̂, ̂, ⋯ , ̂ secondary paths used in Equation (8.15) are the elements of the m-th column of ̂, ̂, ⋯ , ̂ the secondary path matrix in Equation (8.11). Thus, is the main path of the m-th column of ̂ , and it can be expressed as . The main path of the m-th column of is defined as the ̂ ̂ path from the m-th secondary speaker, which generates the largest cancellation sound on the i-th

156 error microphone among all error microphones, to the i-th error microphone. By examining all adaptive filters, the maximum step size of updating is determined by and × ̂ . Since the magnitude of has been equalized, to achieve the similar maximum step () () size for all adaptive filters, the magnitude of main secondary paths have to be equalized. × However, the magnitude of other secondary paths of should be changed correspondingly to keep the coupling effects among the channels unchanged.

represents the FFT transformation of the secondary path which can be calculated as

⋯ (8.16) = ⋯ = FFT ⋮ ⋯ × It is assumed that with is the main path of the m-th column of ̂ ̂ ∈ [̂, ̂, ⋯ , ̂ ] . Then, the magnitude of all main paths can be equalized to a reference value as

̂ ̂ ̂ ⋯ ̂ ̂ ̂ (8.17) ⋯ = ⋮ ̂ ̂ ̂ ⋯ × where is called virtual secondary path matrix, which contains modified impulse × responses of secondary paths; is reference value, which can be set to any desired value. Usually, is set to unit value 1, or the mean magnitude of the main secondary path that has the maximum mean magnitude amongst all main paths from column 1 to column ; with denotes the mean magnitude value of the corresponding secondary path in = 1, 2, ⋯ , the frequency range of interests. In this way, the mean magnitudes of the main secondary paths

157 of secondary path matrix are equalized to the reference value , while the mean magnitudes of other secondary paths in the column are adjusted correspondently to keep the coupling effects unchanged. Then, as shown in Figure 8-2, the virtual secondary path will replace the estimated secondary path of Figure 8-1. The proposed channel equation algorithm is further demonstrated by using a two-input two-output (2I2O) active road noise control system with six reference accelerometers. The reference signals and secondary paths are measured from a test vehicle. All four secondary paths are modeled by using off-line system identification method with 256-tap FIR filters (Kuo &

Morgan, 1996). The magnitude of original and normalized reference signals are shown in

Figures 8-3(a) and 8-3(b), respectively. By comparing the two figures, the equalized reference signals are closer to each other than the original ones. Figures 8-4(a) and 8-4(b) show the magnitude response of the measured secondary paths, which are and . and , , are the main paths of the first and second column of the secondary path matrix. By applying channel equalization algorithm, the main paths of the virtual secondary path are normalized to the mean value of , which means the reference value is set as . Thus, the = [ ] secondary paths of the first column of the virtual secondary path matrix stay the same as that of the estimated secondary path matrix ( and ), while the secondary paths = = of the second column of are modified as shown in Figure 8-5. It is seen that have the same magnitude level as .

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Figure 8-3. Magnitude spectrum s of original and equalized reference signals : (a) Original

reference signals; (b) Equalized reference signals.

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(b) ( ) and ( ) ) Figure 8-4. Magnitude responses of four estimated secondary path transfer functions : (a) S11

) ) ) ( ) and S21 ( ); (b) S22 ( ) and S12 ( ).

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Figure 8-5. Magnitude responses of virtual secondary path transfer functions by using channel

) ) equalization algorithm. (Keys: S ( ) and S ( )) . 22 12

8.4 Numerical Simulation

The proposed channel equalization algorithm applying to multichannel active road noise control system with multiple references is validated numerically by using recorded experimental data. Four-door sport utility vehicle (SUV) with the V-6 engine is used for the experiment. A global control is not attempted at this stage and the sound pressure reductions are mainly for the driver seat location. The frequency range of targeted noise reduction is from 80 to 250 Hz, where

161 has the largest noise energy. To effectively control the low-frequency road noise, a 2I2O ANC system is designed. As the secondary source, two loudspeakers are placed at the left and right sides of driver’s headrest position. As the error sensor, two microphones are positioned adjacent to the driver’s left and right ears. The estimated secondary paths and virtual secondary paths modified by channel equalization algorithm are shown in Figures 8-4 and 8-5, respectively, and discussed in the last section. The reference signals of the control system are the vibration acceleration detected by the accelerometers attached to the vehicle body, such as wheels, sub- frames, suspensions, and so on. In general, the reductions of feedforward ANC system are limited by the coherence function between the reference signals and the interior road noise signals. Therefore, the selection of reference accelerometers is very important. In this study, the number of the accelerometers is determined by principle component analysis (Sutton et al, 1994).

The locations of the accelerometers are determined by investigating the multiple coherence function (Bendat & Piersol, 1980), which measures the linear relationship between the targeted noise and a set of reference signals. It also can be used to predict the maximum sound pressure level reduction that a particular set of reference accelerometers is capable of achieving. Once the multiple coherence function is obtained, the maximum potential noise reduction in decibels can be estimated as , where is the multiple coherence () = −10log (1 − :()) :() function. After the analysis, six locations are selected that give the best multiple coherence value.

Figures 8-6 shows the multiple coherence function between the reference signals and road noise at left error microphone (Error 1) and right error microphone (Error 2). The actual vehicle data of both reference signals and microphone signals are measured when the vehicle is driving on the glen-eagle road at constant speed of 30 mile per hour (mph). The sampling frequency of the experiment is 2048 Hz.

162

1

0.8

0.6

0.4

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0 0 100 200 300 400

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Figure 8-6. Multiple coherence function s between a set of reference signals and targeted road

noises. (Keys: solid black line , error 1; and dotted blue line , error 2).

The simulation model is built by Matlab/Simulink (The MathWorks, Inc.) as the system configurations of Figures 8-1 and 8-2. The control filters used 256 coefficients. For the control simulations, 40,960 sample data corresponding to 20 seconds are us ed, while the last 8192 samples of error signals after the algorithms reach convergence are taken as the steady -state response. Figures 8-7(a) and 8-7(b) show the spectrums of the error microphone signals before and after control at the left and right erro r microphone, respectively. The step size for each implementation is set to the largest value while still maintaining the stability. The solid black line is the baseline response of road noise when the active control system is off , while the dashed blue and dotted red line s are the resultant responses when the control is on for the cases using the conventional FXLMS algorithm and the one with channel equalization technique, respectively. It

163 can be seen that the control system using conventional FXLMS algorithm yielded more reduction at error microphone 1 compared to error microphone 2. This is because the magnitude of is higher the magnitude of , which result in difference convergence of the adaptive filters. In contrast, from the results represented by the dotted red line in Figure 8-7, the FXLMS

algorithm with channel equalization algorithm can yield more balanced noise reduction between

the two error microphones. Also the improved algorithm produced 1-3 dB more reduction in the

control frequency from 110 -250 Hz compared to the conventional one at the error microphone 2.

However, both control methods yields invisible reduction around the frequency of 80 Hz. This is

because the reference signals are poorly correlated with the road noise around the frequency of

80 Hz, which can be investigating at Figure 8-6. It is the limitation of feedforward control system.

5 dB Sound Pressure Level (dBA) Level Pressure Sound

0 100 200 300 400

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(a) Error 1

164

5 dB Sound Pressure Level (dBA) Level Pressure Sound

0 100 200 300 400

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(b) Error 2

Figure 8-7. Comparison of active noise control results between using conventional FXLMS

algorithm and proposed channel equalization algorithm: (a) Error 1; (b) Error 2. (Keys: solid

black line , baseline road noise response; dashed blue line , conventional FXLMS

algorithm; and dotted red line , proposed channe l equalization algorithm) .

In some ANC applications, such as vehicle powertrain noise, the reference signal can be digitally synthesized by signal generator based on some frequency information (tachometer signal). Hence, the magnitude of the reference signals can be easily controlle d. However, in other applications, it is desirable to directly use the reference signal from its source, such as vehicle road noise. In practice, some vibration acceleration signals from the reference accelerometers can be much smaller than others , which significantly degrade the overall performance of ANC system , because the adaptive filters associated with small er reference signals converge much

165 slower than others, as we discussed in Section 8.3. The second simulation is designed that the magnitude of one reference signal is smaller than others. As shown in Figure 8-8, the reference signal from the first accelerometer, represented by the blue line, is assumed to be the one with smaller magnitude. Figure 8-9 shows the control results by using the ANC configuration of

Figures 8-1 and 8-2. It has been seen that the FXLMS algorithm with channel equalization algorithm can achieve much more reduction compared to the conventional one in the frequency range 130 -160 Hz at both error microphone 1 and 2. This is because channel equalization algorithm automatically adjusts the amplitude of reference signals to the save level, such that the adaptive filters associated with the first reference signal can have similar convergence as other adaptive filters. As shown in Figure 8-10, the first reference signal is well correlated with road noises at both error microphone positions in the frequency range 130 – 160 Hz, where it is capable to yield noise reduction.

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Figure 8-8. Magnitude spectrums of assumed reference signals.

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Figure 8-9. Comparison of active noise control results between using conventional FXLMS

algorithm and proposed channel equalizatio n algorithm by using assumed reference signals: (a)

Error 1; (b) Error 2. (Keys: solid black line , baseline road noise response; dashed blue

line , conventional FXLMS algorithm; and dotted red line , proposed channel

equalization algorithm).

8.5 Conclusions

A novel approach of multi -reference multi-channel FXLMS algorithm by using channel equalization algorithm has been demonstrated to control actual road noise. It has been shown that the proposed channel equalization algorithm, which intends to equalize the amplitude levels of reference signals and the main secondary paths, overcome limitation caused by the channel dependent property of the conventional multichannel FXLMS algorithm. This enhancement is able to balance the reduction at all error sensors, and i mprove the overall performance of the control system. The other advantage of the channel equalization is the simplicity of its implementation, which does not increase the computational complexity. Numerical simulation studies are conducted for vehicle inte rior road noise control using experimental data. Simulation results show that the channel equalization algorithm can lead 1 -3 dB additional attenuation at the error microphone that has less reduction by using the conventional FXLMS algorithm, when the magnitude response of physical secondary are quite different. The improvement of channel equalization algorithm becomes more significant when the magnitude levels of reference signals are also quite different.

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Chapter 9. Conclusions and Recommendations

9.1 Conclusions

The goal of this dissertation is to develop a feasible active noise control system for use inside the passenger compartment of vehicle. The proposed active noise control system is composed of reference sensors (tachometer for powertrain noise and accelerometers for road noise), error sensors (microphones), secondary sources (loudspeakers), and controller with advanced control algorithms. Several algorithms have been proposed to overcome the difficulties by using the conventional FXLMS algorithm. Detailed conclusions of the proposed control algorithms and systems are given at the end of each chapter. Here, we summarize the findings of the active control of powertrain and road noise.

In general, powertrain response can be categorized into two types of controllable signals: steady-state and transient responses. The steady-state powertrain response occurs when the engine runs on constant speed, while the transient powertrain response occurs when the engine runs on time-varying speed. In both cases, the powertrain response contains a large number of harmonics that are directly related to the engine speed. To achieve better performance, feedforward control is often used. Feedforward control system requires that the reference signal is well correlated with the targeted noise. To ensure good coherence between the reference signal and powertrain response, tachometer is used to estimate the current engine speed. Then, a reference signal is generated by a sine wave generator based on the estimated engine speed. The active control of steady-state powertrain response has been successfully applied in previous studies. However, there are some difficulties when active control is applied to transient powertrain response, because the amplitude and frequency of the transient powertrain noise vary

169 with time. The most important factor of controlling transient powertrain noise is the convergent speed of the control algorithm. The control algorithm with a fast convergent speed is able to better track the change of transient powertrain noise than the one with a slower convergent speed, such that the performance of the control system is improved. With this in mind, the key factor is to increase the convergent speed of the control algorithm. The most common control algorithm for ANC system is FXLMS algorithm. However, this algorithm suffers slow convergence, especially when the order of the control filter is long. In this dissertation, frequency-domain analysis is conducted to understand the physical mean of the convergent speed of the FXLMS algorithm. Then, virtual secondary path algorithm is proposed to increase the convergent speed of the FXLMS algorithm for the powertrain response control systems. Another advantage of the proposed virtual secondary path algorithm is that the algorithm is simple to implement without causing any computational burden.

There are other improvements made in this dissertation for powertrain noise control. The

FXLMS algorithm inherently has some nonlinear properties. One of them is called sinusoidal interference, which results in overshoot at some sinusoidal components (engine order response) that is initially targeted to be reduced. The overshoot problem usually occurs when the sinusoidal components are close to each other, which is when engine runs at lower speed. This problem plagues the use of the FXLMS algorithm to control multiple engine order responses simultaneously. One solution is to split the adaptive filter, such that the adjacent sinusoidal components of each reference signal can be spaced out farther apart. However, increasing the number of adaptive filter eventually increases the computational cost. The proposed twin-

FXLMS algorithm is to split the adaptive filter into two sets, which significantly suppresses the overshoot with a small increase in computational load.

170

The performance of ANC system can be limited by the calculation power of the hardware.

Therefore, a computational-efficient adaptive algorithm is desired. In chapter 5, time-frequency- domain algorithm is proposed, which required much less computations than the conventional

FXLMS algorithm. Two implementations of the TF-FXLMS algorithm are given. A reasonable reduction and tracking ability have been proved by the computer simulation. However, compared to the time-domain algorithm, the convergent speed for the harmonic like noise (powertrain noise) is relatively slow. Therefore, the reduction performance of TF-FXLMS algorithm is slightly worse than the time-domain algorithm, especially when the time-domain algorithm is enhanced by virtual secondary path algorithm. However, when there is calculation limitation of the hardware, TF-FXLMS algorithm is an alternative solution.

The noise attenuation of the ANC system also depends on the locations of the secondary loudspeakers and error microphones. The study on placement of the secondary loudspeakers and error microphones is beyond the scope of this dissertation. However, by using different set of loudspeakers, we found that more attenuation can be achieved by ANC system when the loudspeakers are closer to the error microphones. The reason is that smaller distance between secondary loudspeakers and error microphones cause less delay of the secondary path. It is seen that up to 25dB noise reduction can be achieved at engine firing order response when the headrest loudspeakers are used and microphone is placed on the ceiling over driver’s head.

Active sound tuning system has been proposed in this dissertation to tune the powertrain response according to the vehicle interior sound quality requirement rather than simply suppress it. Since powertrain response conveys useful information, as such engine power, engine speed, gear change and etc., it is not desired to cancel the noise completely. The performance of the active sound tuning system has been demonstrated by applying to both steady-state and transient

171 powertrain noise. Although only one enhanced and one reduced orders are exemplified, the proposed system is capable of handling more engine orders simultaneously. Also, active sound tuning system can use time-domain, frequency-domain, or subband adaptive algorithms. This technology has the potential to have great impact on the vehicle industry. As the demand for fuel efficiency increases, small, hybrid, and will be more and more popular. As we know it, small cars have relative small engines; hybrid cars partially use its engine; and electric cars do not even have an engine. The consequence is that all of these cars are not able to produce a powerful dynamic sound during acceleration. To achieve a sporty sound by these cars, active sound tuning system has the potential to produce a virtual sound through audio system.

Compared to powertrain noise, road noise is more difficult to be controlled through ANC system, due to its random nature and multiple noise sources. Since road noise is more fatiguing and irritating than having benefit, it is desired to attenuate it in broad frequency range. To achieve the broadband noise control, the feedforward control system is often used. As discussed in chapter 6, the maximum potential noise reduction of sound pressure level is directly related to the coherence value. It means that large coherence value can yield more reduction. Thus, the most important factor of controlling road noise is to find a good reference signal. Noise road is induced as the tire treads meet random irregularities in the road surface. The most common method to detect noise sources is to attach accelerometers to the vehicle body, such as wheels, suspensions, and other vehicle body structures. In general, road noise has many noise sources.

Hence, to detect all independent sources of the noise, many accelerometers are often required. In this dissertation, multiple coherence function and principle components analysis are used to determine the optimal number and placement of the reference accelerometers. Six accelerometers along with their optimal locations are chosen based on these methods.

172

The subband FXLMS algorithm has been proposed to replace the conventional FXLMS algorithm for road noise control. Compared to the conventional FXLMS algorithm, the subband

FXLMS algorithm has two advantages: i) faster convergent speed for colored noise control; ii) less computational requirement. As shown in the simulations, the subband FXLMS algorithm attenuates road noise in a broader frequency range compared to the conventional FXLMS algorithm and pushes the noise reduction by using feedforward control close to the theoretical limit. Thus, to improve the performance of active road noise control system, it is critical to find a set of accelerometers that provide even better multiple coherence function between the reference signals and the targeted road noise.

Feedback control approach is another control strategy for ANC system. Since the bandwidth of reduction by using feedback control approach is limited by the delay of secondary path, it is difficult to be used to control broadband noise. However, the feedback control approach can be combined with the feedforward control approach to obtain a hybrid system. In this way, the feedback control has potential to attenuate the noise components that has poor coherence value and cannot be reduced by the feedforward control alone. As shown in the simulation in chapter 6, the performance of combined feedforward-feedback control system is better than the performance by using feedforward and feedback system separately. As a result, the proposed combined feedforward-feedback control yields more than 6 dBA reduction in the frequency range 100-170 Hz and 3.8 dBA of overall reduction.

As we discussed previously, the conventional FXLMS algorithm causes heavy computational burden, especially for multi-reference multi-channel channel system. Road noise control system is one of the applications that require multiple references and multiple channels.

Because of the limitation of computational power of control hardware, previous studies often

173 reduce the order of the adaptive filter or the number of accelerometers. However, either method significantly reduces the performance of the ANC system. One feasible solution is to develop a computational-efficient adaptive algorithm. Chapter 7 proposed a multi-reference multi-channel

TF-FXLMS algorithm for road noise control. As the demonstration control system (two loudspeakers, two error microphones, and six accelerometers), the computational complexity of the TF-FXLMS algorithm can be reduced to 24% of that of the time-domain algorithm for real multiplications and 27% for real additions. In contrast to powertrain noise control, TF-FXLMS algorithm for road noise control will not sacrifice the attenuation performance. In fact, more reduction is expected by using TF-FXLMS algorithm compared to conventional FXLMS algorithm, because the gradient estimate can be adjusted for each frequency bin.

Channel equalization algorithm is proposed to improve the performance of multichannel active noise control system. In practice, because of the arrangement and sensitivities of the actuators (speakers), transducers (microphones) and physical environment, the magnitude of the main secondary path transfer functions can be very different from each other. This difference will cause difficulty in the overall convergence of the algorithm, which will result in minimal attenuation at some error sensors. The algorithm can be used to balance the reduction at multiple error microphones, in order to improve the overall performance of the control system. Another advantage is that the implementation of the channel equalization algorithm is relatively easy. It is implemented off-line without causing any calculation burden. This algorithm has been successfully applied to both powertrain and road noise control.

Although most of the proposed control algorithms and system are developed for use inside passenger cabin to control powertrain and road noise, they can be extended to control other narrowband or broadband noise in broad range of applications.

174

9.2 Recommendations for Future Studies

The following work has been proposed for the future scope of this dissertation:

• Developing an active sound quality control system (ASQC) for use of vehicle cabin. It is

well known that the perceived quality of sound is not only related to sound pressure level,

but also to the temporal and spectral characteristic of the sound. A sound perceived as

having good quality does not necessarily imply possessing low sound pressure level. The

reason is that the environment, context and expectation play a significant role in

determining the perceived quality. Active sound quality control approach can be an

effective way to enhance the sound quality adaptively.

• Developing active sound quality control algorithm by embedding the psychoacoustic

models into the control algorithm. For example, A-weighted sound pressure level is one

of the attributes of sound quality used widely to predict the response of human auditory

system. The A-weighted sound pressure level is obtained by adjusting the amplitude in

frequency domain based on the original sound pressure level. The idea is to use the A-

weighted sound pressure instead of the original sound pressure to form the cost function,

such that the adaptive algorithm is intended to reduce the A-weighted sound pressure

level rather than the linear sound pressure level. To do so, the advanced sound quality

control algorithms will be developed based on the TF-FXLMS or subband FXLMS

algorithms. Hence, the sound pressure that is sensed by the error microphone can be

easily adjusted in frequency domain or in subband domain based on the A-weighting

function. Similarly, other spectral-based sound quality metrics, such as loudness,

sharpness, roughness etc., will also be implemented to reduce the value of these metrics,

such that the overall sound quality can be improved.

175

• Investigating the effect of noise masking on the active sound quality control system.

Because of the temporal or/and spectral masking, the reduction at one acoustic noise

component may cause another acoustic noise component more audible. Therefore, it is

important to investigate how noise masking affects the performance of the proposed

ASQC system. Then, a strategy will be developed to maximize the ASQC performance

by benefiting from the noise masking phenomenon. Also, the proposed ASQC system can

be used to generate the desired noise masking sound. In some cases, if the acoustic noise

components cannot be reduced by either passive control or active control techniques, the

ASQC system can generate a small amount of pleasing sound to mask the undesired noise

components such that the overall sound quality can be improved.

• Developing a virtual sensor technique for active control of vehicle powertrain and road

noise. Since most of active control system is designed to treat the sound at the

microphone location and it is sometimes impractical to place the microphone close to the

ears of the driver and the passengers, the ANC system may not be optimized to work well

enough. The virtual sensor technique is a possible solution to control the noise at desired

location by using remote error sensors (microphones).

• Combining active noise control with active vibration control for road noise. Active

vibration control can be used to control the main transfer path to reduce the structure-

borne road noise transmission. The idea is to mount actuator parallel to the bushing and

generate cancellation to attenuate the vibration. Then, the active road noise control

system and active vibration control system can be combined effectively to maximize the

noise reduction in the passenger compartment of vehicle.

176

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