Active Control of Impulsive Noise Using Reference Weighted Fxlms Algorithm

Active Control of Impulsive Noise using Reference Weighted FxLMS Algorithm

A thesis submitted to the

Graduate School

of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE

in the Department of Mechanical and Materials Engineering

of the College of Engineering and Applied Science

2017

Rushikesh Ajay Dhakad

Bachelor of Technology, Mechanical Engineering,

Veermata Jijabai Technological Institute, Mumbai, India – 2013

Committee Chair: Teik C. Lim, Ph. D.

Committee Members: Jay H. Kim, Ph. D.

David F. Thompson, Ph. D.

Abstract

Active noise control (ANC) has been an attractive solution for automobile and aerospace industry for interior noise reduction. ANC has been successfully incorporated in industry such as the automotive industry while being subject to several constraints like maximum frequency of cancellation, and evaluation criteria like convergence time, stability, mean noise reduction (MNR) etc. While engine order cancellation and road noise (which is a broadband random noise) control have been thoroughly researched upon and hence considerably commercialized as solutions, a solution for impulsive noise yet remains elusive. This is primarily due to the difficulty in ensuring a stable performance, need to extract a number of complicated parameters for incorporating into the algorithm, high computational complexity and so on. This thesis thus attempts to present a simplified solution in the form of a reference-weighted filtered reference least mean square

(RWFxLMS) algorithm for active control of impulsive noise which does not compromise on performance metrics like stability, convergence time, mean noise reduction, etc.

The first chapter chronicles briefly literature available in the ANC research area in general and later focuses on work done for impulsive noise ANC. The second chapter, while briefly introducing the existing algorithms that inspired the development of the proposed RWFxLMS algorithm, describes the concept of the proposed algorithm. Performance of the proposed algorithm against a variety of impulsive noise data sets is studied while noting observations regarding its convergence and cancellation. The third chapter goes on to categorize previously available algorithms for impulsive noise ANC into three categories and later compares the computational complexity and convergence performance (rate of convergence and error at

convergence) of the proposed RWFxLMS algorithm with the best algorithm from each category.

It also includes a discussion about practical considerations while performing impulsive noise ANC.

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Acknowledgements

I would like to express my gratitude to my supervisor and committee chair Dr. Teik Lim for allowing me to pursue my research for master’s thesis in the active noise control area. I thank him for his continual support and encouragement throughout the research. I also thank Ford Motor

Company and MentorGraphics for their financial support for this research.

I thank my thesis committee members Dr. Jay Kim and Dr. David Thompson for their strict examination and insightful suggestions, which are helpful for the completing this thesis satisfactorily.

I thank Dr. Tao Feng for guiding me from start till end of this study. His critical comments and suggestions ensured that I stay on the right track and complete this study in a timely manner.

I thank Mr. Guo Long, who is a wonderful researcher himself, for being an excellent senior co- worker in the Vibro-Acoustics Lab at University of Cincinnati and for being the peer with whom

I could discuss my queries and ideas related to active noise control. I also thank him for critically reviewing the chapters of this thesis.

I thank the University of Cincinnati and College of Engineering and Applied Sciences for the

Graduate Scholarship and for providing a healthy atmosphere for learning.

I am grateful for the knowledge and wisdom about ANC gained at HALOsonic, Harman

International, Novi during my summer internship in 2017. I am hugely thankful to Mr. Jonathan

Christian, who shared his vast experience of implementing ANC in automobiles, helped me develop a more practical approach of developing ANC algorithms and implementation and also reviewed the ‘practical considerations’ section of this thesis. I also thank Mr. Kevin Bastyr at

Harman, for sharing his insights about ANC from a psychoacoustic perspective.

I thank my roommates here with whom I shared apartments in Cincinnati, for their friendship, help and support. I thank all friends made in the university through groups like IPALs, AID,

BhaktiYoga Club, etc. for making life great on campus.

Finally, I thank my family for their constant love and cheer-leading despite staying 9.5 hours apart on the globe.

Contents

Abstract ...... i Acknowledgements ...... iv Abbreviations ...... xii 1 Introduction ...... 1 1.1 ANC and its relevance in general ...... 1 1.2 First patent, work of Olsen and May and the FxLMS algorithm ...... 1 1.3 Periodic noise (harmonic, engine order, etc.) and random noise (road noise, etc.) ANC ..... 3 1.4 Random and repetitive impulsive noise ...... 4 1.5 Problem statement and outline of concept of proposed algorithm ...... 8 2 Reference Weighted FxLMS Algorithm for Active Control of Impulsive Noise ...... 11 2.1 Introduction ...... 11 2.2 Algorithm development...... 16 2.2.1 Existing NLMS, modified FxLMP and FxGSNLMS ...... 16 2.2.2. Proposed Reference Weighted FxLMS (RWFxLMS) algorithm ...... 18 2.3 Analysis ...... 26 2.3.1 Convergence ...... 26 A. Optimal step size and maximum step size ...... 26 B. Observations related to convergence ...... 28 2.3.2 Computational Complexity...... 30 2.4 Numerical Simulation ...... 31 2.4.1 Random Impulsive Noise case ...... 33 A. Symmetric α-Stable Impulsive Noise model ...... 33 B. Contaminated Gaussian model ...... 41 2.4.2 Repetitive Impulsive Noise case...... 45 A. A sequence of 25 impulses synthesized as the reference excitation ...... 45 B. Repetitive periodic impulses (50 impulses) mixed with background white noise ...... 46 2.5 Conclusion ...... 47 3 Comparative Study of Several Adaptive Algorithms for Impulsive Noise Control ...... 48 3.1 Introduction ...... 48 3.2 Various adaptive algorithms for impulsive noise control ...... 50 3.2.1 Leahy’s Filtered Reference Least Mean P-Norm (FxLMP) Algorithm ...... 50

3.2.2 Akhtar’s Modified Normalized FxLMS (NFxLMS) Algorithm ...... 51 3.2.3 G. Sun’s Enhanced FxLMM Algorithm ...... 52 3.2.4 Y. Zhou’s FxGSNLMS Algorithm ...... 55 3.2.5 Proposed RWFxLMS Algorithm ...... 56 3.3 Results and comparison ...... 59 3.3.1 Computational Complexity...... 59 3.3.2 Convergence ...... 62 A. Symmetric-α-Stable noise (α = 1.5)...... 63 B. Contaminated Gaussian ...... 64 C. Pure repetitive impulses ...... 66 D. Repetitive impulses with background noise ...... 67 E. Weighting of error signal in combination with weighting of reference ...... 68 3.3.3 Tuning parameters ...... 70 3.4 Conclusion ...... 71 4. Conclusion ...... 72 4.1 Practical Considerations ...... 72 4.2 Conclusions ...... 76 4.3 Future Work ...... 77 References ...... 78

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Figures Figure 1 FxLMS algorithm for feedforward active noise control (block diagram) ...... 2

Figure 2 PDF of standard symmetric α-stable process for various α values ...... 12

Figure 3 Exploded view of the tail of PDF of standard symmetric α-stable process for various α values ...... 12

Figure 4 PDF of standard symmetric α-stable process for various β values ...... 13

Figure 5 PDF of standard symmetric α-stable process for various γ values ...... 14

Figure 6 PDF of standard symmetric α-stable process for various δ values ...... 14

Figure 7 Gaussian PDF and its quadratic approximation curve ...... 19

Figure 8 Comparison of convergence of FxGSNLMS which uses Gaussian weight function with that using quadratic approximation of Gaussian PDF as weight function ...... 19

Figure 9 Error signal of FxGSNLMS which uses Gaussian weight function and that using quadratic approximation of Gaussian PDF as weight function ...... 20

Figure 10 Symmetric-α-stable PDFs (α = 1.5) and their quadratic approximation curves; (a) range of x: -5:5; (b) range of x: -500:500 ...... 23

Figure 11 Comparison of convergence of FxGSNLMS which uses Gaussian weight function with that using quadratic approximation of s-α-s PDF as weight function ...... 24

Figure 12 Error signal of FxGSNLMS which uses Gaussian weight function and that using quadratic approximations of s-α-s PDF as weight function ...... 24

Figure 14 Feedforward active noise control configured with the RWFxLMS algorithm ...... 31

Figure 15 Primary path P(z) (solid line) and secondary path S(z) (dotted line) models: (a) Impulse response function; (b) Frequency response function ...... 32

Figure 16(a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.5 ...... 35

Figure 17(a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.6 ...... 35

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Figure 18 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.7 ...... 36

Figure 19 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.8 ...... 37

Figure 20 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.9 ...... 38

Figure 21 Performance comparison of the impulse noise (α = 1.6) in the error signal ANC OFF and ANC ON: (a) time domain; (b) frequency domain...... 40

Figure 22 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise generated using Contaminated Gaussian model – case 1 ...... 42

Figure 23 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise generated using Contaminated Gaussian model – case 2 ...... 43

Figure 24 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise generated using Contaminated Gaussian model – case 3 ...... 44

Figure 25 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for repetitive impulsive noise without any background white noise ...... 45

Figure 26 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for repetitive impulsive noise mixed with background white noise ...... 46

Figure 27 (a) Primary noise; (b) Convergence (MNR) using FxLMS algorithm ...... 49

Figure 28 SISO ANC system (block diagram) for impulsive noise with G. Sun’s enhanced FxLMM algorithm ...... 53

Figure 29 SISO ANC system (block diagram) for impulsive noise with Y. Zhou’s FxGSNLMS algorithm ...... 55

Figure 30 Symmetric-α-stable PDF (α = 1.5) and its quadratic approximation curves; range of (a) x: -5:5 and (b) x: -500:500 ...... 57

Figure 31 SISO ANC system (block diagram) for impulsive noise with proposed RWFxLMS algorithm ...... 58

Figure 32 (a) Normalized computational complexity of different impulsive noise control algorithms against varying length of adaptive filter; (b) zoomed-in plot (a) to show separation between curves for FxLMS and RWFxLMS ...... 61

Figure 33 Comparison of convergence curves obtained using the various ANC algorithms under consideration and proposed RWFxLMS algorithm for s-α-s impulsive noise with α = 1.5 ...... 63

Figure 34 Comparison of convergence curves obtained using the various ANC algorithms under consideration and proposed RWFxLMS algorithm for contaminated Gaussian noise (rimp = 100; pr = 0.005) ...... 64

Figure 35 Comparison of convergence curves obtained using the various ANC algorithms under consideration and proposed RWFxLMS algorithm for contaminated Gaussian noise (rimp = 10; pr = 0.0005) ...... 65

Figure 36 Comparison of convergence curves obtained using Y. Zhou’s FxGSNLMS algorithm and proposed RWFxLMS algorithm for pure repetitive (periodically occurring) impulsive noise ...... 66

Figure 37 Comparison of convergence curves obtained using the various ANC algorithms under consideration and proposed RWFxLMS algorithm for repetitive (periodically occurring) impulsive noise mixed with background Gaussian (white) noise ...... 67

Figure 38 SISO ANC system (block diagram) for impulsive noise with proposed RWFxLMS algorithm reference-weighting of step-size combined with error weighting ...... 68

Figure 39 Comparison of mean noise reduction (MNR) of the proposed RWFxLMS algorithm and its error-weighted-cum-reference-weighted modification for s-α-s impulsive noise with α = 1.1...... 69

Figure 40 (a) Synthesized reference signal; (b) Reference signal recorded using accelerometer mounted on a hub in a car chassis ...... 73

Figure 41 Loudness in sone and loudness level in phon versus 2kHz tone-burst duration [Images of plots taken from (Zwicker, 2013)] ...... 75

Tables

Table 1 Total number of operations per iteration in current simulation study for each of the algorithms used for comparison with proposed RWFxLMS ...... 61

Table 2 Time taken to complete 5 × 104 iterations by each impulsive noise ANC algorithm in MATLAB® ...... 62

Table 3 Comparison of impulsive noise ANC algorithms based on tuning parameters ...... 70

Abbreviations

ANC: active noise control

CG: contaminated Gaussian

FFT: fast Fourier transform

FIR: finite impulse response

FRF: frequency response function

FxGSNLMS: filtered reference (x= reference signal) generalized step-size normalized least mean- square

FxLMM: filtered reference least mean m-estimate

FxLMP: filtered reference least mean p-norm

FxLMS: filtered reference least mean square

FxRMC: filtered reference recursive maximum correntropy

IID: independently identically distributed

ILC: iterative learning control

IRF: impulse response function

LMK: least mean kurtosis

MCC: maximum correntropy criterion

MG-FxRLS: modified gain filtered reference recursive least square

xii

MIMO: multi-input multi-output (in ANC case, multiple speaker, multiple error microphones)

MNR: mean noise reduction

NFxLMS: normalized filtered reference least mean square

NLMS: normalized least mean square

NR: Net Reduction

NSS-FxLMS: normalized step-size filtered reference LMS

NVH: noise, vibration and harshness

PDF: probability distribution function

RC: repetitive control

RI: recursive inverse

RPM: rotations per minute

RWFxLMS: reference-weighted filtered reference least mean square

SA: sign algorithm

SNR: signal-to-noise ratio

S-α-S: symmetric α-stable

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1 Introduction

1.1 ANC and its relevance in general

Active noise control (ANC) (Kuo S. M., 1994) which works on the principle of destructive interference of acoustic waves, has managed to be an attractive tool for noise cancellation at least for the low-frequency audible ranges in sectors like aerospace, automobiles, headsets, etc. Passive control techniques like using sound-absorbing materials, damping material, increasing stiffness, adding mass only lead to increased weight. By replacing these techniques with ANC, the automobile sector in particular benefits by saving significant amount of weight and associated cost, and, needless to say, lighter vehicles give higher fuel efficiency. Rapid advance in semiconductor technology has also made mass production and implementation of a compact controller-cum- amplifier (termed as simply ‘amplifier’ in the supplier circles) possible.

1.2 First patent, work of Olsen and May and the FxLMS algorithm

Paul Leug in (United States Patent No. 2043416 , 1936) laid down the principle of active noise control using the technique now identified as ‘feedforward’ control, wherein the plane-wave acoustic signal detected by a microphone in a duct is electrically modified and send out through a secondary source (a speaker) such that it is exactly out of phase with the detected signal and hence due to destructive interference, leads to cancellation and hence silence. Later in the 1950’s, Olson and May demonstrated and published their misleadingly titled ‘electronic sound absorber’ (Olson,

1953), (Sun G. , 2013). In contrast to Leug’s technique, this works on ‘feedback’ control principle.

A manually adaptive feedforward control for transformer noise was proposed by Conover

(Conover, 1956). This was perhaps for the first time that a non-acoustic signal – here the transformer voltage, full-wave rectified and band-pass filtered – was used as a ‘reference’ signal.

The adaptive noise cancellation least mean square (LMS) algorithm introduced for electrical signals (Widrow, 1960) is now being widely applied in the acoustic noise cancellation in the form of filtered reference LMS (FxLMS) algorithm. To the problem for compensation of secondary path – electroacoustic path between secondary speaker to the ‘error’ microphone - one of the solutions proposed by (Morgan, 1980b) was to place the estimated secondary path in the weight update path of the reference signal, apart from the hypothetical solution of inversion of secondary path which might also lead to instability. The FxLMS algorithm was independently derived by

Widrow (B. Widrow, 1981) for adaptive control and Burgess (Burgess, 1981) for ANC applications.

Figure 1 FxLMS algorithm for feedforward active noise control (block diagram)

1.3 Periodic noise (harmonic, engine order, etc.) and random noise (road noise, etc.) ANC

Discussing further the applications of ANC in vehicle NVH (noise, vibration and harshness) control, we come across research work modifying the FxLMS algorithm in most cases, dealing with the vehicle interior noises such as engine boom and engine orders, basically related to the powertrain (Bravo, 1999); (Inoue, 2004); (Li M. F., 2009); (Duan J. L.-R.-T., 2009); (Duan J. L.,

2011);), and noise related to the tire-road interaction, i.e. road noise (Elliott S. J., 1990); (Sutton,

1994); (Sano, 2001); (Dehandschutter, 1998); (Duan J. L., 2011); (Sun, Li, & Lim, 2012);

(Belgacem, 2012). These applications differ in the nature of noise (engine noise is harmonic while road noise is random) and hence in the type of reference signals in the adaptive feedforward framework, type of algorithm, etc. required to implement ANC. Engine noise being synchronized with the speed of the engine, engine RPM (rotations per minute) signal or raw crank signal is used as reference for engine noise cancellation. Road noise control algorithms use acceleration signals acquired by mounting accelerometers on chassis locations like sub-frame, turret, etc. as reference.

ANC has been applied to reduce vehicle exterior noise as well in the form of active mufflers

(United States Patent No. 5,097,923, 1992), (Hansen, 1996), (Garabedian, 2001).

In order to successfully implement active noise control, engineers have to work within some serious practical constraints. Detailed study of acoustic mechanism involved in active noise control has been done by Elliot (Elliott S. J., 1990), (Elliott & Nelson, 1993). In (Elliott S. J., 1990), it is stated that up to 10 dB reductions in the primary noise in the zone of quiet are produced around the monitor microphone within only one-tenth of the wavelength of frequency. Also, the sources of primary noise and cancelling noise (control loudspeaker) should not be more than one-tenth of

the smallest wavelength of cancellation apart to achieve a 10 dB power attenuation. This practically limits the highest frequency that can be satisfactorily cancelled to create a sizeable zone of quiet.

Smallest frequency in the human hearing range being 20 Hz, the maximum distance of separation from noise source to control source is limited to 1.7m. In a feedforward type of adaptive algorithm implementation, the reference signal must be the precise cause of the section of the sound field that is cancelled by the controller, or in other words, the section of the reference signal that is being processed by the controller to generate cancelling sound at a given point of time should be correlated with the noise at that point to ensure ‘causality’.

1.4 Random and repetitive impulsive noise

Majority of the research and even commercial ANC systems are focused on stationary noise which following Gaussian distribution. ANC for transient noise too has caught attention. Unwanted impact noise at industrial and construction locations like that of pile drivers, punching machines, as well as impact noise due to road cracks, speed-breakers or gaps between concrete blocks are potential problems that ANC can solve. Impact noise can be either repetitive or randomly occurring pulses.

For repetitive (periodic) impact noises, iterative learning control (ILC) and/or repetitive control

(RC) algorithms have been applied by Pinte et al (Pinte G. , 2007); (Pinte G. B., 2009); (Pinte G.

D., 2007); (Pinte G. S., 2010); (Stallaert, 2010). Y. Zhou developed on their work to give an optimal RC algorithm (Zhou Y. Y., 2012a). The drawback of ILC and RC algorithms, which

already involve complex mathematics and control theory in design of non-causal FIR filter based on stable inversion of the plant model, is that it requires trigger signal or period information which may not always be available practically.

There are impulses or impacts which may not occur so predictably. They may be single impacts which are basically transient signals and may vary in amplitude, pulse width, duration of decay, etc. Such noise signals, repetitive or randomly occurring, pose a challenge to the conventional adaptive algorithms due to various reasons. First of all, the high-amplitude impulse in the primary signal and/or error sends them to instability by causing sudden and large update in the adaptive filter weight and thus causing it to diverge, unless they have mechanisms like leakage to deal with it. It may also be difficult to obtain a correlated reference signal for adaptive feedforward control.

There were alternative efforts to design fixed controllers which add damping to the system in order to attenuate the transient noise. Costin and Elzinga came up with a feedback ANC system for tire impact noise (Costin, 1989). Bai et al (Bai M. R., 1995); (Bai M. R., 1996), (Bai M. R., 1997) and

Van Niekerk et al (van Niekerk, 1995) used H2 and/or H-∞ theory to develop an impulsive noise control methodology, which they verified by both simulation and experiments.

On the conventional FxLMS algorithm, several researches have made modifications which either work to minimize p-norm of the error or make the algorithm robust by detecting outliers using thresholds put on error or/and reference signals. Development along these lines started with the

FxLMP algorithm (Leahy, 1995) where the cost function (quantity to be optimized, or rather minimized) is not the mean squared error (second order moment) unlike Gaussian noise but a fractional order moment of the error called the p-norm. This needs accurate estimation of ‘α’

(characteristic exponent) parameter of the signal assuming that the noise follows a symmetric α-

stable distribution (Fama, 1968). Moreover, p-norm calculation increases computational complexity. The first modified FxLMS algorithm which uses thresholds was given by Sun et al in

2006 (Sun X. K., 2006). Threshold parameters determined by performing statistical analysis offline were placed on the reference signal. Akhtar and Mitsuhashi modified and extended this algorithm by putting thresholds on both reference and error signals (Tahir Akhtar, 2009). A more robust Hampel’s three-part M-estimator function was applied by Zou et al. (Zou, 2000) thus giving a LMM (least mean M-estimate) algorithm. Thanigai et al (Thanigai, 2007) applied LMM to control impulsive noise in infant incubators in the form of FxLMM algorithm. In 2011, Wu et al

(Wu, He, & Qiu, 2011), enlightened by the theorem that an α-stable process is a logarithmic order process (Gonzalez, Paredes, & Arce, 2006), arrived at a modified cost function for impulsive noise control – a squared log of the absolute error. They named it FxLogLMS algorithm. Sun, G. et al

(Guohua Sun, 2015) realized that impulses in the reference signal may still lead to instability since it is part of the weight update equation. Hence, an additional Hampel M-estimator was added in the reference signal path to weight update, thus giving an enhanced modified FxLMM algorithm.

Other algorithms that belong to this family are filtered weight filtered reference normalized LMS algorithm by Lifu Wu et al (Wu & Qiu, 2013) and a modified FxLMM algorithm by Peng Li et al

(Li & Yu, 2013).

Besides the threshold based algorithms, researchers have applied a variety of techniques to the impulsive noise ANC problem. Shukri Ahmed et al, 2011 (Ahmad, Kukrer, & Hocanin, 2012) proposed a Robust Recursive Inverse (RI) adaptive algorithm which achieved robustness by choosing weights on the basis of L1 norms of the autocorrelation matrix and the cross-correlation vector. Yoo et al (Yoo, Shin, & Park, 2015) modified the sign algorithm (SA) to give a variable step-size sign algorithm which calculates the mean-square deviation of the SA to obtain step size.

Lu Lu et al (Lu & Zhao, 2017), while replacing the LMS cost function in FxLMS with a least mean kurtosis (LMK) criterion, also incorporated a recursive sampled variance method, i.e., to estimate the MSE, which is part of estimation of the cost function, it adopts a sliding window approach. Lu

Lu et al (Lu & Zhao, 2017) in another research paper applied the maximum correntropy criterion

(MCC) to give a filtered reference recursive maximum correntropy (FxRMC). Zeb et al (Ayesha

Zeb, 2017) gave another threshold-based algorithm that switches between modified gain filtered reference recursive least square (MG-FxRLS) and normalized step-size filtered reference LMS

(NSS-FxLMS) based on a threshold value of mean noise reduction (MNR) to ensure fast convergence in non-stationary environment with MG-FxRLS and save computational cost when under threshold with NSS-FxLMS. The common hurdle to implementing these algorithms practically is the associated computational complexity.

Akhtar and Mitsuhashi came up with an ad hoc basis modified normalized FxLMS algorithm which does not require thresholding (Akhtar M. T., 2010), normalized LMS concept borrowed from (Douglas, 1994). What it requires, though, is computation of the l2-norm of the filtered reference vector from a current available data. In 2013, Zhou came up with the generalized step- size normalized filtered reference Least Mean Square (FxGSNLMS) (Zhou, Zhang, & Yin, 2015).

The research paper summarizes and compares the slopes of ‘score functions’ of most of the threshold based algorithms until then, like Sun X.’s algorithm, Akhtar’s algorithm, FxLogLMS, and FxLMM. A score function is a function of error amplitude that modifies the error factor of the weight update equation. It observes that due to the ‘hard-limiting’ property of Sun’s, Akhtar’s and

Thanigai’s score functions, they lack the tolerance of smoothness. While the FxLogLMS’s score function uses a rather smooth score function, it encounters a ‘dead zone’ when |e(n)| < 1, which will affect system performance. Later, a new generalization is defined in the form of a ‘weight

function’ or a ‘step-size normalized function’, which when multiplied with the error gives the score function at that point. Interpreting the threshold-based algorithms above as step-size normalized

FxLMS, the paper later suggests that the common pattern in the step-size normalized function is that it is roughly similar to probability density function (PDF) of a Gaussian distribution. Hence, if this function is applied directly as the weight function, it would be a ‘soft’ limiter having smoothness and tolerance. It is worth noting that the weight function is a function of the primary signal and not the error signal. Though this approach eliminates the need of estimation of thresholds, cost function selection or complex gradient computation, it uses, in the equation of the

PDF of the Gaussian distribution, an exponential calculation is involved in the weight update equation. Moreover, it erroneously concludes that the FxGSNLMS algorithm is insensitive to the choice of another parameter σ (standard deviation), which shall be examined in subsequent chapters. Hence, there are two parameters to be tuned for - µ (step size) and σ.

1.5 Problem statement and outline of concept of proposed algorithm

To summarize, the existing algorithms for impulsive noise control either involve too many parameters for tuning, or have some complex calculations involved which increases their computational complexity which is linked to processing time or ‘latency’ of the algorithm – time delay between impulse detection in reference signal and transmission of actual cancelling signal through the speaker.

To overcome the computational complexity introduced by the exponential calculation in

FxGSNLMS and also have minimum parameters for tuning, i.e. step size µ, a novel approach is proposed in this study. A polynomial (quadratic, cubic, 6-degree) curve-fit is done on the general form of Gaussian PDF equation. The coefficients are used in the weight function equation to get

very similar results. Hence, an extension, or rather a generalization of this concept of having a weight function proportional to the PDF values of the reference signal was tested. A similar approximation was employed for the PDF curve of a non-Gaussian noise data, in particular, symmetric-α-stable data of predetermined length. Initially, different curve fit coefficients were obtained for noise profiles with different α values. Later, it was observed that acceptable convergence was obtained no matter what set of coefficients is used, for primary noise of different

α values. Therein, it was concluded that coefficients obtained by fitting for noise profile with α =

1.5 gives the best convergence rate, whether primary noise has any α value from 1.5 to 1.9 (also for α = 2.0 which is Gaussian distribution case), thus eliminating the need to have an accurate estimate of α. The algorithm obtained as a result can be termed as ‘reference weighted filtered reference LMS’ or ‘RWFxLMS’ algorithm, since a polynomial function of the reference signal comes in the numerator of the filter weight update equation. The development and procedure for implementation will be discussed in detail in section 2.2.2. of chapter 2.

RWFxLMS does not need threshold estimation – online or offline. While most of the impulsive noise ANC algorithms aim at prevention of divergence of the adaptive filter weight due to outliers in the noise, the proposed algorithm cancels the impulses too. Salient feature of this algorithm is that with not more than one tuning parameters, it gives excellent reduction (upto 20dB) for noise with varying degree of impulsiveness, whether it be repetitive or random impulsive noise.

In chapter 2, the statistical models that have been used to simulate real-world impulsive noise – the symmetric α-stable and the contaminated Gaussian impulsive noise models – are introduced.

Certain existing adaptive algorithms for impulsive noise control namely normalized LMS algorithm, FxLMP which minimizes the p-norm instead of mean-squared error signal and its

modification and generalized step-size normalized FxLMS algorithm are discussed briefly. The reference-weighted filtered reference least mean-square (RWFxLMS) algorithm is introduced, which uses a quadratic function of the reference signal obtained by curve-fitting the probability distribution of an impulsive signal to weight the step-size at that instant. It is observed that this algorithm gives excellent reduction of impulsive noise and convergence without adding any significantly to the computational cost. An observation related to the convergence property of the proposed algorithm is made, and its computational complexity is calculated. Its performance for various types of impulsive noise signals – random impulsive with varying degree of impulsiveness and repetitive impulses - is studied.

Chapter 3 sorts available impulsive noise ANC algorithms into categories - normalization type, threshold type and weighted step-size type. Akhtar’s modified normalized filtered reference least mean square (NFxLMS), G. Sun’s enhanced filtered reference least mean m-estimate (FxLMM) belong to the first two categories respectively, while Y. Zhou’s generalized step-size normalized filtered reference least mean square (FxGSNLMS) and the proposed reference-weighted filtered reference least mean square (RWFxLMS) belong to the third category. A case is made for industrial application of the proposed algorithm by comparing it with other algorithms under consideration, based on computational complexity, convergence performance (rate of convergence and reduction at convergence) and also number of tuning parameters involved. It also includes a discussion about practical considerations while performing impulsive noise ANC.

2 Reference Weighted FxLMS Algorithm for Active Control of Impulsive Noise

2.1 Introduction

One of the main challenges for active noise control algorithms is impulsive noise, which has a significant number of disturbances that are high in amplitude but occur randomly at a low probability. Such a non-Gaussian noise can be modelled in the form of an α-stable distribution

(Shao, 1993). The properties of stable distribution are discussed in (Roll, 1968). An α-stable distribution has four main parameters in the form of characteristic exponent α, skewness parameter

β, scale parameter γ and location parameter δ.

∞ 푙표푔 ∅ (푡) = 푙표푔 [ 푒𝑖푡푥푑퐹(푥)] = 푖훿푡 − 훾|푡|훼 = 푖훿푡 − |푐푡|훼 (2-1) 푒 푥 푒 ∫−∞ When skewness (δ) is zero, the distribution becomes symmetric α-stable.

훾|푡|훼 ∅푥(푡) = 푒 (2-2) Gaussian distribution has α = 2 whereas α = 1 for Cauchy’s distribution. As the data becomes more and more impulsive, α goes on decreasing, or the tail of the distribution function becomes heavier.

Also, scale parameter is roughly associated with variance, since, for Gaussian distribution, γ =

0.5*variance.

Figure 2 PDF of standard symmetric α-stable process for various α values

Figure 3 Exploded view of the tail of PDF of standard symmetric α-stable process for various α values Assuming a standard distribution, i.e. γ = 1, the noise samples are generated in this study using the

MATLAB program ‘stblrnd’ which uses algorithms as formulated in (J.M. Chambers, 1976) and

(Weron, 1995). Another MATLAB function ‘alpha_loglik’ (Paulo, 2012) based on the quantile estimation method proposed in (McCulloch, 1986), can be used to estimate these four parameters of a given data set offline.

Figure 4 PDF of standard symmetric α-stable process for various β values

Figure 5 PDF of standard symmetric α-stable process for various γ values

Figure 6 PDF of standard symmetric α-stable process for various δ values

It is to be noted that the second order moments for the variable do not exist for a stable non-

Gaussian distribution and hence, mean-square error cannot be used as the optimization criterion.

The FxLMS algorithm therefore gives poor performance for noise of a stable non-Gaussian distribution. For such a distribution, only moments of the order lower than α exist. The FxLMP algorithm (R. Leahy, 1995) is thus based on the minimization of such a fractional order moment p, which is error raised to the power p.

Although FxLMP ANC algorithm may show to be more robust to impulsive, it adds substantial computational cost in the form of calculating fractional power for every iteration. It would also need prior estimation of p, which depends on α, for every data set.

Another model for synthesizing impulsive noise considered in this study is the contaminated

Gaussian (CG) model. In CG model (Kim, 1995) impulsive noise is modeled as impulses added into or contaminating Gaussian noise.

푥표(푛) = 푥푔(푛) + 푥𝑖푚푝(푛) (2-3)

푥𝑖푚푝(푛) = 푏(푛)푥푤(푛) (2-4) where 푥푤 and 푥푔 are modelled as independently identically distributed (IID) zero-mean Gaussian noise with standard deviations 𝜎푤 and 𝜎푔 respectively, 푥𝑖푚푝 is the impulsive noise having ‘b’ as a sequence of zeros and ones where probability of occurrence of ones is 푝푟. Impulsiveness is thus

2 2 2 2 determined by the ratio of variances of 푥𝑖푚푝 and 푥푔, i.e. 푟𝑖푚푝 = 𝜎𝑖푚푝 ⁄𝜎푔 = 푝푟 𝜎푤 ⁄𝜎푔 . Hence, reference signal is synthesized as a sum of this impulsive noise and background white noise.

푥(푛) = 푥푤푛(푛) + 푥표(푛) (2-5)

2 2 Signal-to-noise ratio is, thus, defined as 푆푁푅 = 10푙표푔10 𝜎푔 ⁄𝜎푤푛 .

2.2 Algorithm development

2.2.1 Existing NLMS, modified FxLMP and FxGSNLMS

A simplified form of the normalized least-mean-square algorithm has been around in signal processing research since 1967 (Nagumo, 1967). The normalized least mean-square algorithm

(Goodwin, 1984), also called ‘projection algorithm’,

푒푘푿푘 푾푘+1 = 푾푘 + 휇 퐿 2 (2-6) ∑푚=1 푥푚,푘 where k is the time instant, 푾 is the adaptive filter of tap length L, 푿푘is the input (reference) data vector in the memory and 푒푘is the error. This algorithm was further studied for its convergence behavior by (Rupp, 1993); (Slock, 1993). Its advantages over the LMS algorithm – faster convergence for both white and correlated input data and stable behavior over a wide range of step sizes – has been mentioned. Its steady state behavior is independent of reference signal power

(Kuo, 1994). It was Douglas (Douglas, 1994) who derived a generalized class of normalized LMS algorithms which can be expressed as;

푾푘+1 = 푾푘 + 휇푒푘퐹푞(푿푘) (2-7a)

푞−1 |푥푖,푘| 푠푔푛(푥푖,푘) 퐿 푞 푖푓 1 ≤ 푞 < ∞ ∑푚=1|푥푚,푘| [퐹푞(푿푘)]𝑖 = { (2-7b) 1 훿𝑖−푛 푖푓 푞 = ∞ 푥푛,푘 where [퐹푞(. )]𝑖 denotes the ith element of the vector-valued function Fq(.), δj is the Kronneker δ function, and ‘n’ is the index for which x is maximum in the vector Xk. For q = 2, this reduces to the NLMS algorithm (eqn. 2-6). This equation (2-7) can be for any valid q can be viewed as a solution to the optimization problem:

푚푖푛푖푚푖푧푒 ||푾 − 푾 || (2-8a) 푘+1 푘 푝

푇 푠푢푏푗푒푐푡 푡표 푑푘 − 푾푘+1푿푘 = 0 (2-8b) 1 1 where ||.||p denotes the 푳푝 norm, p is obtained by solving ⁄푝 + ⁄푞 = 1. This is clearly the LMP algorithm (Nikias, 1993) which is further applied to the active attenuation of acoustic impulsive noise problem in the form of FxLMP algorithm (R. Leahy, 1995).

푝−1 푾(푛 + 1) = 푾(푛) + 휇푝|푒(푛)| 푠푔푛(푒(푛))풙푓(푛) (2-9)

In equation (2-9), 퐱f is the reference signal 풙 filtered through secondary path. To do away with calculation of the fractional lower order as well as that of the thresholds (c1, c2) as required by

Sun’s and other threshold-based algorithms, and taking into account the peaky nature of the error signal as well, a modified normalized step size FxLMS algorithm is proposed by Akhtar et al

(Akhtar, 2010),

휇̃ 휇(푛) = 2 (2-10) ‖풙 (푛)‖ +퐸 (푛)+훿 푓 2 푒 where 퐸푒(푛), the energy of the residual error signal e(n) can be estimated online using a low-pass estimator as

2 퐸푒(푛) = 휆퐸푒(푛 − 1) + (1 − 휆)푒(푛) (2-11) It is to be noted that the calculation of the 2-norm does involve a considerable number of multiplications in itself, which is equal to the length of the adaptive filter and one less number of additions.

Another algorithm that inspires development of the proposed algorithm is the so-called generalized step-size normalized filtered reference least mean-square (FxGSNLMS) algorithm (Yali Zhou,

2015). The authors arrive at a general form of the threshold-based impulse noise control algorithms

(Sun’s (Xu Sun, 2006), Akhtar’s (Akhtar M. T., 2009), Thanigai’s FxLMM (Thanigai, Kuo, &

Yenduri, 2007) and compare their score functions,

푾(푛 + 1) = 푾(푛) + 휇휓(푛)[푠̂(푛) ∗ 풙(푛)] (2-12) where 휓 is the score function. Then they go ahead to define a weight function, which is ratio of the score function to the error, i.e. 푞(푛) ≜ 휓(푛)/푒(푛). In this way, eqn. 2-12 can be written as

푾(푛 + 1) = 푾(푛) + 휇푞(푛)푒(푛)[푠̂(푛) ∗ 풙(푛)] (2-13) At this stage, by plotting the schematic diagram of the weight functions of the above-mentioned algorithms, an observation is made that they are all in some way similar to the probability distribution function of a Gaussian distribution. Hence, a weight function is arrived at – Gaussian distribution of the reference signal,

1 푥(푛)2 휇(푛) = 휇푞(푛); 푞(푛) = 푒푥푝 (− ) (2-14) √2휋휎 2휎2 where 𝜎 is arbitrarily chosen ‘standard deviation’. It is to be noted that this invariably is a normalized FxLMS algorithm in yet another form. When tuned appropriately for µ and σ, it does provide excellent reduction and convergence characteristics against noise of all degrees of impulsiveness. Its biggest and most obvious drawback is the calculation of exponential of square of reference signal, i.e. exp(-x(n)2) at every iteration, which gives a huge computational burden.

2.2.2. Proposed Reference Weighted FxLMS (RWFxLMS) algorithm

In order to retain the excellent reduction, stability and convergence characteristics of the

FxGSNLMS algorithm and also finding a way around the exponential function calculation, an approximation (quadratic/cubic/degree-6 polynomial) of the Gaussian probability distribution function (PDF) curve was used to check ANC performance for data over a range of α values

(Figure 7).

−8 2 −21 푞푎푝푝푟표푥(푛) = −1.098 × 10 푥(푛) + 4.956 × 10 푥(푛) + 0.001914 (2-15)

Figure 7 Gaussian PDF and its quadratic approximation curve Very competitive results were obtained as can be seen in following Figure 7 and Figure 8.

Parameters of noise used: α = 1.7, β = 0, γ = 1, δ = 0.

Figure 8 Comparison of convergence of FxGSNLMS which uses Gaussian weight function with that using quadratic approximation of Gaussian PDF as weight function

Figure 9 Error signal of FxGSNLMS which uses Gaussian weight function and that using quadratic approximation of Gaussian PDF as weight function (note: error curve for quadratic approximation (yellow) practically lays over the one for FxGSNLMS (orange) and hence hides the latter) Hence, an extension to this concept of having a normalization/weight function proportional to the

PDF values of the reference signal was tested. A similar approximation (quadratic/cubic/degree-6 polynomial curve-fit) was employed for the PDF curve of a non-Gaussian noise data, in particular, symmetric-α-stable data (α = 1.5:1.9) of predetermined length. Equation (2-16a) is obtained by quadratic approximation of PDF curve for α = 1.5 s-α-s data with range of x (Figure 10(a)) through

-5:5 and equation (2-16b) for range of x through -500:500 (Figure 10(b)).

The results obtained were strikingly similar in terms of convergence and steady-state reduction, no matter how high the degree of the polynomial curve fit was. Hence, the quadratic approximation for the weight function given by equations (2-16a, 2-16b), which is computationally least expensive, was chosen for the simulations throughout this study.

Thus, the steps in the procedure for implementing the RWFxLMS algorithm are enlisted below:

i. Observe noise data set – its peak-to-peak range (e.g. -500:500 in this study).

ii. Obtain the parameters α, β, γ, δ using MATLAB’s ‘alpha_loglik’ function. iii. If α falls within 1.5-1.9, generate a noise data set using MATLAB’s ‘makedist’ function

using α = 1.7, β = 0, γ = 1, δ = 0. If α is below 1.5, use 1.2 as α value of the distribution. iv. Generate a PDF of the data set using MATLAB’s ‘pdf’ function and a vector covering the

range of the observed peak-to-peak values.

v. Obtain curve-fit coefficients (quadratic) of the PDF using MATLAB’s ‘fit’ function. vi. Use a quadratic function of the reference signal with the obtained curve-fit coefficients for

the weighting function qRWFxLMS(n) in the weight-update equation (2-13) which can be

rewritten as_

푾(푛 + 1) = 푾(푛) + 휇푞푅푊퐹푥퐿푀푆(푛)푒(푛)[푠̂(푛) ∗ 풙(푛)] (2-13a)

(a)

(b)

(c)

(d)

Figure 10 Symmetric-α-stable PDFs (α = 1.5) and their quadratic approximation curves; (a) range of x: -5:5; (b) range of x: -500:500; (c) zoomed-in plot (b); quadratic approximation curve in (b) and (c)

Figure 11 Comparison of convergence of FxGSNLMS which uses Gaussian weight function with that using quadratic approximation of s-α-s PDF as weight function

Figure 12 Error signal of FxGSNLMS which uses Gaussian weight function and that using quadratic approximations of s-α-s PDF as weight function

−8 2 −18 푞푅푊퐹푥퐿푀푆(푛) = −0.01025 × 10 푥(푛) − 1.501 × 10 푥(푛) + 0.1814 (2-16a) OR;

−8 2 −21 푞푅푊퐹푥퐿푀푆(푛) = −1.499 × 10 푥(푛) + 2.635 × 10 푥(푛) + 0.002249 (2-16b)

This approach gives the same level of cancellation and convergence time while saving a great deal in terms of computational cost, which shall be discussed further in section 2.3.2. The weight function thus obtained can be used for cancellation of impulsive noise of a wide range of α values

(1.5-1.9).

2.3 Analysis

2.3.1 Convergence

Optimal step size and maximum step size

Ardakeni and Abdullah (2010) have derived, for a general secondary path, the following expression:

퐺(푧) 1 + 휇휆 푠2 = 0 (2-17) 푙 표 푧퐿−푧퐿−1

2 퐿−1 푠푖 퐿−𝑖−1 where 푙 = 0,1, … , 퐿 − 1, 퐺(푧) = ∑𝑖=0 2 푧 푎푛푑 휆푙 is the 푙푡ℎ eigenvalue of the autocorrelation 푠표 matrix R of the filtered reference signal. This gives the basic root locus equation which is further used to calculate upper bounds of the step size by the authors. For a secondary path modelled as pure delay, the maximum step size obtained is:

휋 휇푚푎푥 = (2-18) 퐿푃푥′(2훥+1) where 푃푥′ is the power of the filtered reference signal, Δ is the delay due to the secondary path.

Later, Sun et al (Guohua Sun T. F., 2015) studied the nature of the control of repetitive impulse- induced transient noise by FxLMS algorithm. The secondary path is considered as a pure delay.

They observed that the filter coefficient update process is following a geometric series and gave the following expression for the 푙푡ℎ element of the adaptive filter weight vector:

퐾 푘−1 푤푙(퐾) = ∑ 휇퐴𝑖푚푝ℎ푙+훥(1 − 휇퐴𝑖푚푝퐴푆(푧)) 푘=0 2 퐾 휇퐴𝑖푚푝퐴푆(푧)ℎ푙+훥 [1 − (1 − 휇퐴𝑖푚푝퐴푆(푧)) ] = 2 2 1 − (1 − 휇퐴𝑖푚푝퐴푆(푧)) ℎ푙+훥 2 2 퐾 = [1 − (1 − 휇퐴𝑖푚푝퐴푆(푧)) ] (2-19) 퐴푆(푧)

th Δ Here, hl is the l primary path filter coefficient in the z-domain and S(z) = 퐴푠(푧)푧 . Hence, for a stable system,

2 2 0 ≤ |1 − 휇퐴𝑖푚푝퐴푆(푧)| ≤ 1 (2-20a) OR

2 0 ≤ 휇 ≤ 2 2 (2-20b) 퐴푖푚푝퐴푆(푧) This gives the maximum step size:

2 휇푚푎푥 = 2 (2-21) (퐴푖푚푝퐴푆(푧))

It is known that the, for a pure repetitive impulse signal, all the eigenvalues are equal to

2 (퐴𝑖푚푝퐴푆(푧)) . It means that equation (2-21) is similar to the one derived for white noise processes,

2 where step size is bounded as 0 < 휇 < . 휆푚푎푥

Another significant observation noted in the above study is:

The number of samples delay Δ of the secondary path affects the steady-state performance of the algorithm. The non-causal part in the optimal solution due to the delay of the secondary path cannot be cancelled and will appear in the error signal.

(Yali Zhou, 2015) derived, for a monotonously convergent system (i.e. magnitude of the misalignment vector 휀(푛) = 푾표푝푡 − 푾(푛) always decreasing with every iteration), the critical step size which is given by

2퐸{휓(푛)푒(푛)} 0 < 휇 < 휇 = (2-22a) 푐푟 2 2 퐸{휓 (푛)‖푿풇(푛)‖ }

Rewriting score function 휓(푛) in terms of weight function 푞(푛) times 푒(푛),

2퐸{푞(푛)푒2(푛)} 0 < 휇 < 휇 = (2-22b) 푐푟 2 2 2 퐸{푞 (푛)푒 (푛)‖푿풇(푛)‖ }

Weight function can be different for different algorithms and hence, the values of μcr.

Observations related to convergence

It is worthwhile to note the difference in convergence rate for two different impulsive noise sets with the same α-value, depending upon when the first few ‘major impulses’ (amplitude value several hundred times higher than the base Gaussian noise maximum value) arrive in the course of filter update. Within the selected observation window (5e4 samples), a situation arises such that if the ‘major impulse’ arrives rather late, say, 2.5e4 and beyond, the algorithm either refuses to converge or undergoes immediate divergence upon the first arrival. Hence, in this study, for such a data set, the convergence was observed when an impulse of comparable amplitude to the

‘naturally occurring’ one was added to the initial 5000th in the first case and then 5000th and 10000th in the second case (figure 13a below). MSE can be observed to be dropping successively with each added impulse, giving an effect that can be termed as ‘vaccination by impulses’, immunizing the algorithm against impulses arriving further. Observing further in fig. 13b below, the steady-state value of MSE is even lower in each subsequent case. This result with proposed RWFxLMS is nearly consistent with the observation made by Sun et al (Guohua Sun T. F., 2015) that the filter coefficient update process for FxLMS algorithm follows a geometric series at every impact event

(see equation (2-19) above). It also explains why certain sets of noise data which have a late- occurring first ‘major’ impulse will diverge on its arrival.

Figure

Figure 13(a) Time history of an impulsive primary noise signal with two impulses added manually; (b) Convergence (MNR) comparison: primary noise data without and with impulses added manually Another approach for industrial implementation of adaptive algorithms is that the tuning is done once to obtain the optimum adaptive filter vector. Hence, this filter is used to cancel the impulsive noise later in future without undergoing adaptation, provided it is known that nature of the primary noise is not going to change. In this case, convergence rate is no longer a matter of concern,

especially when listener’s comfort is always guaranteed. Such a scenario would be preferred since the adaptive filter has already converged to its optimum value which enables cancellation of any further arriving noise including peaky, high-amplitude impulses, and does not have to go through the sub-optimum start-to-converged stage wherein noise would be still audible. However, real-life application is not as simplistic. In an automobile cabin, for example, frequent changes in the plant

(esp. secondary path) would occur (loading/unloading, change in number of occupants, door/window opening, and so on). Therefore, this idea of a ‘frozen adaptive filter’ cannot be implemented.

2.3.2 Computational Complexity

The FxLMS algorithm requires 2(2M+N) number of operations (additions and multiplications included) per iteration (where M is length of adaptive filter and N is length of estimated secondary path). In the simulations performed for this study, a 128-tap adaptive filter and a 256-tap estimated secondary path have been used. The order of operations O(n) = M for the FxLMS algorithm.

A quick look at the modified adaptive filter weight update equations (2-16a) or (2-16b) above tells that for the quadratic approximation form of the weight function used, three additions and three multiplications are required in addition to those in the original FxLMS algorithm. Hence, the order of operations O(n) does not increase with this modification. This makes the RWFxLMS algorithm highly attractive from the computational cost perspective.

2.4 Numerical Simulation

Figure 14 Feedforward active noise control configured with the RWFxLMS algorithm The frequency response functions (FRFs) and impulse response functions (IRFs) of the general primary path and secondary paths as shown in Figure 15 are taken from (Kuo, 1994). The sampling frequency therein is 2048 Hz. The estimated secondary path S(z) is used in place of the physical electro-acoustic secondary path model. The number of filter taps used to model these paths as P(z) and S(z) is 256. The adaptive filter W(z) is an FIR filter of tap-weight length 128.

Figure 15 Primary path P(z) (solid line) and secondary path S(z) (dotted line) models: (a) Impulse response function; (b) Frequency response function

Although target noise for cancellation is not Gaussian, mean noise reduction (MNR) is still found to be a convenient criterion for performance evaluation of proposed RWFxLMS algorithm and further comparison with other algorithms done in chapter 3. MNR is defined as

퐴 (푛) 푀푁푅(푛) = 퐸{ 푒 } (2-23a) 퐴푑(푛)

퐴푒(푛) = 휆퐴푒(푛 − 1) + (1 − 휆)|푒(푛)| (2-23b)

퐴푑(푛) = 휆퐴푑(푛 − 1) + (1 − 휆)|푑(푛)| (2-23c) where 퐴푒(푛) and 퐴푑(푛) are recursive estimates of the power of the error 푒(푛) and primary noise

푑(푛). 휆 (=0.999) is the forgetting factor. While plotting, MNR is plotted on dB scale as

푀푁푅푑퐵(푛) = 20 푙표푔10 |푀푁푅| (2-24)

2.4.1 Random Impulsive Noise case

Following plots show the error signal reduction using proposed RWFxLMS algorithm as compared to original primary noise when the control is OFF.

A. Symmetric α-Stable Impulsive Noise model

The symmetric α -stable impulsive noise model is used to generate the reference signal, which is passed through the primary path filter to give primary noise. A detailed description of this model can be found in section 2.1 Introduction. Characteristic exponent α which also gives impulsiveness of the signal, is varied through 1.5 to 1.9, i.e. in decreasing order of impulsiveness.

Skewness β = 0, scale parameter γ = 1 and location parameter δ = 0. Sampling rate is 2kHz.

Simulation time is 25 seconds (i.e. number of data points is 5 × 104).

From the following results of the simulation, it can be easily observed that the proposed

RWFxLMS algorithm is highly efficient in control of various degree of impulsiveness of noise.

The error measured in terms of MNR converges to 20dB reduction after about 5 seconds.

Moreover, it also attenuates the highly peaky impulses to a convincing level without going unstable at their occurrence.

Figure 16(a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.5

Figure 17(a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.6

Figure 18 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.7

Figure 19 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.8

Figure 20 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise with α = 1.9

Figure 21 gives a zoomed in time domain plot around occurrence of an impulse in the primary noise. Figure 21(b) is frequency domain plot of primary noise and error signal given by RWFxLMS algorithm obtained by performing FFT (Fast Fourier Transform) around the impulse in the primary noise under consideration. A relatively flat response in frequency domain, apart from the ample amount of peak reduction obtained using proposed algorithm can be observed.

It should however be noted that the Fourier transform or its FFT version is not an accurate method of looking at a transient signal like an impulsive signal in frequency domain, since the accuracy along time axis would drop if accuracy along frequency axis in increased and vice-versa using this transform (Zhu, 2006). Methods like the wavelet transform (continuous wavelet, analytic wavelet),

Hilbert transform, etc. are recommended.

Figure 21 Performance comparison of the proposed RWFxLMS algorithm for impulse noise (α = 1.6) in the error signal ANC OFF and ANC ON: (a) time domain; (b) frequency domain

B. Contaminated Gaussian model

In this section the contaminated Gaussian noise model is used to generate impulsive reference signal which is also used to generate primary signal by passing it through the primary path filter.

A detailed description of this model can be found in section 2.1. The parameters used in case 1

(Figure 22) are: SNR = −40 (dB), impulsiveness rimp = 100; pr = 0.0005, which correspond to small impulsiveness. Case 2 (Figure 23) uses SNR = −35 (dB), impulsiveness rimp = 300; pr =

0.0005, thus giving a more impulsive noise. Case 3 (Figure 24) uses SNR = −40 (dB), impulsiveness rimp = 300; pr = 0.005 which has even more number of impulses due to the increases probability pr.

The stability and convergence of MNR to 20dB reduction level seen in each of the three cases is consistent with that obtained with symmetric α -stable impulsive noise model.

Figure 22 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise generated using Contaminated Gaussian model – case 1

Figure 23 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise generated using Contaminated Gaussian model – case 2

Figure 24 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for impulsive noise generated using Contaminated Gaussian model – case 3

2.4.2 Repetitive Impulsive Noise case

A. A series of 25 impulses synthesized as the reference excitation

Figure 25 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for repetitive impulsive noise without any background white noise

B. Repetitive periodic impulses (50 impulses) mixed with background white noise

푣푎푟(푝푢푟푒 𝑖푚푝푢푙푠푒푠) Parameters: S푁푅 = 10 (dB) (calculated as S푁푅 = 10 log ( ) 10 푣푎푟(푤ℎ𝑖푡푒 푛표𝑖푠푒)

Figure 26 (a) Time history of residual error signal - ANC OFF and ON, and (b) Convergence (MNR) - with proposed RWFxLMS for repetitive impulsive noise mixed with background white noise

2.5 Conclusion

The study develops a novel reference normalized filtered reference least mean-square algorithm for impulsive noise control. It uses second-degree function of the impulsive reference signal obtained by curve-fitting over a probability distribution function curve of a typical impulsive noise dataset to weight the step-size at a given instant. Based on the analysis conducted through several numerical simulations, it can be concluded that the new algorithm provides excellent control in terms of level of cancellation and convergence speed for a wide range of impulsiveness of primary noise in the random impulsive noise case as well as for the pure repetitive impulses and repetitive impulses mixed with white noise cases. The random impulsive noise data was generated with symmetric α-stable noise model as well as contaminated Gaussian noise model. Computational complexity is not affected in terms of order of operations as compared to conventional filtered reference least mean-square algorithm. Further detailed comparison with some existing adaptive algorithms for impulsive noise active control shall be conducted in chapter 3 of this study.

3 Comparative Study of Several Adaptive Algorithms for Impulsive Noise Control

3.1 Introduction

The conventional filtered reference least mean square (FxLMS) algorithm is a feedforward adaptive algorithm in which the reference signal is ‘fed-forward’ through the adaptive filter to cancel out the primary acoustic noise signal. FxLMS algorithm has proven its efficacy for control of acoustic noise when the noise is tonal i.e. it has a certain dominant frequency of oscillation.

This logic can be extrapolated for engine noise containing harmonics or ‘orders’ of a base frequency, e.g. vehicle powertrain noise, wherein engine crank rotation speed is used so as to generate the reference signal using sinewave generators. The equation for filter adaptation/ update in the conventional FxLMS algorithm contains the term which adds to the existing filter weight and is proportional to the reference signal at that instant. In case of a randomly occurring impulse in the reference signal, such as a high-amplitude spike due to an impact, the filter weight undergoes a sudden excessive update. This impedes the convergence of the algorithm and may even lead to divergence.

(a)

(b)

Figure 27 (a) Primary noise; (b) Convergence (MNR) using FxLMS algorithm Therefore, the primary aim of active impulse noise control algorithms is to control this impeding and/or destabilizing update occurring at the arrival of the high-amplitude impulse. Based on how they attempt to achieve this, they can be categorized into following types.

 FxLMP: using pth moment instead of 2nd moment (mean-square)

 Normalization type: Normalize step size with a certain function of the reference signal

and/or error signal {NLMS (Goodwin, 1984), Akhtar’s modified normalized FxLMS

(Akhtar M. T., 2010), Wu’s FxLogLMS (Gonzalez, Paredes, & Arce, 2006)}

 Threshold type: Ignore / limit, in the filter update process, samples of the reference signal

if their amplitude exceeds a certain threshold set by its statistics {Thanigai’s FxLMM

(Thanigai, Kuo, & Yenduri, 2007), modified FxLMM by G. Sun (Guohua Sun, 2015)}

 Weighted step-size type: multiply/ weight step size with certain function of the reference

signal and/or error signal {Zhou’s FxGSNLMS (Yali Zhou, 2015), proposed RWFxLMS}

3.2 Various adaptive algorithms for impulsive noise control

In this section, apart from Leahy’s filtered reference least mean p-norm (FxLMP) algorithm the algorithms with which the proposed RWFxLMS algorithm is compared are described in brief.

3.2.1 Leahy’s Filtered Reference Least Mean P-Norm (FxLMP) Algorithm

The conventional filtered reference least mean square (FxLMS) algorithm (B. Widrow, 1981),

(Burgess, 1981) for adaptive control of acoustic noise is based on minimizing the mean squared error of the error signal (difference between primary noise which is target for cancellation and cancelling noise which is given out by controller through speaker(s)). In impulsive noise case, which does not follow the Gaussian distribution and therefore does not have a mean value, LMS criterion can no longer be used. Instead, the p-th moment of the error can be minimized, to give the FxLMP algorithm.

푚푖푛푖푚푖푧푒 ||푾 − 푾 || (3-1) 푘+1 푘 푝

th Here, 푾푘+1 is the adaptive filter vector at the (k + 1) instant, ||.||p denotes the 푳푝 norm, p is usually chosen to be close to but less than the ‘α’ (characteristic exponent) parameter of the signal, e.g. 1.49 for a signal of α = 1.5. The downside of this algorithm is the increased computational complexity for calculating the fractional order moment for estimating the p-norm. Hence, while this algorithm is theoretically highly robust against impulsive noise, it is hardly ever used in practice.

3.2.2 Akhtar’s Modified Normalized FxLMS (NFxLMS) Algorithm

The normalized FxLMS (NFxLMS) algorithm (Akhtar M. T., 2010) meant to prevent divergence of adaptive filter weight in the presence of impulsive acoustic noise is derived from the normalized

LMS (NLMS) (Goodwin, 1984).

풘(푛 + 1) = 풘(푛) + 휇(푛)푒(푛)풙푓(푛) (3-2a) It can also be seen as a normalized step-size algorithm, wherein the step-size is normalized with the 풍ퟐ-norm of the reference signal, i.e.,

휇̃ 휇(푛) = 2 (3-2b) ‖풙 ‖ +훿 푓 2 where 훿 is the ‖풙 ‖ is the 푙 -norm of the current filtered reference signal vector potentially 푓 2 2 containing an filtered impulsive data sample and 훿 is to avoid division by zero. Hence, Akhtar modified this algorithm to account for the impulsive effect in the residual error signal as well, to give a modified normalized step size FxLMS algorithm (Akhtar 2010).

휇̃ 휇(푛) = 2 (3-3a) ‖풙 (푛)‖ +퐸 (푛)+훿 푓 2 푒

퐸푒(푛), the energy of the residual error signal e(n), can be estimated online using a low-pass estimator as

2 퐸푒(푛) = 휆퐸푒(푛 − 1) + (1 − 휆)푒(푛) (3-3b)

3.2.3 G. Sun’s Enhanced FxLMM Algorithm

In order to control the effect of peaky outliers in the reference and/or error signals, the following class of algorithms uses thresholds on reference and/or error signals. Choice of the threshold levels is based on statistics of the signals, such as 99.9 and 0.01 percentile in Sun’s algorithm (Xu Sun,

2006).

풘(푛 + 1) = 풘(푛) + 휇푒(푛)[푺̂(푛) ∗ 푿푐(푛)] (3-4a)

0, 푥(푛) ≥ 푐2 푥푐(푛) = { 0, 푥(푛) ≤ 푐1 (3-4b) 푥(푛), 표푡ℎ푒푟푤푖푠푒

A more robust criterion in the form of Hampel’s three-part M-estimator to replace the least-square error was introduced by Zou et al. (Zou, 2000) in the least mean M-estimate (LMM) algorithm.

Thanigai et al. (Thanigai, Kuo, & Yenduri, 2007) extended it to the filtered reference M-estimate

(FxLMM) algorithm.

푒2(푛) , 0 ≤ |푒(푛)| ≤ 휉 2 휉2 휉|푒(푛)| − , 휉 ≤ |푒(푛)| ≤ 훥 2 1 𝜌{푒(푛)} = 휉 (3-5) 2 (|푒(푛)|−훥 )2 휉 휉 2 2 (훥1 + 훥2) − + , 훥1 ≤ |푒(푛)| ≤ 훥2 2 2 훥1−훥2 휉 휉2 (훥 + 훥 ) − , 훥 ≤ |푒(푛)| { 2 1 2 2 2 where 𝜌{푒(푛)} is the M-estimate function of the error signal, 휉, Δ1푎푛푑 Δ2 are the threshold parameters based on estimated variance of the ‘impulse-free’ signal. Further, a score-function is

휕휌{푒(푛)} ψ{푒(푛)} defined as ψ{푒(푛)} = , and weighting function as 푞{푒(푛)} = . 휕푒(푛) 푒(푛)

푒(푛), 0 ≤ |푒(푛)| ≤ 휉

휉푠푖푔푛(푒(푛)), 휉 ≤ |푒(푛)| ≤ 훥1 휓{푒(푛)} = (|푒(푛)|−훥2)휉 [ ] 푠푖푔푛(푒(푛)), 훥1 ≤ |푒(푛)| ≤ 훥2 훥1−훥2 { 0, 훥2 ≤ |푒(푛)| (3-6a) Hence, filter weight update equation of FxLMM algorithm is performed using

풘(푛 + 1) = 풘(푛) + 휇푞{푒(푛)}푒(푛)풙푓(푛) (3-6b) G. Sun et al used a simplified M-estimator (Li, 2011) on the reference signal in addition to the error signal in their enhanced modified FxLMM.

Figure 28 SISO ANC system (block diagram) for impulsive noise with G. Sun’s enhanced FxLMM algorithm 푥(푛), 0 ≤ |푥(푛)| ≤ 휉

휉푠푖푔푛(푥(푛)), 휉 ≤ |푥(푛)| ≤ 훥1 푥푐(푛) = (|푥(푛)|−훥2)휉 (3-7a) [ ] 푠푖푔푛(푥(푛)), 훥1 ≤ |푥(푛)| ≤ 훥2 훥1−훥2 { 0, 훥2 ≤ |푥(푛)|

푥푐 gives the ‘trimmed’ reference signal using Hampel’s three-parts re-descending function. Thus, filter weight update equation in G. Sun’s enhanced FxLMM algorithm can be expressed as

풘(푛 + 1) = 풘(푛) + 휇푞{푒(푛)}푒(푛)[푺̂(푛) ∗ 푿푐(푛)] (3-8)

The threshold values 휉, Δ1 and Δ2 are computed online by choosing appropriate confidence levels

휃휉, 휃Δ1and 휃Δ2 as follows.

휃휉(푛) ≜ 푃푟{|푒(푛)| > 휉} (3-9a)

휃훥1(푛) ≜ 푃푟{|푒(푛)| > 훥1} (3-9b)

휃훥2(푛) ≜ 푃푟{|푒(푛)| > 훥2} (3-9c)

However, in the current simulation study, 휃휉, 휃Δ1and 휃Δ2are computed online using the following relationships with the estimated variance (or standard deviation) of the ‘impulse-free’ error signal,

휉(푛) = 25𝜎̂푒(푛) (3-10a)

훥1(푛) = 200𝜎̂푒(푛) (3-10b)

훥2(푛) = 5000𝜎̂푒(푛) (3-10c) where,

2 2 𝜎̂푒 (푛) = 휆𝜎̂푒 (푛 − 1) + 퐶1(1 − 휆)푚푒푑{푨푒(푛)}(3-11a)

퐶1 = 1.483[1 + 5/(푁푤 − 1)] (3-11b)

2 2 2 푨푒(푛) = {푒 (푛), 푒 (푛 − 1), … , 푒 (푛 − 푁푤 + 1)} (3-11c) The above equation for median computation was given by Zou et al. G. Sun et al. (Guohua Sun,

2015) came up with a more accurate form of median computation,

′ 2 2 2 푨푒(푛) = {[푒(푛) − 푢̂(푛)] , [푒(푛 − 1) − 푢̂(푛 − 1)] , … , [푒(푛 − 푁푤 + 1) − 푢̂(푛 − 푁푤 + 1)] } (3-12a) where

푢̂(푛) = 휆푢̂(푛 − 1) + 퐶1(1 − 휆)푒(푛) (3-12b)

A similar process as given through equations (3-10) to (3-12) for computing the threshold through variance estimation for reference signal is involved per iteration.

3.2.4 Y. Zhou’s FxGSNLMS Algorithm

Figure 29 SISO ANC system (block diagram) for impulsive noise with Y. Zhou’s FxGSNLMS algorithm The generalized step-size normalized filtered reference least mean square (FxGSNLMS) algorithm

(Yali Zhou, 2015) avoids the tedious process of choosing the threshold values and proposes a more generalized equation for the variable step size. The authors arrive at a general form of the threshold-based impulse noise control algorithms (Sun’s (Xu Sun, 2006), Akhtar’s (Akhtar M. T.,

2009), Thanigai’s FxLMM (Thanigai, Kuo, & Yenduri, 2007)) and compare their score functions,

푾(푛 + 1) = 푾(푛) + 휇휓(푛)[푠̂(푛) ∗ 풙(푛)] (3-13) where 휓 is the score function. Then they go ahead to define a weight function, which is ratio of the score function to the error, i.e. 푞(푛) ≜ 휓(푛)/푒(푛). In this way, eqn. 3-13 can be written as

푾(푛 + 1) = 푾(푛) + 휇푞(푛)푒(푛)[푠̂(푛) ∗ 풙(푛)] (3-14) At this stage, by plotting the schematic diagram of the weight functions of the above-mentioned algorithms, an observation is made that they are all in some way similar to the probability distribution function of a Gaussian distribution. Hence, a weight function is arrived at – Gaussian distribution of the reference signal,

1 푥(푛)2 휇(푛) = 휇푞(푛); 푞(푛) = 푒푥푝 (− ) (3-15) √2휋휎 2휎2 where 𝜎 is arbitrarily chosen ‘standard deviation’. It is to be noted that this invariably is a normalized FxLMS algorithm in yet another form. Since it multiplies rather than divides by the weight function, it is placed in a separate category for comparison with proposed reference-weighted filtered reference (RWFxLMS) algorithm.

3.2.5 Proposed RWFxLMS Algorithm

In the proposed reference weighted filtered reference least mean square (RWFxLMS) algorithm, the concept of having a weighting function which represents the probability distribution function

(PDF) of an impulsive, non-Gaussian noise is implemented. In a way, this can be said to be eliminating the need for tuning for σ (standard deviation) parameter in Y. Zhou’s RWFxLMS, while enabling the algorithm to achieve the same results, as will be observed in section 3.3 further.

A quadratic approximation of the PDF curve of a representative impulsive noise data set is obtained over a range of values shown by the primary noise. In the current study, a symmetric 훼- stable noise (훼=1.7) is used to plot the PDF curve for which the quadratic approximation coefficients are obtained over a chosen range, i.e., -500:500.

Figure 30 Symmetric-α-stable PDF (α = 1.5) and its quadratic approximation curves; range of (a) x: -5:5 and (b) x: -500:500

Figure 31 SISO ANC system (block diagram) for impulsive noise with proposed RWFxLMS algorithm Analogous to equation 3-15, the weighting function thus obtained as described above is

−8 2 −21 푞푅푊퐹푥퐿푀푆(푛) = −1.499 × 10 푥(푛) + 2.635 × 10 푥(푛) + 0.002249 (3-16)

3.3 Results and comparison

The performance comparison between ANC algorithms is traditionally done using metrics like computational complexity and convergence rate. It is also worthwhile discussing factors such as number of parameters that the users (researchers and/or engineers) have to tune considering the possibility that these algorithms have potential industrial application.

3.3.1 Computational Complexity

The FxLMS algorithm requires 2(2N+M) number of operations per iteration (where M is length of adaptive filter and N is length of estimated secondary path). In the simulations performed for this study, a 128-tap adaptive filter and a 256-tap estimated secondary path have been used. The order of operations O(n) = M for the FxLMS algorithm.

Equation (3-3a) for Akhtar’s modified NFxLMS algorithm indicates that the normalization equation adds N+1 multiplications and N+1 additions per iteration to the original number of operations as per FxLMS equation. Therefore, the total number of operations per iterations increases to 2(3N+M+1). This means that the modified NFxLMS does not significantly increase the order of operations from O(n) = M.

Computational complexity of enhanced FxLMM algorithm (Guohua Sun, 2015) is given by

2(2푁 + 푀) + 2푁푤푙표푔푁푤 표푟 2(2푁 + 푀) + 2푁푤, when it uses the sequential bubble sorting algorithm or the parallel bubble sorting algorithm respectively for the median computation step for the variance estimation process involved per iteration. In this computational cost, the authors have ignored the cost associated with comparing samples multiplications and additions in the

2 2 recursive equation of estimation of 𝜎̂푒 (푛) and 𝜎̂푥 (푛).

The computational complexity of the FxGSNLMS algorithm is apparently significantly high due

푥(푛)2 to the exponential function of squared reference signal, i.e. 푒푥푝 (− ) every iteration. However, 2휎2 it being unknown to the authors as to how exactly the comparison between an exponential operation and a multiplication operation can be quantified, the current study makes a qualitative assumption that exponential function calculation is generally expensive computationally, especially when the power (−푥(푛)2) is fractional in most cases.

It has been already mentioned in section 2.3.2 Computational Complexityof chapter 2 that for the quadratic approximation form of the weight function used in RWFxLMS, three additions and three multiplications are required in addition to those in the original FxLMS algorithm. Therefore, the total computational cost is simply 2(2N+M) + 6. The order of operations, therefore, remains the same as in FxLMS algorithm. More importantly, as the length of the adaptive filter, increases, the difference in number of operations remains more and more insignificant, as will be seen clearly in figure 2(b).

Table 1 Total number of operations per iteration in current simulation study for each of the algorithms used for comparison with proposed RWFxLMS

Algorithm Computational cost per iteration for M (length of estimated secondary path) = 256 and N (length of adaptive filter) = 128 per iteration FxLMS 1024 Akhtar’s modified NFxLMS 1282 G. Sun’s enhanced FxLMM ~2266 Proposed RWFxLMS 1030

The total time required by the processor to complete a certain number of iterations of the filter weight adaptation process through each algorithm can be used as a rough criterion for comparison of the computational load that each of them subjects the processor to. MATLAB® ‘tic-toc’ pair of commands is placed at start and end of the loop of each adaptive algorithm. In current study, the

total number of iterations is 5 × 104. The following table gives processing time for each algorithm.

Multiple runs for the same impulsive noise data set were carried out for every algorithm and the average of five processing times is noted for each algorithm.

Table 2 Time (s) taken to complete 5 × 104 iterations by each impulsive noise ANC algorithm in MATLAB®

Algorithm Akhtar’s G. Sun’s FxGSNLMS RWFxLMS modified enhanced NLMS FxLMM Symmetric-α- 0.884821 2.248274 0.886753 0.800941 stable (α = 1.5) Contaminated 0.822608 1.973172 0.762343 0.756425 Gaussian Pure repetitive NA (not NA (not 0.782901 0.751284 impulses applicable) applicable)

Repetitive 0.773734 1.961716 0.712738 0.679716 impulses with background noise

3.3.2 Convergence

This section contains comparison of convergence curves obtained for various types of impulsive noise (randomly-occurring impulses (symmetric-α-stable and contaminated Gaussian models) and repetitive impulses (without and with background Gaussian noise) using the algorithms discussed in section 3.2 Various adaptive algorithms for impulsive noise controlabove.

A. Symmetric-α-Stable noise (α = 1.5)

Figure 33 Comparison of convergence curves obtained using the various ANC algorithms under consideration and proposed RWFxLMS algorithm for s-훼-s impulsive noise with 훼 = 1.5

The proposed RWFxLMS algorithm gives very comparable reduction of error signal to the algorithms best in each of the classes discussed above. The above plot in Figure 33 shows the best results achieved by all the algorithms for the given noise data set. While Akhtar’s modified

NFxLMS algorithm shows the fastest convergence rate, it does have a higher steady state error

(~2dB of difference) than all the other algorithms.

B. Contaminated Gaussian

Figure 34 Comparison of convergence curves obtained using the various ANC algorithms under consideration and proposed RWFxLMS algorithm for contaminated Gaussian noise (푟𝑖푚푝 = 100; 푝푟 = 0.005)

For this particular data set of contaminated Gaussian type impulsive noise (푟𝑖푚푝 = 100; 푝푟 =

0.005), all the algorithms under consideration perform equally well in terms of net reduction obtained at convergence. This may be explained to the nearly Gaussian nature (very high 훼; ~1.9) of this data set.

A slightly better convergence rate is given by Akhtar’s modified NFxLMS and proposed

RWFxLMS algorithms.

Figure 35 Comparison of convergence curves obtained using the various ANC algorithms under consideration and proposed RWFxLMS algorithm for contaminated Gaussian noise (푟𝑖푚푝 = 10; 푝푟 = 0.0005)

The algorithms are tested for an impulsive noise set with higher impulsiveness (higher peak values of impulses compared to the peak values of background noise) and lesser probability of occurrence of impulse generated in this Contaminated Gaussian noise model, to test the effect of comparatively more randomly occurring impulses. The convergence curves shown in Figure 35 above indicate the best rate of convergence by Akhtar’s modified NFxLMS algorithm. This algorithm incorporates (energy of error) in normalization in addition to 2-norm of reference signal, which is not a case with the other algorithms, which may be the reason for the improved rate of convergence.

C. Pure repetitive impulses

Out of the algorithms for impulsive noise ANC under consideration, only Y. Zhou’s FxGSNLMS and the proposed RWFxLMS algorithms are able to control pure repetitive impulses, i.e. periodically occurring impulses without any background noise contamination. This is a hypothetical case, though, as there will always be some white noise in the background present in the signal.

Both the algorithms trace the same convergence curve, as can be observed in figure 36 below.

Figure 36 Comparison of convergence curves obtained using Y. Zhou’s FxGSNLMS algorithm and proposed RWFxLMS algorithm for pure repetitive (periodically occurring) impulsive noise

D. Repetitive impulses with background noise

In case a set of periodically repetitive impulses is mixed with background white noise, e.g. punching machine, presses, etc., the weighting class of algorithms (FxGSNLMS in yellow and proposed RWFxLMS in purple) performs better, both in terms of error at convergence and convergence rate, than the normalization (Akhtar’s modified NFxLMS) and thresholding (G. Sun’s enhanced FxLMM) types, as seen in Figure 37 below.

E. Weighting of error signal in combination with weighting of reference

In combination with reference weighting, error weighting was tried out to verify if it could improve the rate of convergence. This is inspired by threshold used on both reference and error signals in

G. Sun’s enhanced FxLMM algorithm. The modification however, gave neither improvement nor degradation in convergence rate, as can be seen in Figure 39.

Figure 38 SISO ANC system (block diagram) for impulsive noise with proposed RWFxLMS algorithm reference-weighting of step-size combined with error weighting

−8 2 −21 푞′푅푊퐹푥퐿푀푆(푛) = (−1.499 × 10 푥(푛) + 2.635 × 10 푥(푛) + 0.002249) × (−1.499 × 10−8푒(푛)2 + 2.635 × 10−21푒(푛) + 0.002249) (3-17)

Figure 39 Comparison of mean noise reduction (MNR) of the proposed RWFxLMS algorithm and its error-weighted-cum-reference-weighted modification for s-훼-s impulsive noise with 훼 = 1.1 One reason for this outcome that can be speculated is that the primary noise is highly correlated to the reference in this study. Any spike in the reference would inevitably occur in the primary noise in this case. This makes any thresholding or weighting on error signal (which is difference between primary noise and cancellation signal) redundant.

3.3.3 Tuning parameters

Following is an objective evaluation of the algorithms under study from a practical real-world tuning point-of-view.

Table 3 Comparison of impulsive noise ANC algorithms based on tuning parameters

Algorithm Tuning procedure Real time tuning parameters

FxLMS Step-size 휇 is changed while observing 흁 reduction and convergence curves. Akhtar’s modified Same as FxLMS 흁 NFxLMS

G. Sun’s enhanced Along with 휇, 휉, Δ1 푎푛푑 Δ2 values (refer 흁, 흃, 횫ퟏ 풂풏풅 횫ퟐ FxLMM equations 3-10) are changed. Y. Zhou’s Step-size 휇 and ‘standard deviation’ 𝜎 are 흁 and 흈 FxGSNLMS changed. Proposed Through this study, it is observed that 흁 RWFxLMS algorithm using the coefficients obtained by curve-fitting for a s-훼-s distribution with 훼=1.7 performs optimally for the entire range of impulsiveness under consideration (훼=1.5:1.9). Hence, effectively, only step- size 휇 needs to be changed for tuning.

Considering the significant amount of time that ANC system engineers have to spend tuning the parameters involved in a multi-input, multi-output (MIMO) system such as in a car, where parameters may be separately tuned per speaker and/or per mic, it is highly desirable that the algorithm at the core does not have a lot of tuning parameters. In this aspect, the proposed

RWFxLMS looks attractive, at least after the curve-fit equation used in the step-size weighting function is obtained and the data does not change on its peak values drastically or its impulsiveness characteristics drastically.

3.4 Conclusion

The ease in tuning, when considered together with the low computational cost (section 3.3.1) and excellent performance for all varieties of impulsive noise (section 3.3.2), makes the proposed

RWFxLMS a strong contender for industrial application.

4. Conclusion

In this chapter, a few considerations that need to be made while practically implementing ANC for impulsive noise are discussed, such as nature and characteristics of reference signal, latency and objective measurement of loudness of impulsive noise. Further, conclusions drawn through this study are mentioned – one regarding the convergence of studied ANC algorithms and the other regarding the proposed RWFxLMS algorithm’s advantages over previous algorithms. Finally, recommendations are made as to how this study can be extended in future.

4.1 Practical Considerations

The reference signal used in this study or any previous studies is a pure trail of impulses. In case of an automobile, the impulses in the reference are recorded at accelerometers in the form of impulse responses of the transfer path between the actual point of impact between, say tire and road, and the accelerometer mounting location.

Figure 40 (a) Synthesized reference signal; (b) Reference signal recorded using accelerometer mounted on a hub in a car chassis This not only introduces a time delay between the impact and its recording, but may also add a phase shift varying with frequency depending upon frequency response characteristics of the transfer path. In a standard case like in an automobile with multiple error microphones, multiple speakers and multiple references, the situation becomes even more complicated. Each microphone location (or vehicle occupant location) is receiving input from all the different sources together.

Therefore, the need to weight each reference signal adds to the tuning effort of the algorithm.

Obtaining a set of reference signals highly coherent with the primary road noise which contains

impacts due to cracks, bumps, rubble, etc. is indeed a challenge, for which accelerometers’

(reference sensors’) placement and choosing the best among available locations becomes a crucial problem.

A factor that can degrade the ANC algorithm performance is the delay that is induced as a result of processing time taken by the controller to produce the correct cancellation signal at the loudspeakers, termed as ‘latency’. If this electrical delay is longer than the delay through the vibro- acoustic path from generation of the impact to its arrival at the error microphone, the cancellation signal may not be effective anymore, especially in case of impulsive noises, which are broadband noise lasting only for few milliseconds (Kuo S. M., 1999). In other words, the system would not satisfy causality condition due to high latency. This leaves very little scope for an adaptive algorithm’s computational complexity.

Objective measurement of loudness of impulsive noise needs review, since perceived loudness varies with pulse duration (Zwicker, 2013). Impulses which last only few milliseconds long may not be evaluated on an SPL in dBA scale. It is also important to consider that the temporal resolution of ear / brain is frequency dependent, and hence, different types of impulses may need different resolution times to be considered depending upon their frequency content. Thus, there is a need to characterize a given type impulsive noise (click / road bump / metallic clunk) in terms of loudness (phon/sone) versus duration of impulse.

To give an idea of this, Figure 41 is shown below. It depicts loudness and loudness level plots only for pulses of a single frequency – a stationary type of noise.

Figure 41 Loudness in sone and loudness level in phon versus 2kHz tone-burst duration [Images of plots taken from (Zwicker, 2013)] For the definition of loudness for non-stationary noises, (ISO 532-1:2017(en), 2017), can be referred to.

4.2 Conclusions

Any algorithm developed for controlling impulsive noise gives a convergent value of error only after undergoing adaptation of the adaptive filter for a certain number of impulses, as observed in innumerable iterations performed through this study with different types of impulsive noise. This is valid since the very expectation of cancelling a first-arriving impulse by an adaptive filter, which has not converged to its optimal value yet, goes against the concept of adaptive filtering.

The proposed reference-weighted FxLMS (RWFxLMS) algorithm stands shoulder-to-shoulder with other algorithms which are most efficient in their respective categories in terms of stability and rate of convergence, while incurring considerably low computational cost and being considerably easy to tune.

4.3 Future Work

It still needs to be validated whether the proposed RWFxLMS algorithm, or any other algorithm meant for active control of impulsive noise, stands the test of multiple reference, multiple input and multiple output problem, such as one presented by a four-wheel automobile. Attempts were made during this study for implementing a 2-reference 2-microphone 2-speaker simulated system for left-front and left-rear locations for data recorded in a sports-utility vehicle, without any success.

The proposed RWFxLMS algorithm may be used in combination with more sophisticated algorithms like the delayless subband algorithm which can be applied for broad-band road noise cancellation.

Since this study is entirely based on numerical simulation, its implementation on a test bench could validate the efficacy of this algorithm while also help to understand better its practical limitations, such as low coherence between reference and primary noise, random nature of occurrence and amplitude of impulsive noise, etc.

Pyschoacoustic study of impulsive noise, its active control and / or masking is highly recommended.

References

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