DEVELOPMENT OF NOVEL

DEVELOPMENT OF NOVEL
DEVELOPMENT OF NOVEL TECHNIQUES
TO STUDY NONLINEAR ACTIVE NOISE
CONTROL
Thesis submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
By
Kunal Kumar Das
Roll No. 50602001
Under the Supervision of
Prof. Jitendriya Ku. Satapathy
Department of Electrical Engineering
National Institute of Technology
Rourkela, INDIA
2011
… dedicated to my loving parents
CERTFICATE
This is to certify that the thesis entitled “Development of Novel Techniques to
Study Nonlinear Active Noise Control”, submitted to the National Institute of
Technology, Rourkela (INDIA) by Kunal Kumar Das, Roll No. 50602001 for the
award of the degree of Doctor of Philosophy in Electrical Engineering, is a bonafide
record of research work carried out by him under my supervision and guidance. The
thesis, which is based on candidate’s own work, has not been submitted elsewhere
for a degree/diploma. This thesis fulfills all the prescribed requirements for the
award of Doctor of Philosophy.
Dr. J. K. Satapathy, Ph. D. (Bradford)
Vice Chancellor
BPUT, ROURKELA
(on lien from
Department of Electrical Engineering
National Institute of Technology
ROURKELA)
ACKNOWLEDGEMENTS
The work presented in this dissertation would not have been possible without
the help and support of a large number of people. I first express my heartiest gratitude
to my guide and supervisor Prof. J.K. Satapathy, Vice Chancellor, Biju Patnaik University
of Technology, Rourkela (on lien from Department of Electrical Engineering, National
Institute of Technology, Rourkela) for his valuable guidance, inspiration and
encouragement in the course of the present work. The successful completion of the
work is due to his constant inspiration and motivation. Working with my supervisor is
highly enjoyable, inspiring and rewarding experience.
I humbly acknowledge the creative suggestions and constructive criticism of Prof
Bidyadhar Subudhi, HOD, Electrical Engineering, Prof. Sarat Kumar Patra and Prof.
Susmita Das committee members, while scrutinizing my research results. I express my
sincere thanks to Prof. Pradeepta Kumar Nanda for his valuable comments and advice
and also faculty members of the Department of Electrical Engineering, NIT Rourkela for
constant advice, useful discussions and encouragement in pursuing the research work.
The help and cooperation received from the Dean and Head of the Department
of Electronics and Communication Engineering, ITER, Bhubaneswar are gratefully
acknowledged.
Last but not the least I expresses my special thanks to my parents, wife(Ruby),
daughter(Adya), sisters and brothers for their prayer and all the inspiration for the last
six years for successful completion of this research work.
Kunal Kumar Das
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CONTENTS
Certificate
Acknowledgement
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Contents
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Abstract
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Acronyms and Abbreviations
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Nomenclatures
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List of Figures
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List of Tables
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1. INTRODUCTION
1.1 Background
1.2 Noise Control Techniques
1.2.1 Passive Noise Control
1.2.2 Active Noise Control
1.3 Classification of Noise
1.3.1 Broadband Noise
1.3.2 Narrowband Noise
1.4 Adaptive Active Noise Control
1.4.1 Adaptive Algorithms for ANC
1.4.2 Multiple-Channel ANC
1.4.3 Nonlinear ANC
1.4.4 ANC Applications
1.5 Motivation
1.5.1 Nonlinearity Effect
1.5.2 Secondary Path Effect
1.5.3 Computational Overload
1.5.4 Feedback Effect
1.6 Objective and Scope
1.7 Organization of the Thesis
1.8 Contribution of the Thesis
1.9 Summary
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2. STUDY AND APPLICATIONS OF ACTIVE NOISE CONTROL
2.1 Background
2.2 Broadband Feedforward Control
2.2.1 FXLMS Algorithm
2.2.2 FELMS Algorithm
2.2.3 Feedback Effect
2.2.4 ANC using IIR Filter
2.2.5 Narrowband Feedforward Control
2.3 Feedback Control
2.4 Multichannel Active Noise Control
2.5 Virtual Sensor Technique (VST)
2.6 Frequency Domain Approach
2.7 Active Noise Control in Headset
2.8 ANC in functional Magnetic Resonance Imaging (fMRI)
2.9 ANC in Infant Incubators
2.10 Study on Nonlinear ANC
2.10.1 Adaptive Volterra Filter (AVF) Approach
2.10.2 Adaptive Bilinear Filter (ABF) Approach
2.10.3 Radial Basis Function Approach (RBF) Approach
2.10.4 Fuzzy Logic (FL) Approach
2.10.5 Fuzzy-Neural Approach
2.10.6 Genetic Algorithm (GA) Approach
2.10.7 Particle Swarm Optimization (PSO) Approach
2.11 Summary
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3. NEURAL NETWORK APPROCH TO NONLINEAR ACTIVE NOISE CONTROL
3.1 Background
3.2 Multilayer Artificial Neural Network
3.3 Proposed Neural Network Technique
3.3.1 Linear Secondary Path
3.3.2 Nonlinear Secondary Path
3.4 Development of Neural Filtered-e LMS Algorithm
3.4.1 Linear Secondary Path
3.4.2 Nonlinear Secondary Path
3.5 Simulation and Results
3.6 Summary
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4. LEGENDRE NEURAL NETWORK FOR NONLINEAR ACTIVE NOISE CONTROL
4.1 Background
4.2 Reduced Structure Legendre Neural Network for Nonlinear ANC
4.2.1 Legendre Polynomial
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4.2.2 Legendre Neural Network
4.3 LFXLMS Algorithm
4.3.1 Nonlinear Secondary Path
4.4 LFELMS Algorithm
4.5 LFXRLS Algorithm
4.6 Fast LFXLMS Algorithm
4.7 Simulation and Results
4.8 Summary
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5. FREQUENCY-DOMAIN APPROACH TO MULTICHANNEL ACTIVE NOISE
CONTROL
5.1 Background
5.2 Filtered-x Least Mean Square (FXLMS) Algorithm
5.3 Time Domain Block Filtered-x LMS (BFXLMS) Algorithm
5.4 Frequency Domain Block Filtered-x LMS (FBFXLMS) Algorithm
5.5 Frequency Domain Block Filtered-x NLMS (FBFXNLMS) Algorithm
5.5.1 Reduced Structure FBFXNLMS Algorithm
5.6 Frequency Domain Block Filtered-e LMS (FBFELMS) Algorithm
5.7 Multichannel FBFXNLMS Algorithm
5.7.1 Computational Complexity
5.8 Frequency Domain Implementation of Legendre
Neural Network for Nonlinear ANC
5.9 Frequency Domain Block Legendre FELMS (FBLFELMS) Algorithm
5.10 Simulation and Results
5.11 Summary
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6. CONCLUSION AND SCOPE FOR FURTHER RESEARCH
6.1 Conclusion
6.2 Scope for Further Research
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REFERENCE
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Bio-data of the Candidate
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_______________________________________
ABSTRACT
______________________________________
Active noise control has been a field of growing interest over the past few
decades. The challenges thrown by active noise control have attracted the notice of the
scientific community to engage them in intense level of research. Cancellation of
acoustic noise electronically in a simple and efficient way is the vital merit of the active
noise control system. A detailed study about existing strategies for active noise control
has been undertaken in the present work. This study has given an insight regarding
various factors influencing performance of modern active noise control systems. The
development of new training algorithms and structures for active noise control are
active fields of research which are exploiting the benefits of different signal processing
and soft- computing techniques. The nonlinearity contributed by environment and
various components of active noise control system greatly affects the ultimate
performance of an active noise canceller. This fact motivated to pursue the research
work in developing novel architectures and algorithms to address the issues of nonlinear
active noise control.
One of the primary focus of the work is the application of artificial neural
network to effectively combat the problem of active noise control. This is because
artificial neural networks are inherently nonlinear processors and possesses capabilities
of universal approximation and thus are well suited to exhibit high performance when
used in nonlinear active noise control. The present work contributed significantly in
designing efficient nonlinear active noise canceller based on neural network platform.
Novel neural filtered-x least mean square and neural filtered-e least mean square
algorithms are proposed for nonlinear active noise control taking into consideration the
nonlinear secondary path. Employing Legendre neural network led the development of
a set new adaptive algorithms such as Legendre filtered-x least mean square, Legendre
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filtered-e least mean square, Legendre filtered-x recursive least square and fast
Legendre filtered-x least mean square algorithms. The proposed algorithms
outperformed the existing standard algorithms for nonlinear active noise control in
terms of steady state mean square error with reduced computational complexity.
Efficient frequency domain implementation of some the proposed algorithms have been
undertaken to exploit its benefits. Exhaustive simulation studies carried out have
established the efficacy of the proposed architectures and algorithms.
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ACRONYMS AND ABBREVIATIONS
ABF
AFC
AGA
ANC
ANFIS
AVF
BFXLMS
CRPSO
dB
DFT
DSP
EFNN
EPI
FAGA
FBFELMS
FBFXLMS
FBFXNLMS
FBLFXLMS
FBLFELMS
FELMS
FFELMS
FFT
FFXLMS
FIR
FL
FLANN
fMRI
FNN
FPGA
FSLMS
FULMS
FXLMS
FXNLMS
Adaptive Bilinear Filter
Adaptive Feedback Cancellation
Adaptive Genetic Algorithm
Active Noise Control
Adaptive Neuro Fuzzy Inference System
Adaptive Volterra Filter
Block Filtered-x Least Mean Square
Conditional Reinitialized Particle Swarm Optimization
Decibel
Discrete Fourier Transform
Digital Signal Processing
Enhanced Fuzzy Neural Network
Eco Planner Imaging
FIR Adaptive Genetic Algorithm
Frequency Domain Block Filtered-e Least Mean Square
Frequency Domain Block Filtered-x Least Mean Square
Frequency Domain Block Filtered-x Normalized Least Mean square
Frequency Domain Block Legendre Filtered-x Normalized Least
Mean Square
Frequency Domain Block Legendre Filtered-e Normalized Least
Mean Square
Filtered-e Least Mean Square
FLANN Filtered-e Least Mean Square
Fast Fourier Transform
FLANN Filtered-x Least Mean Square
Finite Impulse Response
Fuzzy Logic
Functional Link Artificial Neural Network
functional Magnetic Resonance Imaging
Fuzzy Neural Network
Field Programmable Gate Array
Filtered-s Least Mean Square
Filtered-u Least Mean Square
Filtered-x Least Mean Square
Filtered-x Normalized Least Mean Square
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FXRBF
GA
Hz
IFFT
IIR
LFELMS
LFXLMS
LFXRLS
LMS
LNN
LNL
LSP
MIMO
MLANN
MLP
MSE
MSE(dB)
MR
NANC
NFELMS
NFXLMS
NICU
NLMS
NN
NSP
PSO
RBF
RMT
RNN
SPE
S/P
P/S
TDNN
VAGA
VC
VFELMS
VFXLMS
VST
WI-FXLMS
Filtered-x Radial Basis Function
Genetic Algorithm
Hertz
Inverse Fast Fourier Transform
Infinite Impulse Response
Legendre Filtered-e Least Mean Square
Legendre Filtered-x Least Mean Square
Legendre Filtered-x Recursive Least Square
Least Mean Square
Legendre Neural Network
Linear Nonlinear Linear
Linear Secondary Path
Multiple Input Multiple Output
Multilayer Artificial Neural Network
Multilayer Perceptron
Mean Square Error
Mean Square Error in decibel
Magnetic Resonance
Nonlinear Active Noise Control
Neural Filtered-e Least Mean Square
Neural Filtered-x Least Mean Square
Neonatal Intensive Care Unit
Normalized Least Mean Square
Neural Network
Nonlinear Secondary Path
Particle Swarm Optimization
Radial Basis Function
Remote Microphone Technique
Recurrent Neural Network
Secondary Path Equalization
Serial to Parallel
Parallel to Serial
Time Delay Neural Network
Volterra Adaptive Genetic Algorithm
Visual Cortex
Volterra filtered-e LMS
Volterra filtered-x LMS
Virtual Sensor Technique
Wiener Initialized Filtered-x Least Mean Square
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NOMENCLATURES
Primary path transfer function
A(z )
Aˆ ( z )
Estimated primary path transfer function between physical and
a (n)
virtual location
Impulse response of primary path
v
aˆ v (n)
B (z )
Bˆ ( z )
B p (z )
Bˆ ( z )
p
Impulse response of filter having transfer function Aˆ v ( z )
Secondary path transfer function
Estimated secondary path transfer function
Secondary path transfer function at physical location
Estimated Secondary path transfer function at physical location
Bˆ v ( z )
Estimated Secondary path transfer function at virtual location
Bˆ _( z )
Adjoint Secondary path filter
Impulse response of secondary path
k th element of secondary path impulse response
b(n)
bk (n)
~
b (n)
b _(n)
~
bk (n)
~
b _(n)
W ( z)
x (n)
x ′(n)
d (n)
dˆ (n)
Impulse response of virtual secondary path
Impulse response of adjoint secondary path
k th element of virtual secondary path impulse response
Impulse response of adjoint virtual secondary path filter
Transfer function of adaptive filter
Reference signal
Filtered Reference signal
Undesired signal
Estimated undesired signal
dˆ p (n)
Estimate of primary disturbance at physical location
dˆv (n)
d p (n)
Estimate of primary disturbance at virtual location
Primary disturbance at physical location
y ( n)
y p (n)
Adaptive filter output
Antinoise at physical location
yˆ p (n)
Estimated antinoise at physical location
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yˆv (n)
Estimated antinoise at virtual location
e( n )
e p (n)
Error signal
Error signal from physical microphone
eˆv (n)
n
i
j
l
p
q
L
P
Q
w (n)
wi , j (n)
Estimated error at virtual location
Time index
An arbitrary variable
An arbitrary variable
An arbitrary variable
An arbitrary variable
An arbitrary variable
Number of reference microphones
Number of secondary loudspeakers
Number of error microphones
Coefficient vector of adaptive filter
Filter weights
x(n)
x ′(n)
X( n )
X′(n)
Y ( n)
E( n )
W ( n)
ξ (n)
N
Reference signal vector
Filtered reference signal vector
µ
µ max
∇ξ ( n )
Px′
E[.]
∆
e′(n)
f (n)
fˆ (n)
F (n)
Fˆ (n)
−1
FFT of reference signal vector
FFT of filtered reference vector
FFT of adaptive filter output vector
FFT of flipped error signal vector
FFT of coefficient vector of adaptive filter
Cost function
Order of adaptive filter
Step size
Maximum allowable step size for FXLMS algorithm
Instantaneous estimate of gradient on MSE surface
Power of the filtered reference signal
Expectation operator
No of samples corresponding to overall delay in the secondary path
Filtered error signal
Impulse response of the feedback path
Estimated Impulse response of the feedback path
Transfer function of the feedback path
Estimated transfer function of the feedback path
z
W f ( z)
Unit delay
Forward path transfer function of IIR filter
Wb ( z )
Feedback path transfer function of IIR filter
x
w f (n)
Weight vector of W f ( z )
w b ( n)
Lw f
Weight vector of Wb ( z )
Lwb
Order of Wb ( z )
u ( n)
u ′(n)
y ( n)
y ′(n)
y ( n)
y ′(n)
I
J
v j (n)
Reference vector
Filtered reference vector
Output signal of adaptive filter
Filtered output signal of adaptive filter
Output signal vector of adaptive filter
Filtered output signal vector of adaptive filter
Number of neurons in input layer
Number of neurons in hidden layer
Synaptic weights of output layer
wij (n )
Synaptic weights of hidden layer
F
u j ( n)
Activation function
Hidden layer node output
c j (n)
Net internal activity level of neural node
M
L p ( x)
Order of secondary path filter
Legendre Polynomial
s P (n)
s ′p (n)
Expanded reference vector
Filtered expanded reference vector
k i (n)
λ
Q i (n)
α i, j
Kalman gain vector
Forgetting factor
Inverse of the autocorrelation matrix
Expanded input signal vector
ψ i (n)
β i, j
Expanded input signal
Expanded input signal vector
γ i, j
Expanded input signal vector
T
Ai
Vector transpose
Number of additions
Number of multiplications
N × N identity matrix
Mi
IN
ON
F2 N
−1
F2 N
Order of W f ( z )
N × N matrix with all zero elements
2N-point FFT operator
2N-point IFFT operator
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LIST OF FIGURES
1.1 Lueg’s active noise control patent (U.S. patent No. 2043416, June 1936)
1.2 Principle of active noise control
1.3 Diagram of the basic arrangement of an ANC in a duct
2.1 Schematic diagram of a broadband feedfoward ANC
2.2 Block diagram of a single channel feedfoward ANC
2.3 Impulse response of primary path
2.4 Impulse response of secondary path
2.5 Block diagram of ANC using adjoint LMS algorithm
2.6 Schematic diagram of ANC with feedback neutralization filter
2.7 Block diagram of ANC with feedback neutralization filter
2.8 Block diagram of ANC (with feedback) using IIR adaptive filter
2.9 Schematic diagram of a narrowband feedfoward ANC
2.10 Schematic diagram of a feedback ANC
2.11 Schematic diagram of a multi-channel ANC for an enclosure
2.12 Schematic diagram of ANC using virtual sensor technique
2.13 Block diagram of ANC using virtual sensor technique
2.14 Block diagram of ANC using frequency-domain FXLMS algorithm
2.15 Schematic diagram of ANC used for headphone or headset
2.16 Block diagram of ANC using filtered-x second
order Volterra adaptive algorithm
2.17 Block diagram of ANC using adaptive output-error bilinear filter
2.18 Block diagram of ANC using RBF
2.19 Structure of the two RBF networks for ANC
2.20 Structure of the two recurrent RBF networks for ANC
2.21 Block diagram of ANC using fuzzy logic
2.22 Structure of the neuro-fuzzy controller used for ANC
2.23 Block diagram of ANC using ANFIS
2.24 Block diagram of ANC using fuzzy-neural network
2.25 Structure of the fuzzy-neural network for ANC
2.26 Block diagram of the genetic ANC system
2.27 Block diagram of PSO-based training of an ANC system
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3.1 Block diagram of ANC using two neural networks
3.2 Block diagram of ANC using neural network
3.3 Neural network controller
3.4 Block diagram of ANC using FELMS algorithm
3.5 MSE(dB) plot for NFXLMS and VFXLMS algorithm (activation function I)
3.6 MSE(dB) plot for NFXLMS and VFXLMS algorithm (activation function II)
3.7 MSE(dB) plot for NFXLMS and VFXLMS algorithm (activation function III)
3.8 MSE(dB) plot for NFELMS and VFELMS algorithm (activation function I)
3.9 MSE(dB) plot for NFELMS and VFELMS algorithm (activation function II)
3.10 MSE(dB) plot for NFELMS and VFELMS algorithm (activation function III)
3.11 MSE(dB) plot for NFXLMS and VFXLMS algorithm (activation function I)
3.12 MSE(dB) plot for NFXLMS and VFXLMS algorithm (activation function II)
3.13 MSE(dB) plot for NFXLMS and VFXLMS algorithm (activation function III)
3.14 MSE(dB) plot for NFELMS and VFELMS algorithm (activation function I)
3.15 MSE(dB) plot for NFELMS and VFELMS algorithm (activation function II)
3.16 MSE(dB) plot for NFELMS and VFELMS algorithm (activation function III)
3.17 Factory Floor Noise
3.18 Buccaneer Jet Cockpit Noise
3.19 MSE(dB) plot for NFXLMS and VFXLMS algorithms( Factory Floor Noise)
3.20 MSE(dB) plot for NFXLMS and VFXLMS algorithms( Jet Cockpit Noise)
3.21 MSE(dB) plot for NFELMS and VFELMS algorithms( Factory Floor Noise)
3.22 MSE(dB) plot for NFELMS and VFELMS algorithms(Jet Cockpit Noise)
3.23 MSE(dB) plot for NFXLMS and VFXLMS algorithms(Factory Floor Noise)
3.24 MSE(dB) plot for NFXLMS and VFXLMS algorithms(Jet Cockpit Noise)
3.25 MSE(dB) plot for NFELMS and VFELMS algorithms(Factory Floor Noise)
3.26 MSE(dB) plot for NFELMS and VFELMS algorithms(Jet Cockpit Noise)
4.1 Legendre polynomial expansion
4.2 Legendre neural network
4.3 Block diagram of ANC using reduced structure Legendre neural network
4.4 Secondary path filter
4.5 Adjoint Secondary path filter
4.6 Virtual Secondary path filter
4.7 Adjoint virtual Secondary path filter
4.8 MSE(dB) plot for LFXLMS, FFXLMS, LFELMS and LFXRLS algorithm
4.9 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm when primary
path is changed at 3000th iteration
4.10 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm
when secondary path is changed at 3000th iteration
4.11 MSE(dB) plot for LFXLMs, FFXLMS and LFELMS algorithm when
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both primary path and secondary path are changed at 3000th iteration
4.12 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm
4.13 Block diagram of LNL nonlinear secondary path model
4.14 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm
4.15 MSE(dB) plot for LFXLMS and fast LFXLMS algorithm
4.16 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(Jet Cockpit Noise)
4.17 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(White Noise)
4.18 MSE(dB) plot for LFXLMS and FFXLMS algorithm when primary path
is changed at 3000th iteration(Jet Cockpit Noise)
4.19 MSE(dB) plot for LFXLMS and FFXLMS algorithm when primary path
is changed at 3000th iteration(White Noise)
4.20 MSE(dB) plot for LFXLMS and FFXLMS algorithm when secondary path
is changed at 3000th iteration(Jet Cockpit Noise)
4.21 MSE(dB) plot for LFXLMS and FFXLMS algorithm when secondary path
is changed at 3000th iteration(White Noise)
4.22 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(Jet Cockpit Noise)
4.23 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(White Noise)
5.1 Block diagram of basic active noise control system
5.2 Block diagram of ANC using FBFXNLMS algorithm
5.3 Block diagram of ANC using reduced structure FBFXNLMS algorithm
5.4 Block diagram of ANC using FBFELMS algorithm
5.5 Comparison computational complexity (a) multiplication (b) addition
5.6 Legendre neural network for nonlinear ANC
5.7 Frequency domain implementation of Legendre neural network for ANC
5.8 details of T block of fig. 5.7 (FBLFXLMS algorithm)
5.9 Details of T block of fig. 5.7 (FBLFELMS algorithm)
5.10 MSE(dB) plot of FBFXNLMS algorithm for multichannel ANC
5.11 MSE(dB) plot of FXNLMS algorithm for multichannel ANC
5.12 MSE(dB) plot of FBFXNLMS algorithm for multichannel ANC
5.13 MSE(dB) plot of FXNLMS algorithm for multichannel ANC
5.14 MSE(dB) plot of FBLFELMS algorithm at error microphone-1
5.15 MSE(dB) plot of FBLFELMS algorithm at error microphone-2
5.16 MSE(dB) plot of FBLFELMS algorithm at error microphone-3
5.17 MSE(dB) plot of FBLFELMS algorithm at error microphone-4
5.18 MSE(dB) plot of FBLFELMS algorithm of all the error microphones
5.19 MSE(dB) plot of LFELMS algorithm of all the error microphones
5.20 MSE(dB) plot of FXNLMS algorithm for white noise
5.21 MSE(dB) plot of FBFXNLMS algorithm for white noise
5.22 MSE(dB) plot of FXNLMS algorithm for factory floor noise
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5.23 MSE(dB) plot of FBFXNLMS algorithm for factory floor noise
5.24 MSE(dB) plot of FXNLMS algorithm for Buccaneer jet cockpit noise
5.25 MSE(dB) plot of FBFXNLMS algorithm for Buccaneer jet cockpit noise
5.26 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise
at error microphone-1
5.27 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise
at error microphone-2
5.28 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise
at error microphone-3
5.29 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise
at error microphone-4
5.30 MSE(dB) plot of FBLFELMS algorithm of all the error microphones (combined)
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LIST OF TABLES
4.1 Computational complexity comparison
5.1 FBFXNLMS algorithm
5.2 FBFELMS algorithm
5.3 Computational complexity per samples
5.4 FBLFXLMS algorithm
5.5 FBLFELMS algorithm
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xv
Chapter 1
Introduction
1.1 Background
Industrialization and ever increasing use of technology in our daily life has led to
exponential increase of acoustical noise pollution. Noise is an irritant and causes mental
strain. It has an adverse psychological effect on living beings and creates an unhealthy
working environment. Continuous exposure to noise has a detrimental effect leading to
temporary or permanent loss of hearing and mood swing. Acoustic noise problem in the
environment is gaining attention due to the tremendous growth of technology that has led
to noisy engines, heavy machinery, pumps, home appliances and a myriad other noise
sources. Exposure to high levels of sound proves damaging to humans from both physical
and psychological aspect. The problem of controlling the noise level in the environment
has been the focus of a tremendous amount of research over the years. Legislations have
been enforced on industries and manufacturers to keep the maximum noise level of their
products under specified limits. However, as long as the quest for larger machinery using
light material continues, the noise pollution level will be on the rise. Due to these reasons,
noise control has gained considerable importance in the recent years. Human desire for a
high-tech but comfortable and noise free living has fueled the development of acoustic
noise control techniques.
Acoustic noise control techniques can be broadly classified into two categories:
passive control technique and active control technique. The combination of passive
control technique with active control technique can yield better overall performance than
for either one alone. In many respects, active noise control (ANC) is a complementary
technology to passive silencing [1].
1.2 Noise Control Techniques
1.2.1 Passive Noise Control
Passive noise control techniques employ sound absorbing material, enclosures,
barriers and silencers to attenuate the undesired noise. These passive techniques are
1
CHAPTER-1
INTRODUCTION
effective for high attenuation over a broad frequency range; but are unable to absorb low
frequency noise. For low frequency noise the passive techniques become relatively larger
and heavier thus considerably increasing the cost. This often makes the passive approach
to reduce low frequency noise very impractical. But the fact is people are more
uncomfortable with low frequency noise rather than high frequency noise because low
frequency noise is not only annoying but produces fatigue, irritation and loss of
concentration, therefore affecting productivity. If low frequency noise is mixed with
speech, it reduces speech intelligibility [1], [2]. In a noise generating system the
amplitude of the low frequency noise is mostly higher than other frequencies. Hence
there is a growing demand for reducing low frequency noise.
1.2.2 Active Noise Control
The design of an active noise controller using a microphone and an electronically
driven loudspeaker to generate a cancelling sound was first proposed and patented using
a purely analog electronic approach in 1930 in France by Coanda [4] and in US by Lueg
in 1936 [5]. The patent outlined the basic idea of ANC, shown in fig.1.1, but at that time
it could not be applied to practice because of a number of factors. Advanced and accurate
electronic instruments (microphone, loudspeakers, digital personal computers) were not
available, digital signal processing and the concept of adaptive systems were not started
then and there was no proper knowledge on noise and various sources of noise. During
the latter half of the 20th century, emergence of digital signal processing made ANC a
viable technique for practical noise reduction. Latter on advances in adaptive systems and
adaptive signal processing which facilitate a time varying system with the ability to adapt
to changing environment, not only revolutionized ANC but also further opened up its
field of applications. Development of high speed special purpose digital signal processors
helped realize practical ANC. More recently fusion of the soft computing approaches like
artificial neural network, fuzzy logic and hybrid techniques (neuro-fuzzy techniques)
with existing ANC techniques have made ANC powerful than ever before. ANC is
currently being researched for use to control noise from jet engines, helicopter, motor
vehicle engines, ventilation systems, generators, transformers, industrial machinery,
traffic, MRI units, torpedoes, headphones and even amplified music.
2
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INTRODUCTION
Fig 1.1 Lueg’s Active Noise Control Patent (U.S. Patent No. 2043416, June 1936) [5].
The ANC is an electroacoustic device that is based on the principle of destructive
interference where the unwanted sound is cancelled by generating an antisound
(antinoise) of equal amplitude and opposite phase. The original unwanted noise and the
antinoise superimpose acoustically, resulting in the cancellation of both sounds. For
example, fig. 1.2 shows the waveforms of a typical unwanted noise (called the reference
noise), the cancelling noise (called the antinoise), and the residual noise that results when
they superimpose. For cancellation of the reference noise the amplitude of antinoise must
be same as that of reference noise but phase should be opposite (1800 out of phase). So
the effectiveness of cancellation of the reference noise depends on the accuracy of the
amplitude and phase of the generated antinoise.
3
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INTRODUCTION
Noise
Residual Noise
Antinoise
Fig 1.2 Basic Principle of Active Noise Control.
As discussed earlier passive noise control approach is effective for controlling
noise over the entire frequency range except at low frequency (below 600 Hz). Hence the
demand is to develop ANC which should be able to control low frequency noise. ANC
take advantage of this situation as lower noise frequency allow lower sampling rate (as
low as 1200 Hz) resulting in comparatively higher sample period (approximately
0.833millisecond for a sampling frequency of 1200Hz). An ANC has to finish all the
computation to generate antinoise within a single sample period and should be ready to
accommodate the next sample of reference signal as soon as it arrives (reference noise
signal samples are available at the rate of one sample every sample period). A larger
signal sample period not only allow the ANC to finish it computation in time but also
facilitates employment of more complicated and computationally intensive adaptive
algorithms for achieving better noise cancellation performance.
So active attenuation is an attractive means to achieve large amounts of noise
reduction electronically, particularly at low frequencies. Another advantage of low
frequencies noise is, it allow plane wave propagation making the job of an ANC easier as
the sound field is not complicated. In essence, active noise control shows real advantages
to control low frequency noise. So ANC has received considerable research interest and
has shown significant potential to control low frequency noise. The creation and superpo4
CHAPTER-1
INTRODUCTION
Duct
Noise
Noise
Source
Reference
Microphone
Residual noise
Antinoise
Secondary
Loudspeaker
Error
Microphone
ANC
FXLMS
Fig.1.3 Diagram of the basic arrangement for an ANC in a duct.
sition of the antinoise for controlling three dimensional reference noise is very complex,
since this involves reconstruction of the whole acoustic event. The discussion in this
thesis restricts itself to one-dimensional ANC in long ducts. One basic arrangement to
cancel noise in a duct is shown in the fig. 1.3.
Working with low frequency noise in ducts has got the following advantages:
firstly, the sound will travel as plane waves upto a certain frequency called cutoff
frequency. The noise of higher frequencies will decay within a short distance from the
source. So the mixing of the reference noise and antinoise is easier. Secondly, the low
frequency noise has a longer wavelength, so that the phase angle changes slowly with
time. This makes the fine control of phase of secondary wave easier. Hence, a stable
interference pattern is possible that results in larger noise reduction. Lastly, the sound
wave travels at much slower speed than the electrical signals, so that a large operation
time for generating the antinoise is available, if secondary source is suitably located.
Acoustic noise can be broadly classified into two types: broadband noise and narrowband
noise.
5
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INTRODUCTION
1.3 Classification of Noise
1.3.1 Broadband Noise
Broadband noise is caused by turbulence and is random in nature. Turbulent noise
distributes its energy evenly across the frequency bands. The examples are the lowfrequency sounds of jet planes and the impulse noise of an explosion.
1.3.2 Narrowband Noise
In narrowband noise most of its energy is concentrated at specific frequencies.
This type of noise is related to rotating or repetitive machines, so it is periodic or nearly
periodic in nature. Examples of narrowband noise include the noise of internal
combustion engines in transportation, compressors as auxiliary power sources and in
refrigerators, and vacuum pumps used to transfer bulk materials in many industries. The
transformer noise which is the hum noise due to the magneto-striction consists of higher
harmonics of the power-line frequency. In another way the noise can be classified into
two types: linear noise and nonlinear noise. The broadband noise is mostly linear. But
there are situations where noise coming from a dynamic system may be nonlinear and
deterministic. Such nonlinear but deterministic noise is referred to as chaotic noise. Some
examples of chaotic noise are Logistic chaotic noise, Lorenz chaotic noise and Duffing
chaotic noise [11]. One practical example of chaotic noise is the fan noise which often
shows chaotic behavior.
1.4 Adaptive Active Noise Control
The acoustic noise source and the environment are time varying, the frequency
content, amplitude, phase, and velocity of the undesired noise are nonstationary (time
varying). So an active noise control system must be adaptive in order to cope with these
changing characteristics. This is the reason why modern active noise control systems
depend heavily on digital signal processing because in the field of digital signal
processing, there are classes of
systems called adaptive systems which have the
capability to vary their coefficients in order to cope with changing environment. Adaptive
systems can be implemented as transversal—finite impulse response (FIR), recursive—
infinite impulse response (IIR), lattice filters, transform-domain filters. Correspondingly
ANC can also be implemented using FIR filter, IIR filter, lattice filters or transform-
6
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INTRODUCTION
domain filters. These yield varieties of ANC different from each other in many aspects.
Performance of each of them is well researched and documented in literature [1].
The coefficients of an adaptive filter are tuned to minimize a predefined cost
function which is generally a function of error signal. The process of adaptation is
automated by DSP adaptation algorithms. Depending on the problem, a large number of
adaptation algorithms have been developed to adapt the system quickly and efficiently.
The pros and cons of available adaptive algorithms have been analyzed mathematically,
through exhaustive computer simulation and also real time implementations. A large
number of researchers are still involved in developing new adaptation algorithms by
exploring advanced digital signal processing techniques and soft computing techniques in
order to optimize the overall performance of the ANC.
1.4.1 Adaptive Algorithms for ANC
The ANC system can be implemented using different adaptive learning algorithms.
The most common algorithm applied to adaptive filters is the least mean-squared (LMS)
algorithm [1]-[3],[7]. The reference noise signal and error signal are used as input to an
adaptive algorithm, which adjusts the adaptive filter coefficients to model (estimate) the
acoustic-channel effects. For ANC, taking into account presence of secondary path, LMS
algorithm is suitably modified to develop an efficient but simple algorithm known as
filtered-x LMS (FXLMS) algorithm which was derived by Widrow [1]. With advances in
digital signal processing, FXLMS algorithm is further modified by many researchers to
improve the ANC overall performance. All the developed algorithms have their relative
merits and demerits in terms of speed of convergence, residual noise, computational
complexity, stability and robustness.
1.4.2 Multiple-Channel ANC
In applications where noise field is complex or the required zone of silence is
quite large, use of single reference microphone, single secondary loudspeaker or an error
microphone is not sufficient to reduce noise to a desirable limit. In this scenario, several
numbers of secondary loudspeakers, error microphones and reference microphones are
employed. The most important aspect is proper positioning of sensors to have overall
global noise suppression and also to optimize the service of each sensor.
7
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INTRODUCTION
1.4.3 Nonlinear ANC
FXLMS algorithm and all its variants are useful for linear ANCs i.e. when the
primary path, secondary path and the reference noise are linear. But practically there are
many sources of nonlinearity which exist in an ANC. So most of the ANCs are found to
be nonlinear in nature hence the present arrangement with linear FXLMS algorithm or all
its variants exhibit performance degradation. There is ample scope for performance
improvement by employing various nonlinear structures and algorithms. Volterra filter is
a nonlinear adaptive filter best suited for this type of problem. The real advantage of
using Volterra adaptive filter lies with the fact that it can be trained by linear type
adaptive algorithms. So FXLMS algorithm and most of its variants can be extended to
train ANC designed using Volterra adaptive filter.
Different neural networks such as multi layer perceptron (MLP) with derivative
based back propagation training algorithm, radial basis function networks, fuzzy logic
and neuro-fuzzy have also been used for nonlinear ANC. Research is in nascent stage for
using derivative free evolutionary techniques such as genetic algorithm and particle
swarm optimization for nonlinear ANC.
ANC systems may soon be available for many noisy environments. As industry
adopts mechatronic design techniques, acoustic considerations can be made from the
outset of the design process. ANC can be expected to be an integral part of vehicle and
industrial design.
1.4.4 ANC Applications
ANC applications can be broadly classified under the following categories
• Duct noise: This is the major application area of ANC because of widespread
industrial applications like heating, ventilating and air conditioning systems. In
air conditioning systems noise is controlled using ANC in the ducts, which is the
transmission path of noise.
• Personal hearing protection: The headphone falls into this classification, where
the loudspeaker generates not only the desired sound but also an antinoise which
cancel the low frequency ambient noise. Headphones also have ear shells which
attenuate high frequency noise.
8
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INTRODUCTION
• Enclosure Noise: Suppression of noise inside an enclosure like the noise inside
the passenger cabin of a propller-driven aircraft is a typical example of enclosure
noise ANC.
• Open field noise: Transformer noise cancellation or creating a zone of silence
near the ears of a person sitting in a chair by providing two loudspeakers to
generate antinoise near the headrest.
• Virtual Microphone: Placing a microphone at the desired place of noise reduction
is not always practical like noise in a duct carrying industrial fluid. Virtual
acoustic sensor techniques have been developed to overcome this problem. Here
virtual acoustic sensors create a zone of silence at the desired location which may
be remote from the physical sensor position.
1.5 Motivation
It has been reported in the literatures that various problems are encountered while
practical implementation of the ANC systems are carried out. Some of these are:
1.5.1 Nonlinearity Effect
In almost all applications of ANC primary path and secondary path exhibit
nonlinear characteristics. In some applications reference noise is also produced by a
nonlinear noise process. This is called nonlinear active noise control (NANC).
Nonlinearity associated with noise process and the paths create problems in linear
adaptation of the ANC. In the recent literatures, several methods have been proposed for
NANC. With advances in digital signal processing and soft computing techniques much
remain to be done to improve the overall performance of NANC.
1.5.2 Secondary Path effect
In order to enable the adaptive filter to learn properly to a desired solution, it is
necessary to compensate for the transfer function of the secondary path, B (z ) , from the
secondary source to the error sensor. Presence of secondary path following the adaptive
filter prevents straight forward applications of the adaptive algorithms. This requires
careful modification of the algorithms to be successfully applied to ANC. This leads to
the modification of least mean square (LMS) algorithm to develop filtered-x least mean
9
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INTRODUCTION
squares (FXLMS) algorithm by Widrow [1]. Development of new algorithms which take
care of secondary path is still an open area of research.
1.5.3 Computational Overload
Filtered-x least mean square (FXLMS) algorithm based ANC needs hundreds of
taps to realize appreciable noise cancellation. As the number of taps increase the
computational complexity of the FXLMS algorithm increases significantly. Also in
FXLMS algorithm the reference signal is required to be filtered through the secondary
path estimate. In practical ANC system, the secondary path estimate also has hundreds of
taps. Hence, involvement of large computational complexity is a major problem with
regards to implementation. A number of research works have been undertaken with an
aim to reduce the computational complexity and make the ANC computation fast.
1.5.4 Feedback Effect
The acoustic feedback from the loudspeaker to the reference microphone, which
causes degradation in the performance of the ANC system, is known as the feedback
effect. Many investigators have proposed different structures to circumvent this problem.
However, their models are practically not implementable or do not provide perfect
cancellation of feedback noise. The online adaptive feedback cancellation (AFC) based
ANC system also shows poor performance if the reference signal is narrowband or
periodic.
From the literature survey on the topic, it has been concluded that attempts have
been made to resolve the associated problems of ANC. However, further scope still exists
to improve on the performance by devising efficient techniques to tackle these problems.
Hence, these observations provide the motivation to undertake research work on these
problems of ANC. The objective of the thesis is to develop novel algorithms using soft
computing and DSP techniques to efficiently mitigate various issues of NANC.
1.6 Objective and Scope
Efficient ANC for real time implementation has been a challenging task as
evident from the literature survey and has opened up new avenues to develop high
performance based ANC for nonlinear environment. The present work primarily focuses
in addressing the following issues.
•
To study the performance of available nonlinear ANC.
10
CHAPTER-1
•
INTRODUCTION
To develop new ANC meant for nonlinear scenario employing soft computing
approaches like artificial neural network.
•
Reduction in computational complexity requirement of the ANC without
compromising with noise cancellation performance.
•
To explore frequency domain implementation of multichannel ANC and
nonlinear ANC techniques.
1.7 Organization of the thesis
The work in this thesis is organized as follows:
Chapter-I: This chapter presents an introduction where various problems linked with
ANC implementation are discussed. A brief literature survey, motivations for doing
research, objective and scope of the thesis and organization of the thesis are also
presented.
Chapter-II: This chapter deals with an exhaustive study of existing linear and nonlinear
ANC schemes. Various factors affecting the noise cancellation performance are
discussed. The role played by ANC and the apparent benefits of employing an ANC in
these applications are analyzed extensively. The problems encountered in the real time
applications of ANC are also highlighted.
Chapter-III: The multilayer perceptron (MLP) is employed as the controller and new
synaptic weight update algorithm is developed. This adaptive algorithm is found to be a
generalized version of FXLMS algorithm. In order to reduce complexity, FELMS
algorithm is also explored by using adjoint secondary path.
Chapter-IV: A novel reduced structure Legendre neural network for nonlinear ANC is
proposed for active mitigation of nonlinear noise processes. Low complexity Legendre
filtered-x LMS (LFXLMS) algorithm, Legendre filtered-e LMS (LFELMS) algorithm,
Legendre filtered-x RLS (LFXRLS) algorithm are developed.
Chapter-V: Frequency domain block algorithms are developed, both for filtered-x and
filtered-e paradigm, basically to reduce computational complexity. The proposed
algorithms are extended to incorporate multichannel ANC systems. The Legendre neural
network developed in the previous chapter is also implemented in frequency domain.
Chapter-VI: The overall conclusion of the thesis and the scope for further research are
outlined in this chapter.
11
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INTRODUCTION
1.8 Contribution of the Thesis
This thesis proposes solutions to a number of burning issues of ANC system and
suggests some novel methods using DSP and soft computing techniques to mitigate them.
The problems dealt in this thesis are secondary path effect, nonlinearity effect,
computational overload and multichannel effect. DSP techniques used in this thesis are
adaptive filtering and discrete transforms like FFT. The soft computing tools used are
Volterra filters, functional link artificial neural network and Legendre neural network.
These are suitably applied to develop efficient ANC systems.
Development of new nonlinear ANC structure and pertinent algorithms is the
principal thrust of this study. The multilayer perceptron is used as the controller and
associated adaptive algorithms are developed. The new neural based adaptive algorithm
is found to be an extended version of FXLMS algorithm. The specific advantage of the
developed ANC relies on the fact that it can incorporate nonlinear secondary path without
using a second neural network for nonlinear secondary path modeling. This is possible by
deriving a time varying virtual secondary path. In an effort to reduce computational
complexity FELMS algorithm is also explored by using adjoint secondary path concept.
To validate the proposed controller and the algorithms, exhaustive simulation study is
carried out. Performances of the proposed algorithms are compared with Volterra based
algorithms. Neural based algorithms are found to be clearly outperforming the Volterra
based algorithms.
A novel Legendre neural network for nonlinear active noise control is also
proposed. Legendre polynomial is used for functional expansion of the reference input of
the controller. Filter bank implementation of the controller is carried out and the
equations relative to weight adaptation are derived. Various block oriented models such
as Linear-Nonlinear-Linear model are successfully used for nonlinear secondary path
modeling in case of the developed Legendre neural network for NANC. Recursive least
square (RLS) algorithm is also explored to develop Legendre filtered-x RLS (LFXRLS)
algorithm. A fast algorithm is also developed which reduces computational complexity
by updating the weight vector once in two iterations without sacrificing the system
performance.
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INTRODUCTION
Frequency domain block algorithms are developed basically to reduce
computational complexity. The proposed algorithms are extended to incorporate
multichannel ANC systems. The nonlinear ANC using Legendre neural network
developed in the present work is also implemented in frequency domain to reduce
computational complexity and is found to be an efficient candidate for nonlinear ANC
systems.
1.9 Summary
This thesis proposes solutions to a number of critical issues of ANC system and
suggests some novel methods using DSP and soft computing techniques to alleviate
them. The problems generally considered in this thesis are nonlinearity effect,
computational overload and adaptive filtering techniques employing both time domain
and frequency domain approaches to study the problems associated with ANC. Further
efficient ANC structure designs based on the platforms of neural network, Volterra
filters, Legendre neural network and functional link artificial neural network have been
developed to achieve better performances.
13
Chapter 2
Study and Applications of Active Noise Control
2.1 Background
Active noise control is an attractive technique for mitigation of undesirable noise,
particularly low frequency noise. ANC should be used in combination with passive noise
control to get an overall noise reduction across the entire audible frequency range.
Researchers have shown interest continuously for the last few decades to develop
efficient ANC. The development was greatly influenced by advances in digital signal
processing and adaptive filtering. More recently domain of ANC benefitted hugely by
applying soft computing techniques. The present research work is particularly focused on
designing better ANC controller employing new architecture/structure and/or developing
novel adaptive algorithms. The effort is concentrated in studying the factors which
greatly influence the real time application of ANC. Some of these important factors are
the steady-state noise reduction capability,
rapid convergence and reducing
computational complexity to reduce response time.
ANC has been in use and currently being researched to be used in diverse field of
engineering and technology. Some of the application areas are, to name a few, heating,
ventilating and air conditioning, headphone, earplug (headset), hearing protection,
hearing aid, infant incubator, transformer noise reduction, functional Magnetic
Resonance Imaging (fMRI), vehicle interior noise reduction, noise reduction in airplane
and locomotives, active voice control, noise reduction in industry and mines and military
etc.
In this chapter an in-depth study of existing ANC techniques is carried out. The
study can be subdivided into two categories
i) Study on linear ANC
ii) Study on nonlinear ANC.
14
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Effort has also been made in this chapter to study some of the important applications of
ANC till date.
2.2 Broadband Feedforward Control
Structurally active noise control can be implemented in two different ways such
as feedforward control and feedback control. Depending on the characteristic of noise to
be cancelled feedforward control is further classified as broadband feedforward control
system and narrowband feedforward control system. The schematic diagram of a
broadband feedforward noise cancellation system in a long duct is shown in fig.2.1.
Duct
Noise
Source
Reference
Microphone
Noise
Residual noise
Antinoise
Secondary
Loudspeaker
Error
Microphone
ANC
FXLMS
Fig 2.1 Schematic diagram of a broadband feedforward ANC system.
Broadband feedforward ANC system use acoustic sensor (reference microphone) to
pickup a coherent reference noise and generates the necessary antinoise (by secondary
loudspeaker) before it propagates past the secondary loudspeaker. Broadband noise
cancellation requires knowledge of the noise source (the reference noise) in order to
generate the antinoise signal. Hence the reference noise is received by a reference
microphone and is fed as an input to the noise canceller. After superposition of noise and
antinoise most of the noise cancel and a small amount of residual noise may remain
which is called as error. This error signal is observed by a microphone called error
15
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
microphone. By the time noise canceller generate the antinoise via secondary
loudspeaker, the reference noise will travel a distance and reach at the position of
secondary loudspeaker. Care should be taken so that there is time alignment between the
reference noise reaching the secondary loudspeaker and the antinoise. Any mismatch in
time alignment will result in causality problem and performance of ANC will be poor.
x(n)
d (n)
A(z)
+
Σ
−
e(n)
dˆ (n)
W (z )
y (n)
B (z )
Bˆ ( z )
x ′(n)
FXLMS
Fig. 2.2 Block diagram of a single-channel feedforward ANC.
The reference noise is distorted by the path while travelling to the secondary loud
speaker position. So the characteristic of this path should be analyzed to achieve good
results. The basic block diagram representing the scheme of ANC of fig.2.1 is shown in
fig.2.2. The path from primary microphone to error sensor is defined as primary path and
is denoted by A(z ) . Similarly the characteristic of the path from secondary loudspeaker to
error microphone is also important and should be analyzed thoroughly. This path is
defined as secondary path and denoted by B (z ) .The secondary path also includes the
D/A(digital to analog) converter, reconstruction filter, power amplifier, loudspeaker,
acoustic path from loudspeaker to error microphone, preamplifier, antialiasing filter, and
A/D (analog to digital) converter [1],[10]. Typical primary path impulse response and
secondary path impulse response estimated practically are shown in the fig.2.3 and fig.2.4
respectively. While designing the active noise canceller, the presence of primary path
16
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
and secondary path must be taken into account as both the paths distort the noise and
antinoise signals.
The acoustic noise source and the environment are time varying, the frequency
content, amplitude, phase, and velocity of the undesired noise are nonstationary (time
varying). So an active noise control system must be adaptive in order to cope with these
Fig 2.3 Impulse response of primary path (V. DeBrunner and D. Zhou [12]).
Fig 2.4 Impulse response of Secondary path (V. DeBrunner and D. Zhou [12]).
17
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
changing characteristics. This is the reason why modern active noise control systems
depend heavily on digital signal processing because in the field of digital signal
processing, there are classes of
systems called adaptive systems which have the
capability to vary their coefficients in order to cope with changing environment. Adaptive
systems can be implemented as transversal—finite impulse response (FIR), recursive—
infinite impulse response (IIR), lattice filters, transform-domain filters. The coefficients
of an adaptive filter are adapted to minimize a predefined cost function which is generally
a function of error signal. The process of adaptation is automated by DSP adaptation
algorithms.
At first glance it seems quite easy to apply directly already available least mean
square (LMS) algorithm to adapt the system to time varying environment. But presence
of a secondary path is found to be a hindrance to do this. Direct use of LMS algorithm
ignoring the presence of secondary path will affect the performance of ANC severely.
This leads to the development of filtered-x LMS (FXLMS) algorithm by Widrow [10].
2.2.1 FXLMS Algorithm
Least mean square (LMS) algorithm would have been sufficient for adaptive filter
weight update if there was no secondary path transfer function in the ANC. But as
mentioned earlier, the secondary path transfer function, B (z ) follows the adaptive filter, so
the conventional LMS algorithm must be modified to ensure convergence. The solution to
this problem is to place an identical secondary path filter in the reference signal path to the
weight update of the LMS algorithm. This realizes the so called filtered-x (FXLMS)
algorithm for ANC. The block diagram of a single channel feedforward ANC using
FXLMS algorithm is shown in fig.2.2. Referring to fig.2.2, x(n) is the reference noise
generated by the noise source at time n. Reference noise travels through the primary path
to reach at the cancellation point. So, d (n) is the noise to be cancelled at the cancellation
point which is obtained by convolving reference noise with primary path as follows
La
d ( n) = ∑ x ( n − k ) a k ( n)
(2.1)
k =0
18
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
where a k (n) is the k th element of the impulse response of the primary path at time n and
La is the order of primary path. y (n) is the control signal generated by the controller or the
adaptive filter, which is given by
N −1
y ( n ) = ∑ x ( n − k ) wk ( n )
(2.2)
k =0
where wk (n) is the k th element of the impulse response of the adaptive filter w (n) at time
n and N is the order of adaptive filter. dˆ (n) is the antinoise generated by the ANC via the
secondary loudspeaker, which is obtained by filtering the controller signal, y (n) , through
the secondary path filter
dˆ (n) =
M −1
∑ y (n − k )b
k
(2.3)
( n)
k =0
where bk (n) is the k th element of the impulse response of the secondary path b(n) at time
n and M is the order of secondary path. The residual noise or error signal collected by the
error microphone after noise cancellation is denoted as e(n) . The residual noise is
expressed as e(n)
e(n) = d (n) − dˆ (n)
= d ( n) − b ( n) * y ( n)
= d (n) − b(n) * [w T (n)x(n)]
(2.4)
where w (n) = [ w0 (n), w1 (n), L , w N −1 (n)]T
is the coefficient vector of adaptive filter W ( z ) at time n, and
x(n) = [ x(n), x(n − 1),L , x(n − N + 1)]T
is the reference signal vector at time n.
The objective of the adaptive filter is to minimize the instantaneous squared error, the cost
function, as given below
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CHAPTER-2
STUDY AND APPLICATIONS OF ANC
ξ ( n) = e 2 ( n)
(2.5)
To achieve this, the most widely used method is the stochastic gradient or LMS algorithm
which updates the coefficient vector in the negative gradient direction on the MSE surface.
The weight update equation is defined as
w (n + 1) = w (n) −
µ
2
∇ξ ( n )
(2.6)
where ∇ξ (n) is an instantaneous estimate of gradient on the MSE surface at time n. This
can be expressed as
∇ξ ( n ) =
∂e 2 (n)
∂e(n)
= 2e(n)
∂w (n)
∂w (n)
Putting value of e(n) from (2.4) we obtain
{
}
∂e(n) ∂[d (n) − b(n) * w T (n) x(n) ]
=
∂w (n)
∂w (n)
But d (n) is independent of w (n) so the gradient becomes
∂e(n)
= −b(n) * x(n) = x′(n)
∂w (n)
So x′(n) is the reference noise signal vector filtered through secondary path filter. Putting
the above value in (2.6) we obtain
w (n + 1) = w (n) + µ e(n)x ′(n)
(2.7)
where µ is the step size which regulates speed of convergence and stability. This is
popularly known as filtered-x LMS (FXLMS) algorithm. This algorithm shows that when
secondary path, B(z ) , follows the adaptive filter, this transfer function must also be placed
in the reference signal path. The basic ANC system described above perform quite well in
reducing broadband as well as narrowband noise in ducts under plane wave conditions. In
this arrangement primary path and secondary path are assumed to be linear in nature and
so represented by linear transfer functions. This type of arrangement is called linear ANC
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STUDY AND APPLICATIONS OF ANC
and is adapted by the relatively simple FXLMS algorithm which requires less
computation.
To implement FXLMS algorithm the secondary path filter B(z ) is to be estimated
first. Many offline and online methods are available for identifying the secondary path
filter. FXLMS algorithm found to be tolerant to errors made in the estimation of Bˆ ( z ) .
FXLMS algorithm generally converge even with a phase estimation error of upto ± 90 0 ,
within the limit of slow adaptation [13].
The maximum allowable step size for FXLMS algorithm is approximately [10]
µ max =
1
Px′ ( N + ∆)
(2.8)
where Px′ = E[ x ′ 2 (n)] is the mean square value or power of the filtered reference
signal x ′(n) , N is the number of adaptive filter coefficients and ∆ is the number of samples
corresponding to the overall delay in the secondary path. The delay in the secondary path
is the most significant factor influencing the convergence behavior of the ANC system,
thus reducing the maximum step size in the FXLMS algorithm.
2.2.2 FELMS Algorithm
E. A. Wan [14] and Popovich [15] introduced an alternative to FXLMS algorithm
by developing adjoint LMS algorithm. Instead of filtering the reference input by estimated
secondary path (this is pre-estimated before the ANC adaptation), they suggested filtering
the error signal by an adjoint secondary path. The block diagram of ANC system using
this technique is shown in fig.2.5. The adjoint LMS algorithm is given below
w (n + 1) = w (n) + µ e′(n)x(n − M + 1)
(2.9)
where e′(n) is the error signal filtered through the adjoint secondary path filter and M is
the order of secondary path filter. Here sufficient number of delays must be added to
reference signal to keep the reference and error signals properly time aligned. The number
of delays is generally order of secondary path filter. In another method, secondary path
equalization (SPE) [1], error signal is filtered through the inverse of secondary path
transfer function and delays are provided to reference signal to keep the reference and
21
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STUDY AND APPLICATIONS OF ANC
error signals time aligned. Both the methods involve filtering the error signal and so the
algorithm is called filtered-e LMS (FELMS) algorithm. V. DeBrunner and D. Zhou [12]
introduced hybrid filtered error LMS algorithm to further enhance the convergence rate.
The FELMS algorithms result in identical residual MSE performance compared to
FXLMS algorithm. The computational complexity of the FELMS algorithm is much less
compared to FXLMS algorithm particularly for multi-input ANCs. However, FELMS
algorithm introduces delays to the update step of the algorithm, which slows down the
speed of convergence.
x(n)
d (n)
A(z )
+
Σ
−
e(n)
dˆ (n)
y (n)
B(z )
W (z )
Adjoint Bˆ ( z )
z − M +1
x(n − M + 1)
FXLMS
e′(n)
Fig.2.5 Block diagram of ANC using adjoint LMS algorithm.
2.2.3 Feedback Effect
The antinoise generated by the secondary loudspeaker to cancel the reference noise
may travel towards the noise source direction. If this antinoise is able to reach the
reference microphone, it will collect antinoise along with reference noise signal. This
results in a corrupted reference signal x(n) . This is called feedback effect which results in
potential instability if the gain of this feedback loop becomes too large. Solution to this
problem is to use directional microphones and loudspeakers, provide a feedback
neutralization filter or use IIR filter.
22
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Duct
Noise
Source
Noise
Antinoise
Reference
Microphone
Secondary
Loudspeaker
+
Σ
-
Error
Microphone
Feedback
Neutralization
Filter
ANC
Update
Algorithm
Fig. 2.6 Schematic diagram of ANC with feedback neutralization filter.
The basic arrangement to neutralize feedback effect in a duct by introducing a
feedback neutralization filter is shown in fig.2.6. The block diagram representation of the
above arrangement is presented in fig. 2.7. Here the feedback component of the reference
microphone signal is cancelled by the output of feedback neutralization filter which
models the transfer function from secondary loudspeaker input to reference microphone
output. The signal actually captured by the reference microphone is the reference noise as
well as the feedback signal given by
u ( n) = x ( n ) + y ( n ) * f ( n )
(2.10)
Finally the input signal provided to the ANC is calculated as follows
x(n) = u (n) − y (n) * fˆ (n)
(2.11)
where f (n) and fˆ (n) are the impulse response of feedback path and estimated feedback
path filter respectively. The success of this technique depends on the accuracy of Fˆ ( z ) in
23
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
x(n)
d (n)
A(z )
+
F (z )
B(z )
u (n)
+
−
Σ
Σ
−
+
Σ
+
Fˆ ( z )
e(n)
dˆ (n)
z −1
y (n)
x(n)
W (z )
Bˆ ( z )
x ′(n)
FXLMS
Fig. 2.7 Block diagram of ANC with feedback neutralization filter.
estimating of F (z ) . The estimation is done offline in the absence of reference noise and is
keep fixed during the ANC operation. But in some cases feedback path vary with time. In
that case Fˆ ( z ) is to be estimated online, along with adaptation of ANC adaptive
filter W (z ) . This create problem as y (n) is highly correlated not only with feedback
antinoise signal (which is essential for adapting Fˆ ( z ) ) but also with the reference noise
(which is essential for adapting W (z ) ). In this situation Fˆ ( z ) will continue to adapt even
after removing the feedback component from the input signal and so create problems in
the adaptation of W (z ) if the convergence rate of Fˆ ( z ) and W (z ) are comparable. This is
the reason why online estimation of feedback path filter is generally avoided.
2.2.4 ANC using IIR Filter
It is well known that IIR adaptive filters have the ability to yield matching
characteristics with fewer filter coefficients compared to FIR adaptive filters. Reduction in
number of filter coefficients leads to reduction in computational complexity for ANC
24
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
implementation. Acoustic feedback in ANC introduces poles in the system. In this
situation IIR adaptive filters can better match the physical system as they have zeros as
well as poles where as FIR filters have only zeros.
The dashed box in the fig.2.8 shows the IIR adaptive filter. The output of the
adaptive filter is calculated as
y (n) = w Tf (n)x(n) + w Tb (n)y (n − 1)
(2.12)
where w f (n) is the weight vector of W f (z ) at time n and w b (n) is the weight vector of
Wb (z ) at time n. x(n) and y (n − 1) are defined as below
x(n)
Noise
Source
+
Σ
+
d (n)
A(z )
Σ
−
+
F (z )
W f (z )
B(z )
+
e(n)
dˆ (n)
y (n)
Σ
+
Wb (z )
Bˆ ( z )
Bˆ ( z )
FXLMS
x ′(n)
y ′(n)
FXLMS
Fig. 2.8 Block diagram of ANC (with feedback) using IIR adaptive filter.
x(n) = [ x(n) x(n − 1) . . . x(n − Lw f − 1)]T
y (n) = [ y (n − 1) y (n − 2) . . . y (n − Lwb − 2)]T
Lw f and Lwb are length of weight vector of W f (z ) and Wb (z ) respectively.
25
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Define a new overall weight vector
 w ( n) 
w ( n) =  f

 w b (n) 
And a new reference vector
 x( n) 
u ( n) = 

 y ( n) 
Then IIR adaptive filter output can be written as
y (n) = w T (n)u(n)
(2.13)
Now proceeding in the similar line as for the FXLMS algorithm, the weight update
equation for IIR adaptive filter can be written as
w (n + 1) = w (n) + µ u ′(n)e(n)
(2.14)
where u ′(n) is the u(n) filtered through the estimated secondary path filter Bˆ ( z ) . This
algorithm can be partitioned to derive two separate weight update equation for w f (n) and
w b (n) as described below
w f (n + 1) = w f (n) + µ x ′(n)e(n)
(2.15)
w b (n + 1) = w b (n) + µ y ′(n − 1)e(n)
(2.16)
where x ′(n) = b(n) * x(n) and y ′(n) = b(n) * y (n − 1)
This algorithm for IIR adaptive filter is called filtered-u recursive LMS algorithm. The
only drawback of this algorithm is that even though experimentally it works well, the
stability and global convergence is not guaranteed.
2.2.5 Narrowband Feedforward Control
In applications where the reference noise is produced by rotating or reciprocating
machines, reference noise is generally periodic (or nearly periodic). This refers to
periodic noises generated by engines, compressors, motors, fans, and propellers. In this
case, direct observation of the mechanical motion (such as speed) of such noise sources is
26
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
possible by using an appropriate sensor. So, here reference microphone can be replaced
by a nonacoustic sensor (such as tachometer). This sensor provides an electrical reference
signal with the same fundamental frequency as the reference noise emitted. Because all of
the repetitive noise occurs at harmonics of the machine’s basic rotational frequency, the
control system can model these known noise frequencies and generate the antinoise
signal. This type of control system is desirable in a vehicle cabin, because it control
engine noise but will not affect vehicle warning signals, radio signals or speech, which
are not normally synchronized with the engine rotation.
Duct
Noise
Residual noise
Noise
Source
Antinoise
Nonacoustic
Sensor
Secondary
Loudspeaker
Signal
Generator
Error
Microphone
ANC
FXLMS
Fig. 2.9 Schematic diagram of narrowband feedforward ANC.
Unfortunately the antinoise produced by secondary loudspeaker propagate in both
downstream and upstream direction. So antinoise not only cancels the reference noise but
also radiates upstream and reach the reference microphone resulting in a corrupted
reference signal. This effect is called feedback effect which introduces poles in the
response of the model and thus results in potential instability if the gain of this feedback
loop becomes too large. The principal advantage of narrowband feedforward ANC
systems is due to the use of nonacoustic sensor (e.g. tachometer) to generate reference
input, feedback effect is eliminated. Fig. 2.9 depicts the schematic diagram of a
narrowband ANC system.
27
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
2.3 Feedback Control
Olson and May [6] first proposed feedback ANC system which uses a carefully
designed amplifier matched to the response of error microphone and secondary
loudspeaker. The feedback ANC systems use only an error microphone and the active
noise controller try to control the noise without having any knowledge about the
upstream reference input. Since feedback ANC uses only an error microphone, it also
avoids the secondary-to-reference feedback problem inherent in many broadband feedfoDuct
Noise
Source
Noise
Residual noise
Antinoise
Secondary
Loudspeaker
Error
Microphone
Feedback
ANC
Fig.2.10 Schematic diagram of feedback ANC.
rward ANC systems. Schematic diagram of feedback ANC is shown in the fig. 2.10.
Feedforward active noise control is found to be robust and stable in comparison to
feedback active noise control [1].
2.4 Multichannel Active Noise Control
Controlling noise inside an enclosure or a large dimension duct is difficult as the
noise field is complicated. Another example is transformer noise cancellation. Use of one
reference microphone, one secondary loudspeaker and one error microphone is not
enough to control such type of complicated noise pattern. So multiple number of
reference microphones, secondary loudspeakers and error microphones are employed
which is called multiple channel ANC. Multiple channel ANC has large number of
primary paths, secondary paths and adaptive filters. All the adaptive filters are generally
28
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
adapted independently. Multiple error FXLMS algorithm developed by Elliott and
colleagues is generally used for multiple channel ANC [1][2]. Schematic diagram of a
multiple channel ANC for an enclosure is shown in the fig. 2.11.
Enclosure
Secondary
Loudspeaker
Reference
Microphone
Error
Microphone
1
1
1
2
2
2
P
Q
Noise
Source
L
ANC
Fig. 2.11 Schematic diagram of multi-channel ANC for an enclosure.
2.5 Virtual Sensor Technique (VST)
All the ANCs analyzed above can be put within one bracket of the so called local
active noise control system. Local ANC put all its effort to minimize undesired noise near
the error microphone (physical sensor). While this result in a small localized zone of
silence being created around the error microphone, the surrounding zone of silence is
quite small. Additionally, the noise level outside the zone of silence is likely to be higher
than the original noise alone. Virtual acoustic sensors have been developed to overcome
the problems associated with local ANC. Virtual acoustic sensors shift the zone of quiet
to a desired location that is remote from the error microphone, as shown in fig. 2.12 [16].
29
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Duct
Noise
Source
Reference
Microphone
Noise
Residual noise
Secondary
Loudspeaker
Error
Microphone
Virtual Sensor
ANC
FXLMS
Fig. 2.12 Schematic diagram of ANC using virtual sensor technique.
In this figure, the zone of silence has been moved from the error microphone to the
virtual sensor region. The block diagram for the above scheme is shown in fig. 2.13. This
technique is also reported as remote microphone technique (RMT) in [17]. In virtual
sensor technique the error signal at the virtual location, eˆv (n) , is estimated using the error
signal available from the error microphone (physical microphone), e p (n) . To accomplish
this task virtual sensor technique requires a preliminary identification stage in which a
second physical microphone is temporarily placed at the virtual location. The secondary
path transfer functions at the physical and virtual locations, Bˆ p ( z ) and Bˆ v ( z ) respectively,
are estimated during the preliminary identification stage along with the primary path
transfer function between the physical and virtual locations, Aˆ v ( z ) . A block diagram of
the virtual sensor technique is given within the dotted box of fig. 2.13. Referring fig.
2.13, first an estimate of the primary disturbance, dˆ p (n) , at the physical microphone is
calculated using
dˆ p (n) = e p (n) − y p (n)
(2.17)
30
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
x(n)
A(z )
W (z )
y (n)
y p (n)
B p (z )
yˆ p (n)
Bˆ p ( z )
+
−
+
d p (n)
Σ
e p (n)
+
Σ
dˆ p (n)
Bˆ v ( z )
Aˆ v ( z )
x ′(n)
Bˆ v ( z )
yˆv (n)
+
ˆ
+ dv (n)
Σ
FXLMS
eˆ p (n)
Fig. 2.13 Block diagram of ANC using virtual sensor technique.
where e p (n) is the actual error signal measured by the error microphone (physical sensor)
and y p (n) is obtained by filtering y (n) through the estimated secondary path at physical
microphone, Bˆ p ( z ) . Then an estimate of the primary disturbance, dˆ v (n) , at the virtual
sensor is calculated as follows
dˆ v (n) = aˆ v (n) * dˆ p (n)
(2.18)
where aˆ v (n) is the filter impulse response corresponding to the transfer function Aˆ v ( z ) .
Finally an estimate of the virtual error signal at the virtual sensor position is calculated as
follows (this can also be verified from the block diagram)
ev (n) = dˆ v (n) + yˆ v (n)
(2.19)
where yˆ v (n) is the estimated secondary disturbance at the virtual sensor position. Thus an
estimate of the virtual error signal has been calculated from the physical error signal. The
31
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
adaptive filter having transfer function W (z ) can be updated using the traditional FXLMS
algorithm as below
w (n + 1) = w (n) + µ ev (n) x′(n)
(2.20)
where x ′(n) is reference input filtered through Bˆ v ( z ) .
2.6 Frequency Domain Approach
Frequency domain adaptive filters have three major advantages over time-domain
adaptive filters
(i) The potential saving in the computation by using Fast Fourier Transform (FFT)
(ii) More accurate estimation of the gradient due to the averaging of the samples in
the whole data block and
(iii) Rapid convergence by using normalized step sizes for each frequency bin
x(n)
d (n)
A(z )
+
Σ
−
dˆ (n)
F
F
T
X(n)
W (z )
I
F
F
T
Y(n)
y (n)
e(n)
B (z )
Bˆ ( z )
x ' ( n)
F
F
T
X′(n)
E(n)
Complex
LMS
F
F
T
Fig.2.14 Block diagram of ANC using frequency–domain FXLMS algorithm.
ANC can be implemented in the frequency domain to take benefits of the above
mentioned advantages. Block diagram for frequency domain implementation of ANC is
shown in the fig.2.14. A block of reference signal samples are transformed to the
frequency domain signal X(n) using FFT. X(n) is then filtered through the frequency
32
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
domain adaptive filter W (z ) to produce frequency domain output Y(n) . Then by IFFT of
Y(n) we get the time-domain output signal vector y (n) . The residual error e(n) and
filtered reference signal x ′(n) are also transformed to frequency-domain by FFT to get
E(n) and X′(n) respectively. Now the adaptive filter coefficients are updated by the
complex LMS algorithm expressed as [1]
W (n + N ) = W (n) + µ X′(n)E(n)
(2.21)
where N is the input signal block length. This is called frequency-domain FXLMS
algorithm. Instead of filtering the signal sample by sample, frequency-domain FXLMS
algorithm processed the signal block by block. This create N sample of delay in the
adaptation process causing difficulty in controlling broadband random noise because of
the causality constraint. But for periodic reference noise, the effect of block processing
delay can be tolerated.
2.7 Active Noise Control in Headset
Headset is generally used for blocking outside noise reaching the ear in highly
noisy environment such as flying aircraft or in a subway train. Headphone is also used for
listening prerecorded music or voice communication in aviation or military. But in case
of a highly noisy environment, outside noise leakage corrupt the desired signal. ANC can
be used to cancel the leakage outside noise in headset. In H. F. Olson and E. G. May [6]
system a microphone is placed to collect noise and loudspeaker is used to generate
antinoise. Shiang-Hwua Yu and Jwu-Sheng Hu [18], Ying Song, Yu Gong, and Sen M.
Kuo [19], Thomas Schumacher, Hauke Kr¨uger, Marco Jeub, Peter Vary, Christophe
Beaugeant [20] and Cheng-Yuan Chang and Sheng-Ting Li [21] developed ANC heatset
basically using feedback control. Huge change in the primary path and secondary path is
observed when the position of the headset is changed and possesses the real challenge in
this application. Romain Serizel, Marc Moonen, Jan Wouters, and Søren Holdt Jensen
[22] and Romain Serizel, Marc Moonen, Jan Wouters, and Søren Holdt Jensen [23] use
ANC technique in hearing aid where as W. S. Gan and S. Kuo [24] developed a
headphone having an integrated audio and active noise controller. Here the headphone
speaker playing the music also generates antinoise to cancel the outside leakage noise.
33
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Noise
Speaker
Controller
Microphone
Fig. 2.15 Schematic diagram of ANC used for headphone or headset.
2.8 ANC in functional Magnetic Resonance Imaging (fMRI)
Magnetic Resonance Imaging (MRI) was widely used in medical and clinical
researches during the past 20 years. functional MRI (fMRI) has also been applied in
neuroscience and psychology studies. High speed echo planar imaging (EPI) technique is
usually applied in fMRI study to acquire high temporal resolution signals [25]. However,
fast switch of magnetic gradients during EPI acquisition induced loud acoustic noise,
which will up to 100 dBL or even louder in higher magnetic field. This acoustic noise can
interfere in the communication between staff and volunteers or patients, impair the
hearing ability, and suppress brain activations in fMRI study. People usually used passive
component, such as earmuff or earplug, to prevent hearing damage from the MRI noise.
However, the noise reduction was limited by the design and the material of the passive
components and the passive components can also hardly solve the communication
problem between staff and volunteers. The other choice for reducing the MRI noise is
active noise cancellation (ANC) system. Active Noise Control (ANC) of fMRI acoustic
noise using the conventional FXLMS approach results in poor cancelation performance
and slow convergence due to its broadband nature and the need for high order adaptive
filters. High order adaptive filters are needed to effectively model the long acoustic
impulse responses. Existing methods to improve the performance of FXLMS based
broadband ANC systems are either computationally expensive or need elaborate
implementation. Casper K. Chen, Tzi-Dar Chiueh and Jyh-Horng Chen [25] introduced a
34
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
new neural-network architecture for reducing the acoustic noise level in magnetic
resonance (MR) imaging processes. The proposed neural network (NN) consists of two
cascaded time-delay NN’s (TDNN’s). This NN is used as the predictor of a feedback
active noise control (ANC) system for reducing acoustic noises. Experimental results
with real MR noises show that the proposed system achieved an average noise power
attenuation of 18.75 dB. Kuan-Hung Cho, Tzi-Dar Chiueh, Ching-Po Lin, Casper K.
Chen, Jyh-Horng Chen [26] experimented an ANC for fMRI using an IIR filter. They
have showed that the ANC system could provide extra 9.4 dB reduction at 937 Hz, which
is the main frequency of the EPI noise. Besides, the fMRI results showed that there was
apparent enhancement in both activated pixel numbers and the activation strength in
visual cortex (VC). Govind Kannan, Ali A. Milani, Issa M.S. Panahi, Nasser Kehtarnavaz
[27] developed Wiener initialized FXLMS (WI-FXLMS) algorithm and demonstrated the
effectiveness of this approach for the active noise control of functional MRI acoustic
noise and several other realistic noise sources.
2.9 ANC in Infant Incubators
Intense level of noise inside infant incubators like neonatal intensive care unit
(NICU) is due to the operation of various medical instruments. These loud noises can
cause serious psychological effects in infants such as changes in heart rate, blood
pressure, oxygenation, respiration, intestinal peristalsis, and glucose consumption.
Lichuan Liu, Shruthi Gujjula and Sen M. Kuo [29] used ANC to reduce this noise. They
have done real time experiments with single channel, multi channel and pseudo-multi
channel ANC. Multi channel ANC, 1 × 2 × 2 (1 reference microphone, 2 secondary speaker
and 2 error microphone), yield better noise reduction results but require more
computational complexity which is a problem for real time implementation. They
proposed pseudo multi channel ANC with same number of sensors as in multichannel
ANC ( 1 × 2 × 2 ). Here after the digital output y(n) is converted to analog signal, it is
amplified to drive both loudspeakers and two analog signals picked up by two error
microphones are mixed by an analog mixer to get an single analog error signal. So here
computational complexity remains same as single channel ANC.
35
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Other Applications of ANC
ANC has been used for other applications like transformer noise cancellation by
X. Qiu and C. H. Hansen [30] and vehicle interior noise by Ji-guang Jiang , Deng-feng
Wang, Yue Zeng, Jun-ting Wang, Xiao-lin Cao [31].
2.10 Study on Nonlinear ANC
The ANC systems discussed till now use an adaptive filter which is either a finite
impulse response (FIR) filter or an infinite impulse response (IIR) filter. The adaptive
algorithm derived for ANC using FIR adaptive filter is FXLMS algorithm. The ANC
using IIR structure as the adaptive filter is based on filtered-u least mean square
(FULMS) algorithm. Both the filter structures, FIR and IIR filter, have a linear form in
the sense that the output has a linear relationship with filter coefficients and reference
input. This enable the adaptive filters, using either FIR or IIR structure, to perform noise
reduction when the primary path and secondary paths are linear. But real time
implementation reveals that nonlinearity exists in ANC. Some of the areas which lead to
the appearance of nonlinearity in the ANC are listed below
•
The reference noise at the cancellation point may exhibit nonlinear distortion due
to the nonlinear transfer function of the primary path, for example, primary noise
propagating in a duct with very high sound pressure.
•
The secondary path between the speaker and the error microphone may exhibit
nonlinear behavior, for example when the amplitude of the antinoise is greater
than the saturation limit of the speaker or the frequency of the antinoise is less
than the cutoff frequency of the speaker.
•
The reference noise may itself exhibit nonlinearity. The noise from a dynamic
system may be a nonlinear and deterministic (chaotic rather than stochastic,
white, or tonal) noise process. For example, fan noise often shows chaotic
behavior.
•
The reference and error sensors may be saturated in real world applications if the
noise level exceeds the dynamic range of the sensors. The loudspeaker, for
example, can excite both the frequency of interest and its associated harmonics.
•
Nonlinearity arising out of specific applications (personal hearing aid, long duct).
36
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Careful examination of a practical ANC system revealed that the nonlinearity
must be taken into account in order to achieve better overall performance. This analysis
paved the way for the development of a new class of nonlinearity tolerated ANC
henceforth called nonlinear active noise control (NANC) system.
Nonlinearity sneaks into the ANC systems because of one or more or combination
of above mentioned factors. This leads to a class of new ANC better known to the
researchers as nonlinear active noise control (NANC). Under this circumstances if the
existing ANC techniques (which use FIR or IIR filter as the adaptive filter) discussed
earlier are employed, performance degradation is noticed. The obvious reason being
application of a linear technique for an ANC which is actually nonlinear in nature. The
solution to this problem is to explore various available nonlinear filtering techniques.
These methods need proper analysis and be suitably modified to apply conveniently to
the problem of nonlinear ANC. The following methods reported in the literature have
been applied to address the issue of nonlinear ANC.
•
Adaptive Volterra Filter (AVF)
•
Adaptive Bilinear Filter (ABF)
•
Mutli-Layer Artificial Neural Network (MLANN)
•
Radial Basis Function (RBF)
•
Recurrent Neural Network (RNN)
•
Fuzzy Logic (FL)
•
Fuzzy Neural Network (FNN)
•
Functional Link Artificial Neural Network (FLANN)
•
Genetic Algorithm (GA)
•
Particle Swarm Optimization (PSO)
While application of MLANN and FLANN have been analyzed in detail in the
subsequent chapters, other methods mentioned above are discussed briefly here. Different
variations of the above mentioned techniques exist. The variations generally include
modifications in the update algorithm. However, none of them provide global minimum
so far as the error minimization is concerned in the sense that if one works well for a
given application fails to achieve the same in the other application. These methods can
37
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
be evaluated based under the following characteristics embedded into the algorithm
(i)Mean Square Error (MSE) achieved after convergence (ii) Speed of convergence (iii)
Computational complexity requirement (iv) Stability
2.10.1 Adaptive Volterra Filter (AVF) Approach
In [32][33] V. John Mathews explains how Volterra series expansion can be used
to build adaptive filters to deal with polynomial model of nonlinearity. This is found to be
very useful for practical applications which involve nonlinear processing of signals.
Filters involving Volterra series expansion are attractive in adaptive filtering applications
because the expansion is a linear combination of nonlinear functions of the input signal.
In another way, the output of this filter is linear with respect to filter coefficients.
Li-Zhe Tan and Jean Jiang [34] reported an adaptive filtered-x algorithm for ANC
using second order Volterra filter. The developed algorithms can be used as alternatives
in the case where the standard filtered-x LMS algorithm does not perform well e.g. for
nonlinear ANC. The block diagram of ANC using second order Volterra filter is shown
in the fig. 2.16.
x (n)
d ( n)
Nonlinear
Primary path
+
y ( n)
W1 ( z )
Σ
×
W2 , 0 ( z )
Σ
×
W2,1 ( z )
Σ
e( n )
Σ
B( z )
-
z −1
M
z
M
−1
×
W2, N −1 ( z )
L
Bˆ ( z )
M
LMS / RLS
Fig.2.16 Block diagram of ANC using filtered-x second order Volterra adaptive
algorithm.
38
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Consider a second order Volterra filter described be the following input and output
relationship
N −1
N −1
N −2
i =o
i =0
i =0
y (n) = ∑ w1 (i, n) x(n − i ) + ∑ w2,0 (i, n) x 2 (n − i ) + ∑ w2,1 (i, n) x(n − i ) x(n − 1 − i )
(2.22)
+ K + w2, N −1 (0, n) x(n − N + 1)
where x(n) and y (n) represents the filter input and output respectively, N is the memory
span and w1 (i, n), w2,0 (i, n), w2,1 (i, n), K , w2, N −1 (0, n) are filter coefficients. The above
equation is implemented as a number of filters operating in parallel. Unlike FXLMS
algorithm where one adaptive FIR filter is used this method use N+1 number of FIR
adaptive filters. The coefficient vector of the adaptive filters involved are defined below
w 1 (n) = [ w1 (0, n), w1 (1, n), .. , w1 ( N − 1, n)]
w 2, 0 (n) = [ w2,0 (0, n), w2, 0 (1, n), .. , w2,0 ( N − 1, n)]
w 2,1 (n) = [ w2,1 (0, n), w2,1 (1, n), .. , w2,1 ( N − 2, n)]
w 2, 2 (n) = [ w2, 2 (0, n), w2, 2 (1, n), .. , w2, 2 ( N − 3, n)]
LLLLLL
w 2, N −1 (n) = [ w2, N −1 (0, n)]
The order of first filter is N, next starting from second filter the filter order reduces from
N to 1.
The input to the adaptive filters are defined below
x 1 (n) = [ x(n), x(n − 1), ..., x(n − N + 1)]
x 2, 0 (n) = [ x 2 (n), x 2 (n − 1), ..., x 2 (n − N + 1)]
x 2,1 (n) = [ x(n) x(n − 1), x(n − 1) x(n − 2), ..., x(n − N + 2) x(n − N + 1)]
x 2, 2 (n) = [ x(n) x(n − 2), x(n − 1) x(n − 3), ..., x(n − N + 3) x(n − N + 1)]
KKKKKKK
x 2, N −1 (n) = [ x(n) x(n − N + 1)]
The adaptive filters output are obtained by filtering the inputs through the corresponding
adaptive filter and then are added to get the output of the controller as follows
y (n) = y1 (n) + y 2,0 + L + y 2, N −1
(2.23)
As shown in the figure residual error signal is given by
e( n ) = d ( n ) − b ( n ) * y ( n )
(2.24)
39
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
where b(n) is the impulse response of the secondary path transfer function B(z ) .
Tan and Jiang [34] used the standard FXLMS algorithm to develop filtered-x
second order Volterra LMS algorithm and update the adaptive filter. Similar to FIR filter
here the goal of the adaptive algorithm is to minimize the instantaneous square error
using the steepest descent algorithm. Weight update formula for the adaptive filters are
written as follows
w 1 (n + 1) = w 1 (n) + µ1 e(n)x1′ (n)
w 2,,i (n + 1) = w 2,i (n) + µ 2 e(n)x ′2,i (n)
(2.25)
where i=0,1,2,. . .,N-1., µ1 is the step size for first adaptive filter and µ 2 is the step size
for all other adaptive filters.
Here it can be noticed that the input to first adaptive filter is x(n) with its delayed
samples. The input to second adaptive filter is x 2 (n) with its delayed samples (squared
terms) and to all other adaptive filters are x(n) x(n − 1), x(n) x(n − 2) etc. with their delayed
samples(cross-terms). So the first adaptive filter represents linearity of the ANC where as
all other filters represent nonlinearity of the system.
They have also used FXRLS algorithm to develop filtered-x second order Volterra RLS
algorithm as follows
k (n) =
λ−1 P(n − 1)x′1 (n)
(2.26)
T
1 + λ−1 x ′1 (n) P(n − 1)x ′1 (n)
w 1 (n + 1) = w 1 (n) + k (n)e(n)
(2.27)
P(n) = λ−1 P(n − 1) − λ−1 k (n)x1′ (n) P(n − 1)
(2.28)
where λ (0 << λ ≤ 1) is the forgetting factor, k (n) is the gain vector and x′1 is the filtered
input signal for the first adaptive filter. Similarly update equations for other adaptive
filters are given below
k ( n) =
λ−1 P(n − 1)x′2,i (n)
(2.29)
T
1 + λ−1 x′2,i (n) P(n − 1)x′2,i (n)
w 2,i (n + 1) = w 2,i (n) + k (n)e(n)
(2.30)
P(n) = λ−1 P(n − 1) − λ−1k (n)x′2,i (n) P (n − 1)
(2.31)
40
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
The algorithms gave better results in terms of MSE for nonlinear ANC. The filtered-x
second order Volterra RLS algorithm shows better convergence speed than its LMS
counterpart.
Subsequently, Li-Zhe Tan and Jean Jiang [35] reported another paper giving
exhaustive theoretical analysis of the above algorithm and named it as Volterra FXLMS
(VFXLMS) algorithm. Here the analysis is not limited to second order Volterra filters
but higher order filters are included. Inclusion of higher order Volterra filters obviously
increases the number of adaptive filters which may enhance the MSE performance of the
ANC but raises the computational complexity of the algorithm substantially. They
implemented the algorithm in multichannel structure. This structure can be called as filter
bank structure [36] in order to avoid confusion with multichannel ANC. They have
shown by computer simulation that VFXLMS algorithm gives better MSE result
compared to the FXLMS in the following two situations
1) The reference noise sensed by a reference microphone is a nonlinear and predictable
noise process, while the secondary path transfer function of an ANC system has
nonminimum phase.
2) The primary path exhibits nonlinear behavior.
2.10.2 Adaptive Bilinear Filter (ABF) Approach
In [32], [33] V. John Mathews and G. L. Sicuranza discussed that Bilinear filters
is a popular class of adaptive filter described by the following input and output
relationship
N −1
N −1
N −1 N −1
i =0
i =1
i = 0 j =1
y (n) = ∑ w1 (i, n) x(n − i ) + ∑ w2 (i, n) y (n − i ) + ∑∑ wi , j (n) y (n − j ) x(n − i )
(2.32)
where the filter coefficients w1 (i, n), w2 (i, n) and wi , j (n) can have different length but for
simplicity they are assumed to have equal length, N. x(n) and y (n) are input and output of
the filter. w1 (i, n) are the feedforward filter coefficients corresponding to input x(n) ,
w2 (i, n) are feedback filter coefficients corresponding to output y (n) and wi , j (n) are
cross filter coefficients corresponding to cross terms x(n) y (n − 1) . In spite of the
simplicity, this is an important nonlinear model since a large class of nonlinear systems
41
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
can be approximated with arbitrary precision using bilinear system models with finite
number of filter weights.
In [37], [38] Sen M. Kuo and Hsien-Tsai Wu used adaptive bilinear filters for
nonlinear ANC. There are two types of adaptive bilinear filters: equation-error methods
and output-error methods [32], [37], [38]. The output-error algorithm computes the filter
output using a truly recursive model since the output signal y (n) is fed back to generate
the adaptive estimate of the desired signal d (n) . The equation-error algorithm calculates
the output signal using the input signal x(n) and the desired signal d (n) , thus is not a
truly recursive estimator. In addition, the equation error method results in biased steadystate solutions, whereas the output-error method provides unbiased estimates.
Furthermore, the desired signal d (n) , is not available for real-time ANC systems in
practical applications, thus the equation-error filter cannot be directly used in them.
Therefore they used the adaptive output error bilinear filter for ANC systems. The block
diagram of ANC using adaptive bilinear filter is shown in the fig. 2.17, below.
Noise
Source
x(n)
d (n)
A(z )
y (n − 1)
Σ
e(n)
−
dˆ (n)
z −1
W (z )
y (n)
Bˆ ( z )
+
B(z )
Bˆ ( z )
y ′(n − 1)
Bilinear
LMS
x ′(n)
Fig. 2.17 Block diagram of ANC using the adaptive output-error bilinear filter.
The bilinear filter coefficients can be rearranged to form the following vectors.
Feedforward filter coefficient vector w 1 (n) = [ w1 (0, n), w1 (1, n),K, w1 ( N − 1, n)]
Feedback filter coefficient vector w 2 (n) = [ w2 (0, n), w2 (1, n),K , w2 ( N − 1, n)]
Cross filter coefficient vector w i , j (n) = [ w0,1 (n), w0, 2 (n), K , w N −1, N −1 (n)]
42
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
All the filter coefficient vectors can be combined to form a single vector
w (n) = [w 1 (n), w 2 (n), K , w i , j (n)]
The input vector for the three filter coefficient vectors can also be rearranged to form the
following vectors
i) x 1 (n) = [ x(n), x(n − 1), K , x(n − N + 1)]
ii) x 2 (n) = [ y (n − 1), y (n − 2), K , x(n − N )]
iii) x 3 (n) = [ x(n) y (n − 1), L, x(n) y (n − N ), x(n − 1) y (n − 1) ,L
, x(n − 1) y (n − N ) , L , x(n − N ) y (n − N ) ]
Similar to the filter coefficient vector three signal vectors are combined to form a
generalized signal vector
x(n) = [x 1 (n), x 2 (n), x 3 (n)]
Therefore the output signal from the bilinear filter can be written as
y ( n) = w ( n ) x T ( n)
The error signal of the ANC is obtained as
e( n ) = d ( n ) − y ( n ) * b ( n )
(2.33)
Similar to the adaptive FIR filter, the goal of the adaptive algorithm for the
bilinear filter is to minimize the instantaneous square error using the steepest descent
algorithm. So following the same procedure as FIR filter, the update algorithm for the
adaptive bilinear filter for ANC can be obtained as
w (n + 1) = w (n + 1) + µ x ′(n) e(n)
(2.34)
From the definition of w (n) , equation (2.34) can be partitioned as following three
independent vector equations
w 1 (n + 1) = w 1 (n) + µ1 x1′ (n) e(n)
(2.35)
w 2 (n + 1) = w 2 (n) + µ 2 x′2 (n) e(n)
(2.36)
w i , j (n + 1) = w i , j (n) + µ 3 x ′3 (n) e(n)
(2.37)
The simulation results provided by Sen M. Kuo and Hsien-Tsai Wu [37],[38]
proved that bilinear filters not only have better MSE performance but also faster
convergence speed compare to FIR and Volterra filters under strong nonlinearity
situation. But because of feedback the stability of the bilinear filter is always doubtful.
43
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
2.10.3 Radial Basis Function (RBF) Approach
Riyanto T. Bambang, Lazuardi Anggono and Kenko Uchida [39] reported an
ANC using radial basis function neural network. As shown in the fig. 2.18 and fig. 2.19,
it consist of two stages, first, the nonlinear secondary path is identified using a radial
basis function network and its learning algorithm. Secondly another radial basis function
Noise
source
x(n)
d (n)
Primary path
y (n)
Secondary path
+
dˆ (n)
-
e(n)
Σ
RBF model for
Secondary path
The controller
based on RBF
FX-RBF algorithm
Fig. 2.18 Block diagram for ANC using RBF.
x(n)
wi
z
wi
−1
z
−1
Σ
Σ
z −1
z −1
M
M
z −1
z −1
Controller RBF network
Secondary path model RBF network
Fig. 2.19 Structure of the two RBF networks for ANC.
44
dˆ (n)
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
network is used for the controller which generates the antinoise. The algorithm developed
turned out to be filtered-x version of RBF algorithm and so named FX-RBF algorithm.
Real time implementation was conducted to evaluate the proposed ANC system. The
centre of Gaussian functions for both model and controller network were trained using Kmeans clustering algorithm. Riyanto T. Bambang [40], [41] designed an ANC using
recurrent radial basis function network where the strategy was analogous to the previous
described work but instead of RBF network recurrent RBF network was used. The
recurrent RBF network used is shown in fig. 2.20. The delayed and weighted output of
each hidden node is feedback to the same node.
x(n)
wi
z
wi
−1
z
−1
dˆ (n)
Σ
Σ
z −1
z −1
M
M
z −1
z −1
Controller recurrent RBF
Secondary path model recurrent RBF
Fig. 2.20 Structure of the two recurrent RBF networks for ANC.
Jiang Lifei [42] developed an ANC based on the analysis of the characteristics of
practical noises in a driver’s cab, using radial basis function neural network. The training
algorithm was also provided in the paper with results. Tokhi and Wood [55] also used
RBF for nonlinear ANC.
2.10.4 Fuzzy Logic (FL) Approach
Since the introduction of fuzzy sets by Zadeh in 1965, many researchers have
applied this theory to diverse engineering topics. In general, the knowledge base and rule
45
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
base are designed from expert experience, and then an inference engine is selected to
make up a fuzzy system. C. Y. Chang and K. K. Shyu [43] reported a simple architecture
to apply fuzzy logic to nonlinear ANC. They proposed seven rules to form a fuzzy FIR
filter, which acts as an antinoise filter to cancel undesired noise. The developed algorithm
resembles FXLMS algorithm so it is called fuzzy filtered-x algorithm. This method tunes
the free parameters automatically and changes the IF–THEN rules adaptively to minimize
the residual noise as new information becomes available.
Traditionally, active noise controllers are designed on the basis of a mathematical
description of the plant and its linearized model. However, it is difficult to control
undesired noise in a nonlinear duct plant. The conventional filtered-x based ANC systems
often require hundreds of weights to control undesired noise. Therefore, resulting
numerical errors, such as the round-off and quantization error are inevitable. Therefore
self-tuned fuzzy-based ANC system can process both the numerical data and linguistic
information to adapt the ANC system. Another advantage of the proposed algorithm
includes the reduction of system complexity and the property of nonlinear compensation.
Cheng-Yuan Chang and Kuo-Kai Shyu [44] proposed a self-tuning fuzzy filteredu algorithm which instead of complex designing procedures of traditional algorithms uses
few mathematical transfer functions to design the ANC system. They have also provided
a fuzzy-based self-tuning algorithm is to adjust the free parameters of the fuzzy filtered-u
algorithm. In addition, the proposed method protects ANC systems against unstable poles
which occur in conventional filtered-u design.
x(n)
+
d(n)
A(z )
Σ
−
dˆ(n)
y (n)
Fuzzy
Controller
B(z )
Fuzzy
Filtered-X
Fig. 2.21 Block diagram of ANC using fuzzy logic.
46
e(n)
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
2.10.5 Fuzzy-Neural Approach
Huynh Van Tuan and Duong Hoai Nghia [45] reported a fuzzy-neural network for
feedback active noise controller. The fuzzy –neural model they have used is shown in the
fig. 2.22 where the term node, G, represent a Gaussian membership function to express
the input fuzzy linguistic variables and “rule nodes, R” represent the fuzzy rules. Node,
N, performs the normalization of the firing strengths coming from pervious layer. They
also developed fuzzy neural-based filtered-x least-mean-square algorithm and proved the
convergence of the algorithm using a discrete Lyapunov function.
G
dˆ (n)
N
R
G
w
G
y (n)
G
R
N
R
N
Σ
z −1
z −1
z −1
Fig. 2.22 Structure of neuro-fuzzy controller used for ANC.
Jian Liu, Jinwei Sun and Guo Wei [46] proposed a new narrowband ANC system
using an ANFIS (adaptive neuro-fuzzy inference system) as an adaptive controller. For
the purpose of computational cost reduction, the nonlinear premise parameters in the
ANFIS are kept fixed and only its linear consequent parameters are adjusted based on a
gradient descent method. The block diagram is shown in fig. 2.23.
Cosine
Wave
Generator
x(n)
ANFIS
y (n)
Secondary
path
dˆ (n)
d (n)
+
−
Fig. 2.23 Block diagram of ANC using ANFIS.
47
e(n)
Σ
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Navid Azadi, Abdolreza Ohadi [47] proposed an enhanced multi-channel active
fuzzy neural network noise controller in a rectangular enclosure. The block diagram of
the ANC system they used and the fuzzy neural network they employed is shown in the
fig.2.24 and fig. 2.25 respectively. They proposed a multi-channel enhanced fuzzy neural
network (EFNN) error back propagation algorithm while taking into account the
secondary path effect in derivation of equations. The results provided shown that
enhanced FNN algorithm outperformed FXLMS algorithm when there is a highly
nonlinear primary path in the ANC system.
d (n)
x(n)
A(z )
e(n)
+
Σ
−
dˆ (n)
Fuzzy Neural
Network
y (n)
B(z )
Fig. 2.24 Block diagram of ANC using fuzzy-neural network.
Layer 2
Fuzzification
Layer 3
Rule Layer
Layer 1
Input
x(n)
Π
M
M
M
Layer 4
Layer 5
Defuzzification
y (n)
/
x(n − N + 1)
Π
Fig. 2.25 Structure of fuzzy-neural network controller for ANC.
48
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
2.10.6 Genetic Algorithm (GA) Approach
Fabrizio Russo and Giovanni L. Sicuranza [48], [49] investigated the performance
of genetic optimization for nonlinear active noise control based on nonlinear Volterra
filters. They showed two advantages of using genetic algorithm (GA) for nonlinear ANC
problem. i) While standard filtered-x algorithms may converge to local minima, GA may
handle this problem efficiently. ii) This class of algorithms does not require the preidentification of the secondary paths because unlike the class of FXLMS algorithm,
estimated secondary path transfer function is not required in adaptation process. Here GA
is used to optimize the filter coefficients of the nonlinear Volterra filter. The block
diagram they suggested is reproduced in the fig. 2.26 below.
x(n)
+
d(n)
A(z )
Σ
−
dˆ(n)
y (n)
Nonlinear
Filter
e(n)
B(z )
Population of
binary strings
Set of
coefficients
01001010…0
10110100…1
…
01100101…1
String
Decoding
Performance
Evaluation
Genetic
Operator
Fig.2.26 Block diagram of the genetic ANC system. (Fabrizio Russo and Giovanni L.
Sicuranza [48]).
49
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Cheng-Yuan Chang and Deng-Rui Chen [50] proposed an adaptive genetic
algorithm (AGA) for an ANC system to eradicate the problem of FXLMS algorithm
converging to local minimum. The conventional ANC system implements the FXLMS
algorithm to update the coefficients of the linear finite-impulse response (FIR) filters. For
nonlinear ANC using nonlinear Volterra filters require Volterra FXLMS (VFXLMS)
algorithm. The proposed method replace FXLMS algorithm with FIR adaptive genetic
algorithm (FAGA) for FIR filters and VFXLMS algorithm with Volterra adaptive genetic
algorithm (VAGA) for Volterra filters. The proposed AGA method does not require preidentifying the secondary path for the ANC operation thereby making the system immune
to secondary path identification error. Performance of the FAGA and VAGA are
compared with FXLMS and VFXLMS algorithms.
2.10.7 Particle Swarm Optimization (PSO) Approach
Nirmal Kumar Rout, Debi Prasad Das and Ganapati Panda [51] presented a new online
ANC algorithm using PSO-based training.
Particle swarm optimization (PSO) is a
nongradient but simple evolutionary computing-type algorithm. Conventionally PSObased algorithm is shown to be ineffective to regain convergence in the case of
occurrence of an abrupt change in primary and/or secondary paths (time varying primary
path and/or secondary path). To cope with this problem, in the paper, the conventional
PSO algorithm is modified to introduce a new conditional reinitialized particle swarm
optimization (CRPSO) algorithm which is used to optimize the weights of an FIR filter.
The added advantage of the algorithm is that it doesn’t require pre-estimation of
secondary path, thereby, making the system immune to secondary path identification
error. Performance of the PSO is compared with FXLMS algorithm and GA based
algorithms. The strategy adopted by Nirmal Kumar Rout, Debi Prasad Das and Ganapati
Panda [51] is represented by the block diagram below, fig 2.27. Nithin V. George and
Ganapati Panda presented a robust evolutionary active noise control system using
Wilcoxon norm and particle swarm optimization algorithm [80].
50
CHAPTER-2
STUDY AND APPLICATIONS OF ANC
Fig. 2.27 Block diagram of PSO-based training of an ANC system (Nirmal Kumar Rout,
Debi Prasad Das and Ganapati Panda [51]).
2.11 Summary
A comprehensive study of existing ANC schemes have been carried out in this
chapter. The study categorize the available ANC schemes into two categories, i) study of
linear ANC ii) study of nonlinear ANC. Literature is rich and well documented for linear
ANC where as research is continuing for nonlinear ANC. Various sources, responsible
for introducing nonlinearity into the ANC system are highlighted. Strategies adopted for
applications of soft computing techniques like multi-layer artificial neural network, radial
basis function network, recurrent radial basis function network, fuzzy logic, neuro-fuzzy
technique, etc. and evolutionary techniques like genetic algorithm and particle swarm
optimization, etc. for nonlinear ANC have been presented. The necessity of ANC in
diverse field of engineering and technology are discussed. The role played by ANC and
the apparent benefits of employing an ANC in these applications are analyzed
extensively. The problems encountered in the real time applications of ANC are also
highlighted. Some of the broad conclusions made from the study are as follows
51
CHAPTER-2
•
STUDY AND APPLICATIONS OF ANC
Extensive documentation exists with reference to linear ANC, so analysis can be
done easily.
•
The very introduction of nonlinearity even in a carefully designed ANC degrades
the quality of noise cancellation quite significantly and hence must be taken into
consideration to improve the ANC performance.
•
More indepth investigation is needed to develop improved techniques for
nonlinear ANC to yield better noise control performance.
•
Research can be focused on one of the following subcategories
Development of new architecture or structure for the ANC controller.
Development of new algorithms which will support the new structures.
Modifications in the existing schemes of ANC with specific emphasis to
structural minimization without sacrificing performance.
Development of new strategy to reduce computational complexity
requirement from the implementation point of view.
Customizing the developed schemes for different application areas.
Utilize the recent knowledge of DSP, soft computing and evolutionary
techniques for achieving better performance.
52
CHAPTER 3
Neural Network Approach to Nonlinear Active
Noise Control
3.1 Background
Active noise control (ANC) has attracted a lot of research interest because of
rapid increase of acoustical noise pollution and insufficiency of passive techniques for
noise control. In an ANC, noise is deliberately introduced with an objective to cancel
another undesirable noise. ANC employ the superposition principle, where the undesired
noise is reduced by adding another noise called antinoise with the same amplitude but
opposite polarity. Antinoise is generated by actuators such as loudspeaker. Linear
adaptive FIR filter along with the filtered-x LMS (FXLMS) algorithm is the most
common strategy applied in both feed-forward and feedback ANC due to its ease in
implementation [2]. It has been mentioned earlier (detail discussion in this chapter) that
there are a number of sources of nonlinearity in an ANC. This gives rise to a situation
where linear FIR filter with FXLMS algorithm will show performance degradation and
even fail in some situations. This is due to the fact that a linear type adaptive system has
been used for approximating a system which exhibit nonlinear characteristics. Thus
avenues are wide open to use adaptive systems with nonlinear approximation capability
like artificial neural network, fuzzy logic and polynomial filter etc. This chapter is
entirely devoted to the study of artificial neural network for nonlinear ANC.
3.2 Multilayer Artificial Neural Network (MLANN)
It has already been established that multilayer artificial neural network based
ANCs have significant performance improvement over the conventional linear ANCs
based on FXLMS or FXRLS adaptive algorithms [52]-[61]. The basic objective of the
present work is primarily aimed at developing new ANC in the MLANN domain having
reduced structural complexity so that these can be easily implemented in real-time. The
53
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
learning algorithms developed to update the weights of different structures utilize the
backpropagation algorithm concept. However, with reference to presence of secondary
path certain modifications have been included in the weight adaptation equation.
In [52] Snyder and Tanaka proposed feedforward control of vibration using a
neural network-based control system, with the aim to derive an architecture which might
be capable of supplanting the commonly used FIR filter with FXLMS algorithm. Fig.3.1
shows the ANC developed by Snyder and Tanaka. They have employed two multilayer
perceptron (MLP) networks, one network was exclusively employed to model the
nonlinear secondary path. This network is trained offline using standard backpropagation
algorithm while the ANC is not in operation. Once converged its weights are frozen and
are, latter on, used for stable operation of the ANC. The other network is called the
controller network which is used to produce the antinoise. At the first glance this problem
may seem trivial, as application of the standard backpropagation algorithm can update the
weights. But owing to the presence of the tapped delay line input to the secondary path
model, the standard backpropagation algorithm cannot be used directly in this
arrangement as it must backpropagate through a tapped delay line. Therefore, the
standard backpropagation algorithm is modified resulting in a formulation of a new
algorithm which enables stable adaptation of the neural controller. The algorithm was
shown to be simply a generalization of the linear filtered-x LMS algorithm.
Martin Bouchard, Bruno Paillard, and Chon Tan Le Dinh [53] presented an
improved training algorithm for the neural network in ANC. They used the same neural
network structure as Snyder and Tanaka [52] but introduced new heuristical training
algorithms with the objective to develop faster convergence speed (by using nonlinear
recursive-least squares algorithms) and/or lower computational loads (by using an
alternative approach to compute the instantaneous gradient of the cost function). The
block diagram of ANC using neural network is shown in fig. 3.1
In [54] Martin Bouchard introduced a heuristic procedure for the development of
recursive-least-squares algorithms based on the filtered-x and the adjoint gradient
approaches. He also used the same structure for ANC, as in [52] and [53], employing
two multilayer feedforward neural networks. But he developed new recursive-leastsquares algorithms for the training of the neural network controller in the two network ca54
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
Disturbance
Signal
+
B (z )
Reference
Noise
-
Σ
Error
Signal 1
Secondary
path Model
Controller
+
z-1
z-1
z-1
z-1
z-1
z-1
Error
Signal 2
Error
Backpropagation
Fig. 3.1 Block diagram of ANC using two neural networks.
scaded structure. It has been seen from the results that these new algorithms yielded
better convergence performance than previously published algorithms.
Cheng-Yuan Chang and Fang-Bor Luoh [56] and Cheng-Yuan Chang and ShingTai Pan [60] presented a new architecture for nonlinear ANC using only one multilayer
feedforward network. The ANC they developed is shown in fig.3.2. Here the neural
network is used for ANC controller where as the secondary path is modeled using a linear
adaptive FIR filters. The network has only three layers, input layer, hidden layer and
output layer. Output layer has only one node where as other layers have more than one
nodes. The activation function of both the hidden layer nodes and output layer node are
taken as linear function, basically to avoid premature saturation of back propagation
algorithm.
They developed
a
new
neural
55
network
update
algorithm
using
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
backpropagation and gradient descent methods. The ANC architecture and the developed
algorithm are simple and easy to implement but the major drawback of this technique is it
employ a FIR filter for secondary path modeling. As discussed earlier, in actual
implementation of ANC there exist a number of factors which contribute to nonlinearity
of secondary path. So using adaptive FIR filter for secondary path modeling degrades the
performance of the ANC.
3.3 Proposed Neural Network Technique
x(n)
d (n)
A(z )
e(n)
+
Σ
−
dˆ (n)
y (n)
Neural
Network
B (z )
Update
Algorithm
Fig 3.2 Block diagram of ANC using neural network.
In this chapter, a feedforward nonlinear active noise control system employing a
multilayer neural network is developed. The multilayer neural network involved is shown
in the fig.3.3. Two separate update algorithms are derived for two situations. First
algorithm is derived when secondary path is modeled as a FIR filter (which assume
secondary path to be linear). Another algorithm is developed when secondary path is
modeled as difference equation representing the nonlinear secondary path. Both the
algorithms are found to be extended version of filtered-x LMS algorithm.
Filtered-e LMS is an algorithm for ANC well known for its low computational
complexity requirement. Both the developed algorithms are modified to accommodate
filtered-e LMS algorithm which results in reduced computational complexity.
Performances of the proposed algorithms are validated through extensive computer
simulations.
56
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
Hidden layer
Input layer
wij (n )
u j (n)
F
x(n)
v j (n)
Output layer
y (n)
x(n − 1)
F
M
Σ
M
x(n − I + 1)
F
Fig 3.3 Neural network controller.
The neural network which will be used as the controller for NANC is shown in
fig. 3.3. The network has three layers, input layer, hidden layer and output layer. Output
layer has only one neuron but hidden layer may have many neurons. Neurons in the
hidden layer have nonlinear activation function where as the neuron in the output layer
has linear activation function. Total number of neurons of input layer and neurons in the
hidden layer are I and J respectively. The network has IJ numbers of hidden layer
synaptic weights represented by wij (n) where i=0,1,2,3,……,I-1and j=0,1,2,3,……,J-1.
The network has J number of output layer synaptic weights v j , j=0,1,2, . . . J-1.
The net internal activity level c j (n) for jth hidden layer neuron at nth instant is
I −1
c j (n) = ∑ x(n − i ) wij (n)
for j=0,1, 2, . . . ,J-1
(3.1)
i =0
Considering F as the activation function, the output of the jth hidden layer neuron at nth
instant is calculated as
u j (n) = F(c j (n))
(3.2)
The final output from the network, at nth instant, is computed considering the node in the
output layer to be a summing unit as given by
57
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
J −1
y (n) = ∑ v j (n)u j (n)
(3.3)
j =0
The final output of the neural network is passed through the secondary path to generate
the antinoise, dˆ (n) . The equation for the antinoise generated by the ANC is given by
dˆ (n) =
M −1
∑b
k
( n) y ( n − k )
(3.4)
k =0
where bk is kth coefficient of impulse response of secondary path model and M is the
order of secondary path filter. For the ANC, d (n) is the undesired noise to be cancelled at
the zone of silence. After the noise cancellation process is over the residual noise is
sensed by the error microphone. This residual noise called error signal, at nth instant, is
calculated as follows
e(n) = dˆ (n) − d (n)
(3.5)
The error signal e(n) actuates a control mechanism, the purpose of which is to apply
corrective adjustments to the synaptic weights. This objective is achieved by minimizing
a cost function or index of performance, ξ (n) . A commonly used cost function based on
the mean-squared-error criterion has been applied here.
1
2
ξ (n) = [e 2 (n)]
1
= [d (n) − dˆ (n)] 2
2
M −1
1
= [d (n) − ∑ bk (n)u (n − k )] 2
2
k =0
(3.6)
Now the synaptic weight update equation can be derived. The update equations are
formulated separately for two situations
•
secondary path is assumed linear (modeled by FIR filter)
•
secondary path is assumed nonlinear (modeled by a difference equation
having higher order terms and cross terms)
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CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
3.3.1 Linear Secondary Path
When the secondary path is linearly modeled, the synaptic weight update
equations for output layer and hidden layer are derived separately.
Output layer synaptic weight update
Using back-propagation and gradient descent methods the synaptic weights of
output layer are updated by adding a negative gradient of the cost function with respect to
the weights of interest
v j (n + 1) = v j (n) − µ ∆v j (n)
j=0, 1, 2, . . . , J-1
(3.7)
where µ is the learning rate (step size)
Above equation can also be written as
v (n + 1) = v (n) − µ ∆v (n)
(3.8)
where v (n) = [v0 (n), v1 (n),L , v J −1 (n) ] is the output layer synaptic weight vector.
The gradients of cost function with respect to output layer synaptic weights is defined as
follows
∆v j (n) =
∂ξ (n)
∂v j (n)
(3.9)
Putting (3.6) in the above equation we get
∆v j (n) =
=
∂ξ (n)
∂v j (n)
1 ∂[e 2 (n)]
2 ∂v j (n)
= e( n )
∂[d (n) − dˆ (n)]
∂v j (n)
Since undesired noise d (n) is independent of the neural network synaptic weights, the
following relation is obtained
∂ξ (n)
∂dˆ (n) ∂y (n)
= − e( n )
∂v j (n)
∂y (n) ∂v j (n)
(3.10)
Taking the expressions for dˆ (n) and y (n) from (3.4) and (3.5) respectively and putting in
the above equation it is found that
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CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
M −1
∂ξ (n)
= −e(n) ∑ bk (n)u j (n − k )
∂v j (n)
k =0
(3.11)
where M is the order of secondary path model and bk (n) is the kth coefficient of
secondary path model impulse response. Putting (3.11) in the (3.7), update equation for
output layer synaptic weights are as follows
M −1
v j (n + 1) = v j (n) + µ e(n) ∑ bk (n)u j (n − k )
(3.12)
k =0
Hidden layer synaptic weight update
Similar to output layer synaptic weights, the hidden layer synaptic weights are
updated as
wij (n + 1) = wij (n) − µ ∆wij (n)
(3.13)
where i=0, 1, . . . , I-1 and j=0, 1, . . . , J-1
The update equation for hidden layer weights can also be written as follows
w (n + 1) = w (n) − µ ∆w (n)
(3.14)
where w (n) the hidden layer synaptic weight matrix, at nth instant, defined by
 w00
w
w ( n) =  10
 M

 wI −10
w01
w11
M
wI −11
w0 J −1 
w1 J −1 
L
M 

L wI −1 J −1 
L
L
The gradient of the cost function with respect to hidden layer synaptic weights is
∆wij (n) =
=
∂ξ (n)
∂wij (n)
1 ∂[e 2 (n)]
2 ∂wij (n)
= e( n )
∂[d (n) − dˆ (n)]
∂wij (n)
Since the desired signal is independent of the hidden layer synaptic weights
∂[dˆ (n)]
∆wij (n) = −e(n)
∂wij (n)
M −1
∂dˆ (n) ∂y (n − k )
= − e( n ) ∑
k = 0 ∂y ( n − k ) ∂wij ( n)
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CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
M −1
= −e(n) ∑ bk (n)
k =0
∂y (n − k )
∂wij (n)
(3.15)
Assuming that synaptic weights are adapted slowly it can be written that
∂y (n − k )
∂y (n − k )
≈
∂wij (n − k )
∂wij (n)
=
∂y (n − k ) ∂u j (n − k )
∂u j (n − k ) ∂wij (n − k )
Taking the expressions for y (n − k ) and u j (n − k ) from (3.3) and (3.2) and putting in the
above equation it is found that
∂c j (n − k )
∂y (n − k )
= v j (n − k )F′(c j (n − k ))
∂wij (n)
∂wij (n − k )
Taking the value of c j (n − k ) from (3.1) and putting in above equation it is observed that
∂y ( n − k )
= v j ( n − k ) F′(c j ( n − k )) x( n − i − k )
∂wij ( n)
(3.16)
Putting values of (3.16) and (3.15) in (3.13) the update equation for hidden layer synaptic
weights can be expressed as given below
M −1
wij (n + 1) = wij (n) + µe(n) ∑ bk (n)v j (n − k )F′(c j (n − k )) x(n − i − k )
(3.17)
k =0
where, i=0, 1,2, …… , I-1. and j=0, 1, 2,…….., J-1
3.3.2 Nonlinear Secondary Path
The equations expressed in the previous section are valid when secondary path
is modeled as linear FIR filter. To take care of the nonlinearity in the secondary path, the
secondary path can be modeled as a difference equation having higher order terms and
cross terms. An example is given below
dˆ (n) = y (n) + 0.8 y (n − 1) + 0.6 y 2 (n − 2) + 0.4 y (n − 2) y (n − 3) + 0.2 y (n − 3) y (n − 5)
142
4 43
4 14444444244444443
square term
cross terms
where y (n) and dˆ (n) are input and output of the secondary path filter model.
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CHAPTER-3
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For this type of nonlinear secondary path model the synaptic weight update equation
derived in the previous section have to be modified to develop a complete set of new
equations.
Output layer synaptic weight update
The synaptic weights of output layer are updated by adding a negative gradient of
the cost function with respect to the weights of interest
v j (n + 1) = v j (n) − µ ∆v j (n)
j=0, 1, 2, . . . , J-1
(3.18)
where µ is the learning rate (step size) and
∆v j (n) =
∂ξ (n)
∂v j (n)
The gradient of the cost function with respect to output layer synaptic weights is
∆v j (n) =
=
∂ξ (n)
∂v j (n)
1 ∂[e 2 (n)]
2 ∂v j (n)
Putting the expression of e(n) in (3.5) it is found that
∂ξ (n)
∂[d (n) − dˆ (n)]
= e( n )
∂v j (n)
∂v j (n)
Since the undesirable noise d (n ) is independent of neural network synaptic weights
∂ξ (n)
dˆ (n)
= − e( n )
∂v j (n)
∂v j (n)
∂dˆ (n) ∂y (n − k )
k = 0 ∂y ( n − k ) ∂v j (n )
M −1
= − e( n ) ∑
(3.19)
Assuming that the weights are adapted slowly the second term of the summation can be
written as
∂y (n − k ) ∂y (n − k )
≈
∂v j (n − k )
∂v j (n)
= u j (n − k )
Putting this value in equation (3.19) it is found that
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
M −1
∂ξ (n)
∂dˆ (n)
= − e( n ) ∑
u j (n − k )
∂v j (n)
k = 0 ∂y ( n − k )
(3.20)
The first term of the summation is represented as follows
∂dˆ (n)
∂dˆ (n) ∂dˆ (n)
∂dˆ (n)
∂dˆ (n)
=[
,
,
,L,
]
∂y (n − k )
∂y (n) ∂y (n − 1) ∂y (n − 2)
∂y (n − M + 1)
~
~
~
~
= [b0 (n), b1 (n), b2 (n),L, bM −1 (n)]
~
= b ( n)
(3.21)
k=0,1, 2, . . . ,M-1
~
~
~
~
Here a FIR filter can be defined with b0 (n), b1 (n), b2 (n),L, bM −1 (n) as its coefficients.
~
~
But here b0 (n),L, bM −1 (n) are varying with respect to time. But still a filter with time
varying filter coefficients can be defined as
~
~
~
~
~
b (n) = [b0 (n), b1 (n), b2 (n),L, bM −1 (n)]T
(3.22)
~
This filter, represented as b (n) , is called virtual secondary path filter [62].
Putting (3.21) and (3.22) in (3.20) the following is obtained
M −1
~
∂ξ (n)
= −e(n) ∑ bk (n)u j (n − k )
∂v j (n)
k =0
(3.23)
Finally putting (3.23) in weight update equation (3.18) results in the following
M −1
~
v j (n + 1) = v j (n) + µ e(n) ∑ bk (n)u j (n − k )
j=0, 1, 2, . . . , J-1
(3.24)
k =0
Hidden layer synaptic weight update
Update equation for hidden layer synaptic weights can be written as
wij (n + 1) = wij (n) + µ e(n)
∂dˆ (n)
∂wij (n)
(3.25)
Proceeding in the similar line as in the case of linear secondary path the following is
obtained
∂dˆ (n) M −1 ∂dˆ (n)
∂y (n − k )
=∑
∂wij (n) k =0 ∂y (n − k )(n) ∂wij (n)
M −1
=
~
∑b
k
(n)v j (n − k )F′(c j (n − k )) x(n − i − k )
k =0
Finally putting (3.26) in (3.25) it is found that update equation becomes
63
(3.26)
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
M −1
~
wij (n + 1) = wij (n) + µ e(n) ∑ bk (n)v j (n − k )F′(c j (n − k )) x(n − i − k )
(3.27)
k =0
This algorithm is termed as neural filtered-x LMS (NFXLMS) algorithm.
3.4 Development of Neural Filtered-e LMS Algorithm
Adjoint LMS algorithm developed by E. A. Wan [14] is a simple alternative to
FXLMS algorithm which reduces computational complexity specifically for multichannel
ANC. Unlike FXLMS algorithm where a secondary path model is placed in the reference
signal path, here an adjoint of secondary path model can be placed in the error path as
shown in the fig. 3.4. In the figure Bˆ _ ( z ) is adjoint of secondary path model Bˆ ( z ) . This
algorithm is called filtered-error LMS or filtered-e LMS (FELMS) algorithm.
FELMS algorithm adaptive equation is
w (n + 1) = w (n) + µ x(n − M + 1)e′(n)
(3.28)
where delay(equal to the order of secondary path model) is provided to reference signal
to compensate for the delay in the error path. e′(n) is the filtered error signal generated by
filtering the error signal by the estimated adjoint secondary path filter denoted by Bˆ _( z ) .
Adjoint secondary path filter is obtained by simply writing the filter coefficients in the
reverse order. On the basis of FELMS algorithm the synaptic weight update algorithm
equations of ANC using neural network can also be modified to obtain neural filtered-e
LMS (NFELMS) algorithm.
x(n)
+
d (n)
A(z )
Σ
e(n)
−
y (n)
W (z )
B(z )
Bˆ _( z )
z − M +1
x(n − M + 1)
e′(n)
FXLMS
Fig 3.4 Block diagram of ANC system using FELMS algorithm.
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CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
Here again two situations are considered for discussion,
•
secondary path is assumed linear (modeled by FIR filter)
•
secondary path is assumed nonlinear (modeled by a difference equation having
higher order terms and cross terms)
3.4.1 Linear Secondary Path
Output layer synaptic weights
The synaptic weight update equation for output layer weights is obtained by
modifying (3.12) as follows.
M −1
v j (n + 1) = v j (n) + µ u j (n − M + 1) ∑ bM −1− k (n)e(n − k )
k =0
= v j (n) + µ u j (n − M + 1)e′(n)
j = 0, 1, 2, . . ., J-1
(3.29)
where e′(n) is the error signal filtered through the estimated adjoint secondary path filter.
Hidden layer synaptic weights
Similarly the synaptic weight update equation for output layer synaptic weights is
derived by modifying (3.17) as follows.
M −1
wij (n + 1) = wij (n) + µ v j (n − M + 1)F(c j (n − M + 1)) ∑ bM −1− k (n)e(n − k )
k =0
= wij (n) + µ v j (n − M + 1)F(c j (n − M + 1))e′(n)
(3.30)
i=0,1,. . . , I-1 and j=0,1, . . .,J-1
3.4.2 Nonlinear Secondary Path
Nonlinearity in secondary path is frequently encountered in ANC systems which
must be taken into account. Virtual secondary path concept can be employed to deal with
nonlinear secondary path. Using the filtered error based algorithm developed in [62] the
adjoint virtual secondary path filter coefficient vector is defined as
~
~
~
~
b_ (n) = [bM −1 (n), bM −2 (n − 1) ,L, b0 (n − M + 1)]
Output layer synaptic weights
Using adjoint virtual secondary path filter, the update equation for output layer
synaptic weights is derived from (3.24) as follows
M −1
~
v j (n + 1) = v j (n) + µ u j (n − M + 1) ∑ bM −1− k (n − k )e(n − k )
k =0
65
j=0,1,. . . , J-1
(3.31)
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
~
where bM −1−k (n − k ) is the kth coefficient of adjoint virtual secondary path filter at (n − k ) th
instant . The above equation can also be written as
v j (n + 1) = v j (n) + µ u j (n − M + 1)e′(n)
~
where e′(n) = b_ (n) * e(n)
Hidden layer synaptic weights
The update equation for output layer synaptic weights is derived from (3.27)
using adjoint virtual secondary path filter as follows
M −1
~
wij (n + 1) = wij (n) + µ v j (n − M + 1)F(c j (n − M + 1)) ∑ bM −1−k (n − k )e(n − k )
(3.32)
k =0
i=0,1, . . . ,I-1 and j=0,1, . . ., J-1
The above equation can also be written as
wij (n + 1) = wij (n) + µ v j (n − M + 1)F(c j (n − M + 1))e′(n)
(3.33)
i=0,1, . . . ,I-1 and j=0,1, . . ., J-1
3.5 Simulation and Results
Extensive simulation work has been done for various nonlinear situations,
activation functions and some selected results are presented here to demonstrate the
effectiveness and performance of the proposed algorithms. In all the experiments the
mean-square error (MSE), defined by
MSE = 10 log10 E (e 2 (n))
(3.34)
has been obtained through simulations in MATLAB 7.6.0 (R2008a) environment to
assess the performance and validate the proposed algorithms. Here, e 2 (n) is the square
of the error at nth iteration, and E(.) is the expectation operator. In each of the
experiments, fifty independent trials are conducted and the average MSE(dB) is
computed to obtain smoother convergence characteristics.
3.5.1 Experiment I
The first experiment considered a nonlinear ANC with nonlinear secondary path.
The nonlinear primary path from noise source to error microphone is considered as in[62]
d (n) = 0.8x(n − 6) + 0.6x(n − 7) − 0.2x(n − 8) − 0.5x(n − 9) −
0.1x(n − 10) + 0.4x(n − 11) − 0.05x(n − 12) + x(n − 6)3
66
(3.35)
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
The nonlinear secondary acoustic path from secondary source to error microphone
considered is
y(n) = u (n) + 0.35u (n − 1) + 0.09u (n − 2) − 0.05u (n)u (n − 1)
+ 0.4u (n)u (n − 2)
(3.36)
The reference noise is generated by filtering a uniformly distributed white noise
through a lowpass filter of order 10 and cutoff frequency 350Hz. Simulations are done
for proposed NFXLMS algorithm and are compared with Volterra filtered-x LMS
(VFXLMS) algorithm. VFXLMS algorithm is the standard algorithm for nonlinear ANC.
The structure of the MLP chosen is 10-3-1. The memory size of the second-order
adaptive Volterra filter is 10. The step size for NFXLMS algorithm is chosen to be 0.03.
However for VFXLMS algorithm the step sizes chosen are 0.003 and 0.0003 for linear
and nonlinear coefficients respectively. Average mean square error (MSE) of fifty
independent run of the algorithm is plotted with respect to number of iteration. Three
different nonlinear activation functions are used for hidden layer neurons where as for all
the experiments output layer neuron has linear activation function. Fig. 3.5 shows the
performance when the activation function of the neurons in the hidden layer is
u j ( n) =
1− e
−c j (n)
1+ e
−c j ( n )
(activation function-I)
(3.37)
Fig.3.6 shows the performance when the activation function of the neurons in the hidden
layer is
u j ( n) =
1
1+ e
−c j ( n )
(activation function-II)
(3.38)
Fig. 3.7 shows the performance when the activation function of the neurons in the hidden
layer is
u j ( n) =
e
c j (n)
−e
−c j (n)
e
c j (n)
+e
−c j ( n )
(activation function-III)
(3.39)
In all the experiments hidden layer synaptic weights and output layer synaptic
weights of neural controller are initialized as uniformly distributed random number in the
range -0.5 to 0.5. All the weights of Volterra filter are also initialized to uniformly
distributed random number in the range -0.5 to 0.5. The steady state MSE(dB) obtained
by NFXLMS and VFXLMS algorithms is -26dB and -22dB respectively. The proposed
67
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
NFXLMS algorithm yield lower steady state MSE(dB) compared to standard VFXLMS
algorithm for all the activation functions.
3.5.2 Experiment II
The nonlinear primary acoustic path from noise source to error microphone and
nonlinear secondary acoustic path from secondary source to error microphoneare are
same as that of simulation-I. The reference noise, MLP structure and weight initialization
scheme also remain same as experiment-1 but the adaption algorithms are based on
filtered-error method. MSE plot for NFELMS algorithm and VFELMS algorithm are
shown in fig. 3.8, fig. 3.9 and fig. 3.10 for the three activation functions respectively. The
step size for NFELMS algorithm is 0.03, while for VFELMS algorithm are 0.003 and
0.0003 for linear coefficients and nonlinear coefficients respectively. The steady state
MSE(dB) obtained by NFELMS and VFELMS algorithms are -22dB and -18dB
respectively. The proposed NFELMS algorithm yield much lower steady state MSE(dB)
compared to VFELMS for all the activation functions.
0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iterations
1.4
1.6
1.8
2
4
x 10
Fig. 3.5 MSE(dB) plot for NFXLMS and VFXLMS algorithms (activation function-I).
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CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Nubmer of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.6 MSE(dB) plot for NFXLMS and VFXLMS algorithms (activation function-II).
-5
NFXLMS
VFXLMS
-10
MSE(dB)
-15
-20
-25
-30
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.7 MSE(dB) plot for NFXLMS and VFXLMS algorithms (activation function-III).
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CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
0
NFELMS
VFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.8 MSE(dB) plot for NFELMS and VFELMS algorithms (activation function I).
0
NFELMS
VFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.9 MSE(dB) plot for NFELMS and VFELMS algorithms (activation function-II).
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CHAPTER-3
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0
NFELMS
VFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.10 MSE(dB) plot for NFELMS and VFELMS algorithms (activation function-III).
3.5.3 Experiment III
The reference noise signal is chosen to be a logistic chaotic type, which is
generated using the recursive equation [64]
x(n + 1) = λ x(n) [1 − x(n)]
(3.40)
where λ = 4 and x(0) =0.9 are chosen. This nonlinear noise process is then normalized to
have unity signal power. In this experiment the primary path transfer function considered
is
A( z ) = z −5 − 0.3z −6 + 0.2 z −7
(3.41)
and the secondary path transfer function is chosen to be the non minimum-phase model
defined by
B( z ) = Bˆ ( z ) = z −2 + 1.5 z −3 − z −4
(3.42)
Simulations are done for proposed NFXLMS algorithm and are compared with
VFXLMS algorithm. The structure of the MLP chosen is 10-3-1. The memory size of the
second-order adaptive Volterra filter is 10. The step size for NFXLMS algorithm is 0.02
and for VFXLMS algorithm are 0.003 and 0.0003 for linear and nonlinear coefficients
71
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
respectively. Average mean square error (MSE(dB)) of fifty independent runs of the
algorithms are plotted with respect to number of iteration. Similar to experiment-I, three
different nonlinear activation functions are used for hidden layer neurons where as output
neuron has linear activation function. Fig. 3.11, fig. 3.12 and fig. 3.13 show the
performance of ANC for activation function I, II, III respectively. The steady state
MSE(dB) obtained for NFXLMS and VFXLMS algorithms are -25dB and -22dB
respectively.
3.5.4 Experiment IV
The primary path from noise source to error microphone and secondary path
from secondary source to error microphoneare are same as that of experiment-III. The
reference noise, MLP structure and weight initialization scheme also remain same as
experiment-III but the adaption algorithms are based on filtered-error method. MSE plot
for NFELMS algorithm and VFELMS algorithm are shown in fig. 3.14, fig. 3.15 and fig.
3.16 for the three activation function respectively. The step size for NFELMS algorithm
is 0.03, and for VFELMS algorithm are 0.003 and 0.0003 for linear coefficients and
nonlinear coefficients respectively. The steady state MSE(dB) obtained for NFELMS and
VFELMS algorithms is -21dB.
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0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.11 MSE(dB) plot for NFXLMS and VFXLMS algorithms (activation function-I).
0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.12 MSE(dB) plot for NFXLMS and VFXLMS algorithms (activation function-II).
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0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.13 MSE(dB) plot for NFXLMS and VFXLMS algorithms (activation function-III).
0
NFELMS
VFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.14 MSE(dB) plot for NFELMS and VFELMS algorithms (activation function-I).
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
0
NFELMS
VFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.15 MSE(dB) plot for NFELMS and VFELMS algorithms (activation function-II).
0
NFELMS
VFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.16 MSE(dB) plot for NFELMS and VFELMS algorithms (activation function-III).
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
3.5.5 Experiment on Real Time Signals
To evaluate the performance of the proposed ANC structures and update
algorithms in practical situations all the above experiments are repeated on some real
time signals. Noise data from a Signal Processing Information Base (SPIB) are used.
SPIB database have been provided by the Rice University [79]. The first signal is factory
floor noise acquired by recording samples from 1 / 2′′ B&K condenser microphone on to
digital audio tape (DAT). This noise was recorded near plate-cutting and electrical
welding equipment. The second signal considered is a Buccaneer Jet cockpit noise.
Buccaneer noise acquired by recording samples from 1/2" B&K condenser microphone
onto digital audio tape (DAT). The Buccaneer was moving at a speed of 450 knots, and
an altitude of 300 feet. The sound level during the recording process was 116 dBA. The
detail information about both the noise are given below and are also plotted in fig. 3.17
and fig. 3.18 respectively.
Sampling rate :19.98 KHz
A/D: 16 bit
Pre-filter : Anti aliasing filter
Pre-emphasis : None
Filter : None
Duration-235sec
Length (uncompressed) : approx 9Mb
Taken from:
NOISE-ROM-0 signal.021
NATO: AC243/(Panel 3)/RSG-10
ESPRIT: Project No. 2589-SAM
Produced by:
Institute for Perception-TNO, The Netherlands
Speech Research Unit, RSRE, United Kingdom
Copyright:
TNO, Soesterberg, The Netherlands, Feb 1990
For more information:
Institute for Perception-TNO,
76
CHAPTER-3
NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
PO-box 23,
3769 ZG Soesterberg,
The Netherlands.
4
4
Factory Floor Noise
x 10
3
2
A m plitude
1
0
-1
-2
-3
-4
0
0.5
1
1.5
2
2.5
Samples
3
3.5
4
4.5
5
6
x 10
Fig. 3.17 Factory Floor Noise.
4
2.5
Jet Cockpit Noise
x 10
2
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
-2
0
0.5
1
1.5
2
2.5
Samples
3
3.5
Fig. 3.18 Buccaneer Jet Cockpit Noise.
77
4
4.5
5
6
x 10
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
3.5.6 Experiment V (Real time signal)
In this experiment two real life noise signals are considered. The primary path,
secondary path, neural network structure and adaptive Volterra filter structure same as
that of experiment-I. Activation functions for hidden layer neurons are activation
function-I where as output neuron has linear activation function. Fig. 3.19 and fig. 3.20
show the MSE(dB) plot of the proposed NFXLMS and VFXLMS algorithms for factory
floor noise and Buccaneer jet cockpit noise respectively. The step size in case of factory
floor noise for NFXLMS algorithm is 0.05 and for VFXLMS algorithm are 0.003 and
0.0003 for linear and nonlinear coefficients respectively. The step size in case of
Buccaneer jet cockpit noise for NFXLMS algorithm is 0.01 and for VFXLMS algorithm
and are 0.001 and 0.0001 for linear and nonlinear coefficients respectively. The steady
state MSE(dB) obtained for NFXLMS and VFXLMS algorithms with factory floor noise
are -28dB and -25dB and with Buccaneer jet cockpit noise are -25dB and -22 dB .
3.5.7 Experiment VI (Real time signal)
The proposed NFELMS and VFELMS algorithms are tested on two noise data
collected from real world environment. MSE(dB) plots for factory floor noise and
Buccaneer jet cockpit noise are shown in fig. 3.21 and fig. 3.22 respectively. The primary
path and secondary path, neural network structure and adaptive Volterra filter structure
are same as that of experiment-II. Activation functions for hidden layer neurons are
activation function-I where as output neuron has linear activation function. The step size
in case of factory floor noise for NFELMS algorithm is 0.04 and for VFELMS algorithm
are 0.003 and 0.0003 for linear and nonlinear coefficients respectively. The step size in
case of Buccaneer jet cockpit noise for NFELMS algorithm is 0.01 and for VFELMS
algorithm are 0.001 and 0.0001 for linear and nonlinear coefficients respectively. The
steady state MSE(dB) obtained for NFELMS and VFELMS algorithms with factory floor
noise are -27dB and -20dB and with Buccaneer jet cockpit noise are -23dB and -18 dB
respectively.
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0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.19 MSE(dB) plot for NFXLMS and VFXLMS algorithms( Factory Floor Noise).
0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.20 MSE(dB) plot for NFXLMS and VFXLMS algorithms( Jet Cockpit Noise).
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0
NFELMS
VFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.21 MSE(dB) plot for NFELMS and VFELMS algorithms( Factory Floor Noise).
0
NFXLMS
VFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
0.2
0.4
0.6
0.8
1
1.2
Number of Iteration
1.4
1.6
1.8
2
4
x 10
Fig. 3.22 MSE(dB) plot for NFELMS and VFELMS algorithms(Jet Cockpit Noise).
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3.5.8 Experiment VII (Real time signal)
This experiment is conducted considering two real time signals as reference
noise. The primary path and secondary path, neural network structure and adaptive
Volterra filter structure are same as that of experiment-III. Activation functions for
hidden layer neurons are activation function-I where as output neuron has linear
activation function. The step size in case of factory floor noise for NFXLMS algorithm is
0.05 and for VFXLMS algorithm are 0.003 and 0.0003 for linear and nonlinear
coefficients respectively. The step size in case of Buccaneer jet cockpit noise for
NFXLMS algorithm is 0.01 and for VFXLMS algorithm are 0.001 and 0.0001 for linear
and nonlinear coefficients respectively. Fig. 3.23 and fig. 3.24 show the MSE(dB) plot of
the proposed NFXLMS and VFXLMS algorithms for factory floor noise and Buccaneer
jet cockpit noise respectively. The steady state MSE(dB) obtained for NFXLMS and
VFXLMS algorithms with factory floor noise are -30dB and -26dB and with Buccaneer
jet cockpit noise are -33dB and -27 dB .
3.5.9 Experiment VIII (Real time signal)
In this experiment two real time reference signals are considered. The primary
path and secondary path, neural network structure and adaptive Volterra filter structure
are same as that of experiment-IV. MSE(dB) plots for factory floor noise and Buccaneer
jet cockpit noise are shown in fig. 3.25 and fig. 3.26 respectively. The step size in case of
factory floor noise for NFELMS algorithm is 0.04 and for VFELMS algorithm are 0.003
and 0.0003 for linear and nonlinear coefficients respectively. The step size in case of
Buccaneer jet cockpit noise for NFELMS algorithm is 0.01 and for VFELMS algorithm
are 0.001 and 0.0001 for linear and nonlinear coefficients respectively. The steady state
MSE(dB) obtained for NFELMS and VFELMS algorithms with factory floor noise are 26dB and -22dB and with Buccaneer jet cockpit noise are -33dB and -27 dB respectively.
In all the experiments, considering real time signals as reference noise, it has
been observed that the proposed neural based algorithms exhibit lower steady state
MSE(dB) performance compared to standard Volterra based algorithms with comparable
convergence time.
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0
VFXLMS
NFXLMS
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
-45
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.23 MSE(dB) plot for NFXLMS and VFXLMS algorithms(Factory Floor Noise).
0
VFXLMS
NFXLMS
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
-45
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.24 MSE(dB) plot for NFXLMS and VFXLMS algorithms(Jet Cockpit Noise).
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
0
VFELMS
NFELMS
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
-45
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.25 MSE(dB) plot for NFELMS and VFELMS algorithms(Factory Floor Noise).
0
VFELMS
NFELMS
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
-45
0
1000
2000
3000
4000
5000
Number of Iteration
6000
7000
8000
Fig. 3.26 MSE(dB) plot for NFELMS and VFELMS algorithms(Jet Cockpit Noise).
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NEURAL NETWORK APPROACH TO NONLINEAR ACTIVE NOISE CONTROL
6 Summary
This chapter focuses on developing a MLP based neural network controller for
nonlinear ANC. Separate algorithms were developed for nonlinear ANC, firstly when
secondary path is assumed linear and then secondly when secondary path is assumed
nonlinear. When secondary path is nonlinear, the developed algorithm is modified using
virtual secondary path filter concept. By computer simulation it has been found that the
developed algorithms are performing well for both linear secondary path and nonlinear
secondary path. The performance of developed algorithms is also evaluated for real time
reference signals. The proposed algorithms outperformed VFXLMS algorithm in terms
of steady state MSE(dB). In order to take advantage of low computational complexity
offered by filtered error LMS algorithm, both the developed algorithms are suitably
modified. Performance of the modified algorithms are also analyzed from computer
simulations and compared with that of VFELMS algorithm. The neural based ANC have
resulted in improved performance in comparison to the Volterra based ANCs.
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Legendre Neural Network for Nonlinear Active
Noise Control
4.1 Background
Based on the rapid progress of high-speed and low-cost computing devices such
as digital signal processor (DSP), digital active noise control (ANC) techniques have
been receiving much attention because of the better performance over conventional
passive methods. Passive methods of noise control are effective over a broad frequency
range except at the lower end (below 600 Hz). On a contrary active noise control is
suitable for low frequency noise only and it is not implementable for higher frequencies.
So very often passive and active noise control methods are employed simultaneously in
order to get an overall noise suppression. In some applications where only low frequency
noise is present employing only ANC is sufficient.
Presence of inherent nonlinearity in the ANC makes the antinoise generation
process very complex. In [35] L. Tan and J. Jiang proposed a nonlinear ANC using
adaptive Volterra filter and developed Volterra FXLMS (VFXLMS) algorithm. They
demonstrate that the developed algorithm can improve control performance over the
linear standard filtered-x LMS algorithm under the following conditions.
i) The reference noise sensed by a reference microphone is a nonlinear and predictable
noise process, while the secondary path transfer function of an ANC system has
nonminimum phase.
ii) The primary path exhibits nonlinear behavior
The drawback of this algorithm is high computational complexity. In [34] L. Tan
and J. Jiang proposed a truncated second order Volterra structure for NANC which is
computationally more efficient.
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
Y. H. Pao [63] proposed an alternate neural network structure called functional
link artificial neural network (FLANN) with an object to reduce the training time of the
neural network and to improve convergence speed. Unlike MLANN where the nodes have
nonlinear activation functions, in case of FLANN the links have nonlinear functions. In
[64] D. P. Das and G. Panda employed FLANN to develop filtered-s LMS (FSLMS)
algorithm (here this algorithm is termed as FLANN-FXLMS or FFXLMS algorithm) for
nonlinear ANC. FFXLMS algorithm outperform VFXLMS algorithm in terms of steady
state mean square error and has less computational requirement. D. P. Das and G. Panda
[64] also proposed fast FFXLMS algorithm with a objective to reduce computational
requirement. In [62] D. Zhou and V. DeBrunner compared the VFXLMS algorithm and
FFXLMS algorithm and represented both by a generalized function expansion equation.
They have extended the algorithms to deal with nonlinear ANC with nonlinear secondary
path (NSP) by introducing virtual secondary path concept. They have also introduced
adjoint virtual secondary path to employ filtered-e based algorithms. Block oriented (such
as, Wiener, Hammerstein and Linear-Nonlinear-Linear structure) representation of
nonlinear secondary path was discussed by them. Basically algorithms for nonlinear ANC
can now be classified as nonlinear ANC with linear secondary path (LSP) and nonlinear
ANC with nonlinear secondary path (NSP).
In this chapter Legendre neural network (LNN) is used for nonlinear ANC. The
adaptive algorithm for Legendre neural network is also developed. The algorithm is found
to be simple and easy to implement and have low computational complexity. In order to
reduce computational complexity further a reduced structure Legendre neural network has
been proposed. The reduced structure Legendre neural network reduces the computational
complexity without sacrificing the performance. Weight update algorithms for reduced
structure LNN based on Filtered-x least mean square (FXLMS), Filtered-e least mean
square (FELMS) and Filtered-x recursive least square (FXRLS) are developed. These are
named as reduced structure LFXLMS algorithm, LFELMS algorithm and LFXRLS
algorithm respectively. These algorithms require less computation compared to FFXLMS
algorithm and are modified to deal with NSP which rests upon virtual secondary path
concept. [66]-[72] reported different strategies to derive fast LMS algorithms. Based on
these papers, in order to reduce computational complexity further, a faster version of
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
reducecd structure LFXLMS algorithm is also developed this chapter in which unlike
LFXLMS algorithm, weights are updated in every alternative iterations.
4.2 Reduced Structure Legendre Neural Network for
Nonlinear ANC
4.2.1 Legendre Polynomial
The Legendre polynomials are denoted by L p (x) , where p=0,1, 2, …, P, P is the
order of expansion and x is the argument of the polynomial. L p (x) constitute a set of
orthogonal polynomials as solutions to the differential equation
d 
dy 
(1 − x 2 )  + n(n + 1) y = 0.

dx 
dx 
The zero and the first order Legendre polynomials are, respectively, given by
L0 ( x) = 1
and L1 ( x) = x .
The higher order polynomials are given by
L2 ( x ) =
1
(3 x 2 − 1)
2
1
L3 ( x) = (5 x 3 − 3 x)
2
1
L4 ( x) = (35 x 4 − 30 x 2 + 3)
8
K
The recursive formula to generate higher order Legendre polynomials is expressed as
LP +1 ( x) =
1
[(2n + 1) xL p ( x) − nLP −1 ( x)]
(n + 1)
(4.1)
Some of the important properties of Legendre polynomials are that (i) they are
orthogonal polynomials, (ii) they arise in numerous problems especially in those
involving spheres or spherical coordinates or exhibiting spherical symmetry and (iii) in
spherical polar coordinates, the angular dependence is always best handled by spherical
harmonics that are defined in terms of Legendre functions.
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
In case of an ANC Legendre polynomial of a reference noise signal sample x(n) is
computed and rearranged to form a vector as follows
1
1
L( x(n)) = [1, x(n), (3x 2 (n) − 1)], (5 x 3 (n) − 3x(n)),L,
2
2
1
{( 2n + 1) x(n) L p ( x(n)) − nLP −1 ( x(n))}]
(n + 1)
Reference noise signal vector at nth instant consist of x(n) and it’s delayed samples and is
defined as x(n) = [ x(n), x(n − 1), x(n − 2),L, x(n − N + 1)]
Legendre polynomial of x(n) can be obtained by expanding each sample of x(n) as
1
L(x(n)) = [1, x(n) , 3 x 2 (n) − 1) ,L
2
1
1, x(n − 1) , 3 x 2 (n − 1) − 1) ,L
2
1
1, x(n − 2) , 3 x 2 (n − 2) − 1) ,L
2
L
(4.2)
1
1, x(n − N + 1) , 3 x 2 (n − N + 1) − 1) ]
2
x(n)
L( x(n))
z −1
L( x(n − 1))
z −1
x ( n − 2)
M
LEGENDRE
EXPANSION
x(n − 1)
M
z −1
L( x(n − N + 1))
x(n − N + 1)
Fig. 4.1 Legendre polynomial expansion.
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
4.2.2 Legendre Neural Network
The elements of above Legendre polynomial expansion of x(n) are rearranged
and partitioned to form P new vectors. Taking first element of each row from above
equation a vector s 0 (n) = [1,1,1,1,L] is formed. Taking the second element of each row
from above equation a second vector s1 (n) is formed as shown below
s1 (n) = [ x(n) , x(n − 1) , L, x(n − N + 1)] .
Similarly third vector is formed
s 2 ( n) = [
1
1
1
(3 x 2 (n) − 1) , ), (3 x 2 (n − 1) − 1) , L, ), (3 x 2 (n − N + 1) − 1)] .
2
2
2
Similar manner P new vector, s 0 (n) s1 (n) , s 2 (n) , . . ., s P (n) can be formed. The vectors
s 0 (n) , s1 (n) , s 2 (n) ,L, s P (n) represent 0th, 1st, 2nd ,. . . , P th order Legendre expansion.
Using these vectors Legendre Neural Network is formed as shown in fig.4.2 where
w 0 (n), w 1 (n),L, w P (n) are adaptable weight vectors. All the vectors can be combined to
form a single vector s(n) .
s(n) = [s 0 (n) , s1 (n) , s 2 (n) ,L, s P (n)]
x(n)
s 0 ( n)
z −1
w 0 ( n)
x(n − 1)
z −1
x ( n − 2)
M
z −1
LEGENDRE
EXPANSION
s1 ( n )
w 1 ( n)
Σ
M
w P (n)
s P (n)
x(n − N + 1)
Fig. 4.2 Legendre neural network.
89
y (n)
CHAPTER-4
LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
x(n)
d (n)
A(z )
+
e(n)
dˆ (n)
LEGENDRE EXPANSION
Memory
s 1 ( n)
s 2 ( n)
−
y (n)
w 1 ( n)
+
w 2 ( n)
+
M
B(z )
M
s P (n)
w P (n)
L
s′P (n)
Bˆ ( z )
M
Update
Algorithm
Fig. 4.3 Block diagram of ANC using reduced structure Legendre neural network.
4.3 LFXLMS Algorithm
Structure of the ANC using Legendre Neural Network is shown in fig. 4.3 where
the input vector x(n) = [ x(n) x(n − 1) .....x(n − N + 1)] is transformed into an output vector
s(n) given by s(n) = L(x(n)) . The nonlinear function L(x(n)) represents a set of the
orthogonal basis functions, implemented in the ‘‘Legendre expansion’’ block. Here the
N-dimensional input pattern x(n) is enhanced to an N(P+1)-dimensional enhanced
pattern
s(n) = [s 0 (n) s 1 (n) . . . s P ] .
where
s 0 (n) = L0 (x(n)) = [111.. . . N number of 1]
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
s1 (n) = L1 (x(n)) = [ x(n) x(n − 1) . . . x(n − N + 1)]
1
s 2 (n) = L2 (x(n)) = [ (3 x 2 (n) − 1)
2
1
1
(3 x 2 (n − 1) − 1) . . . (3 x 2 (n − N + 1) − 1)]
2
2
.......................
Comparing to FLANN where trigonometric functions are used in the functional
expansion, LNN uses Legendre orthogonal functions. The major advantage of LNN over
FLANN is that the evaluation of Legendre polynomials involves less computation
compared to that of the trigonometric functions. Therefore, LNN offers faster training
compared to FLANN. Corresponding to P+1 expanded input vectors the network has
P+1 number of adaptive filters w 0 (n), w1 (n),L, w P (n) , operating in parallel. This
approach is called filter bank implementation [36]. The order of each adaptive filter is N.
Employing filter bank implementation output of LNN at time n is obtained by summing
outputs of all the adaptive filters.
P
P
i =0
i =0
y ( n) = ∑ y i ( n) = ∑ s i ( n) w i ( n)
T
(4.3)
where w i (n) is the weight vector of ith adaptive filter at nth instant. Estimated desired
signal dˆ (n) is obtained by filtering LNN output by the estimated secondary path B(z ) .
Error at time n is defined as e(n) = d (n) − dˆ (n) . A popularly used cost function based on
the mean-squared-error criterion is chosen here.
1
2
ξ (n) = [e 2 (n)]
1
= [d (n) − dˆ (n)]2
2
Using the FXLMS algorithm the weight vectors of each adaptive filter is updated as
w i (n + 1) = w i (n) + µe(n)s′i (n)
(4.4)
where s′i (n) is the input signal, s i (n) , filtered through the estimated secondary path and µ
is the step size which control convergence and stability. This algorithm is called
Legendre FXLMS (LFXLMS) algorithm. In Legendre neural network for Pth order
Legendre expansion P+1 number of adaptive filters operate in parallel. But it is observed
that the input vector for first adaptive filter always contain 1, so it not dependant on
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reference noise signal and it don’t carry any information about the reference noise signal.
Intuitively it is concluded that removing first adaptive filter from the network may not
affect the network performance. This has been confirmed from extensive simulation work
that removing the first adaptive filter did not degrade the network performance. This new
structure is called as reduced structure Legendre neural network. The advantage of
reduced structure Legendre neural network is to reduce the computational complexity. It
has been seen that for 2nd order Legendre expansion, the saving in computational
complexity is 33%, while for 3rd order expansion saving is 25%. From simulation study it
has been observed that 3rd order expansion is sufficient to obtain noise reduction.
Increasing the order of expansion does not result in further mean square error reduction.
The output of reduced order Legendre neural network can now be written as
P
P
i =1
i =1
y ( n) = ∑ y i ( n ) = ∑ s i ( n) w i ( n)
T
In LFXLMS algorithm all the expanded input vectors have to be separately
filtered through the estimated secondary path. Here also dropping the first input vector
results reduction in number of filtering required so computational requirement is further
reduced.
4.3.1 Nonlinear Secondary Path
For nonlinear active noise cancellation with nonlinear secondary path the update
equation for weight vectors is written as
1 ∂ξ (n)
w i (n + 1) = w i (n) − µ
2 ∂w i (n)

∂dˆ (n) 
= w i (n) + µ E e(n)

∂w i (n) 

where i=1,
2, … P
As in the classic LMS algorithm, we can use the instantaneous value to approximate the
ensemble mean, yielding
w i (n + 1) = w i (n) + µe(n)
∂dˆ (n)
∂w i (n)
Note that
∂dˆ (n) M −1 ∂dˆ (n) ∂y (n − m)
=∑
⋅
∂w i (n) m=0 ∂y (n − m) ∂w i (n)
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where M is the memory size of the NSP. Assuming that for small step size, w i (n) is
slowly varying it can be written that
∂y (n − m) ∂y (n − m)
≈
∂w i (n − m)
∂w i (n)
(4.5)
T
But y (n − m) = w i (n − m)s i (n − m)
Putting this value in (4.5) we get
∂y (n − m)
= s i ( n − m)
∂w i (n)
The update equation of w i (n) can now be written as
∂dˆ (n)
s i ( n − m)
m = 0 ∂y ( n − m)
M −1
w i (n + 1) = w i (n) + µe(n) ∑
(4.6)
The first term of summation of (4.6), is found to be
 ∂dˆ (n)
∂dˆ ( n)
∂dˆ ( n)
=
,
,
∂y (n − m)  ∂y (n) ∂y (n − 1)
∂dˆ (n)
∂dˆ ( n)
, ... ,
∂y( n − 2)
∂y ( n − M + 1)



(4.7)
The elements of above vector are found to be time varying. This vector is called virtual
~
secondary path, denoted by b (n) and defined as follows [62]
[
~
~
~
~
b (n) = b0 (n) , b1 ( n),L , bM −1 ( n)
]
Putting (4.7) in (4.6) the weight update equation now becomes
M −1
~
w i (n + 1) = w i (n) + µe(n) ∑ bm (n)s i (n − m)
m =0
w i (n + 1) = w i (n) + µe(n)~si ' (n)
(4.8)
where ~si ' is the expanded input signal filtered through virtual secondary path. This update
algorithm is LFXLMS algorithm for nonlinear secondary path.
4.4 LFELMS Algorithm
Adjoint LMS algorithm was developed by Wan [14] and provides a simple
alternative to the FXLMS algorithms. In adjoint LMS, the error signal (rather than the
input signal) is filtered through an adjoint secondary path filter. This algorithm is
alternatively termed as filtered error least mean square (FELMS) algorithm. FELMS
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algorithm drastically reduces computational complexity of FXLMS algorithm for
multichannel ANC. The degree of saving in computational complexity increases with
increase in number of channels. Saving in computational complexity can also be achieved
for our LFXLMS algorithm by using the technique of FELMS algorithms. Structure of
Legendre neural network is such that a number of adaptive filters operate in parallel
which is called as filter bank approach. The key for the application of FELMS algorithm
is to develop the adjoint secondary path. In case secondary path is linear one (LSP)
adjoint secondary path is obtained by writing the coefficients of secondary path in reverse
order. Thus adjoint secondary path can be written as follows
b _(n) = [bM −1 (n) , bM − 2 (n) , . . . , b0 (n)]
(4.9)
The block diagram of secondary path filter and adjoint secondary path filter are shown in
fig. 4.4 and fig. 4.5 respectively.
si (n)
b0 (n)
z −1
b1 (n)
×
z −1
z −1
bM −2 (n)
×
+
bM −1 (n)
×
+
×
si′ (n)
+
Fig. 4.4 Secondary path filter
e( n )
bM −1 (n)
z −1
×
bM −2 (n)
z −1
×
z −1
b1 (n)
×
b0 (n)
×
e′(n)
+
+
Fig. 4.5 Adjoint secondary path filter
94
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When secondary path is nonlinear (NSP) the adjoint secondary path cannot be
formed directly by writing the coefficients in reverse order. First the virtual secondary
path filter is obtained and then adjoint virtual secondary path filter can be derived from it.
Since virtual secondary path is a time varying filter so its adjoint version can be obtained
by not only reversing the filter coefficients but also using delayed filter coefficients. The
adjoint virtual secondary path is defined as follows [62]
~
~
~
~
b _(n) = [bM −1 (n) bM − 2 (n − 1) ... b0 (n − M + 1)]
(4.10)
The block diagram of secondary path filter and adjoint secondary path filter are shown in
fig. 4.6 and fig. 4.7 respectively.
si (n)
~
b0 (n)
z −1
~
b1 (n)
×
z −1
z −1
~
bM − 2 (n)
×
+
~
bM −1 (n)
×
+
×
si′ (n)
+
Fig. 4.6 Virtual secondary path filter
e( n )
z −1
~
bM −1 (n)
×
~
bM − 2 (n − 1)
z −1
×
z −1
~
b1 (n − M + 2)
×
~
b0 (n − M + 1)
×
e′(n)
+
+
Fig. 4.7 Adjoint virtual secondary path filter
95
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The weight update equation can be written as
w i (n + 1) = w i (n) + µe′(n)s i (n − M + 1)
(4.11)
where e' (n) is the error filtered through the adjoint secondary path for LSP and adjoint
virtual secondary path for NSP. M is the length of virtual secondary path.
4.5 LFXRLS Algorithm
Generally recursive least square (RLS) algorithm is employed to enhance speed of
convergence but at the cost of increase in computational complexity [7], [8]. RLS
algorithm for Legendre neural network is used to develop Legendre filtered-x recursive
least square (LFXRLS) algorithm. The summary of the LFXRLS algorithm is as follows
The weight update equation for the adaptive filters is given below
w i (n + 1) = w i (n) + µe(n)k i (n)
(4.12)
where, i = 0,1,...., P is the number of adaptive filters.
The individual Kalman gain vector is defined below
k i ( n) =
z i ( n)
s i ( n) z i ( n ) + 1
(4.13)
z i (n) = λ−1Q i (n − 1)s Ti (n)
(4.14)
and the inverse of the autocorrelation matrix
(4.15)
Q i (n) = λ−1 [Q i (n − 1) − k i (n)z i (n)]
where 0 ≤ λ < 1 is the forgetting factor, which weights the recent data more heavily in
order to accommodate nonstationary signals.
4.6 Fast LFXLMS Algorithm
The filter-bank implementation of the LFXLMS algorithm is shown in fig. 4.3.
In such a scheme, the residual error sensed by the error microphone, is expressed by
e( n ) = d ( n ) + b ( n ) ∗ y ( n )
(4.16)
In (4.16), b(n) represents the impulse response of the secondary path transfer function
and y (n) is the output of the Legendre neural network, which is computed as
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P
y ( n) = ∑ y i ( n)
(4.17)
i =1
where
yi (n) = sTi (n)w i (n)
(4.18)
and i = 1,2,....P, P is the order of function expansion. Using LFXLMS algorithm the
weight update equations at time n − 1 and n , respectively, are written as
w i (n) = w i (n − 1) − µ e(n − 1) s′i (n − 1)
(4.19)
w i (n + 1) = w i (n) − µ e(n) s′i (n)
(4.20)
where s′i (n) is the input signal vector, s i (n) filtered through the secondary path filter.
Output of the controller
The output of the controller at time n − 1 and n , is written as
yi (n − 1) = s i (n − 1)w Ti (n − 1)
(4.21)
yi (n) = s i (n)w Ti (n) , i = 1, 2,L, P
(4.22)
Inserting (4.19) into (4.22) yields
T
yi (n) = s i (n)w Ti (n − 1) − µe(n − 1)s i (n)s′i (n − 1)
(4.23)
From (4.21) and (4.23) we obtained
0
 yi (n − 1) s i ( n − 1)  T


=
w
(
n
−
1
)
−
 y ( n)   s ( n)  i
 µe( n − 1)s (n)s′T (n − 1) 
i
i
 i
  i



(4.24)
Consider the first term on the right hand side of (4.24)
 α i ,1
s i (n − 1) T
 s (n)  w i (n − 1) = α
 i

 i,0
α i,2 
α i ,1 
w Ti, 0 (n − 1)
 T

 w i ,1 (n − 1) 
(4.25)
where
α i , 0 = [si (n) si (n − 2) . . . si (n − N + 2) ]
α i ,1 = [si (n − 1) si (n − 3) . . . si (n − N + 1) ]
(4.26)
α i , 2 = [si (n − 2) si (n − 4) . . . si (n − N ) ]
and
[
w i , j (n) = wi , j (n) wi , j +2 (n) . . . wi , j + N −2 (n)
]
T
for j=0, 1
97
(4.27)
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
Adding and subtracting α i ,1w i ,1 (n − 1) to the first row and α i ,1w i , 0 (n − 1) to the second row
on the right hand side of (4.25), the following expression is obtained
α i ,1 ( w i , 0 ( n − 1) + w i ,1 ( n − 1)) + (α i , 2 − α i ,1 ) w i ,1 ( n − 1) 
s i ( n − 1)  T
w
(
n
−
1
)
=
α ( w ( n − 1) + w ( n − 1)) + (α − α ) w ( n − 1)  (4.28)
 s ( n)  i
i ,1
i ,1
i ,0
i ,0
 i

 i ,1 i , 0

As the N / 2 element vector, α i ,1 (w i , 0 (n − 1) + w i ,1 (n − 1)) , in (4.28) is common to both the
rows, it requires only a one time computation. Thus, (4.28) requires 1.5 NP
multiplications for two time steps, which means 0.75 NP multiplications on an average
for each time step.
M 1 = 0.75 NP
Therefore, this saves 0.25 NP multiplications.
Now, consider the second term on the right-hand side of (4.23). Let
ψ i (n) = s i (n)s′i T (n − 1)
(4.29)
Further ψ i (n) can be computed with less number of computations as follows:
ψ i (n) = ψ i (n − 2) + [si (n) si′ (n − 1) + si (n − 1) si′ (n − 2)]
− [si (n − N ) si′ (n − N − 1) + si (n − N − 1) si′ (n − N − 2)]
(4.30)
Since the term [si (n − N ) si′ (n − N − 1) + si (n − N − 1) si′ (n − N − 2)] in (4.30) has already
been computed at time n − N ,ψ i (n) requires only 2 P multiplications for two time steps,
where P is the total number of FIR adaptive filters.
The terms α i , 2 − α i ,1 and α i ,1 − α i , 0 in (4.28) require only one addition each, as all
terms except the first term in these summations have already been computed at n − 2 time
step. Therefore, the number of multiplications required per time step, to compute (4.28),
is given by 0.5 P ( N + 2) .
M 1 = 0.5P( N + 2)
(4.31a)
In order to compute (4.30), we required 1.5 P additions per sample. Therefore, the total
number of additions required per sample to compute the output of the controller is equal
to A1
A1 = 0.5P( N + 5)
(4.31b)
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Secondary Path Filtering
The terms si′ (n − 1) and si′ (n) can be computed in a similar manner as done in case
of yi (n − 1) and yi (n) were computed in (4.24) and are shown below
si′ (n − 1) = s i (n − 1)b T (n)
i = 1,2,L P
(4.32)
si′ (n) = s i (n)b T (n)
(4.33)
 si′ (n − 1) β i ,1 (b 0 + b1 ) + (β i , 2 − β i ,1 )b 1 
 s ′ (n)  = β (b + b ) − (β − β )b 
1
i ,1
i,0
0
 i
  i ,1 0
(4.34)
where
β i , 0 = [s i (n) s i (n − 2) . . . s i (n − M + 2) ]
β i ,1 = [s i (n − 1) s i (n − 3) . . . s i (n − M + 1) ]
(4.35)
β i , 2 = [s i (n − 2) si (n − 4) . . . s i (n − M ) ]
where M is the order of the secondary path filter transfer function and
[
b j = b j b j + 2 . . . . b j + M −2
]
T
(4.36)
Equation (4.34) requires 1.5MP multiplications for two time steps, since the term
β i ,1 (b 0 + b 1 ) has to be computed only once. Therefore, the number of multiplications
required per time step is 0.75MP,
M 2 = 0.75MP .
(4.37a)
The term b 0 + b 1 can be precomputed and stored. Hence, it requires no additions and the
terms β i , 2 − β i ,1 and β i ,1 − β i , 0 requires one addition each. Therefore, the total number of
additions required per sample is given by
A2 = 0.5P( M + 2)
(4.37b)
Weight Update
Substituting (4.33) into (4.32), we obtain
w i (n + 1) = w i (n − 1) − µe(n − 1)s′i (n − 1) − µe(n)s′i (n)
(4.38)
 γ Ti,1 
 γ Ti, 0 
w i , 0 (n + 1) w i ,0 (n − 1)
 w (n + 1)  =  w (n − 1)  − µe(n − 1)  T  − µe(n)  T 
 i ,1
  i ,1

 γ i ,2 
 γ i ,1 
(4.39)
where
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
γ i , 0 = [si′ (n) si′ (n − 2) . . . si′ (n − N + 2) ]
γ i ,1 = [si′ (n − 1) si′ (n − 3) . . . si′ (n − N + 1) ]
(4.40)
γ i , 2 = [si′ (n − 2) si′ (n − 4) . . . si′ (n − N ) ]
and w i ,0 (n) and w i ,1 (n) are defined as in (4.27)
Rewriting (4.25) as
 γ Ti,1 γ Ti,0  e(n − 1)
w i , 0 (n + 1) w i ,0 (n − 1)
 w (n + 1)  =  w (n − 1)  − µ  T T  

 i ,1
  i ,1

 γ i , 2 γ i ,1   e(n) 
(4.41)
Adding and subtracting µ γ Ti,1e(n) to the first row and µ γ Ti,1e(n − 1) to the second row on
the right-hand side of (4.41), the following equation is obtained
w i , 0 (n + 1) w i ,0 (n − 1)  γ Ti,1 µ (e(n − 1) + e(n) − ( γ i ,1 − γ i ,0 ) T µe(n) 

 w (n + 1)  =  w (n − 1)  −  T
T
 i ,1
  i ,1
  γ i ,1 µ (e(n − 1) + e(n) − ( γ i , 2 − γ i ,1 ) µe(n − 1)
(4.42)
By following a similar analysis as we did for (4.34), the weight update (4.28) requires
0.75NP multiplications per iteration,
M 3 = 0.75 NP
(4.43a)
The number of additions required to compute (4.42) is given by A3
A3 = 0.5P(2 N + 3)
(4.43b)
From (4.31a), (4.37a), and (4.43a), the number of multiplications per sample is given by
M 1 + M 2 + M 3 = P(1.25 N + 0.75M + 1), compared to the standard FFXLMS, which
requires (2P +1)(2N +M) multiplications. From (4.31b), (4.37b), and (4.43b) the total
number of additions is given by A1 + A2 + A3 = 0.5P(3N + M + 10), whereas for the
standard FFXLMS, this number is (2P+1)(2N+M-2). Note that while comparing these
algorithms, the multiplication due to step size ( µ ) has not been taken into account.
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
Table 4.1.
Computational Complexity Comparison
per sample for computation of
sample for computation of
Number of additions required per
Number of multiplications required
Algorithm
LFXLMS
LFELMS
FFXLMS
Fast LFXLMS
Controller
Output
N(P-1)
N(P-1)
N(2P +1)
0.5P(N+2)
Secondary
path filtering
M(P-1)
M
M(2P+1)
0.75MP
N(P-1)
N(P-1)
N(2P +1)
0.75NP
Total
(2N+M)(P-1)
2N(P-1)+M
(2P+1)(2N +M)
P(1.25N+0.75M+1)
Controller
Output
(N-1)(P-1)
(N-1)(P-1)
(N-1)(2P +1)
0.5P(N+5)
Secondary
path filtering
(M-1)(P-1)
M-1
(M-1)(2P+1)
0.5P(M+2)
N(P-1)
N(P-1)
N(2P +1)
0.5P(2N+3)
(2N+M-2)(P-1)
(2N-1)(P-1)+M-1
(2P+1)(2N+M-2)
0.5P(3N+M+10)
Weight
update
Weight
update
Total
4.7 Simulation and Results
Extensive simulation work has been done for various nonlinear ANC and some
selected results are presented to validate the proposed algorithms. The performance of the
proposed LFXLMS algorithm, LFELMS algorithm and LFXRLS algorithm are compared
with FLANN based algorithm (FFXLMS algorithm). A number of nonlinear ANC with
linear secondary path and nonlinear secondary path are tested. Simulation result of fastLFXLMS is also compared with LFXLMS algorithm. Mean square error (MSE) in dB
defined by
MSE(dB) = 10 log10 {E (e 2 (n)}
(4.44)
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is plotted for each simulation. In each of the experiments, hundred independent trials are
conducted and the average MSE(dB) is plotted to obtain smoother convergence
characteristics.
4.7.1 Experiment I
In the first experiment a nonlinear ANC with the primary path transfer function
defined below is considered [64]
A( z ) = z −5 − 0.3 z −6 + 0.2 z −7
(4.45)
The secondary path considered is a non-minimum-phase filter with transfer function
B( z ) = z −2 + 1.5 z −3 − z −4
(4.46)
Secondary path is assumed to be perfectly estimated i.e. Bˆ ( z ) = B ( z )
The reference noise is the logistic chaotic noise generated by the following equation [11]
x(n + 1) = λx(n)[1 − x(n)]
(4.47)
where λ = 4 and x(0) = 0.9 are used. This noise process is then normalized to have unit
signal power. MSE(dB) for proposed LFXLMS, LFELMS and LFXRLS algorithms are
plotted and compared with that of FFXLMS algorithm. The step size for LFXLMS,
LFELMS, FFXLMS, LFXRLS algorithms are 0.0004, 0.0003, 0.0004, 0.005 respectively
and forgetting factor for LFXRLS algorithm considered is 0.99. MSE(dB) plots for all the
algorithms are shown in the fig. 4.8. The steady state MSE(dB) obtained by LFXLMS,
LFELMS, LFXRLS and FFXLMS algorithm are -30dB, -29 dB, -32 dB and -26 dB
respectively. The proposed algorithms results in lower steady state MSE(dB) compared to
FFXLMS algorithm which indicates LNN based algorithms perform better than FFXLMS
algorithm.
In order to verify tracking capability of the developed algorithms the primary path
transfer function and/or secondary path transfer function are varied after the algorithms
entered into convergence region. In the first test primary path is changed (the undesired
noise at the cancellation point is changed from d (n) to − d (n) ) at 3000th iteration. The
MSE(dB) plot is shown in fig. 4.9, which confirms that all the algorithms successfully
converged even after variation in primary path. In the second test the secondary path
transfer
function
is
changed
from
102
B( z ) = z −2 + 1.5 z −3 − z −4
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
B( z ) = 0.9 z −2 + 1.1z −3 − 1.1z −4 at 3000th iteration and the MSE(dB) plots obtained are
shown in the fig. 4.10. In the third test both the primary path and secondary path are varied
at 3000th iteration and the MSE(dB) plots are shown in the fig. 4.11. From all the three
tests it can be concluded that the proposed algorithms are able to track variation in the
primary and secondary paths.
4.7.2 Experiment II
In order to analyze the performance of the proposed algorithms in case of
nonlinear active noise canceller, nonlinear primary path as well as nonlinear secondary
path is considered here. In this experiment the nonlinear primary path is defined by the
following primary to desired signal relationship [62]
d ( n) = x( n) + 0.8 x (n − 1) + 0.3 x( n − 2) + 0.4 x( n − 3) − 0.8 x ( n) x( n − 1) +
(4.48)
0.9 x( n) x( n − 2) + 0.7 x( n) x( n − 3)
Similarly nonlinear secondary path considered has the following input to output
relationship
dˆ ( n) = y ( n) + 0.35 y ( n − 1) + 0.09 y ( n − 2) − 0.5 y ( n) y ( n − 1) + 0.4 y ( n) y ( n − 2)
(4.49)
Reference signal is considered to be white noise. MSE(dB) for the proposed LFXLMS and
LFELMS algorithms are obtained using virtual secondary path and compared with that of
FFXLMS algorithm. The step size for LFXLMS, LFELMS and FFXLMS algorithms are
0.0002, 0.0001 and 0.0002 respectively The steady state MSE(dB) obtained by LFXLMS,
LFELMS and LFXRLS algorithm are -14dB, -12 dB and -12 dB respectively. The
proposed LFXLMS algorithm outperformed FFXLMS algorithm in terms of steady state
MSE(dB). But the vital advantage of the proposed algorithms is their low computational
complexity.
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-5
LFXLMS
FFXLMS
LFELMS
LFXRLS
-10
MSE(dB)
-15
-20
-25
-30
500
1000
1500
2000
Number of Iteration
2500
Fig. 4.8 MSE(dB) plot for LFXLMS, FFXLMS, LFELMS and LFXRLS algorithm.
LFXLMS
FFXLMS
LFELMS
0
-5
MSE(dB)
-10
-15
-20
-25
-30
1000
2000
3000
Number of Iteration
4000
5000
Fig. 4.9 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm when primary
path is changed at 3000th iteration.
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-5
LFXLMS
FFXLMS
LFELMS
-10
MSE(dB)
-15
-20
-25
-30
1000
2000
3000
Number of Iteration
4000
5000
Fig. 4.10 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm when secondary
path is changed at 3000th iteration.
0
LFXLMS
FFXLMS
LFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
1000
2000
3000
Number of Iteration
4000
5000
Fig. 4.11 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm when both
primary path and secondary path are changed at 3000th iteration.
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
LFXLMS
FFXLMS
LFELMS
2
0
-2
MSE(dB)
-4
-6
-8
-10
-12
-14
-16
500
1000
1500
2000
Number of iteration
2500
Fig. 4.12 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm.
4.7.3 Experiment III
Another experiment is conducted on the nonlinear active noise canceller. In this
experiment the nonlinear primary path is defined by the following input to output relation
[62]
d ( n) = x ( n − 5) + 0.8 x ( n − 6) + 0.3 x ( n − 7) + 0.4 x ( n − 8) + 0.2 x ( n − 5) x ( n − 6) −
0.3 x ( n − 5) x ( n − 7) + 0.4 x ( n − 5) x ( n − 8)
(4.50)
The nonlinear secondary path considered is in cascade form. It consist of cascading of
three blocks, Linear ( l1 )-Nonlinear ( N )-Linear ( l 2 ) (LNL) shown in fig. 4.13.
y (n)
z1 (n)
l1
N
z 2 ( n)
dˆ (n)
l2
Fig. 4.13 Block diagram of LNL nonlinear secondary path model.
The blocks l1 , N , l2 are defined as follows
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
l1 = 1 − 0.6 z −1 + 0.05 z −2
(4.51)
N ( z1 ) = 3.3 tanh[0.3z1 (n)]
(4.52)
l2 = 1 + 0.2 z −1 + 0.05 z −2
(4.53)
The reference noise considered here is white noise. The step size for LFXLMS, LFELMS
and FFXLMS algorithms are 0.0003, 0.002 and 0.0003 respectively. Proceeding in the
same manner as in [62] the MSE(dB) obtained by LFXLMS, LFELMS and FFXLMS
algorithm are plotted in the fig. 4.14, which suggest equivalent steady state MSE(dB) for
all the three algorithms. But here it should be noted that the proposed algorithms have
lower computational complexity requirement.
5
LFXLMS
FFXLMS
LFELMS
MSE(dB)
0
-5
-10
-15
-20
500
1000
1500
2000
Number of Iteration
2500
Fig. 4.14 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm.
4.7.4 Experiment IV
The primary path, secondary path and input signal are same as same that of
experiment I. But adaptive algorithm for weight update is fast LFXLMS algorithm and
LFXLMS algorithm. Step size for both the algorithms is 0.01. MSE(dB) plot for the
algorithms are shown in fig.4.15. Both the algorithms yield identical results but real
advantage of fast LFXLMS algorithm is lower computational complexity.
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CHAPTER-4
LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
0
Fast LFXLMS
LFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
0
500
1000
1500
2000 2500 3000
Number of Iteration
3500
4000
4500
5000
Fig. 4.15 MSE(dB) plot for LFXLMS algorithm and Fast LFXLMS algorithm.
4.7.5 Experiment on Real Time Signal
In order to assess the performance of the developed algorithms in a real time
environment a few experiments are conducted on real time signals as reference noise.
Two reference signals considered are Buccaneer Jet cockpit noise (used in previous
chapter) and white noise. White Noise is acquired by sampling high-quality analog noise
generator (Wandel & Goltermann) [79]. It exhibits equal energy per Hz. bandwidth.
4.7.6 Experiment V (Real Time Signal)
In this experiment two reference signals picked up from real world environment
are considered. Fig.4.16 and fig.4.17 shows MSE(dB) plots of LFXLMS, FFXLMS and
LFELMS algorithms for Buccaneer jet cockpit noise and white noise respectively. The
primary path and secondary path are identical to the experiment-1. Step size(Buccaneer
jet cockpit noise) for LFXLMS is 0.0005, FFXLMS is 0.0005 and LFELMS is 0.0004.
The step size (white noise) for LFXLMS is 0.0004, FFXLMS is 0.0005 and LFELMS is
0.0003. The steady state MSE(dB) obtained by LFXLMS, FFXLMS and LFELMS
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
algorithm in case of jet cockpit noise are -27dB, -25dB and -27dB and in case of white
noise are -23dB, -22dB and -22dB respectively.
4.7.7 Experiment VI (Real Time Signal)
In order to verify the tracking capability of the developed algorithms the primary
path transfer function is varied after the algorithms entered into convergence region. The
primary path is changed (the undesired noise at the cancellation point is changed
from d (n) to − d (n) ) at 3000th iteration. Fig.4.18, fig.4.19 shows MSE(dB) plots of
LFXLMS and FFXLMS algorithms for Buccaneer jet cockpit noise and white noise
respectively. Step size (Buccaneer jet cockpit noise) for LFXLMS is 0.0005 and
FFXLMS is 0.0005. The step size (white noise) for LFXLMS is 0.0004 and FFXLMS is
0.0005. The steady state MSE(dB) obtained by LFXLMS and FFXLMS algorithms
confirms that all the algorithms successfully converged after variation in primary path.
4.7.8 Experiment VII (Real Time Signal)
Experiment-VI is conducted once again but this time secondary path rather than
primary path is varied after the ANC entered into steady state region. The secondary path
transfer
function
is
changed
from
B( z ) = z −2 + 1.5 z −3 − z −4
to
B( z ) = 0.9 z −2 + 1.1z −3 − 1.1z −4 at 3000th iteration and the MSE(dB) plots obtained are
shown in the fig.4.20 and fig.4.21 for Buccaneer jet cockpit noise and white noise
respectively. From the above two tests it can be concluded that the proposed algorithms
are able to track variations in the primary and secondary paths.
4.7.9 Experiment VIII (Real Time Signal)
Performances of the proposed algorithms are assessed in nonlinear environment.
Here both the primary and secondary paths are considered nonlinear and their transfer
functions are identical to experiment-II. Fig.4.22 and fig.4.23 shows MSE(dB) plot for
LFXLMS, FFXLMS and LFELMS algorithms for Buccaneer jet cockpit noise and white
noise respectively. The steady state MSE(dB) obtained for LFXLMS, FFXLMS and
LFELMS algorithms in case of Jet Cockpit Noise are -9dB, -4dB, -7dB and in case of
white noise are -10dB, -5dB and -9dB.
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
LFXLMS
FFXLMS
LFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
500
1000
1500
2000
Number of Iteration
2500
Fig. 4.16 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(Jet Cockpit
Noise).
0
LFXLMS
FFXLMS
LFELMS
-5
MSE(dB)
-10
-15
-20
-25
-30
0
500
1000
1500
2000
Number of Iteration
2500
3000
Fig. 4.17 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(White Noise).
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CHAPTER-4
LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
LFXLMS
FFXLMS
0
-5
MSE(dB)
-10
-15
-20
-25
-30
1000
2000
3000
Number of Iteration
4000
5000
Fig. 4.18 MSE(dB) plot for LFXLMS and FFXLMS algorithm when primary path is
changed at 3000th iteration(Jet Cockpit Noise).
LFXLMS
FFXLMS
0
MSE(dB)
-5
-10
-15
-20
-25
-30
0
1000
2000
3000
Number of Iteration
4000
5000
Fig. 4.19 MSE(dB) plot for LFXLMS and FFXLMS algorithm when primary path is
changed at 3000th iteration(White Noise).
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
LFXLMS
FFXLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
1000
2000
3000
Number of Iteration
4000
5000
Fig. 4.20 MSE(dB) plot for LFXLMS and FFXLMS algorithm when secondary path is
changed at 3000th iteration(Jet Cockpit Noise).
LFXLMS
FFXLMS
0
-5
MSE(dB)
-10
-15
-20
-25
-30
0
1000
2000
3000
4000
Number of Iteration
5000
6000
Fig. 4.21 MSE(dB) plot for LFXLMS and FFXLMS algorithm when secondary path is
changed at 3000th iteration(White Noise).
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
-1
LFXLMS
FFXLMS
LFELMS
-2
-3
MSE(dB)
-4
-5
-6
-7
-8
-9
-10
500
1000
1500
2000
Number of Iteration
2500
Fig. 4.22 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(Jet Cockpit
Noise).
LFXLMS
FFXLMS
LFELMS
-1
-2
-3
MSE(dB)
-4
-5
-6
-7
-8
-9
-10
-11
500
1000
1500
2000
Number of Iteration
2500
Fig. 4.23 MSE(dB) plot for LFXLMS, FFXLMS and LFELMS algorithm(White Noise).
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LEGENDRE NEURAL NETWORK FOR NONLINEAR ANC
4.8 Summary
This chapter proposes a computationally efficient reduced structure Legendre
neural network for nonlinear active noise cancellation. Update algorithms Legendre
FXLMS (LFXLMS) and Legendre FELMS (LFELMS) for the proposed network are
derived. RLS algorithm is also employed to develop Legendre FXRLS (LFXRLS)
algorithm to obtain faster convergence but at the cost of increase in computational
complexity. A fast version of LFXLMS algorithm called fast LFXLMS algorithm is also
developed which reduces computational complexity by almost 25%. Extensive simulation
are conducted for various reference noises and the MSE(dB) plots are obtained.
Experiments are also conducted considering real time reference signals such as
Buccaneer jet cockpit noise and white noise. The experimental results presented here
demonstrates the superior performance of the proposed algorithms in terms of MSE(dB)
and computational complexity compared to standard FFXLMS algorithm.
114
CHAPTER 5
Frequency-Domain Approach to Multichannel
Nonlinear Active Noise Control
5.1 Background
Active noise control has been a field of growing interest over the past few
decades. Traditionally ANCs are realized by adaptive filtering in time domain. Adaptive
filtering in frequency domain is an attractive alternative to time domain adaptive filtering.
The theory for frequency domain adaptive filtering is well developed and literature is
thick. Frequency domain filters have primarily two advantages compared to time domain
implementation. The first advantage is the potentially large scale savings in the
computational complexity. The Fast Fourier Transform (FFT) is an efficient
implementation of the Discrete Fourier Transform (DFT) which provides this savings. A
second advantage is that DFT generate signals that are approximately uncorrelated
(orthogonal). As a result a time varying step size can be used for each adaptive weight,
thereby allowing faster convergence. Another secondary advantage of frequency domain
adaptation is more accurate estimation of gradient due to the averaging of samples in a
whole data block.
Several researchers have implemented the ANC in frequency domain using
different variations of the FXLMS algorithm. Q. Shen and A. S. Spanias [73], G. A.
Clark, S. K. Mitra, and S. R. Parkar [74] proposed block implementation of the FXLMS
algorithm, both in time and frequency domain, which is exact implementation of FXLMS
algorithm. Sen M. Kuo, Mansour Tahernezhadi and Li Ji [75], M. R. Asharif, T.
Takebayashi, T. Chugo and K. Murano[76], Reichard and Swanson[77], D. P. Das, G.
Panda and S. M. Kuo[78] proposed different methods for frequency domain
implementation of active noise cancellation.
In this chapter, a simple and computationally efficient frequency domain
algorithm for multichannel ANC is proposed. In addition, normalized LMS [3], [7]
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CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
algorithm is employed to facilitate variable step size. This new algorithm is termed as
multichannel frequency domain block filtered-x normalized LMS (FBFXNLMS)
algorithm. E. Wan [14] proposed adjoint LMS algorithm which led to the development
FELMS algorithm which is an efficient alternative to FXLMS algorithm with an
advantage of reduction in computational complexity for multiple input ANCs. A
frequency domain implementation of FELMS algorithm has been proposed and is termed
as frequency domain block filtered-e least mean square (FBFELMS) algorithm.
Legendre neural network for nonlinear ANC, developed in the previous chapter is also
implemented in frequency domain using Fast Fourier Transform (FFT). The developed
algorithms are termed as frequency domain block Legendre filtered-x least mean square
(FBLFXLMS)
and frequency domain block Legendre filtered-e least mean square
(FBLFELMS) algorithm.
5.2 The Filtered-x Least Mean Square (FXLMS) Algorithm
While developing the time domain block FXLMS algorithm, the FXLMS
algorithm is first outlined. The FXLMS is the most common algorithm applied in both
feedforward and feedback ANC due to its ease in implementation [2]. The basic ANC
system is shown in fig 5.1 where the path from the noise source to the cancellation point
is defined as primary path and it has a transfer function, A(z ) . ANC also has secondary
x(n)
d (n)
A(z )
+
Σ
_
W (z )
y (n)
B (z )
dˆ (n)
Bˆ ( z )
x' ( n)
FXLMS
Algorithm
Fig. 5.1 Block diagram of the basic active noise control system.
116
e(n)
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
path which is defined as the path leading from the adaptive filter output to error sensor
that measures the residual noise and it has a transfer function, B (z ) . Most available ANC
algorithms including FXLMS, require online or offline identification of secondary path.
In an ANC system the secondary path transfer function, B (z ) follows the
adaptive filter. Therefore, to ensure convergence, the conventional LMS algorithm is to
be suitably modified. The most appropriate modification is by placing an estimate of this
secondary path transfer function, Bˆ ( z ) in the reference signal path to the weight update
of the LMS algorithm. Hence the algorithm is referred to as filtered-x LMS (FXLMS)
algorithm. Referring to fig. 5.1, the residual noise signal at nth time instant is expressed as
e(n) = d (n) − dˆ (n)
(5.1)
= d (n) − y (n) * bˆ (n)
(5.2)
= d (n) − w (n) * x(n) * bˆ (n)
(5.3)
where x(n) = [ x(n) x(n − 1) . . . . x(n − N + 1).
d (n) = noise to be cancelled at the canceling point,
dˆ (n) = output of the ANC,
b(n) = the impulse response of the secondary path transfer function,
bˆ (n) = estimate of the impulse response of the secondary path transfer function,
and * denotes linear convolution operation.
w (n) = estimate of the weight vector at nth instant,
= [ w0 (n) w1 (n) .... wN −1 (n)]T
The weight update equation in FXLMS algorithm is given by
w ( n + 1) = w ( n) + µ c( n)
(5.4)
where µ= convergence coefficient.
c( n ) = x ' ( n ) * e( n )
(5.5)
x ' ( n ) = x ( n) * b ( n)
(5.6)
The FXLMS algorithm is thus described by (5.1) through (5.6), which involves three
convolution operations to compute x′(n), y (n), and dˆ (n) .
117
Out of these three linear
CHAPTER-5
convolution
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
operations,
two
are
performed
in
actual
implementation
to
evaluate x ′(n), and y (n) . But in case of ANC system the third convolution is actually not
computed as the signal dˆ (n) corresponds to output of the speaker and is used as an
antinoise signal. In the next section the time-domain block filtered-x least mean square
(BFXLMS) algorithm is derived in detail.
5.3 The Time Domain Block Filtered-x LMS (BFXLMS)
Algorithm
In block filtering instead of computing the ANC output sample by sample a block
of output is computed simultaneously and this is possible by using overlap save or
overlap add method. Out of these two methods the overlap-save method has been chosen
as it is computationally more efficient. Now using overlap-save method the FXLMS
algorithm is implemented by redefining (5.5) as follows
c( n ) = x ' ( n ) * e( n )
where e(n) = [e(n − N + 1) . . . . e(n − 1) e(n)]
Computation of
c(n) actually involve cross correlation, but writing e(n) in time
increasing order the cross correlation operation is converted to convolution operation. In
block FXLMS the computation of the three linear convolution operations (for
obtaining x ′(n), y (n), and c(n) ) can be implemented in a simple manner as:
  x( n )   I N 

y (n + N ) = [ O N I N ]
*   w ( n) 

x(n + N ) O N 

(5.7)
  x ( n)   I N  ˆ 
x' (n + N ) = [ O N I N ]
 *   b( n )
x(n + N ) O N 

(5.8)
  x' ( n)   I N 

c (n + N ) = [ O N I N ]
*  e( n + N ) 

x' (n + N ) O N 

(5.9)
where the matrix I N is an N × N identity matrix and the matrix O N is an N × N matrix
with all zero elements and x(n + N ) = [ x(n + N ) x(n + N − 1) . . . . x(n + 1)].
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5.4 Frequency Domain Block Filtered-x LMS (FBFXLMS)
Algorithm
Fast Fourier Transform can be used to efficiently compute the associated linear
convolutions in BFXLMS algorithm. It is shown in the previous section that the
BFXLMS algorithm essentially consists of three linear convolutions defined in (5.7) to
(5.9). FFT based implementation of all the three linear convolutions can be done by
defining F2 N and F2−N1 as the 2N-point FFT and IFFT operators respectively. The linear
convolution in (5.7) may be implemented using the 2N point FFT and IFFT as
 x ( n) 
X(n + N ) = F2 N 

x( n + N ) 
(5.10)
I 
W(n) = F2 N  N  w (n)
O N 
(5.11)
y (n + N ) = [O N
(5.12)
I N ]F2−N1 [X(n + N ) ⊗ W (n)]
where ⊗ denotes point-by-point multiplication. Similarly, (5.8) may be implemented
using the 2N point FFT and IFFT as
I 
B(n) = F2 N  N b(n)
O N 
(5.13)
x' (n + N ) = [O N
(5.14)
I N ]F2−N1 [X(n + N ) ⊗ B(n)]
The FFT-based implementation of (5.9) can be written as
 x' ( n) 
X′(n + N ) = F2 N 

 x' ( n + N ) 
(5.15)
I 
E(n + N ) = F2 N  N e(n + N )
O N 
(5.16)
c(n + N ) = [O N
(5.17)
TN ]F2−N1 [X' (n + N ) ⊗ E (n + N )]
The weight update equation of FBFXLMS algorithm becomes
w (n + 1) = w (n) + µ c(n + N )
(5.18)
Here weights are updated in time domain where as all the linear convolution operations
required for executing the algorithm is done in frequency domain.
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CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5.5 Frequency Domain Block Filtered-x Normalized LMS
(FBFXNLMS) Algorithm
The conventional linear adaptive filters based on the normalized LMS (NLMS)
algorithm obtain rapid convergence for highly correlated signals, by improving a large
eigenvalue spread. With similar reasoning, the proposed approach relies upon the NLMS
algorithm in order to solve the slow convergence problem. Accordingly FXLMS
algorithm is modified to yield filtered-x normalized least mean square (FXNLMS)
algorithm. The weight update equation for FXNLMS algorithm is given by
w (n + 1) = w (n) +
µ
T
x( n ) x( n ) + ε
x ' ( n ) e( n )
(5.19)
where ε is a small constant. Incorporating block implementation and subsequently
employing FFTs and IFFTs to reduce computational load the weight update equation
(5.19) becomes
w ( n + N ) = w ( n) +
µ
T
x( n + N ) x( n + N ) + ε
c( n + N )
(5.20)
This equation represents the frequency domain block filtered-x normalized LMS
(FBFXNLMS) algorithm.
5.5.1 The Reduced Structure FBFXNLMS algorithm
Further reduction in computational complexity can be achieved by carefully
observing the block diagram of fig.5.2 and discarding some steps. To complete the
weight update process the reference signal is multiplied with the estimated secondary
path in frequency domain. After multiplication the resulting signal undergoes two
transformations; first from frequency domain to time domain and then again back to
frequency domain. The steps followed are described as follows. (i) From frequency
domain to time domain by a 2N point IFFT (ii) The first N samples are deleted (iii)
Overlapping with previous N samples is done (iv) The signal is again transformed to
frequency domain by a 2N point FFT. Thus process is to transform a frequency domain
signal to time domain and again transforming the time domain signal to frequency
domain. If these two transformations are removed, the above mentioned four steps
become redundant but still the performance is preserved. So the resulting signal after
reference signal is multiplied with the estimated secondary path in frequency domain is
120
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
directly used for weight update. By removing these steps, an approximate version is
obtained [78]. In this way one FFT and one IFFT computations can be saved. The
complete algorithm is shown in table 5.1. The block diagram of ANC using the
FBFXLMS algorithm is shown in fig. 5.3.
x(n)
d (n)
P(z )
+
∑
−
dˆ (n)
B(z )
I
F
F
T
F
F
T
Two
block
S/P
 x ( n) 
x( n + N )


[
O
N
I
N
]
P/S
y (n)
FFT
Bˆ ( z )
IN 
O N 
IN 
O N 
F
F
T
w (n)
IFFT
[
O
N
I
N
]
Memory
P/S
w (n + N )
∑
x' ( n)
 x ' ( n) 
x' ( n + N ) 


µ
S/P
Two
block
T
x( n + N ) x(n + N ) + ε
c( n + N )
FFT
[
O
N
I
N
]
IFFT
E (n + N )
X' ( n + N )
F
F
T
 I N  S/P
O N 
Fig.5.2 Block diagram of ANC using FBFXNLMS algorithm.
121
e(n)
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
x(n)
d (n)
P(z )
+
∑
e(n)
−
ˆ
d ( n)
B(z )
S/P
Two
block
I
F
F
T
F
F
T
 x ( n) 
x( n + N )


[
O
N
I
N
]
P/S
y (n)
FFT
Bˆ ( z )
IN 
O N 
IN 
O N 
F
F
T
w (n)
Memory
w (n + N )
∑
µ
T
x( n + N ) x(n + N ) + ε
c( n + N )
[
O
N
I
N
]
IFFT
E (n + N )
X' ( n + N )
F
F
T
 I N  S/P
O N 
Fig.5.3 Block diagram of ANC using reduced structure FBFXNLMS algorithm.
122
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5.6 Frequency Domain Block Filtered-e LMS (FBFELMS)
Algorithm
Filtered-e least mean square (FELMS) algorithm is widely used as an alternative
to FXLMS algorithm in multiple input multiple output (MIMO) ANC which reduces
computational complexity and memory requirements. Adjoint LMS algorithm for ANC
was developed by E. A. Wan [14] which is a FELMS algorithm where error signal rather
than input signal is filtered through adjoint of secondary path filter. Incorporating block
processing and using FFT and IFFT, FELMS algorithm can be implemented in frequency
domain as follows.
 x( n ) 
X(n + N ) = F2 N 

 x( n + N ) 
(5.21)
I 
W(n) = F2 N  N  w (n)
O N 
(5.22)
y (n + N ) = [O N
(5.23)
I N ]F2−N1 [X(n + N ) ⊗ W (n)]
where ⊗ denotes point-by-point multiplication. Similarly filtering of error block through
estimated secondary path may be implemented using the 2N point FFT as follows
I 
B(n) = F2 N  N b r (n)
O N 
(5.24)
I 
E(n + N ) = F2 N  N e(n + N )
O N 
(5.25)
E' (n + N ) = [E(n + N ) ⊗ B(n)]
(5.26)
where b r (n) is obtained by flipping b(n) .
The FFT-based implementation of weight adaptation can be written as
c(n + N ) = [O N
TN ]F2−N1 [X(n + N ) ⊗ E' (n + N )]
w (n + N ) = w (n) + µc(n + N )
(5.27)
(5.28)
The complete algorithm is shown in table 5.2.
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CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
x(n)
d (n)
P(z )
+
e(n)
∑
−
ˆ
d ( n)
B(z )
S/P
Two
block
I
F
F
T
F
F
T
[
O
N
I
N
 x ( n) 
x( n + N )


]
P/S
y (n)
S/P
IN 
O N 
FFT
IN 
O N 
FFT
w (n)
Memory
w (n + N )
∑
F
F
T
IN 
O N 
µ
c( n + N )
[
O
N
I
N
]
IFFT
X( n + N )
E′(n + N )
Fig.5.4 Block diagram of ANC using FBFELMS algorithm.
124
Bˆ ( z )
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
Table 5.1
FBFXNLMS algorithm
Controller Output:
 x ( n) 
X(n + N ) = F2 N 

x( n + N ) 
I 
W(n) = F2 N  N  w (n)
O N 
y (n + N ) = [O N
I N ]F2−N1 [X(n + N ) ⊗ W (n)]
Filtering through secondary path:
I 
B( n) = F2 N  N b( n)
O N 
x' (n + N ) = [O N
I N ]F2−N1 [X(n + N ) ⊗ B(n)]
Weight update:
 x' ( n) 
X′(n + N ) = F2 N 

 x' ( n + N ) 
I 
E(n + N ) = F2 N  N e(n + N )
O N 
c(n + N ) = [O N
w ( n + N ) = w ( n) +
TN ]F2−N1 [X' (n + N ) ⊗ E (n + N )]
µ
T
x( n + N ) x( n + N ) + ε
125
c( n + N )
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
Table 5.2
FBFELMS algorithm
Controller Output:
 x ( n) 
X(n + N ) = F2 N 

x( n + N ) 
I 
W(n) = F2 N  N  w (n)
O N 
y (n + N ) = [O N
I N ]F2−N1 [X(n + N ) ⊗ W (n)]
Filtering through secondary path:
I 
B ( n) = F2 N  N  b r ( n)
O N 
I 
E(n + N ) = F2 N  N e(n + N )
O N 
E' (n + N ) = [E(n + N ) ⊗ B(n)]
Weight update:
c(n + N ) = [O N
TN ]F2−N1 [X(n + N ) ⊗ E' (n + N )]
w ( n + N ) = w ( n ) + µ c( n + N )
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CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5.7 Multichannel FBFXNLMS Algorithm
Noise control in a large dimension duct or enclosure requires a multiple channel
ANC systems. Multiple channel ANC engage several secondary loudspeakers to control
noise at multiple error microphone locations. It is assumed that in a multiple channel
ANC, L number of reference microphones, P number of secondary loudspeakers and Q
numbers of error microphones are employed. So in total LP numbers of adaptive filters
are present and their impulse responses are represented as w lp (n) and PQ number of
secondary paths is represented as b pq (n) . Applying multiple error LMS algorithm,
proposed by Elliott [1], [2], multiple channel ANC problem can be solved by applying
FBFXNLMS to all possible single channel paths in the multiple channel system. The
weight update equation can be written as
w lp (n + 1) = w lp (n) +
µ
T
x l (n + N ) x l (n + N ) + ε
Q
∑c
lq
( n)
(5.29)
q =1
For 1<l<L and 1<p<P and
c lq (n + N ) = [O N
TN ]F2−N1 [ X′l (n + N ) ⊗ E q (n + N )]
(5.30)
X′l (n + N ) = X l (n + N ) ⊗ B pq
(5.31)
I 
B pq = F2 N  N b pq
O N 
(5.32)
I 
E q (n + N ) = F2 N  N e q (n + N )
O N 
(5.33)
where b pq is the impulse response of the secondary path extending from pth loudspeaker
and qth error microphone.
5.7.1 Computational Complexity
In case of FXNLMS algorithm to obtain N samples of controller outputs, LPN2
multiplications and LPN(N-1) additions are required. For filtering N samples of reference
signal through the secondary path of length N, LPQN2 multiplications and LPQN(N-1)
additions are required. For weight update, LP(Q+1)N2 multiplications and LP(Q+1)N2
additions are required. Therefore the total number of multiplications required is
127
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
2LP(Q+1)N2 and the total additions required is NLP(Q+1)(2N-1). For FBFXNLMS
algorithm, single channel ANC using overlap save method, the N-point FBFXNLMS
algorithm involves the computation of (i) six 2N-point FFTs, (ii) three 2N point complex
multiplications and (iii) N number of weight updates. For real-valued input data, total
number of real multiplications is 12Nlog2N+24N and the real additions is
24Nlog2(N)+13N.
In case of multichannel ANC the number of 2N point FFT/IFFT required for (i)
input signal transform is L, (ii) adaptive filter output signal transform is P, (iii) adaptive
filter transform is LP, (iv) secondary path transfer function transform is PQ, (v) error
signal transform is Q, (vi) transform of product of filtered input signal and error is LP. So
total number of FFT is L+P+2LP+PQ+Q. Each FFT requires 2Nlog2(N) real
multiplications and 4Nlog2(N) real additions . Also LP,LPQ, LPQ number of 2N point
frequency domain complex multiplications are required for computing adaptive filter
output, filtered input signal, weight update respectively. Each 2N point complex
multiplication involves 8N real multiplications and 4N real additions. Also the number of
real additions required for weight update is LPN+2LPQN. So total real multiplications
required is (L+P+2LP+PQ+Q)2Nlog2(N) +(LP+2LPQ)8N and real additions required is
(L+P+2LP+PQ+Q)4Nlog2(N)+(LP+2LPQ)4N+LPN
+2
LPQN.
Computational
complexity for multichannel FXNLMS and multichannel FBFXNLMS algorithms with
L=2, P=2, Q=2 are tabulated in table 5.3 and plotted in fig. 5.5.
Table 5.3.
Computational Complexity per sample
N
Number of multiplication
FXNLMS
FBFXNLMS
Number of addition
FXNLMS
FBFXNLMS
32
768
340
756
532
64
1536
376
1524
604
128
3072
412
3060
676
256
6144
448
6132
748
512
12288
484
12276
820
1024
24576
520
24564
892
128
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
14000
FXNLMS
FBFXNLMS
12000
12000
10000
10000
Number of Additions
N u m b e r o f M u lt ip lic a t io n s
14000
8000
6000
8000
6000
4000
4000
2000
2000
0
0
100
200
300
N
400
500
FXNLMS
FBFXNLMS
0
600
(a)
0
100
200
300
N
400
500
600
(b)
Fig.5.5 Comparison of Computational Complexity (a) Multiplications (b) Additions.
From table 5.3 and fig 5.5 saving in computational requirement can be observed.
5.8 Frequency Domain Implementation of Legendre Neural
Network for Nonlinear ANC
The Block diagram of the nonlinear ANC using Legendre neural network (LNN)
as the controller, developed in the previous chapter, is shown in fig.5.6. Here the
reference noise vector x(n) = [ x(n) x(n − 1) .....x(n − N + 1)] is transformed into an output
vector s(n) given by s(n) = L(x(n)) . The nonlinear function L(x(n)) represents a set of
the orthogonal basis functions, implemented in the ‘‘Legendre expansion’’ block. Here
the N-dimensional input pattern x(n) is enhanced to an NP-dimensional enhanced pattern
s(n) given by
s(n) = [s 0 (n) s 1 (n) . . . s P −1 ] .
(5.34)
= [ L0 (x(n)) L1 (x(n)) .... LP −1 (x(n))]
where
s 0 (n) = L0 (x(n)) = [111.... N number of 1]
s1 (n) = L1 (x(n)) = [ x(n) x(n − 1) . . . x(n − N + 1)]
1
1
1
s 2 (n) = L2 (x(n)) = [ (3 x(n) 2 − 1)
(3 x(n − 1) 2 − 1) . . . (3 x(n − N + 1) 2 − 1)]
2
2
2
........
129
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
x(n)
d (n)
A(z )
+
Σ
e(n)
dˆ (n)
LEGENDRE EXPANSION
Memory
s1 ( n )
w 1 ( n)
−
y (n)
+
Σ
B(z )
+
s 2 ( n)
w 2 (n)
+
Σ
+
M
M
s P (n)
M
w P (n)
L
s ′p (n)
Bˆ ( z )
M
Update
Algorithm
Fig.5.6 Legendre Neural Network for Nonlinear ANC.
For reduced structure LNN the s 0 (n) is discarded. Employing filter bank implementation
output of LNN at time n is
P
P
i =1
i =1
y ( n) = ∑ y i ( n ) = ∑ s i ( n) w i ( n )
(5.35)
where w i (n) is the weight vector of ith adaptive filter at time n. Estimated desired
signal dˆ ( n) is obtained by filtering LNN output through secondary path B(z ) . Error at
time n is defined as e(n) = d (n) − dˆ (n) . Considering a cost function of ξ (n) =
1
E[e 2 (n)]
2
and using the FXLMS algorithm the all the weight vectors can be updated separately as
w i (n + 1) = w i (n) + µe(n)s i ' (n)
for i = 1,2, ... , P
(5.36)
where s i ' (n) is the input signal s i (n) filtered through the estimated secondary path and µ
is the step size which control convergence and stability.
130
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
Exploiting the properties of FFT the above algorithm can be implemented in
frequency domain with block filtering which reduces the computational burden of the
algorithm. In this case all the convolutions and correlations are done in frequency
domain. This algorithm is termed as frequency domain block Legendre FXLMS
(FBLFXLMS) algorithm. Defining F2 N and F2−N1 as the 2N-point FFT and IFFT the linear
convolutions are implemented as follows
 s ( n) 
S i (n + N ) = F2 N  i

s i (n + N )
(5.37)
I 
Wi (n) = F2 N  N  w i (n)
O N 
(5.38)
y i (n + N ) = [O N
(5.39)
I N ]F2−N1 [S i (n + N ) ⊗ Wi (n)]
where ⊗ denotes frequency-by-frequency bin multiply or in more general way point-bypoint multiplication of two vectors. The I N is an N × N identity matrix, the matrix O N is
an N × N matrix with all zero elements. Similarly filtering of reference signal through
the estimated secondary path can be done in frequency domain as follows
I 
B(n) = F2 N  N b(n)
O N 
(5.40)
s i ' (n + N ) = [O N
(5.41)
I N ]F2−N1 [S i (n + N ) ⊗ B(n)]
Calculation of ci (n + N ) , which are required for weight update are done in the following
steps.
 s ' (n) 
S 'i (n + N ) = F2 N  i

s i ' (n + N )
(5.42)
I 
E(n + N ) = F2 N  N e(n + N )
O N 
(5.43)
c i (n + N ) = [O N
[
I N ]F2−N1 S i' (n + N ) ⊗ E(n + N )
]
(5.44)
The weight vectors are updated as follows
w i ( n + N ) = w i ( n) + µ c i ( n + N )
(5.45)
131
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
The block diagram of ANC using Legendre neural network in frequency domain and
updated by FBLFXLMS algorithm is shown in the fig. 5.7 and fig. 5.8. The complete
algorithm is briefed in table 5.4
In the reduced structure FBFXNLMS algorithm two FFTs are saved by removing
two FFTs in the calculation of X' (n + N ) . Here also two FFTs can be saved by removing
two FFTs in the calculation of S 'i (n + N ) . For Pth order Legendre expansion in total 2P
number of FFTs can be saved.
5.9 Frequency Domain Block Legendre FELMS (FBLFELMS)
Algorithm.
In an ANC using Legendre neural network each of the expanded reference signal
vectors s i (n) (i=1,2,…, P)is to be filtered through the estimated secondary path. Number
of filtering can be reduced by applying FELMS algorithm where only error signal is to be
filtered through the estimated secondary path. Taking advantage of this fact
implementation of ANC using Legendre neural network has been tried in frequency
domain by employing FELMS algorithm. The resulting algorithm is named as frequency
domain block Legendre FELMS algorithm (FBLFELMS) algorithm.
Referring to the previous section the steps of FBLFELMS algorithm is described
as follows. Defining F2 N and F2−N1 as the 2N-point FFT and IFFT, the linear convolutions
are implemented as given below
 s ( n) 
S i (n + N ) = F2 N  i

s i (n + N )
(5.46)
I 
Wi (n) = F2 N  N  w i (n)
O N 
(5.47)
y i (n + N ) = [O N
(5.48)
I N ]F2−N1 [S i (n + N ) ⊗ Wi (n)]
where ⊗ denotes frequency-by-frequency bin multiply or in more general way point-bypoint multiplication of two vectors. The I N is an N × N identity matrix, the matrix O N is
an N × N matrix with all zero elements. Similarly another convolution can be done as
132
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
I 
B(n) = F2 N  N b r (n)
O N 
(5.49)
I 
E(n + N ) = F2 N  N e(n + N )
O N 
(5.50)
E′(n + N ) = [E(n + N ) ⊗ B(n)]
(5.51)
and cross correlation can be done as
 s ' (n) 
S 'i (n + N ) = F2 N  i

s i ' (n + N )
c i (n + N ) = [O N
(5.52)
[
I N ]F2−N1 S i' (n + N ) ⊗ E′(n + N )
]
(5.53)
So the weights are updated as follows
w i ( n + N ) = w i ( n) + 2 µ c i ( n + N )
(5.54)
The block diagram of ANC using Legendre neural network in frequency domain and
updated by FBLFELMS algorithm is shown in the fig. 5.7 and fig. 5.9. The complete
algorithm is briefed in table 5.5.
x(n)
d (n)
A(z )
+
Σ
e(n)
dˆ (n)
LEGENDRE EXPANSION
Memory
s 1 ( n)
−
y (n)
y1 (n)
Σ
Ts
B(z )
y 2 (n)
s 2 ( n)
Σ
Ts
M
M
y P (n)
s P (n)
Ts
Fig.5.7 Frequency domain implementation of Legendre Neural Network for ANC.
133
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
s i (n)
S/P
Two
block
I
F
F
T
F
F
T
[
O
N
I
N
y i (n)
]
P/S
F
F
T
 I N  S/P
O N 
 s i ( n) 
s (n + N )
 i

FFT
IN 
O N 
Bˆ ( z )
IN 
O N 
F
F
T
w i (n)
Memory
w i (n + N )
∑
µ
c i (n + N )
[
O
N
I
N
]
IFFT
E (n + N )
S i ' (n + N )
Fig.5.8 Details of Ts block of fig. 5.7 (for FBLFXLMS Algorithm).
134
e(n)
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
s i (n)
S/P
Two
block
I
F
F
T
F
F
T
[
O
N
I
N
]
P/S
y i (n)
 s i ( n) 
s (n + N )
 i

FFT
IN 
O N 
w i (n)
Memory
w i (n + N )
∑
µ
c i (n + N )
[
O
N
I
N
IFFT
]
IN 
O N 
Bˆ ( z )
FFT
E′(n + N )
S i (n + N )
F
F
T
 I N  S/P
O N 
Fig.5.9 Details of Ts block of fig. 5.7 (for FBLFELMS Algorithm).
135
er (n)
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
Table 5.4
FBLFXLMS algorithm
Controller Output:
 s ( n) 
S i ( n + N ) = F2 N  i

s i ( n + N ) 
I 
Wi (n) = F2 N  N  w i (n)
O N 
y i (n + N ) = [O N
I N ]F2−N1 [S i (n + N ) ⊗ Wi (n)]
Filtering through secondary path:
I 
B(n) = F2 N  N b(n)
O N 
s i ' (n + N ) = [O N
I N ]F2−N1 [S i (n + N ) ⊗ B(n)]
Weight update:
 s ' (n) 
S 'i (n + N ) = F2 N  i

s i ' (n + N )
I 
E(n + N ) = F2 N  N e(n + N )
O N 
c i (n + N ) = [O N
[
I N ]F2−N1 S i' (n + N ) ⊗ E(n + N )
w i ( n + N ) = w i ( n) + µ c i ( n + N )
136
]
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
Table 5.5
FBLFELMS algorithm
Controller Output:
 s ( n) 
S i (n + N ) = F2 N  i

s i (n + N )
I 
Wi (n) = F2 N  N  w i (n)
O N 
y i (n + N ) = [O N
I N ]F2−N1 [S i (n + N ) ⊗ Wi (n)]
Filtering through secondary path:
I 
B(n) = F2 N  N b r (n)
O N 
I 
E(n + N ) = F2 N  N e(n + N )
O N 
E′(n + N ) = [E(n + N ) ⊗ B(n)]
Weight update:
 s ' (n) 
S 'i (n + N ) = F2 N  i

s i ' (n + N )
c i (n + N ) = [O N
[
I N ]F2−N1 S i' (n + N ) ⊗ E′(n + N )
w i ( n + N ) = w i ( n) + 2 µ c i ( n + N )
137
]
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5.10 Simulation and Results
To analyze the performance of the proposed frequency domain multichannel ANC
algorithms extensive simulation experiments studies have been carried out. Results of
some of the experiments are shown here. In all the experiments mean square error (MSE)
in dB defined by
MSE(dB) = 10 log 10 ( E (e 2 (n)))
(5.55)
is obtained through simulation. MSE in dB plotted is average of twenty independent runs
of the experiments to get a smoother curve.
5.10.1 Experiment I
In this experiment, a multichannel ANC with one reference microphone, two
loudspeakers and two error microphones are considered (1 × 2 × 2) . Memory size N is
chosen to be 10. In this experiment the two linear primary path transfer functions
considered are described by [62]
A11 ( z ) = z −5 − 0.3z −6 + 0.2 z −7
A11 ( z ) = z −5 − 0.4 z −6 + 0.1z −7
and the four secondary path transfer functions considered are minimum-phase FIR
models described below
B11 ( z ) = z −2 + 0.5 z −3
B12 ( z ) = z −2 + 0.6 z −3
B21 ( z ) = 0.9 z −2 + 0.4 z −3
B22 ( z ) = 0.9 z −2 + 0.3z −3
Reference noise is taken as white noise. MSE(dB) is obtained for the proposed
frequency domain multichannel FBFXNLMS algorithm taking step size µ=0.09. For
comparison with corresponding time domain algorithm, MSE(dB) for FXNLMS
algorithm with µ=0.05 is also obtained. Simulation results are plotted in fig.5.10 and
fig.5.11 respectively. Steady state MSE(dB) achieved by FBFXNLMS algorithm and
FXNLMS algorithm is found to be same -27dB. From the results, it is evident that the
proposed algorithm offers identical performance as the time domain FXNLMS algorithm
138
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
for multichannel ANC but the real advantage of the proposed algorithm is large saving in
computational complexity.
-5
FBFXNLMS
-10
MSE(dB)
-15
-20
-25
-30
-35
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.10 MSE(dB) plot of FBFXNLMS algorithm for multichannel ANC.
-5
FXNLMS
-10
MSE(dB)
-15
-20
-25
-30
-35
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.11 MSE(dB) plot of FXNLMS algorithm for multichannel ANC.
139
CHAPTER-5
FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5
FBFXNLMS
0
MSE(dB)
-5
-10
-15
-20
-25
-30
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.12 MSE(dB) plot of FBFXNLMS algorithm for multichannel ANC.
0
FXNLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.13 MSE(dB) plot of FXNLMS algorithm for multichannel ANC.
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5.10.2 Experiment II
In this experiment, experiment I is repeated considering logistic chaotic noise as
the reference noise. The logistic chaotic noise is generated using the recursive equation
[62], [64]
x ( n + 1) = λ x ( n) [1 − x( n)]
(5.56)
where x (0) = 0.9 , and λ =4.
This nonlinear noise process is then normalized to have unity signal power. MSE(dB) is
obtained for the proposed frequency domain multichannel FBFXNLMS algorithm and
time domain FXNLMS algorithm. The step sizes used for FBFXNLMS and FXNLMS
algorithm are µ=0.09 and µ=0.07 respectively. Simulation results for FBFXNLMS and
FXNLMS algorithm are plotted in fig.5.12 and fig.5.13 respectively. Steady state
MSE(dB) achieved by FBFXNLMS algorithm and FXNLMS algorithm is found to be
same -21dB. From the results, it is evident that the proposed algorithm offers identical
performance as the time domain FXNLMS algorithm for multichannel ANC.
5.10.3 Experiment III
In this experiment, a nonlinear multichannel ANC is considered. The ANC has
one reference microphone, two secondary loudspeakers and four error microphones
(1 × 2 × 4) . The 2nd order Legendre neural network is employed for nonlinear ANC and is
updated by the proposed frequency domain block Legendre FELMS (FBLFELMS)
algorithm. Memory size N is chosen to be 10. The reference noise is taken as logistic
chaotic noise defined in experiment II. This nonlinear noise process is then normalized to
have unity signal power. In the experiment four linear primary path transfer functions
considered are described below [62]
A11 ( z ) = [ z −5 − 0.3z −6 + 0.2 z −7 ]
A12 ( z ) = [ z −5 − 0.2 z −6 + 0.1z −7 ]
A13 ( z ) = [ z −5 − 0.3z −6 + 0.1z −7 ]
A14 ( z ) = [ z −5 − 0.2 z −6 + 0.2 z −7 ]
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
The eight secondary path transfer functions are non-minimum-phase FIR models and are
described by the following equations.
B11 ( z ) = [ z −2 + 1.5 z −3 − z −4 ]
B21 ( z ) = [ z −2 + 1.7 z −3 − z −4 ]
B31 ( z ) = [ z −2 + 1.8 z −3 − z −4 ]
B41 ( z ) = [ z −2 + 1.9 z −3 − z −4 ]
B12 ( z ) = [ z −2 + 1.5 z −3 − z −4 ]
B22 ( z ) = [ z −2 + 1.2 z −3 − z −4 ]
B32 ( z ) = [ z −2 + 1.1z −3 − z −4 ]
B42 ( z ) = [ z −2 + 1.0 z −3 − z −4 ]
Step size for FBLFELMS algorithm taken is µ=0.0009. MSE(dB) obtained at
each of the four error microphones and the overall MSE(dB) that combine results of the
four error microphones are plotted in fig.5.14 , fig.5.15 , fig.5.16 , fig. 5.17, fig.5.18
respectively. For comparison with corresponding time domain algorithm combined
MSE(dB) using time domain LFELMS algorithm is plotted in fig. 5.19. The step size and
memory length for LFELMS algorithm are 0.0005 and 10 respectively. Steady state
MSE(dB) achieved at all the error microphones is found to be same -25dB. From fig.
5.17 and fig.5.18 it is observed that overall MSE(dB) of FBLFELMS algorithm and
LFELMS algorithm is -25dB. From the results, it is evident that the proposed algorithm
offers identical performance as the time domain LFELMS algorithm for multichannel
ANC but the real advantage of the proposed algorithm is large saving in computational
complexity.
142
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
10
Error Microphone-1
5
0
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
-40
0
1000
2000
3000
4000
Number of Iteration
5000
6000
Fig.5.14 MSE(dB) plot of FBLFELMS algorithm at error microphone-1.
10
Error Microphone-2
5
0
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
-40
0
1000
2000
3000
4000
Number of Iteration
5000
6000
Fig.5.15 MSE(dB) plot of FBLFELMS algorithm at error microphone-2.
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
10
Error Microphone-3
5
0
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
-40
0
1000
2000
3000
4000
Number of Iteration
5000
6000
Fig.5.16 MSE(dB) plot of FBLFELMS algorithm at error microphone-3.
10
Error Microphone-4
5
0
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
-40
0
1000
2000
3000
4000
Number of Iteration
5000
6000
Fig.5.17 MSE(dB) plot of FBLFELMS algorithm at error microphone-4.
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
0
FBLFENLMS
-5
MSE(dB)
-10
-15
-20
-25
-30
0
1000
2000
3000
4000
Number of Iterations
5000
6000
Fig.5.18 MSE(dB) plot of FBLFELMS algorithm of all the error microphones
(combined).
10
LFELMS
5
0
-5
MSE(dB)
-10
-15
-20
-25
-30
-35
-40
0
1000
2000
3000
4000
Number of Iteration
5000
6000
Fig.5.19 MSE(dB) plot of LFELMS algorithm of all the error microphones (combined).
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
5.10.4 Experiment on Real Time Signal
In an attempt to assess the performance of the proposed algorithms on signals
picked up from real life environments, experiments are conducted once again. The
reference signals considered are buccaneer jet cockpit noise, factory floor noise and white
noise [79].
5.10.5 Experiment IV (Real Time Signal)
This experiment is conducted on three reference noise signals collected from real
life environment. The setup is same as experiment I and the signals used are buccaneer jet
cockpit noise, factory floor noise and white noise. The MSE(dB) plots of FXNLMS
algorithm and FBFXNLMS algorithm for buccaneer jet cockpit noise, factory floor noise
and white noise are shown in fig. 5.20 - fig. 5.25. The step sizes used for FBFXNLMS
and FXNLMS algorithm for white noise are 0.09 and 0.07 respectively. Similarly step
sizes for factory floor noise and Buccaneer jet cockpit noise are 0.06, 0.05 and 0.06, 0.05
respectively. From the results, it is evident that the proposed algorithm offers identical
performance as the time domain FXNLMS algorithm for multichannel ANC.
5.10.6 Experiment V (Real Time Signal)
Experiment-III is repeated here considering the Buccaneer jet cockpit noise as
reference signal. MSE(dB) for FBLFELMS algorithm obtained at each of the four error
microphones and the overall MSE(dB) that combine results of the four error microphones
are plotted in fig.5.26 - fig.5.30 respectively. Step size for FBLFELMS algorithm taken is
µ=0.0009. From the results, it is evident that the proposed algorithm offers good
performance as for multichannel ANC but the real advantage of the proposed algorithm is
large saving in computational complexity.
146
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
0
FXNLMS
-10
MSE(dB)
-20
-30
-40
-50
-60
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.20 MSE(dB) plot of FXNLMS algorithm for white noise.
0
FBFXNLMS
-10
MSE(dB)
-20
-30
-40
-50
-60
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.21 MSE(dB) plot of FBFXNLMS algorithm for white noise.
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
0
FXNLMS
-10
MSE(dB)
-20
-30
-40
-50
-60
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.22 MSE(dB) plot of FXNLMS algorithm for factory floor noise.
0
FBFXNLMS
-10
MSE(dB)
-20
-30
-40
-50
-60
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.23 MSE(dB) plot of FBFXNLMS algorithm for factory floor noise.
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
0
FXNLMS
-10
MSE(dB)
-20
-30
-40
-50
-60
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.24 MSE(dB) plot of FXNLMS algorithm for Buccaneer jet cockpit noise.
0
FBFXNLMS
-10
MSE(dB)
-20
-30
-40
-50
-60
0
500
1000
1500
2000
2500
Number of Iterations
3000
3500
4000
Fig. 5.25 MSE(dB) plot of FBFXNLMS algorithm for Buccaneer jet cockpit noise.
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0
Error Microphone-1
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
0
2000
4000
6000
8000
Number of Iteration
10000
12000
Fig.5.26 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise at error
microphone-1.
0
Error Microphone-2
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
0
2000
4000
6000
8000
Number of Iteration
10000
12000
Fig.5.27 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise at error
microphone-2.
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
0
Error Microphone-3
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
0
2000
4000
6000
8000
Number of Iteration
10000
12000
Fig.5.28 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise at error
microphone-3.
0
Error Microphone-4
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
0
2000
4000
6000
8000
Number of Iteration
10000
12000
Fig.5.29 MSE(dB) plot of FBLFELMS algorithm for Buccaneer cockpit noise at error
microphone-4.
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
0
FBLFENLMS
-5
-10
MSE(dB)
-15
-20
-25
-30
-35
-40
0
2000
4000
6000
8000
Number of Iterations
10000
12000
Fig.5.30 MSE(dB) plot of FBLFELMS algorithm of all the error microphones
(combined).
5.11 Summary
This chapter deals with frequency domain implementation of multichannel ANC.
Frequency domain implementation is made possible by block processing and using FFTs.
Frequency domain block filtered-x normalized LMS (FBFXNLMS) algorithm is
developed for noise mitigation in multichannel ANC. The proposed algorithm employed
normalized LMS algorithm to facilitate variable step size control. Normally
computational complexity requirement for frequency domain implementation is lower
than its time domain counterpart. Here also computational complexity analysis of the
developed algorithm is found to be much lower than time domain FXNLMS algorithm
for multichannel ANC. Frequency domain block filtered-e LMS (FBFELMS) algorithm
is developed which implement FELMS algorithm in frequency domain. New algorithms
were developed for nonlinear ANC using Legendre neural network in the previous
chapter. In this chapter frequency domain implementation of Legendre neural network for
nonlinear ANC is also carried out. Detailed mathematical formulation of the algorithm
for filter bank implementation is presented. Frequency domain Legendre filtered-e LMS
152
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FREQUENCY DOMAIN APPROACH TO MULTICHANNEL NONLINEAR ANC
(FBLFELMS) is also developed which deals with frequency domain implementation of
LFELMS algorithm. The validity of the proposed algorithms is demonstrated through
extensive computer simulations. Performance of the proposed algorithms is also
evaluated on signals collected from real life environment such as Bunnaneer jet cockpit
noise, white noise and factory floor noise. The performance of proposed frequency
domain algorithms is found to be equivalent to their time domain counterparts but the real
advantage is in huge computational complexity reduction capability.
153
Chapter 6
Conclusion and
Scope for Further Research
6.1 Conclusion
The work described in the thesis is primarily concerned with the study and development
of novel nonlinear active noise control systems. The main focus of this research work is to
design novel ANC based on conventional feedforward neural network topology. In particular,
development of weight update algorithms for the proposed nonlinear ANC is the main challenge
of this dissertation. Further, certain modifications in the existing and proposed algorithms have
been incorporated to make them suitable for multichannel ANC. Development of adaptive
algorithms for multilayer artificial neural network for nonlinear ANC with nonlinear secondary
path model is one of the principal contribution of this dissertation. Development of adaptive
algorithms for Legendre neural network in filter bank implementation is another major
contribution of this thesis work. This dissertation work has also contributed immensely to the
issues of
frequency domain implementation of nonlinear ANC. Some of the major
achievements of the present study are mentioned below.
One of the focus areas was on developing a MLP based neural network controller for
nonlinear ANC. Two separate adaptive algorithms were developed for nonlinear ANC
considering two situations (when secondary path was assumed linear or nonlinear). For nonlinear
secondary path, adaptive algorithm was modified using virtual secondary path filter concept. By
computer simulations it was observed that the proposed algorithms (using various nonlinear
activation functions) outperformed standard VFXLMS algorithm in terms of steady state
MSE(dB) with faster speed of convergence. In order to take advantage of low computational
complexity of filtered error LMS algorithm, both the developed algorithms were suitably
modified to develop NFELMS algorithm. Performance of the modified algorithms were also
analyzed by computer simulation on real life signals and compared with that of VFELMS
algorithm. Finally it has been concluded that the proposed adaptive algorithms for MLP based
ANC controller are superior to the standard Volterra based algorithms.
154
CHAPTER-6
CONCLUSION AND SCOPE FOR FURTHER RESEARCH
Another primary area of focus was to explore Legendre neural network for nonlinear
ANC. The adaptive algorithm for Legendre neural network used as controller of ANC was first
developed. The algorithm was found to be simple and easy to implement and has low
computational complexity. In order to reduce computational complexity further a reduced
structure Legendre neural network was proposed. The reduced structure Legendre neural network
with its reduced computational complexity was found out to be performing well with reference to
steady state MSE(dB) level. LFXLMS algorithm, LFELMS algorithm and LFXRLS algorithm
were developed for reduced structure LNN based on FXLMS algorithm, FELMS algorithm and
FXRLS algorithm respectively. The developed algorithms require less computation compared to
FFXLMS algorithm. The developed algorithms were modified to deal with nonlinear secondary
path (NSP) which relies upon virtual secondary path concept. In order to reduce computational
complexity faster version of LFXLMS algorithm was also developed.
The third zone of focus has been frequency domain implementation of multichannel
ANC.
Conventionally
computational
complexity
requirement
of
frequency
domain
implementation is lower than its time domain counterpart. So FBFXNLMS algorithm was
developed for multichannel ANC using efficient DSP tools like FFT and IFFT. The developed
algorithm uses NLMS algorithm to facilitate variable step size. Observing that FELMS algorithm
is an efficient alternative to FXLMS algorithm with an advantage of reduction in computational
complexity frequency domain block filtered-e least mean square (FBFELMS) algorithm was
proposed. The reduced structure Legendre neural network for nonlinear ANC was also
implemented in frequency domain. Two new adaptive algorithms, frequency domain block
Legendre filtered-x least mean square (FBLFXLMS) algorithm and frequency domain block
Legendre filtered-e least mean square (FBLFELMS) algorithm were developed. Analysis of the
results of computer simulation on synthetic data and real life data revealed identical performance
(MSE(dB) and speed of convergence) with reference to the proposed frequency domain
algorithms and their corresponding time domain algorithms but with lower computational
complexity.
155
CHAPTER-6
CONCLUSION AND SCOPE FOR FURTHER RESEARCH
6.2 Scope for Further Research
To conclude the thesis, following are some pointers for further work.
Multichannel implementation of neural ANC could be a research area, which should be
explored. Multichannel implementation requires simultaneous operation of many neural
networks which increases the complexity of the system. Study can be done to effectively
manage the networks to optimize the overall performance of the ANC.
The implementation aspects in FPGA or hybrid FPGA and DSP combined platforms of
some of these algorithms and structures should also be investigated. Implementing the
whole system in FPGA or hybrid FPGA and DSP through many challenges.
Focus be concentrated to design efficient filter structures based on new topologies to
address the ANC problems.
Different novel methods based on evolutionary and bio-inspired techniques be
developed with the basic objective to optimally train the weights of the adaptive filter
structures. Emergence of evolutionary and bio-inspired techniques and their highly
effective variants can really change the world of ANC.
156
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Path,” IEEE International Conference on Signal Processing, Communication,
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Publication
[P1]
Kunal Kumar Das and Jitendriya Kumar Satapathy, “New Neural Network
Algorithms for Nonlinear Active Noise Cancellation with Nonlinear Secondary
Path,” IEEE International Conference on Signal Processing, Communication,
Computing and Networking Technologies, Kumaracoil, Thuckalay, Tamilnadu,
India, pp. 286-290, July 2011.
[P2]
Kunal Kumar Das and Jitendriya Kumar Satapathy, “AdjointLMS Algorithm
Based on Functional Expansion for Nonlinear Active Noise Cancellation,” IEEE
National Conference on Computational Intelligence, Control, and Computer
Vision in Robotics & Automation, Rourkela, pp.213-217, March 2008.
[P3]
Kunal Kumar Das and Jitendriya Kumar Satapathy, “ Novel Algorithms Based on
Legendre Neural Network for Nonlinear Active Noise Control with Nonlinear
Secondary Path,” International Journal of Computer Science and Information
Technologies, vol. 3(5), pp. 5036 - 5039, 2012.
[P4]
Kunal Kumar Das and Jitendriya Kumar Satapathy, “Frequency-Domain Block
Filtered-x NLMS Algorithm for Multichannel ANC,” IEEE First International
Conference on Emerging Trends in Engineering and Technology, Nagpur,
pp.1293-1297, July 2008.
[P5]
Kunal Kumar Das and Jitendriya Kumar Satapathy, “Legendre Neural Network
for Nonlinear Active Noise Cancellation with Nonlinear Secondary Path,” IEEE
International Conference on Multimedia Signal Processing and Communication
Technologies, Aligarh, India, pp.40-43, December 2011.
165
BIO-DATA OF THE CANDIDATE
Name of the candidate
:
Kunal Kumar Das.
Father’s Name
:
Prasanta Kumar Das.
Date of Birth
:
10th September 1975.
Permanent Home Address
:
AT-Purunahatsahi,
(Near-Binod Bhaban)
PO-Baripada,
Dist-Mayurbhanj,
ORISSA-757001.
Phone
:
+91-9437218383.
E-Mail
:
[email protected]
2012
:
PhD dissertation submitted, NIT
Rourkela.
2001
:
M.E. in Electronic Systems &
Communication Engineering,
REC, Rourkela.
1997
:
B.E. in Electrical Engineering,
UCE, Burla.
PROFESSIONAL EXPERIENCE
:
14 years of teaching and research
experience.
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