ACTIVE NOISE
CONTROL PRIMER
AlP Series in
Modern Acoustics and Signal Processing
ROBERT T. BEYER, Series Editor-in-Chief
Physics Department, Brown University
EDITORIAL BOARD
YOICHI ANDO, Faculty of Engineering, Kobe University, Kobe, Japan
FLoYD DUNN, Bioacoustics Research Lab, University of Illinois,
Urbana , Illinois
JOHN ERDREICH, Ostergaard Associates, West Orange, New Jersey
CHRIS fuLLER, Department of Mechanical Engineering, Virginia Polytechnic
Institute, Blacksburg, Virginia
WILLIAM HARTMANN, Department of Physics, Michigan State University, East
Lansing, Michigan
IRA HIRSCH, Central Institute for the Deaf and the Department of Psychology,
Washington University, S1. Louis, Missouri
HERMAN MEDWIN, Naval Postgraduate School, Monterey, California
JOANNE L. MILLER, Department of Psychology, Northeastern University, Boston, Massachusetts
LARRY ROYSTER, Department of Mechanical and Aerospace Engineering,
North Carolina State University, Raleigh, North Carolina
JULIA DOSWELL ROYSTER, Environmental Noise Consultants, Raleigh, North
Carolina
WILLIAM A. VON WINKLE, New London, Connecticut
BOOKS IN THE SERIES
Producing Speech, Contemporary Issues for Katherine Safford Harris, edited
by Fredericka Bell-Berti and Lawrence 1. Raphael
Signals, Sound, and Sensation, by William M. Hartmann
Computational Ocean Acoustics, by Finn B. Jensen, William A. Kuperman,
Michael B. Porter, and Henrik Schmidt
Pattern Recognition and Prediction with Applications to Signal Characterization, by David H. Kil and Frances B. Shin
Oceanography and Acoustics: Prediction and Propagation Models, edited by
Alan R. Robinson and Ding Lee
Handbook of Condenser Microphones, edited by George S.K. Wong and Tony
F.W. Embleton
Seismic Wave Propagation and Scattering in the Heterogeneous Earth, by
Haruo Sato and Michael C. Fehler
Active Noise Control Primer, by Scott D. Snyder
ACTIVE
NOISE CONTROL
PRIMER
Scott D. Snyder
University of Adelaide, Australia
With 75 Illustrations
AlP
PRESS
,
Springer
Scott D. Synder
Department of Mechanical Engineering
University of Adelaide
Adelaide, South Australia 5005
Australia
Series Editor:
Robert T. Beyer
Physics Department
Brown University
Providence, RI 02912
USA
Library of Congress Cataloging-in-Publication Data
Snyder, Scott D.
Active noise control primer / Scott D. Snyder.
p. cm. - (Modern acoustics and signal processing)
Includes bibliographical references and index.
ISBN 978-1-4612-6437-8
ISBN 978-1-4419-8560-6 (eBook)
DOI 10.1007/978-1-4419-8560-6
1. Noise control. 2. Acoustical engineering. 1. Title. II. AlP series in modern
acoustics and signal processing
TD892 .S58 2000
620.2'3-dc21
99-040962
Printed on acid -free paper.
© 2000 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 2000
Softcover reprint of the hardcover Znd edition 2000
AlP Press in an imprint ofSpringer-Veriag New York, Inc.
AII rights reserved. This work may not be translated or copied in whole or in patt without the
written permission of the publisher (Springer Science+Business Media, LLC), except for brief
excerpts in connection with reviews or scholafly analysis. Use in connection with any form of
information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
the former are not especially identified, is not ta be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by
anyone.
Production managed by Jenny Wolkowicki; manufacturing supervised by Jeffi-ey Taub.
Typeset by KP Company, Brooklyn, NY, from the author's files.
9 8 7 6 5 4 3 2 1
ISBN 978-1-4612-6437-8
To
Gill, Tom and Isaac
Series Preface
Soun is noght but air y-broke
-Geoffrey Chaucer
end of the 14th century
Traditionally, acoustics has formed one of the fundamental branches of physics.
In the twentieth century, the field has broadened considerably and has become
increasingly interdisciplinary. At the present time, specialists in modem acoustics can be encountered not only in physics departments, but also in electrical
and mechanical engineering departments, as well as in mathematics, oceanography, and even psychology departments. They work in areas spanning from musical instruments to architecture to problems related to speech perception. Today,
six hundred years after Chaucer made his brilliant remark, we recognize that
sound and acoustics is a discipline extremely broad in scope, literally covering
waves and vibrations in all media at all frequencies and at all intensities.
This series of scientific literature, entitled Modem Acoustics and Signal Processing (MASP), covers all areas of today's acoustics as an interdisciplinary
field. It offers scientific monographs, graduate-level textbooks, and reference
materials in such areas as architectural acoustics, structural sound and vibration,
musical acoustics, noise, bioacoustics, physiological and psychological acoustics, speech, ocean acoustics, underwater sound, and acoustical signal processing.
Acoustics is primarily a matter of communication. Whether it be speech or
music, listening spaces or hearing, signaling in sonar or in ultrasonography, we
seek to maximize our ability to convey information and, at the same time, to
minimize the effects of noise. Signaling has itself given birth to the field of
signal processing, the analysis of all received acoustic information or, indeed,
all information in any electronic form. With the extreme importance of acoustics
for both modem science and industry in mind, AlP Press, now an imprint of
Springer-Verlag, initiated this series as a new and promising publishing venture.
We hope that this venture will be beneficial to the entire international acoustical
community, as represented by the Acoustical Society of America, a founding
vii
viii
Series Preface
member of the American Institute of Physics, and other related societies and
professional interest groups.
It is our hope that scientists and graduate students will find the books in this
series useful in their research, teaching, and studies. As James Russell Lowell
once wrote, "In creating, the only hard thing's to begin." This is such a beginning.
Robert T. Beyer
Series Editor-in-Chief
Preface
Active noise control has become one of the most popular research topics in
the "engineering" domain, with hundreds of journal papers covering dozens of
associated topics reaching the academic press each year. However, despite this
research effort, the number of practical, commercial implementations reaching
the marketplace has been extremely slim. Apart from active headsets, and the
odd air conditioning and vehicle implementation, it is difficult to think of practical examples.
There are a large number of reasons for this lag between the commercial and
academic worlds. Active noise control systems are very complex, usually requiring the designer to achieve some synergy between microelectronics, transducer
technology and physical acoustics; having the skills to do this requires significant experience. Noise problems which are truly amenable to active control
solutions are not as widespread as many people think. I cannot, for example,
quiet your neighbour's dog, or stop traffic noise from entering the house built
next to a superhighway, or, in most cases, even provide a practical solution to
the problem of the noisy refrigerator. Even in instances where it does work, the
frequency range over which control can be achieved is usually quite limited. If
I dwell too long on all of these thoughts, I will be tempted to mutter a statement
along the lines "an expert in a useless field!"
However, having said all of these nasty things, I will say that when active
noise control works, it really works. There is almost a feeling of disbelief in the
audience when, for example, you reduce the level of the fundamental tone in a
commercial leaf vacuum by 30 dB, or the low frequency engine noise in a
vehicle cabin by a similar amount. The trick is to know when to apply the
technology, what problems are amenable.
This brings us to this book. This book grew out of a set of manuals and
papers a colleague of mine, George Vokalek, and I wrote to support an "active
control development kit." The aim of the kit was to provide the microelectronics
required for commercial designers to implement active noise control systems in
their various products. The problem was to give the designers some indication
of how active noise control actually (physically) worked, where it could be
IX
x
Preface
applied, and what results could be reasonably expected without going into pages
of mathematical expressions. This book was my attempt at a solution. Since that
time, I have found it to be a useful introduction for new graduate students,
senior-level students undertaking active noise control projects, and secondary
and tertiary teachers looking for new ideas to aid the instruction of fundamental
physics. For those who are interested, there is an "experimental kit" which supports this text, available from the Michigan-based company Arbor Scientific:
www.arborsci.com.
In keeping with the aims, this book is short and descriptive, almost totally
without mathematical expressions. As the title indicates, it is meant to be a
"primer," an introductory text. It assumes that the reader has essentially no
knowledge of acoustics, signal processing, or noise control. Hopefully, after
reading the book, this will change.
Scott D. Snyder
Contents
Series Preface .................................................................................... vii
Preface ............................................................................................... ix
1. Introduction ...............................................................................
1
Welcome to the World of Active Noise Control! ................................
Chapter Summary.. ..... ... ..... ...... ......... ....... ....... .... ... ... .... ... ..... ... ....... ... ...
Do I Have to Read the Whole Book? .... ....... ....... ....... ..... .......... ..... .....
What Is Active Noise Control? .............................................................
Adaptive Feedforward Active Control Noise .......................................
Advanced Reading ......... ..... ............. .... ............ .... ...... ... .... ..... .... ....... .....
I
2
3
3
4
5
Background: Fundamentals of Sound ...................................
7
What Is Sound? .....................................................................................
What Is Noise? ......................................................................................
Quantifying Sound Levels ......... ... ... ....... ....... .... ..... ... ... .... ... ..... ....... ... ...
Sound Waves .........................................................................................
Frequency Analysis ..... ............. ..... ........... ... ... ....... ... ..... .... ... .... ........ .....
Sine Waves ............................................................................................
Fourier Analysis ...... ... .................. ... .... ....... ..... .... ... ... ... .... ..... ... ....... ... ...
Harmonics .... ............ ... ... ........ .... ........ ...... ... ......... ... ... .... ... ..... ... ....... ... ...
1. Anything that Rotates ......... ........ ..... ... ....... .... ... ... ........ ... ......... .....
2. Many Devices Which Use Mains Electricity..............................
3. Devices Which Are Driven Beyond Their Capabilities ..............
Human Perception of Sound .. ... .... ...... .......... ......... ... ..... ....... .... ... .... .....
Acceptable Sound Levels .. ... ..... ... ........ ......... ...... ... ... ....... ..... .......... ... ...
7
8
8
10
14
15
19
21
22
24
25
27
30
2.
3. Fundamentals of Noise Control.............................................. 36
Prerequisite Discussion: Power and Impedance ....... ..... ....... .... ... ... ... ...
The Magnitude of Acoustic Power ..... .... ... ..... .... ... ... ..... ... ... ..... ....... .....
Decibel Units for Acoustic Power ........................................................
36
40
40
Xl
xii
Contents
Power, Pressure, and Hearing Loss ......................................................
Real and Imaginary Impedance ............................................................
What Is Noise Control? .........................................................................
What Is Passive Noise Control? ...........................................................
What is Active Noise Control? .............................................................
4.
5.
6.
7.
41
42
43
43
44
Free Space Noise Control ........................................................ 46
Passive Noise Control Approaches .......................................................
Active Control Approaches ...................................................................
46
51
Enclosed Space Noise Control.................................................
67
Where Does the Sound Come From? ...................................................
How Does the Sound Get Out Again? .................................................
How Does the Sound Field Arrange Itself? ....... ....... ... ... ..... ... .......... ...
Passive Noise Control Approaches .......................................................
Active Noise Control Approaches ........................................................
67
68
69
74
76
Control of Sound Propagation in Ducts ................................
81
Sound Fields in Ducts ...........................................................................
Modes in Ducts ... .......... ....... ..... ........... ...... ....... ......... ... ........ ....... ...... ...
Impedance in Ducts ....... ..... ... ... ......... ..... .... .... ... ........ ............ ... .......... ...
Passive Noise Control Approaches .......................................................
Sidebranch Resonator ..... ... ............ ... ... .... ...... ........ ..... ............... ....... .....
Expansion Chamber .... ... .................... ............. ........ ...... ............ ....... ......
Helmholtz Filter .....................................................................................
Dissipative Passive Control Techniques ...............................................
Active Noise Control Approaches ........................................................
Reference and Error Signal Quality.... .... .... .......... .... ... ........ ... .... ... .......
Reference Sensor/Control Source Separation Distance ........................
Control Source Position in the Duct .......... .... ..... ... ............. ..... ....... ......
Duct Response Characteristics .. ... ..... ......... .... ..... ..... ..... ........ ... .... .........
81
82
83
84
85
86
87
88
89
90
91
92
92
Active Noise Controller Overview..........................................
95
Some Important Facts ....... ... ..... ........ ..... .... ......... ... ... ... .... ..... ... .... .... ..... 96
Digital System Requirements .... ... ..... ... .... ..... ...... ... .......... ............ ......... 96
Controller Output (Digital Control Filter) Requirements .... ........... ... ... 104
Adaptive Algorithm Requirements ....................................................... 107
8.
Controller Fundamentals ......................................................... 113
General Control System Outlines and Definitions .... ........... .... ....... ..... 114
Physical System Limitations ......... ... ... ....... .... ..... ... ... ..... ....... ... ........ ..... 119
Interfacing a Digital System ................................................................. 121
Contents
xiii
Background .................................................... ........................................ 121
Required Additions for Digital Control................................................ 124
Overview of the Controller ................................................................... 127
Controller Component 1: The Digital Filter ......................................... 128
What Is a Digital Filter? ................................................................... 128
Specifying the "Appropriate" Digital Filter ..................................... 132
Specifying the Digital Filter Length ................................................. 134
Controller Component 2: The Adaptive Algorithm ............................. 135
Background: Making Use of Adaptive Signal Processing ............... 135
Gradient Descent Algorithms ............................................................ 136
Evaluating the Gradient ..................................................................... 139
The Convergence Coefficient ............................................................ 140
Quantization Errors and Leakage ...................................................... 143
Slowing Down the Algorithm to Improve Performance .................. 145
Controller Component 3: Cancellation Path Modeler .......................... 146
Selecting the Sample Rate ................................................................ 150
So What Is the Optimum Sample Rate? ........................................... 154
Index .................................................................................................. 157
1
Introduction
Welcome to the World of Active Noise Control!
Active noise control is an exciting area of engineering research and development. When applied correctly to problems which are amenable, active noise
control technology can provide astonishing results. This is particularly true
for low-frequency noise problems, where traditional passive noise control
techniques often require a lot of size, mass, and money.
The key phrases in the sentences above are applied correctly and problems
which are amenable. Just as correct application of active noise control can
produce incredible results, incorrect application can be incredibly disappointing. Unfortunately, for the uninformed user the latter outcome is far
more likely than the former. This is a problem which has dogged the technology in its push for commercial acceptance; active noise control is not a universal panacea. This is also the reason for writing this book.
The application of active noise control technology, or even the consideration of its application, is generally a complex task. The correct application of active noise control requires an integration of physics, engineering,
and digital signal processing. While one does not need to be an expert in
all areas to make a system work, a basic knowledge is required in all areas
to make a system work well, or to rationalize why a system has failed to
work.
Enter the Active Noise Control Primer. The idea behind this book is to
present all of the basic knowledge required to assess the potential of an active
noise control system for a given problem, and to provide some guidance with
system setup and elementary tuning. The book is written from the standpoint
of teaching someone who has no knowledge of the field, but has a reasonable
grasp on basic physics and mathematics. The book does not contain detailed
equations and mathematical descriptions of acoustics, vibrations, signal processing, and the like. There are other books, which fall into the Research
Monograph category, that the reader can consult for this information (several
of these are referenced at the end of this chapter). The book does cover, in a
descriptive fashion, the areas which are important:
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
2
1. Introduction
• basic acoustics;
• basics of human perception of sound;
• sound power and related concepts;
• the fundamentals of passive noise control strategies for several classes of
problem (required for assessing whether passive control is a better option
than active control);
• the fundamentals of active noise control strategies for several classes of
problems;
• the basics of digital systems;
• the basics of adaptive controllers (to facilitate elementary operation and
tuning); and
• a more detailed description of the active noise control adaptive control
system (to facilitate "better" tuning).
Chapter Summary
Following this Introduction, the chapter contents are:
Chapter 2. Background: Fundamentals of Sound. This chapter provides a
brief discussion on the fundamental concepts which will be required knowledge for understanding active noise control. Included in this discussion
are topics of frequency, waves, wavelengths, Fourier analysis, harmonic
signals, and human hearing.
Chapter 3. Fundamentals of Noise Control. This chapter provides a general
discussion on the topic of "noise control," both passive and active, from
the standpoint of the flow of energy.
Chapter 4. Free Space Noise Control. This chapter looks at the problem of
controlling noise radiating into a free space environment (basically, an
environment where there are no walls to impede the propagation of sound
waves). Both active and passive approaches are considered.
Chapter 5. Enclosed Space Noise Control. This chapter looks at the problem
of controlling unwanted sound fields in enclosed spaces (rooms, vehicles,
etc.). Again, both active and passive approaches are considered.
Chapter 6. Control of Sound Propagation in Ducts. This chapter looks at the
last general group of problems, those which involve sound propagating in
a duct (such as an air-conditioning system or car exhaust). As before, both
active and passive approaches are considered.
Chapter 7. Active Noise Controller Overview. This chapter provides an introduction to adaptive feedforward active noise controllers, their operation,
and tuning. Included in this chapter is a basic discussion of digital systems
and their particular requirements, as well as a "heuristic" description of the
controller operation.
Chapter 8. Controller Fundamentals. This chapter provides a more detailed
description of the adaptive feedforward active noise controller, providing
Do I Have to Read the Whole Book?
3
information which is important for advanced controller tuning. Mathematics are kept to a minimum.
Do I Have to Read the Whole Book?
The question which often arises when one is in Chapter 1 is, do I have to read the
whole book (to get enough information about active noise control to "do something")? The answer is "yes, if you want to get the maximum benefit from your
acquisition." However, if you are short of time, consider the following:
1. If you are using this text in conjunction with an active noise control
demonstration/experimentation kit, the first chapter you may wish to
read is the Controller Overview, Chapter 7. This will provide you with
enough background information in a short space of time to understand
the adaptive feedforward active noise controller sitting in front of you.
2. If you are using this text to gain a basic knowledge of acoustics and noise
control, then you may want to concentrate on Chapters 2 and 3, on Fundamentals of Sound and Fundamentals of Noise Control.
3. If you have a specific target in mind for active noise control, and already
know a bit about acoustics, then first read Chapter 3, Fundamentals of
Noise Control. After that, try to fit your problem into one of the general
categories of free space/enclosed space/duct noise control, and read the
appropriate Chapter 4,5, or 6. If you are interested in simply experimenting with active noise control, without any specific target in mind, select
Chapter 4.
4. If you are planning to do more "advanced" work in active noise control,
such as part of a student project, it is worthwhile working through Chapter 8, Controller Fundamentals.
Enough with preliminary comments: let's proceed.
What Is Active Noise Control?
Having purchased a book to explore active noise control, it is sensible that
the first question be: "What is active noise control?"
Active noise control is a technique which aims to "cancel" unwanted noise
by introducing an additional, electronically generated, sound field. Although
the idea can be traced back over 60 years, commercial implementation has
really only been possible for something more like 10-20 years. This is largely
due to recent advances in microprocessor and actuator technology.
At present, active noise control accounts for a minute portion of all noise
control implementations. Practically all noise control involves traditional
"passive" techniques, techniques which are discussed in Chapters 3-6. There
are several reasons for this, which include:
4
1 . Introduction
1. Active noise control is only useful for certain types of problems. As will
be discussed, these are generally low-frequency problems, usually tonal,
with either simple or contained sound fields. It should be stated, however, that the sorts of problems which are amenable to active noise control are not uncommon in real life.
2. Active noise control is more complicated than passive noise control, in
that it involves the integration of electronics, transducers (loudspeakers,
microphones, etc.), and acoustics.
3. There are not a great number of noise control practitioners who have
experience in active noise control.
One of the aims of this book is to make active noise control more accessible, enabling the technology to spread.
Adaptive Feedforward Active Noise Control
The controller descriptions contained in this book are for a certain type of
active noise control system, referred to as an adaptive feedforward active
noise control system. An outline of an adaptive feedforward active noise control implementation for attenuating sound propagation in an air conditioning
duct is shown in Figure 1.1.
As shown in Figure 1.1, there are four basic components in an adaptive
feedforward active noise control system:
A reference microphone, which provides a measurement, referred to as the
reference signal, of the impending noise some time before it arrives at the
controller. In an air-conditioning duct, the noise must travel away from the
fan and move down the duct. Therefore, if a microphone is placed "upFan noise
source
f l\. V1\ V1\
~n.wanted
\folse
Reference
microphone
Cancellation
Residual
noise
><><X
.... .........
~
Control
source
....;..-
-
Error
microphone
Control
system
FIGURE 1.1. Main components in a typical adaptive feedforward active noise control
system.
Advanced Reading
5
stream" of the control system, it can perform this required a priori sampling
of the noise.
A control system, which is responsible for taking the reference signal measurement of the impending noise and calculating what is required to cancel
it. The control system must be fast enough to do this calculation before the
unwanted noise arrives.
A control source, which is used to generate the canceling sound field. In this
case, the control source is a loudspeaker.
An error microphone, which is used to sample what noise actually remains
after the cancellation operation. This measurement is referred to as the
error signal because it provides an indication of how much the controller
was "in error" when it derived the canceling waveform. If the calculation
was perfect, then the unwanted noise would have been completely canceled and the "error" would be zero.
While it is notionally straightforward to envisage the job of the control
system ("invert" the measured signal, so that it cancels the sound wave when
it arrives), the actual calculation procedure is quite difficult. This is because
the change in the noise disturbance as it propagates down the duct must be
accounted for, as must be change in the signal as it passes through the reference microphone and the change in the controller output as it passes through
an amplifier and loudspeaker. These various changes will themselves change
over time, as microphones age, as air speeds and temperatures change, and
even as fungus grows on the loudspeakers and microphones! Based upon
these considerations, it is apparent that the controller must be self-tuning, or
adaptive. That is, it must be able to adjust its calculation procedure to suit
the current environment in which it is operating. The error signal measurement is actually used by the controller to do this adaptation, via the use of an
adaptive algorithm. The adaptive algorithm attempts to adjust the output
calculation such that the sound field measurement at the error microphone is
"0," meaning that there is complete cancellation at the error microphone.
In order to make an active noise control system work, or even to determine
if it is physically capable of solving a target problem, it is necessary to have
a good knowledge of the "acoustics" of the problem. Do not be misled by the
concepts of "adaptive control" and "an algorithm that tunes the controller to
get the best result." The controller can only work within the bounds defined
by the fundamental physics. It is therefore appropriate to begin our more
detailed discussions with chapters that address the fundamentals of acoustics
as required for active noise control.
Advanced Reading
If after reading the Primer you wish to consider active noise control in more
depth (equations and all), there are several books that delve into explicit
detail. These include:
6
1. Introduction
C.H. Hansen and S.D. Snyder (1997). Active Control of Noise and Vibration. E&FN
Spon: London.
C.R Fuller, SJ. Elliott, and P.A. Nelson (1996). Active Control of Vibration. Academic
Press: London.
S.M Kuo and D.R. Morgan (1996). Active Noise Control Systems. Wiley: New York.
P.A. Nelson and SJ. Elliott (l992).Active Control of Sound. Academic Press: London.
2
Background:
Fundamentals of Sound
What Is Sound?
Sound is the sensation produced at the ear by very small pressure fluctuations present in the surrounding medium (which we will assume to be air).
This sensation is produced in response to the pressure fluctuation-induced vibration of the ear drum. The fluctuations in the surrounding air
constitute a sound field. The pressure fluctuations themselves are usually
referred to as sound pressure or acoustic pressure.
We will discuss ways of quantifying a sound field, of putting a number
to "how big" it actually is, shortly. However, to arrive at some idea about
the size of the fluctuations, note that the sound field pressure fluctuations are placed on top of (static) atmospheric pressure. This is similar to
saying that the ocean floor is continually subject to the "average," or
static, pressure from the weight of the ocean, along with the much smaller
pressure fluctuations which result from a change in ocean depth due to
the waves on the surface. If you were to fill a 1 liter softdrink bottle with
water and balance it on one finger, the pressure on your finger is similar in
magnitude to atmospheric pressure. In a healthy person, this pressure is
present on both sides of the ear drum. If you now take a standard holepunch and extract a small paper circle from a piece of photocopy paper,
the pressure caused by placing this paper circle on your finger is similar
in amplitude to the pressure fluctuations present during a loud conversation. The conclusions are that (1) sound pressure fluctuations are pretty
small, and (2) our ears are pretty sensitive to be able to detect the pressure
fluctuations.
You might now be wondering about the limits of human hearing. If you
were to take the above paper circle and cut it into something like 40,000
tiny pieces, a healthy young child would be able to detect a sound pressure fluctuation that is similar in amplitude to the pressure caused by one
of these pieces placed on the end of your finger.
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
7
8
2. Background: Fundamentals of Sound
What Is Noise?
Noise is unwanted sound. What constitutes noise is therefore subjective; what
is "music" to your ears might be "noise" to your parents, and what is an
inconsequential byproduct of one group's activities (such as a rock concert or
auto race) may be an outrageous production of noise to another group. One
type of sound which is generally agreed upon as being noise is that which
comes from industrial processes and, in particular, sound from machines:
punches, saws, sausage making machines, etc. However, even this may be
music to the factory owner's ears, as it may represent money being made.
Why is it important to do something to limit the unpleasantness of noise?
For centuries it has been recognized that noise is a hindrance to (or perhaps a
symptom of non-) peaceful coexistence between humans. In more modern
times, noise has been recognized as a serious health hazard, with significant
financial compensation paid to those whose hearing has been impaired by
their job or circumstances. This recognition, and the resultant penalties for
noncompliance with accepted standards, has provided the impetus for an
entire area of commercial activity and research: noise control. This text is
really a spin-off from this field.
Quantifying Sound Levels
Being able to communicate a problem is a prerequisite to fixing it. We must
therefore be able to quantify sound levels, to assign them a number which
describes "how big," before we can determine an appropriate course of action.
Sound is most often described in terms of pressure. Commonly, it is the
amplitude of the pressure fluctuation which is of interest. Pressure is not the
only quantity which could be used to describe sound, but it is the most convenient because (1) it is simple to measure, and (2) it is a scalar quantity, and
does not require a direction to be associated with it. Possible alternatives
such as velocity are not scalar quantities, and so to use them as the basis for
quantifying a sound field would complicate matters.
Pressure P is defined as the force F acting on a given area A (P =FIA). The
standard (SI) unit of measurement for pressure is the Pascal, abbreviated as Pa.
The quantity of 1 Pa is defined to be equal to a force of 1 Newton applied over
an area of 1 square meter.
The human ear can detect an incredible range of pressure fluctuations in
the ambient environment, from approximately 0.00002 Pa (threshold of hearing for a healthy young person) to 100 Pa or more (pain begins around 60 Pa).
These pressure fluctuations sit on top of the atmospheric pressure of approximately 101 kilopascals (101,000 Pa). The ear is also incredibly sensitive to
relative changes in the acoustic pressure fluctuations. For example, a change
in level from 0.0001 Pa to 0.0005 Pa will be as noticeable as a change from 1
Pa to 5 Pa.
Quantifying Sound Levels
9
The range and sensitivity of the ear led workers at the Bell Laboratories to
define a new unit to quantify acoustic pressure. The unit was originally called
the "Bel" in honor of their founder (Alexander Graham Bell), but was later
modified to become the decibel (deci = 10, so a decibel is equal to 10 Bels).
The decibel is commonly abbreviated as "dB." The decibel scale is a logarithmic scale. This means that the units increment proportional to the logarithm
of the quantity of interest. Sound pressure, denoted L p, in decibels is defined
by the relationship
In this expression, p is the amplitude of the pressure fluctuation in Pascals,
and Pref is a reference amplitude. This latter quantity is defined as the threshold of hearing, which is 20 microPascals (0.00002 Pa). Substituting this value
into the expression, a more convenient form is obtained:
Table 2.1 lists some common settings and scenarios for a range of sound
pressure levels.
Note that when expressed in the decibel scale, a sound pressure increment
of 20 dB is actually an increase in pressure amplitude by a factor of 10. For
example, a pressure amplitude of 1 Pa is 20 dB greater than a pressure amplitude of 0.1 Pa. Does this mean that if the sound level is increased by 20 dB we
will perceive it to be ten times as loud? Or, if the sound level is decreased by
20 dB, will we perceive it to be one-tenth as loud? The answer is no. Tests on
humans have revealed the subjective assessment results in Table 2.2.
TABLE 2.1. Examples of typical settings for a range of sound pressure levels.
Sound
Pressure
Level (dB)
140
120
100
80
60
40
20
o
Examples
Artillery noise at gunner's position
Rock concert (front rows), front of heavy
industrial presses
Industrial settings
Shouting, next to a busy road
Speech levels, restaurants, shopping malls
Quiet residential
Recording studio
Threshold of hearing for a healthy young person
Subjective
assessment
Pain
Very noisy
Noisy
Quiet
Very quiet
10
2. Background: Fundamentals of Sound
TABLE 2.2. Subject assessment of sound level
changes.
Change in sound
pressure level (dB)
6
10
20
40
Subjective assessment
Barely perceptible
Clearly perceptible
Half/twice as loud
Much quieter/louder
It is interesting to consider what the above results imply for the world of
advertising. A company might say that their new and improved product, for
example, a dishwasher, is so quiet because sound pressure has been reduced
by 50%. Well, a 50% reduction in sound pressure is a drop of 6 dB (try this
calculation, for example, for the change of 0.1 Pa to 0.05 Pa, and for 0.01 Pa
to 0.005 Pa). From the above table, we see that this reduction is barely perceptible. However, 50% certainly sounds nice on the surface. The moral of the
story is, beware of companies quoting sound reductions as percentages rather
than as decibels.
Sound Waves
We have already stated that a sound field is simply a group of small pressure
fluctuations. The next point to consider is, how did they get there? Further,
what happens to the pressure fluctuations as time progresses?
Sound pressure fluctuations are most commonly generated by something
which is vibrating. An example of this is a loudspeaker, where the sound is
"generated" by the vibrating loudspeaker cone. However, vibrations are not
the only way to generate the required pressure fluctuations. They may, for
example, be generated by some aerodynamic phenomena. This situation is
commonly encountered with large saw blades, where the forced motion of the
air around the teeth is responsible for significant noise generation.
Let us consider the case of a vibrating surface in a bit more detail, as it is
the simplest to visualize. As shown in Figure 2.1, if a vibrating surface placed
in air is at equilibrium (neither bent in or out, but rather in the middle at its
resting position) the air next to it will be at its equilibrium, atmospheric,
pressure. Suppose now the vibrating surface moves in an outward direction.
Initially the air immediately in front of it is not moving, and so is compressed
somewhat. The number of air molecules in front of the vibrating surface is the
same, but the volume available to them has decreased owing to the presence
of the outward-curved surface. The result: a pressure fluctuation is born! As
the vibrating surface reverses direction and heads inward, it passes once again
through equilibrium and then beyond. In the process, the air is "stretched"
(rarefaction, as the volume available to the air molecules is increased). An-
Sound Waves
Air is compressed
in front of surface
Air at equilibrium
1. Surface at
equilibrium
11
Air is rarefied
in front of surface
2. Surface
moves outward
3. Surface
moves inward
FIGURE 2.1. Generation of pressure fluctuations by a vibrating surface.
other pressure fluctuation is born, this time being a small reduction in the
equilibrium value.
If we were to measure the pressure in front of the vibrating surface, say with
a microphone, we would see a series of equilibrium/compression/equilibriumlrarefaction levels, as shown in Figure 2.2. This looks very much like a
wave. In fact, this type of plot is referred to as a waveform.
Once the vibrating surface generates a pressure fluctuation, what happens
to it? Does it just sit there, waiting to be "undone" by subsequent panel
motion? The answer is no. The pressure fluctuation moves away from the
panel. This is analogous to what happens when you throw a stone into a pond.
The water in front of the stone is pushed in and then bulges out, and this
motion travels away from the point of contact as a wave.
This is precisely how sound pressure fluctuations move away from the
source: as waves.
M
ure
3
3
FIGURE 2.2. Time history of pressure in front of the vibrating surface, showing alternat-
ing areas of (1) equilibrium, (2) compression, and (3) rarefaction.
12
2. Background: Fundamentals of Sound
While sound may travel as waves, the waves do not resemble the plot in Figure
2.2. That is, they do not look like a vibrating piece of guitar string. Rather than
moving as a bending wave (with displacement perpendicular to the direction of
travel), sound pressure fluctuations travel as longitudinal, or compression, waves.
This type of wave can also be seen, for example, in light springs which are shaken
on their ends (a "slinky"). If you could see an acoustic (sound) wave it would
look something like that which is shown in Figure 2.3, with areas of compression
and rarefaction moving away from the sound source.
It is intuitively obvious that sound pressure must travel as a compressional
wave; you cannot "bend" air, you can only compress and rarefy it. However, it
is worthwhile putting a more scientific rationale to this thought, as many
types of disturbances travel as waves and the rationale can be applied to all.
Consider a bending wave propagating along a piece of string, as shown in
Figure 2.4. How is it that the wave can travel? If we were able to look at little
"pieces" of the string, we would find that each piece is sliding against its
neighbor and so acting to push it up or down. This action is referred to as
shear. For a bending wave to travel, the medium in which it is traveling
(string, air, whatever) must be able to support shear stress. That is, the individual medium elements must be able to slide against one another without
having the whole thing fall to pieces. Metal bars and pieces of string can
support shear stress, but air cannot. Technically speaking, the viscosity of
most gases and liquids is too low to support shear stress, and therefore bending waves cannot exist to any significant degree.
If bending waves cannot travel in air, then can compression waves (acoustic waves) travel in things like metal bars? Yes, they can. Practically anything
can be compressed to some degree, which is the requirement for propagation
of a compressional wave.
So, sound pressure fluctuations travel away from their source as waves. Is
there any particular speed with which they travel? The answer is yes, although the speed will change with variations in temperature, pressure, and
constituents of the medium in which the wave will propagate. The speed of
FIGURE 2.3. Typical acoustic (compressional) wave, showing areas of compression and
rarefaction.
Sound Waves
13
Bending wave
in a string
Itl
string "elements"
One element "pushes" the
next element up/down by
sliding against it
FIGURE 2.4. A bending wave propagates by having one "element" slide up/down against
the next "element"; this is a "shear" motion.
sound, which is normally denoted by the character c, can be calculated using
the formula
.JEtpmeters per second
c=
where E is the Young's modulus for solids or the bulk modulus for gases, and
p is the density of the medium. Basically, E is a measure of how hard it is to
compress a given material. For something like metal or glass, E is a (very)
large number, while for air it is a relatively small number. The density of the
medium is simply a measure of "how heavy" it is for a given volume. For
steel, the density is high, while for air it is low.
If we study the above relationship long enough, we come to the conclusion
that compression waves travel quickly in hard (stiff) light materials, and slowly
in soft, heavy materials. Consider lead and steel. Steel is much harder, and
somewhat lighter, than lead. As a result, we expect compression waves to
travel faster in steel: the approximate speeds are 5200 meters per second (mI
s) for steel, and 1210 mls for lead. The speed of sound in air is comparatively
slow, at approximately 344 mls.
As mentioned above, the precise speed of sound is dependent upon both pressure and temperature. As the temperature increases, so too does the speed of sound.
Further, if sound waves are traveling in the "wind," then the speed of sound increases by an amount equal to the wind speed if it is flowing with the wind, and
decreases by an amount equal to the wind speed if it is flowing against the wind.
The speed of light is almost instantaneous compared to the speed of sound.
This becomes obvious when, for example, you sit in a sporting stadium watching a bat-and-ball game (such as baseball). You will see the person hit the ball
before you hear the ball strike the bat. The greater the disparity between sight
and sound, the cheaper the seats!
14
2. Background: Fundamentals of Sound
You can also use the speed of sound to soothe frightened children in a
thunder storm. Once the lightning flashes, count the seconds until the thunder arrives. For approximately every 5 seconds counted, the storm is a further
mile away, so "no need to worry" (this strategy for soothing children can
backfire sometimes if the storm is close).
Frequency Analysis
If you are a witness to a robbery, and the police asked you to describe the
voice of the assailant, you would certainly include some mention of "pitch."
For example, you might say that he or she had a "deep, low" voice or a "high,
squeaky" voice. In fact, after the notions of "loud" and "soft," the description
of "high" or "low" pitch is probably the most common way to subjectively
qualify the characteristics of sound and to communicate "what something
sounds like."
What you are in fact doing when you describe a sound as high or low
pitched is making a statement about the frequency content of the sound field.
Frequency content is important for a whole host of reasons, ranging from how
a human perceives a given sound pressure level, to what equipment is required to reproduce sound, and to what techniques can be used to control
unwanted noise. It is therefore important that we be able to quantify frequency, to assign it a number in the same way we did with sound pressure
amplitude so that we can communicate and work with the notion of "low
pitch/high pitch" in a scientific way.
So what actually is frequency? Frequency is a measure of how many times
an acoustic wave moves from compression to rarefaction and back again in a
given period of time (the standard time increment is 1 second).
Consider Figure 2.5. Here a sound wave is propagating past a given point,
and the sound pressure is being monitored with a microphone. In this instance, the time between successive peaks is constant (we will discuss this
~
::l
CJ)
CJ)
~
a.. L...--+-+---t-"'T"\ \ _L..--\----?
Direction of wave propagation
FIGURE 2.5 . Measurement of an acoustic wave with a frequency of 137 Hz.
Sine Waves
15
Time
FIGURE 2.6.
A more "typical" acoustic (sound) pressure measurement: what a mess!
point in more depth shortly). Over the course of I second, we find that there
are 137 transitions from compression to rarefaction and back again. Therefore, the frequency of the acoustic wave is 137 Hertz (abbreviated Hz).' The
units of Hertz are also known as cycles per second. A wave with a frequency of
137 Hz will cycle between maximum/minimum/maximum amplitudes 137
times in I second.
This notion of measuring frequency by counting transitions between compression/rarefaction/compression works when the acoustic wave is "nice and
smooth" (we will quantify this description shortly), as was the case in Figure
2.5. However, in the real world, it is rather unusual to encounter such a "pure"
acoustic wave. It is entirely possible, maybe even probable, that the timehistory of the sound pressure measurement will look something like that
which is shown in Figure 2.6. Subjectively, this is a mess! How do we assign
a frequency to this short of signal? The answer is, we don't. At least not a
single frequency. Rather than talk about the "frequency" of the acoustic wave,
we talk about the "frequency content." To expand upon this, it is necessary to
discuss the concept of a sine wave and the work of Fourier.
Sine Waves
The concept of a sine wave and the work of Fourier are absolute cornerstones
of scientific and engineering thought, and indeed of technology as we know
it. Practically any technical advancement, from the "simple" generation of
electricity to the extraordinary feats of space probes, would lack description
without them.
1 The units of Hertz were named in honor of German physicist Heinrich Rudolph Hertz,
who was the first person to produce electromagnetic waves artificially.
16
2. Background: Fundamentals of Sound
Consider the arrangement shown in Figure 2.7. Here we have a bar pinned
at one end and rotating in a circle. At any point in this rotation, we can
calculate the value of the sine of the angle between the horizontal and the
bar position. From simple trigonometry, the sine of the angle is the ratio of
the length of a line drawn straight up or down (perpendicular) between the
horizontal and the tip of the bar (denoted as x, or the "opposite"), and the
length of the bar itself (the "hypotenuse"). This value of sine will be positive
if the x line is pointing straight up and negative if the x line is pointing
straight down.
If we plot the value of the sine of the angle starting from an angle of
o degrees (when the bar is lying flat) and rotating completely around the
circle (through 360 degrees), we get a plot which looks something like that
which is shown in Figure 2.7. But we have encountered this shape before: it is
exactly the same shape as the plot of sound pressure amplitude for our "nice
and smooth" acoustic wave in Figure 2.5. Hence we call this "nice and smooth"
wave, the pressure amplitude of which varies over time in exactly the same
way as the sine of an angle varies as the bar is rotated, a sine wave. This sort
of wave is also referred to as tonal, or harmonic (although harmonic tends to
be more general, as we will discuss shortly).
Let us now expand our thought experiment to include some measure of
time. Suppose instead of plotting the sine of the angle for all bar positions in
the circle of rotation, we "spin" the bar at a certain speed, quantified by the
number of revolutions which take place in 1 second, and plot the sine of the
angle for all bar positions during that 1 second period. As shown in Figure
2.8, if the bar undergoes, say, four complete revolutions in 1 second, then we
expect to see the sine wave pattern repeated four times. Technically, the bar is
spinning at 4 Hz, or four cycles (complete revolutions) per second. If the bar
speeds up, to, say, eight revolutions per second then we expect to see eight
repetitions of the sine pattern over the 1 second interval; the bar is now
spinning at 8 Hz. The plots for 4 Hz and 8 Hz have the same general shape
:; ~h
'jg
Q)
.,.:. ____-'-___--' ? i
a.
g>
(J)
0
V>R>J'
Angle
:,). Rotate the bar completely
around the circle (through
360 degrees).
FIGURE 2.7. The plot of the sine of the angle between the rotating bar and the horizontal
edge.
Sine Waves
~
II
' slow"
4 Hz
17
=
Q)
Ol
C
tU
time (s)
'5
~
'iii
8.
a.
Q)
c
U5
t::
"fast" =
8Hz
~________~______~ X
~ Rotate the bar completely
time (s)
around fast and slow
Q)
c
U5
FIGURE 2.8. Two plots of the sine of the angle for all bar positions over a 1 second period
for two different rotational speeds.
(a sine wave) but differentJrequencies. They go through different numbers of
cycles during a 1 second period.
In classifying a sound pressure wave, we need to include mention of both
its shape and frequency. For example, the sound pressure is a "sine wave at
100 Hz," or it is "tonal at 200 Hz." In the first of these, the general shape of the
pressure (measurement) is a sine wave, and the number of cycles it undergoes
per second is 100. In the second case, the general shape of the pressure measurement is again a sine wave (or, better, sinusoidal), and the number of cycles
it undergoes is 200. We would then go on to say that the first case of 100 Hz
is a lower frequency than the second case of 200 Hz and, conversely, the
second case of 200 Hz is a higher frequency than the first case of 100 Hz.
What type of frequency do we normally deal with when discussing sound?
Are they 4 Hz or 40 Hz or 400,000 Hz? We will defer the answer to this
question until after we discuss the work of Fourier.
Two additional descriptors of a sine wave or waves which are of importance are amplitude and phase; see Figure 2.9. The amplitude of a sine wave
describes exactly "how large" it is . There are two measures of amplitude
which are commonly cited: peak and root mean square, or RMS . Peak amplitude is simply the maximum (pressure) amplitude of the sine wave. It is the
"top of the hump" in the plot. In a mathematical description of pressure field,
it is the peak amplitude which is often used as the descriptor.
The RMS amplitude provides a measure of the average sound pressure
level over time. It is the measure which is used by noise control practitioners
to assess the potential for hearing damage in a given environment. Referring
to Figure 2.9, if the amplitude of the sound pressure measurement were simply averaged over time, the result would be zero. This is because for each
18
2. Background: Fundamentals of Sound
Peak amplitude
RMSamplitud
Phase =900
.l, Phase =180
j
0
-r---¥---t---t---y-~ time (s)
t Phase =0
Phase =270
0
0
FIGURE 2.9. Amplitude and phase of a sine wave.
"bit" of acoustic wave compression (positive amplitude) there is an equal,
canceling "bit" of acoustic wave rarefaction (negative amplitude).2 However,
this certainly does not mean that the end result is zero insofar as your ear, and
perception, are concerned. In order to obtain an average amplitude which
reflects the effect of the sound field on the listener, the following procedure
could be adopted:
1. Square the amplitude of each "bit" of the pressure measurement. This
turns all points on the wave into positive numbers.
2. Get the average of the squared amplitudes (add them together and divide
by the number of measurements).
3. Take the square root of the result.
The resulting number from this procedure is the RMS average.
For a sine wave, the RMS amplitude is equal to 0.707 times the peak
amplitude. For sound pressure measurements which are not sinusoidal, the
three step procedure described above must be followed (in some form) to
calculate the RMS amplitude.
The other descriptor of interest here is phase. To see what "phase" actually
refers to, recall that the sine wave plot was formulated by taking the sine of
the angle that a rotating bar makes with the horizontal as it spins around (refer
back to Figure 2.7). We could therefore classify each location on the sine
wave plot by the associated (bar) angle. The positions of vertical up (90
degrees), horizontal to the left (180 degrees), vertical down (270 degrees),
and horizontal to the right (0/360 degrees) are marked on Figure 2.9. This
form of description is the phase. We might, for example, say that the "phase is
90 degrees," meaning that the curve is at its peak.
How can we have "negative pressure" in real life? Remember that the acoustic pressure is simply a small amplitude perturbation of the background (atmospheric) pressure;
regions of negative acoustic pressure, that occur in the rarefaction part of the sound
field, are simply slight reductions in the atmospheric pressure and are still "positive" in
absolute terms.
2
Fourier Analysis
19
Phase is most commonly used as a relative descriptor, rather than an absolute descriptor. For example, if there are two sine waves, and one is at the top
of the curve (90 degrees) and the other is at the beginning of the curve (0
degrees), we would say that the relative phase is 90 degrees. These two curves
will remain 90 degrees apart forever.
In active noise control, it is common to talk about signals being "in phase" or
"out of phase." If two signals are "in phase," then the phase difference is 0 degrees. If two signals are "out of phase," then the phase difference is 180 degrees.
Finally, this comparison between the plot of the sine of an angle and the
pressure in a wave, and the conclusion that they are, in fact, the same shape,
has been done using hand-waving and eye-balling. Is it possible to prove
mathematically, in a rigorous fashion, that the shapes are the same? The answer is yes. Technically you have to solve the "wave equation," which provides a mathematical description of the sound pressure at any point in space
as a function of time. If you do this, then you will see "sine" terms appear.
Trust me (I sell used cars, too). Most academic textbooks on acoustics work
through this calculation.
Fourier Analysis
As was mentioned, there are very few "pure" sinusoidal acoustic waves. So you
might be wondering, what is the use of this type of quantification?
The true value of the notion of a "sine wave" becomes apparent when considering the idea of a Fourier transform. Back in the nineteenth century, Joseph
Fourieil arrived at the conclusion that any steady-state waveform can be described as the sum of a number of sine waves with differing amplitudes and
phases. So, for example, the complex waveform shown in Figure 2.10 is actually
-r---'I--+--i-+-~-+-+-+~f-r-~ time (s)
FIGURE 2.10. A complex waveform that is the sum of two sine waves with different
amplitudes and phases.
31t is interesting to note that the early nineteenth century French mathematician J.B.
Fourier, the "discoverer" of the Fourier Transform, did not work with noise at all; he
was interested in the transfer of heat between objects.
20
2. Background: Fundamentals of Sound
the sum of two independent sine waves. More complex waveforms will be the
sum of even more sine waves. Even "random noise" can be described as the sum
of sine waves. The description of random noise is similar to that of white light, in
that it is the sum of "all" sine waves in the frequency range of interest.
This notion that any waveform can be described as the sum of a group of
sine waves is a powerful tool for studying and quantifying sound. It provides
us with a mathematical way of explaining a variety of phenomena, ranging
from why you can identify high- and low-pitched components in a complex
sound field (such as coming from an orchestra), to why a recording sounds
"tinny" when played through poor quality loudspeakers, and to why active
noise control works in some instances and not others (all of these will be
answered later in this book). In fact, sound fields are most commonly described in terms of their spectrum, which is the variation in amplitude (and
possibly phase) of the components of the waveform, ordered in terms of (sine
wave) frequency (see Figure 2.11) . The analysis of a waveform in terms of its
constituent sine waves is referred to as spectral analysis,frequency analysis,
or Fourier analysis. 4 The sine waves which make up a given waveform are
referred to as the frequency components of the signal. For example, we might
say that a sound pressure field has "significant 120 Hz and 150 Hz components," meaning that the set of sine waves that make up the measured waveform of the sound field include sine waves with frequencies of 120 Hz and
150 Hz, and that the amplitudes of these sine waves are relatively large.
Fourier not only arrived at the amazing conclusion that all waveforms can
be described by the sum of sine waves, but also developed a mathematical
way of working out what the frequencies, amplitudes, and phases of the sine
Sine waves at these frequencies
are the main components of the
sound pressure field
Q)
u
-HlHlfUI-Il-ft-il-'rit-tt-tthHIHt-~ time (s)
.€
a.
E
<t:
Frequency
Measured waveform
(time series data)
Spectrum
FIGURE 2.11. A complex waveform and its spectrum.
It is interesting to note that the human ear performs a sort of "physical" Fourier
analysis in its processing of sound, with different parts of the ear responding to the
different frequency components in the sound field.
4
Harmonics
21
waves are. This technique is known as the Fourier Transform. There is also an
"opposite" technique which is used to turn a group of sine waves into a
waveform. This latter technique is called the Inverse Fourier Transform.
While the Fourier transform has been a cornerstone technique in many
areas of engineering and mathematical science since its inception, the "practical" use of spectral analysis has truly blossomed in the last 30 years as a
result of the development of technique for fast calculation of the Fourier
transform: the Fast Fourier Transform, or FFT. 5 The FFT calculates the amplitude and phase of frequency components which are evenly spaced across the
spectrum. For example, the FFT might calculate the amplitude and phase of
the 200 Hz, 202 Hz, 204 Hz, ". frequency components. What happens if the
actual frequency is not exactly one of those in the FFT calculation (such as
201 Hz for the example above)? Essentially, it is placed in the bins of the
closest frequency. The size of the bin for a given stated frequency encompasses the range of frequencies that lie from half-way between the bin of
interest and the one below it, to half-way between the bin of interest and the
one above it. For the example above, the 202 Hz bin encompasses frequency
components between 201 Hz and 203 Hz. The frequency which defines a
given bin lies in the center of the bin, and is referred to as a center frequency.
The results returned by the FFT for a given bin include contributions from all
frequencies which lie in the bin.
Harmonics
In the study of sound and vibration, and important concept is that of harmonics. Harmonics are frequency components which are integer multiples of some
"fundamental" harmonic frequency. For example, if 100 Hz is the fundamental harmonic frequency, then 200 Hz is the second harmonic, 300 Hz is the
third harmonic, 400 Hz is the fourth harmonic, etc. If harmonics are plotted
in terms of their frequency, they appear as evenly spaced "spikes" as shown in
Figure 2.12.
If a waveform is periodic (that is, if it is made up of a pattern that repeats
itself every cycle), then the spectrum of the waveform will contain a series of
harmonics. The simplest example is that of a sine wave, that contains a single
harmonic (the fundamental). If the waveform is perfectly square, then it
contains all odd-numbered harmonics (1, 3, 5, etc.), as shown in Figure 2.13.
If the waveform is some other periodic shape, it will be constructed from
some other combination of harmonics. Note that the number of times the
pattern repeats itself every second will be the frequency of the fundamental
harmonic.
The Fast Fourier Transform was first outlined in-depth only a quarter of a century
ago, in The Fast Fourier Transform, by E. Oran Brigham (Prentice-Hall, Englewood
Cliffs, NJ).
5
22
2. Background: Fundamentals of Sound
Note the even spacing
of harmonic peaks
Frequency
Spectrum
FIGURE 2.12. Atypical plot of harmonics.
The sources of a large number of practical noise problems produce sound
fields which are comprised of harmonics (that is, harmonic sound fields) .
Examples of these include the foHowing.
1. Anything that Rotates
Consider the generation of a sound field by a fan, as depicted in Figure 2.14.
Every time a fan blade goes past a given point, a pressure pulse is created,
resulting in sound generation. The base frequency of the sound in Hertz is
equal to the rotational speed of the fan (revolutions per second) multiplied by
the number of blades. So, for example, if a fan rotates 60 times per second, and
there are four blades, then there are 240 pressure pulses created each second.
The fundamental frequency is 240 Hz. However, while the sound pressure
waves generated by the movement of the fan blades will be periodic, they will
r-
r--
r--
r--
r--
time (s)
"--
"--
..... .....
Measured waveform
(time series data)
I..-
OJ
"C
~
a.
E
«
Frequency
Spectrum
FIGURE 2.l3. A square wave and its spectrum.
Harmonics
/
+-'--+-+-+-+---'I--f-+-+-+-.....--~ time (s)
A pressure pulse occurs
each time a blade passes
FIGURE 2.14.
23
Measured waveform
(time series data)
Sound field generation by a fan.
not in general be perfectly sinusoidal. Instead, they will be "distorted" sine
waves, comprised of multiple harmonics (at 480 Hz, 720 Hz, etc.).
The question which arises here is, why must the frequency content of the
periodic waveform generated by the fan be harmonic? Why can't it simply be
any combination of frequencies? This is a surprisingly difficult question to
answer in a physical way, and is one that baffles students at all levels. By
definition, if a given waveform is "periodic," then the shape of the wave must
repeat itself on a cycle-to-cycle basis; see the waveform in Figure 2.14 for an
example. It is straightforward to appreciate that if a given frequency component is to be present in a wave which is periodic, then the frequency component must start out each cycle of the wave at exactly the same phase position.
If it does not, then two cycles in the waveform would not have an identical
shape, and so the wave would not be periodic. Following on from this, if a
given frequency component is to start each cycle in the (total) periodic wave
at the same phase position, then the frequency component itself must move
through an integer number of cycles during the course of one wave period.
For example, if the periodic waveform in Figure 2.14 repeats itself 240 times
per second, then all frequency components present in the waveform must
have moved through an integer number of cycles in 11240 of a second. Frequencies which have the potential to do this include 240 Hz (one complete
cycle in that time), 480 Hz (two complete cycles in that time), 720 Hz (three
complete cycles in that time), etc. However, these are simply the harmonics of
the 240 Hz fundamental frequency.
For the example above, all of the frequencies which are harmonics of the
240 Hz fundamental frequency have the potential to exist in the waveform
which repeats itself 240 times per second. However, the precise harmonics
which are actually present, and their relative amplitudes and phases, will vary
depending upon the precise shape of the periodic waveform.
This concept of harmonics is straightforward to grasp if there are multiple
sound sources, operating at 240 Hz, 480 Hz, etc. In this case, we could say
24
2. Background: Fundamentals of Sound
that the frequency content (from the individual sound sources) is driving the
shape of the periodic waveform. What is harder to grasp, and what is far, far
more common, is for the shape of the waveform to drive the frequency content. In order for a given periodic waveform to exist, it must include multiple
harmonics. That is the only possible way to obtain a repeating shape on a
cycle-to-cycle basis. Additional sound sources are not required to produce
this shape. Consider, for example, a loudspeaker which is outputting a single
frequency. If you place your finger on the loudspeaker you will distort the
shape of the sound wave. You haven't introduced additional sound sources,
but you have altered the harmonic content of the sound field simply by altering the shape of the periodic waveform.
This may all sound like a "mathematical" concept aimed purely at describing the shape of a wave. What makes it all so important, though, is how
"physical" things respond when the wave impinges upon them. To a physical
object, the source of the wave is irrelevant. All that a physical object sees is
the multiple harmonically related frequencies which make up the wave. Your
ear is the classic example. If you put your finger on the loudspeaker and
distort the shape of the output sound wave, what do you hear? Usually, it is
"buzzing." You aren't physically driving the speaker at the frequency of the
buzzing. However, the buzzing will be a harmonic of the frequency at which
you are driving the loudspeaker, a harmonic frequency which is mathematically necessary to produce a wave with the distorted shape. If you change the
position where you touch the loudspeaker, and so change the distorted "shape"
of the output sound wave, then you will also change the harmonic content of
the signal. As a result, it will sound different.
The question of why physical objects respond to different (sine wave)
frequency components in different ways is beyond the scope of this text. It is
a fundamental component of the study of Mechanical Engineering. It will
suffice here to say "that is the way it is," and that everyone has a salient
example: their ear.
Returning to our discussion of sources of harmonics, fans are not the only
pieces of rotating machinery that produce harmonics. Other examples include pumps (for example, in a swimming pool), engines, aircraft engines
(propeller aircraft can be especially painful), compressors (think of the large
units used by road crews!), and in fact anything that rotates. To a listener, the
"low"- frequency components (we will classify these shortly) sound like a
"rumble," while the high-frequency components sound like a "rattle."
2. Many Devices Which Use Mains Electricity
Mains electricity, the sort of electricity which comes from a wall socket,
provides alternating electrical current. The electricity is delivered in the form
of a wave, with a fixed frequency (60 Hz in places like the United States, 50
Hz in places like Australia). Many appliances which plug into the wall "latch
on" to the current frequency and, as a result, operate at this frequency. In turn,
Harmonics
ure
25
Cf!v?Y
coming out
FIGURE 2.15.
Sound field generation by a transformer.
the appliances may generate sound fields which are periodic, with frequency
components that are harmonics of the mains frequency (50 Hz or 60 Hz). This
is true for many low-cost electrical motors and fans. However, it is also true for
a large number of nonrotating devices such as transformers.
Transformers vibrate in response to changes in their internal magnetic
field. As this magnetic field is generated by the electricity entering the transformer, it is intuitively obvious that the vibrations must be, in some way,
related to the frequency of the electric current. They are, in fact, harmonically
related. The strength of the magnetic field increases and decreases in response to the current "wave." It also changes direction, from "north" to "south"
(to adopt the terminology commonly used to describe the two ends of a bar
magnet). However, to the metal components of the transformer, whether the
field is north or south has no bearing upon the physical response. They only
see "strong" and "weak." (Recall that while north and south magnets attract,
and north and north magnets repel, both north and south magnets will attract
steel. The concept of "repelling" is only important if there are two magnetic
fields.) The result is that the fundamental frequency of vibration is twice the
frequency of the current. This can be either 120 Hz (in the United States) or
100 Hz (Australia). We call the pattern of the resultant vibration rectified.
That is, there is no "negative" component, as shown in Figure 2 .15.
As before, while being periodic, neither the vibration nor the resulting
sound pressure field is perfectly sinusoidal (it is rectified). Therefore, it contains harmonics. Most people are familiar with the "buzz" caused by the
higher frequency harmonics in the sound pressure field generated by a transformer and/or power supply in an appliance.
3. Devices Which Are Driven Beyond Their Capabilities
A third common situation which leads to harmonic sound field generation
occurs when devices are driven beyond their capabilities. A common example of this is when low-quality or poorly chosen loudspeakers are used to
reproduce sound. If the loudspeaker is driven to the point where it "hits the
26
2. Background: Fundamentals of Sound
stops," where the cone travels the maximum possible distance and is then
physically restrained, the waveform produced by the loudspeaker will be
distorted. This distortion results in the introduction of harmonics into the
signal. (The degree to which this happens when the loudspeaker is not driven
to its stops is often referred to as harmonic distortion by manufacturers. This
provides a measure of the quality of the sound production.) Most people
would be familiar with the "buzz" produced by a small or low-cost loudspeaker which is driven to its limits by sound with a large "bass" component.
This is the generation of high-frequency harmonics. The generation of harmonics and periodic sound fields will take on greater significance after we
discuss how human beings perceive sound.
Side Story. Have you ever noticed that a train appears to change frequency as
it comes toward you and eventually passes you? (Steam or diesel trains that
is. The trains which produce a loud harmonic sound field.) This is due to a
phenomenon on known as Doppler shift. As the train approaches, it is producing pressure fluctuations at a given rate (say, 30 fluctuations per second for
argument's sake). If the sound source was standing still, subsequent peaks
and troughs of the acoustic waves would arrive at fixed intervals of time (a
new peak would arrive every 1130 of a second). However, when the train in
coming toward you, it is decreasing the distance over which subsequent waves
have to travel. This means that subsequent waves will take less time to arrive.
The result is that new peaks arrive at time intervals that are shorter than 1130
of a second. Maybe, say, 32 waves arrive in the space of a second, rather than
30. To the human receiving the sound, it appears that the frequency is now 32
Hz, not the original 30 Hz. The frequency appears (and is, in fact) increasing!
The amount of increase depends upon the proportional change in distance
between you and the train, that occurs between the subsequent generation of
sound waves. This proportional distance change increases as the train gets
closer. The result is that the train noise increases in frequency as the train gets
closer. The converse is also true: the train noise decreases in frequency as the
train goes past and gets farther away. This change in frequency is the Doppler
shift.
This shift in frequency also applies to light (in fact, the concept of a Doppler shift is more often applied to light than sound). Light can be split into a
spectrum in the same way as sound: red light is a "low"-frequency sine wave,
and blue light is a "high" -frequency sine wave. If light is being emitted by an
object that is moving away from you, then the light is made just a little bit
more "red" (the frequency drops just a little). Scientists can use this phenomenon to tell that the Universe is expanding. By looking at the spectra of light
from distant stars, and comparing it to the spectra of light from closer stars of
similar composition, it is apparent that distant stars have spectra with components that are lower in frequency than components found in the spectra of
closer stars (the spectrum experiences a "red shift"). The conclusion is that
the stars are moving away from us and the Universe is expanding.
Human Perception of Sound
27
Unfortunately, this is a proportional effect, and it is impossible to know
exactly how fast the expansion rate is, as this would require a knowledge of
the actual distance between Earth and the distant star. While scientists have
come up with a set of steps to estimate the distance, the accuracy of the
procedure is still open to debate. Why is this unfortunate? Because in order to
answer the "big question" of whether the Universe will expand forever or
eventually stop, or even will eventually collapse back on to itself, scientists
need to know the actual speed of expansion. Too bad.
Side Story. An interesting phenomena occurs when two sine waves with almost the same frequency and same amplitude are put together. Over the course
of many cycles, the peaks of the two sine waves come together and draw apart.
As a result, the pressure amplitudes will add together, cancel out, add together, cancel out, setting up a low-frequency "beating" phenomena. As the
two frequencies get closer together the beating will slow, eventually stopping when the frequencies are identical. This is a very convenient way to
match two frequencies, and is a phenomenon which is exploited by most
guitar players to tune their instruments.
Human Perception of Sound
Based upon the discussion in the previous sections, we now have a "tool"
which can be used to quantify characteristics associated with sound and its
perception: the tool of frequency analysis and the associated concept of frequency components.
The first question to ask is, do human beings actually perceive frequency
components in a sound pressure field or is the concept of frequency components just a mathematical tool? The answer is yes, human beings do perceive
frequency components in a sound field. Think of an orchestra. A person can
usually hear the "low frequency" of the bass instruments, such as a tuba or a
double bass, along with the high pitched notes of a violin or flute; we are
perceiving different frequency components in a single sound field. This conclusion is even more salient when the sound field comes from a single source,
such as a recording of the orchestra played from one loudspeaker. Here there
is only one point of sound field generation, but we can hear a variety of
instruments.
The next question to answer is, what is the range of frequencies that a
human being can actually hear? The human hearing range is usually cited as
being from 20 Hz to 20,000 Hz (the latter is usually denoted as 20 kiloHertz,
or 20 kHz), as shown in Figure 2.16. This upper limit, however, tends to be
somewhat high in practice; 16 kHz-18 kHz is probably more realistic.
There are two important points to note regarding the range of human hearing.
The first is that, while the range of frequencies that a human can perceive may
range from a few tens of Hertz to tens of thousands of Hertz, for a number of
28
2. Background: Fundamentals of Sound
Speech range: 125 Hz - 6000 Hz
Most crucial: 500 Hz - 2000 Hz
)
(
Range of human hearing (frequency in Hz)
I
20
Low frequencies
I
1000
Mid-range frequencies
20000
High frequencies
FIGURE 2.16. The range of human hearing.
practical reasons it is useful to consider 1000 Hz to be the middle of the range.
Humans do not hear/perceive frequencies in a "linear" fashion. For example, it is
straightforward to tell the difference between 40 Hz and 60 Hz, somewhat more
difficult to tell the difference between 1000 Hz and 1020 Hz, and almost impossible to tell the difference between 15,000 Hz and 15,020 Hz. Reality is better
reflected if the frequency range is divided up logarithmically, in the same vein as
sound pressure levels. This will be discussed further shortly.
The second important point to note is the range of frequencies required for
speech recognition. This is important for a number of reasons, ranging from
the assessment of the quality of a telephone to assessing the "importance" of
a person's hearing damage (that is, can they still communicate?). The overall
range of frequencies used in speech is something like 200 Hz-6000 Hz. However, the most important range of frequencies is more like 300 Hz-2000 Hz.
The typical male voice has its peak energy output around 350 Hz, while the
typical female voice peaks around 700 Hz. 6
One other interesting point to note is the relationship between "pitch,"
which is the subjective assessment of frequency, and the actual value of frequency. In general, humans perceive frequencies ranging from tens of Hertz
to a few hundred Hertz as "low pitched," frequencies from the mid-hundreds
of Hertz to a few thousand Hertz as "mid range," and frequencies above a few
thousand Hertz as "high pitched." However, if a signal contains several harmonically related frequency components, then it is usually the lowest-frequency fundamental harmonic which determines the pitch. If the fundamental
harmonic is removed (say, by active noise control) the pitch remains the
same. The ear "inserts" the missing fundamental frequency component! However, if other harmonics are removed, the pitch will change.
6 It is interesting to note that this frequency of peak energy output is changing, particularly for females. Regular surveys conclude that the female voice is becoming lower in
frequency, moving toward the frequency of the male voice. This trend poses interesting
questions for sociologists studying the "role" of women in society. It is well known that
babies prefer higher-pitched voices, while "power" figures in society are associated
with lower-pitched voices. Acoustics provides fuel for debate!
Human Perception of Sound
29
Humans are not, in general, good judges of pitch. Research has found that
humans tend to perceive frequencies below approximately 500 Hz to be higher
than they actually are, and frequencies above approximately 500 Hz to be
lower than they are. The difference between human perception and the actual
frequency decreases as the sound pressure level is increased.
It was mentioned that humans do not perceive frequency in a linear fashion.
For this reason, acoustics and noise control "professionals" tend to work with
sound split into frequency "bands" which are logarithmically related (as opposed to the linearly related bands of the Fast Fourier Transform, at, for example,
200 Hz, 202 Hz, 204 Hz, etc.) There are a variety of possible ways to construct
these bands. The industry standard bands, which are those agreed to by the International Standards Organization, are based upon increments in frequency of 10",
where n is a single decimal number ranging from 1.4 to 4.3. The most common
band, called an octave band, ranges across three of the above increments (for
example, n = 1.4, 1.5, and 1.6 comprise the 31.5 Hz octave band). The next most
common band is the one-third octave band, which uses just one increment.
The preferred frequency bands in the range from 100 Hz to 1250 Hz are listed
in Table 2.3. Note that n defines the center (or middle) frequency of the band, and
that the upper and lower limits placed on the band lie half-way between adjacent
bands. Note also that the band details repeat themselves with a multiplication of
ten. For example, the limits on the 1000 Hz one-third octave band are ten times
those on the 100 Hz band. The limits on the 10,000 Hz one-third octave band
would be ten times those of the 1000 Hz band, and so on. Finally, octave bands
can be viewed as comprised of three one-third octave bands, as indicated above.
For example, the 125 Hz octave band comprises the 100 Hz, 125 Hz, and 160 Hz
one-third octave bands. The frequency limits of an octave band range from the
lower band limit on the lowest constituent one-third octave band and the upper
band limit on the highest constituent.
TABLE 2.3. A samEle of Ereferred octave and one-third octave bands.
One-third
Octave
octave band
Lower
band
Band
band
centre
centre
number
limit
(tOn)
freguency
frequenc~
88
100
20
113
125
125
21
141
160
22
176
299
23
225
250
250
24
283
315
25
353
400
26
440
500
500
27
565
630
28
707
800
29
880
1000
1000
30
1130
1250
31
Upper
band
limit
113
141
176
225
283
353
440
565
707
880
1130
1414
30
2. Background: Fundamentals of Sound
One of the most important phenomena in noise control is that, while human beings can detect frequencies in the range of 20 Hz to 20 kHz, they do
not perceive all frequencies equally. Even more importantly, the potential for
various frequency components to damage human hearing is not uniform across
the frequency range. These two facts inherently govern almost all areas of
noise control and noise control legislation.
What do we mean when we say "humans perceive different frequencies
differently"? Basically, if a sound pressure field has a given amplitude, say of
1 Pa (94 dB), then it will appearlouder to the listener if thefrequency is 1000
Hz than if the frequency is 100 Hz. Further, it will appear much louder if the
frequency is 1000 Hz than if the frequency is 30 Hz. The amplitudes are the
same, but the perception is different.
This difference in perception has lead to the development of a set of standard "weighting curves" that can be applied to a given sound pressure measurement to make it better reflect human perception of the sound. These curves
have the somewhat dubious titles of "A" (for sound levels below 55 dB), "B"
(for sound levels in the range 55 dB-85 dB), and "e" (for sound levels above
85 dB). By far the most commonly encountered of these is the "A" curve, not
because most sound fields are below 55 dB, but because the A-weighting
curve most closely predicts the damage that can be caused to hearing by a
given sound field. That is, the potential for a sound field to damage human
hearing is a function of the frequency content of the field, and this function
closely matches the A-weighting curve.
A plot of the A-weighting curve is given in Figure 2.17, with some actual
numbers listed in Table 2.4. Note that the low frequencies are heavily discounted in their contribution to the A-weighted average sound pressure level,
while the mid-range frequency components are left essentially unaltered. This
has important implications for active noise control, as will be discussed later.
How can the A-weighting curve be applied to a measurement? One way is
as depicted in Figure 2.18. First, break the measurement into one-third octave
bands, and then add or subtract the number of decibels shown in the weighting curve on a band-by-band basis. For example, referring to Table 2.3, the
measurement in the 100 Hz one-third octave band would have 19.1 dB removed from it, the measurement in the 125 Hz one-third octave band would
have 16.1 dB removed from it, etc. Finally, add the weighted levels together
to get a weighted RMS (average) level. The weighting average is denoted
with an "A"; for example, 85 dB(A). Professional measuring equipment does
this sort of procedure automatically.
Acceptable Sound Levels
The susceptibility of an individual to bodily damage from some environmental condition is difficult, if not impossible, to assess analytically, and will
vary from individual to individual. A common example of this is smoking. It
Acceptable Sound Levels
31
o
.......
CD
"0
-
'-'
....
o
-10
()
co
-
-; -20
A-weighting curve
c
..c:
Ol
.Ci.i
S -30
-4 0 t.....L....J......J....J.....L....L.....J.......L....J'-'--'--'-J.....L...L...J....L...L....L.....J.--L-L.J.......J
31.5
63
125
FIGURE 2.17.
500
2000
8000
250
1000
4000
Frequency (Hz)
The A-weighting curve.
has been known for years that smoking is damaging to an individual's health,
and yet most people know someone who has smoked heavily for decades with
little or no apparent bodily damage. Conversely, there are individuals who
apparently develop lung cancer in response to relatively small amounts of
passive smoking. Hearing damage due to environmental noise is much the
same. Criteria for "what is acceptable" in terms of an individual's exposure to
sound levels, and how to modify the allowable exposure time in response to
increasing or decreasing sound levels, are derived from empirical data. The
subjectivity that is inherent in the assessment of empirical data has meant
TABLE 2.4. A sam}2le of A-weighting values.
One-third
One-third
A-weighting
octave
octave
band
band
correction
- 19.1 dB
1000
100
- 16.1 dB
1250
125
- 13.4 dB
1600
160
- 10.9 dB
2000
200
- 8.6 dB
2500
250
- 6.6 dB
3150
315
4000
400
-4.2 dB
5000
- 3.2 dB
500
6300
- l.9 dB
630
- 0.8 dB
8000
800
A-weighting
correction
0.0 dB
0.6 dB
1.0 dB
1.2 dB
1.3 dB
1.2 dB
1.0 dB
0.5 dB
- 0.1 dB
- 1.1 dB
32
2. Background: Fundamentals of Sound
RMS value =
87 dB
Gl
"0/\
/1
I....J
TIme
"Raw"
measurement
:::l
.t::
~
~
~Frequency>
<{
_~ RMS value
84 dB(A)
=
Apply
A-weighting
FIGURE 2.18. Applying the A-weighting curve.
that different criteria have been adopted in different countries. Noise control
professionals must be fluent in the regulations of their specific areas. The
specific values that will be given in this section will not be correct for all
countries. However, the general framework of the criteria and the "ballpark"
numbers will be globally applicable.
So what is an acceptable level of sound (noise)? Criteria used to answer
this are based on the following criteria:
1. the risk of hearing damage to an individual must be small;
2. the reduction in work efficiency due to background noise must be small;
3. speech is possible (where "necessary"); and
4. noise levels in the community must be acceptable.
A cynical person might call these the "motherhood statements of noise control."
Consider, first, criterion 1: The risk of hearing damage to an individual
must be small.
What kind of damage? The most common form of permanent damage is the
loss of perception (an inability to hear low-to-moderate sound levels) in the
mid- to high-frequency range. This includes part of the range required for
speech recognition. While this is a stereotypic trait of aging men (a condition
known as presbycusis), it can also affect young and old men and women who
are exposed to excessive noise levels.
Most people would at some time have suffered from a "temporary threshold shift," where perception is lost for a short period of time. This may occur,
for example, after attending a (very) loud concert or nightclub and is often
accompanied by a ringing in the ears (tinnitus). The effects of occasional
exposure to loud noise are largely reversed after a period of time away from
the noisy environment. However, persistent exposure (even weekly at a loud
nightclub) can lead to permanent damage.
How much reduction in the ability to hear is too much? This is often
answered by reference to the ability to hear speech. It is generally accepted
that if an individual's perception of sound is reduced by less than 25 dB from
that of a healthy young person, then the hearing damage is "acceptable"
Acceptable Sound Levels
33
(insofar as hearing damage goes-if you loose only one of ten toes, is this
acceptable?). Once the hearing loss is greater than 25 dB then the individual
has serious impairment. If the loss of hearing perception is in the range of 25
dB-92 dB, then the individual is said to be hearing impaired with the degree
of impairment equal to 1.5 percentage points for every decibel of perception
loss over the 25 dB threshold. If the hearing loss is over 92 dB, then the
individual is deaf.
So how much noise can a person be subject to over the course of his or her
working life and still hope to hear without impairment? Based upon the idea
that perception of speech is the most important criterion, then assessment of
human data suggests that exposure to 80 dB(A) for 8 hours per day, 250 days
per year, for a 40 year working life is an acceptable limit. However, most
countries have legal limits that are greater than this. Two common legal limits
are 90 dB(A) exposure for 8 hours per day, and 85 dB (A) exposure for 8 hours
per day. For the former limit, data suggests that 25% of the population will
suffer some form of permanent hearing damage after 30 years of daily exposure, and 40% will develop permanent damage after 40 years. If the latter
limits were adopted, then the percentage of the population which would suffer permanent hearing damage in each instance would drop by something like
10 percentage points.
What happens if you are forced to work in a noisy environment? The most
sensible thing to do is to wear hearing protection. However, if you must enter
a noisy environment, then the allowable exposure time must be reduced. The
generally adopted formulas are based upon the idea of cutting the exposure
time in half (daily or weekly) every time the sound level goes up by x dB(A),
where x is a country-specific or possibly industry-specific (such as the Navy)
number. Perhaps the most common number for x is 3 dB(A). This means that if
an individual was working in a country with a legal limit of 90 dB(A) exposure and his or her working environment was actually 93 dB(A), they could
legally only work there without hearing protection for 4 hours per day instead
of 8, or perhaps 20 hours per week instead of 40. If the level was 96 dB(A),
then the time would be further halved to 2 hours per day or 10 hours per week.
A discussion of how to arrive at the number "3 dB" for the exposure time
halving criterion will be taken up in the next chapter.
Consider the dilemma of a nightclub bartender. It is not uncommon to have
sound levels of 105 dB(A)-110 dB(A) in the vicinity of a bar at a noisy
nightclub. Based upon the lower end of this range, the allowable time of
exposure should be halved five times (105-90 = 15, which is five times 3
dB(A) above the legal limit). This means that the bartender can legally work
for slightly more than 1 hour per week, and even then will stand a significant
risk of permanent hearing damage if the job becomes a life-long profession.
The bartender should be wearing hearing protection (ear muffs). However,
then he or she would not be able to hear the patrons ordering drinks. There
is the dilemma. (Most bartenders will tell you that they can't hear the
patrons anyway and usually lip-read drink orders. So the actual dilemma is
34
2. Background: Fundamentals of Sound
more likely to be based upon appearance, as a bartender with ear muffs might
"look silly.")
How much can hearing protection help? The ability of hearing protection,
such as ear muffs, to reduce the exposure of the ear to noise is very much
dependent upon the quality of the protection device and the quality of the fit.
Perhaps the most commonly encountered hearing protection devices are foam
ear inserts (which look like oversized cigarette butts). These are "squished"
and inserted into the ear, where they expand to fill the ear canal. These can
provide 10 dB(A) or more reduction of the sound levels at the ear drum.
However, the quality of the fit and the cleanliness of foam insert (which becomes dirty after one or two fittings) playa huge part in determining the
actual reduction. It is possible for the noise reduction to become 0 dB(A) in
some cases. This user-specific quality of foam inserts has seen them labeled
"not legally acceptable for hearing protection" in some countries.
High quality ear muffs can provide 20 dB(A) or more attenuation of the
sound pressure levels at the ear. Specifications are normally available from
the manufacturer.
Consider now criterion 2: That reduction in work efficiency due to
background noise must be small, along with criterion 3: That speech is
possible (where "necessary").
There are two aspects of efficiency reduction which must be considered here:
the reduction in efficiency due to the reduced inability for workers to communicate, and a reduction in efficiency due to individual worker stress. The
former the ability for workers to communicate, yields a set of allowable sound
pressure levels that vary as the distance between the talker and listener increases (interested readers are referred to American Standard ANSI S3.14 for
more information). However, to generalize, in environments where constant
communication is required (such as a foreman's office), the background noise
levels should be kept under 70 dB(A).
The latter criterion relating to levels of stress in an individual to background noise levels is difficult to quantify. A good definition of an individual's
stress is the difference between what they are asked to cope with, and what
they perceive they can actually cope with (thus what is stressful for one person is not necessarily stressful for another). It is possible to say that too little
noise will often reduce the efficiency of individuals performing manual tasks
(that is, they end up wanting to go to sleep!). However, too much noise also
leads to a reduction in efficiency, and irritability which continues after the
work shift is completed.
Finally, consider criterion 4: That noise levels in the community must be
acceptable.
The limits placed on sound pressure levels "in the community" are dependent
upon what "the community" actually is. Generally speaking, recommended
Acceptable Sound Levels
35
levels inside areas of meetings and study (relatively "quiet" areas) are in the
range of 30-45 dB(A). Noisier areas, such as bus stations, may be slightly
higher than this, and quiet areas, such as hospitals, may be slightly lower.
Specification of the acceptable level of noise at the boundaries of domestic dwellings is also dependent upon a number of factors, including time of
day, zoning of the dwelling, characteristics of the noise, etc. However, a level
of 40 dB(A) is a good starting point.
3
Fundamentals of Noise Control
Active and passive noise control are two approaches to a common problem: how to
get rid of unwanted noise. Active noise control aims to attenuate unwanted sound
by introducing an electronically generated "canceling" sound field. Passive noise
control aims to attenuate unwanted sound by modifying (structurally) the characteristics of the environment in which the sound source operates. In many ways, the
two approaches are complementary, rather than alternative. To arrive at this conclusion, it is necessary to investigate the conditions under which each noise control
technique performs well and the conditions under which they do not.
The purpose of this chapter is to provide some general background information pertinent to the investigation of noise control system performance
and to understanding the physical mechanisms which are employed by a
given noise control technique. The investigation will concentrate on broad
"physical" characteristics related to the placement of components and will
not look at the questions relating to the electronic part of the active noise
control system. This latter topic will be the subject of two subsequent chapters (7 and 8). We will begin by looking at the questions of "what are active
and passive noise control." We will then compare the techniques as applied to
common problems: control of sound radiating into free space, control of sound
propagating in a duct, and control of sound fields in an enclosed space.
It should be noted that it is not possible to provide anything more than a
superficial discussion of passive noise control in the space provided. Interested readers should consult a dedicated text for more thorough information. 1
Prerequisite Discussion: Power and Impedance
Before we can discuss active and passive noise control and the physical phenomena which make them work, it is necessary to discuss the concepts of
sound power and impedance.
Consider the following question, which has a pertinent acoustic analogy:
why is it that we can plug a 25 watt light bulb into a socket, and then a 100
I For example, D.A. Bies and C.R. Hansen (1996). Engineering Noise Control, 2nd ed.
E&F Spon: London.
36
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
Prerequisite Discussion: Power and Impedance
37
watt bulb into the same socket, and draw out different amounts of energy?
After all, the socket is the same, as is the wiring and the electricity available
to the socket from the mains. Further, why is it we can plug a portable heater
into the same wall socket and draw out 20 times more electrical energy?
Again, the socket and connection are the same.
Consider another related question, which is easier to answer. Why is it that
we can take a garden hose, connect it to a tap, and get either a dribble of water
out of it or a raging gush? We have the same hose, the same tap, the same pipes
connected to the house, and yet different amounts of water flowing out of the
hose. The answer is simple: we turn on the tap a small amount for the dribble
or a large amount for the gush. In engineering terms, we could say we altered
the impedance of the system, which in turn altered the amount of flow.
In engineering and science, it is common to discuss phenomena (such as
acoustics, electricity, etc.) that involve the movement of something (sound
waves, electrons, etc.) in terms of three quantities: a quantity relating to a
potential difference, a quantity relating to flow, and a quantity relating to
impedance (or its inverse, admittance). A common potential difference quantity is pressure: there is a difference in water pressure between the house pipes
on one side of the tap (high pressure) and the garden hose on the other side of
the tap (low pressure), and so there exists the potential to do some work (by
tapping into the flow of water as it moves from high pressure to low pressure
when we turn on the tap). Another common potential difference quantity is
voltage: there is a difference in voltage between wires on one side of the wall
socket (high voltage) and the light bulb on the other side of the socket (low
voltage), and so there exists the potential to do some work (by tapping into
the flow of electricity when we connect the light bulb). A third common
potential difference quantity, and one that will be important when discussing
one aspect of active noise control, is force. If you place your finger on an
object, such as a table or a loudspeaker cone, and push, then there is a difference between the force on one side of the object (where you are pushing) and
the other (where there is no pushing), and so there exists the potential to do
some work as the object moves.
Common flow quantities related to the above potential difference quantities are flow rate for the water flow (how many gallons or liters flow away from
the source per second), current for electricity (how many electrons flow away
from the source per second), and velocity for the force case.
What determines how much flow actually occurs? An impedance quantity
("impedance" from the word "impede," or to stop from moving). If a system
has a high impedance, then the flow is reduced. If a system has a low impedance, then the flow is increased. For example, when the tap is opened only a
small amount then the impedance "seen" by the water in the pipe as it looks
at the tap is high. As a result, only a small amount of water is allowed to pass
over a given period of time. As the tap is opened further the impedance is
reduced. More water is now allowed to pass, until we reach the point where we
have a gush.
38
3. Fundamentals of Noise Control
In the case of electricity, as the supply of electrons "looks" out the wall
socket and possibly down a wire (say, leading to the light bulb), it sees some
resistance to movement. In the case of the 25 watt bulb, the resistance is high
and so relatively few electrons can pass through the wire in a given period of
time. In the case of the 2000 watt portable heater the resistance is low and so
a relatively large number of electrons can move down the wire in a given
period of time.
In the case of the force input, we can talk about mechanical impedance.
The mechanical impedance of, for example, a table top (which moves a relatively small amount when you push it) is greater than the mechanical impedance of a loudspeaker cone (which moves a larger amount).
In all of these cases, we can write a mathematical relationship
flow = (potential difference)/(impedance).
What, you may ask, does this discussion have to do with acoustics and sound?
The generation of a sound field involves the same three quantities as described above:
pressure (sound pressure), which is the potential difference quantity;
volume velocity (displacement of a volume of air over a given period of time),
which is the flow quantity; and
acoustic impedance (also referred to as radiation impedance), which is the
impedance quantity.
Hint. When thinking about pressure, volume velocity, and impedance, it is
useful to consider a "personal" analogy. Pretend that you are the sound field,
and that you are trying to get through the environment in front of you. View
the environment as a jungle. As you try to pass through the jungle you have
the ability to provide a certain amount of pressure against what is in front of
you, be it trees, grass, etc. This is analogous to sound pressure, or potential
difference. How fast you will travel (analogous to volume velocity) is a function of the things in front of you trying to impede your progress (trees, grass,
etc.) The density of the things, and how much they hinder your progress, is
analogous to impedance.
Example. It is common to classify sound sources as either "constant pressure" (constant potential difference) or "constant velocity" (constant flow).
An example of a constant pressure sound source is a fan. The blades of the fan
rotating at a certain speed will compress the air by the same amount regardless of the air flow. Using our "jungle analogy," this is equivalent to saying
you will flex your muscles and push with a constant force regardless of what
is in front of you. If the impedance is low, such as a little light grass, you will
travel relatively quickly for your constant push. If, however, the impedance is
high, such as when you hit a tree, the flow will slow (or even stop). You will
simply provide a constant push. This is the same for the sound pressure field
Prerequisite Discussion: Power and Impedance
39
leaving the fan. For a given acoustic pressure (corresponding to a given fan
rotational speed), if the acoustic impedance is low then the flow of acoustic
waves away from the fan will be high. However, if the acoustic impedance is
high then the flow of acoustic waves away from the fan will be low.
An example of a constant volume velocity source is a loudspeaker. Regardless of what is in front of it (within reason), it will continue to displace a
constant volume of air. This is analogous to saying you are going to make
your way through the jungle at a constant speed regardless of what is in front
of you. Pretend you are tied to a rope, being pulled along at a constant speed.
If the impedance is low, just a little light grass, the pressure you must exert on.
the environment to get through at the given speed is relatively small. If,
however, the impedance is large (vines, tress, etc.), then you must exert a
much greater pressure to keep moving. This is the same for the sound pressure
field leaving a loudspeaker. For a given volume velocity (speaker cone displacement), if the acoustic impedance is high then the sound pressure levels
will also be high. However, if the acoustic impedance is low then the sound
pressure levels will also be low.
While it is useful to be able to relate the three quantities pressure/volume
velocity/impedance using the mathematical relationship stated in the previous paragraph, more important relationships from the noise control perspective are those that relate the above quantities to power. Power is the flow of
energy per unit of time. It is the quantity which "does something," which acts
to change the environmental circumstances in which we find ourselves. Power
is most commonly defined in units of watts (W), which describe the flow of
energy in Joules per second. For example, a 25 watt light bulb has 25 Joules
of energy flowing through and away from it every second, whereas a 100 watt
light bulb has 100 Joules of energy flowing through and away from it every
second. Note that in the case of a light bulb, the majority of this energy flow
is commonly in the form of heat, not light.
Power is normally defined by the mathematical relationship
power =(potential difference) x (flow).
Of course, the mathematical relationship between potential difference, flow,
and impedance can be used to describe power in terms of different quantities.
A common description of power for acoustics work is
power =(impedance) x (flow).2
An important observation at this point is that acoustic power is related to
sound pressure squared, and not simply sound pressure. This form of relationship can be stated using the impedance relationships
power =(potential difference)2J(impedance).
40
3. Fundamentals of Noise Control
From a noise control perspective, why is sound, or acoustic, power important? Acoustic power, being the flow of energy away from a sound source,
determines how loud the noise will be away from the immediate vicinity of
the sound source. That is, sound power provides a measure of how much
energy will flow away from the sound source and potentially show up to
bother you or me.
As with sound pressure, the sound power radiated from a noise source can
be divided into frequency components (energy flowing away from the source
at, say, 100 Hz, 200 Hz, whatever). It can therefore also be A-weighted to
better reflect what potential this energy flow has for damaging human hearing. It is the resulting A-weighted acoustic power measurements that are commonly cited on domestic appliances. For example, your new dishwasher may
be labeled "60 dBA"-this is a measure of A-weighted sound power in decibel units.
There are several questions that arise at this point in the discussion, including; (1) What is the magnitude of acoustic power? (2) What is the formula
for a decibel measurement of acoustic power? and (3) How does acoustic
power relate to acoustic pressure?
The Magnitude of Acoustic Power
Acoustic power is an incredibly small quantity compared to other forms of
energy flow which spring to mind (such as 100 watts from a light bulb). Most
commonly encountered domestic sound sources, such as refrigerators, have
acoustic power outputs in the range of milliwatts (1 milliwatt is 111000 of a
watt). If you really splashed out and purchased a high-end sound system, with
a loudspeaker/amplifier combination rated at hundreds of watts, you might
be able to produce 1 watt of acoustic power. Maybe. As with the light bulb, a
large portion of the power is dissipated as heat.
Decibel Units for Acoustic Power
As was described in the previous chapter, the decibel scale is a logarithmic
scale. The numbers in the decibel scale increment proportional to a squared
quantity (such as pressure squared) expressed as a power of 10. There is a
small but important difference between the decibel measurement of acoustic
power and the decibel measurement of acoustic pressure. That difference arises
because acoustic power is inherently a "squared" quantity. We saw that acoustic power can be related directly to pressure squared, and not just pressure, via
the impedance. As a result, if sound pressure increases by a factor of 10 then
the decibel measure of pressure increases by 20 dB (this was shown in the
previous chapter). However, if the sound power increases by a factor of 10, the
decibel measure increases by 10 dB, not 20 dB as in the pressure case. Simi-
Power, Pressure, and Hearing Loss
41
lady, if the sound power doubles then the decibel measure increases by 3 dB,
not 6 dB, as in the case of sound pressure.
Decibel units for sound power are derived from a base measurement of 10-12
watts (that is, 0.000000000001 watts!). This amount is given a value of 0 dB.
Therefore, acoustic power W in decibels is calculated by
W (decibels) = 10 log W (watts) + 120 dB.
Power, Pressure, and Hearing Loss
An interesting question which relates the discussion here to that in the previous chapter on acceptable noise levels is this: What is responsible for damaging your hearing? Is it the flow of energy into your ear (related to sound
power), or is it the potential difference caused by the incident sound pressure
across the ear? Further, what difference does the answer make?
The answer to the first question is, we really don't know for sure whether it
is pressure, power, or some combination of the two that damages hearing. As
with most biological criteria (such as damage to hearing due to noise, damage
to your brain due to head impact, etc.), the true mechanism of damage in an
individual case is a complex interaction of multiple variables mediated by
some genetically determined susceptibility (read these long words as an academic way of saying "it's hard to tell for sure"). Therefore, the criteria are
derived by plotting "points" (people) on a graph and drawing a line to provide some average answer. For example, you might take 10 people who have
worked for 30 years in different environments with different background noise
levels, measure the hearing loss of each person, and try to derive some mathematical relationship between hearing loss and background noise level. By
this, we mean person 1 was subject to 95 dB for most of his working life and
has lost 40% of his hearing, person 2 was subject to 70 dB for most of her
working life and has lost 5% of her hearing, etc. The resulting graphical
relationship between the points (level versus hearing loss) should provide
you with an indication of the quantity which is responsible for hearing damage, pressure, or power.
How could you arrive at some conclusion about what damages hearing by
plotting points on a graph, and what difference does it make? Basically, if
sound pressure was responsible for hearing damage, then you would assume
that every time the sound pressure doubled the risk of hearing damage would
also double. Noting that a doubling of sound pressure is equivalent to a 6 dB
increase, this means (roughly) that if person A worked in an area that was 6 dB
louder than the area person B worked in, and both worked for exactly the
same amount of time, then person A would be twice as likely to develop
hearing damage. The data on the graph should reflect this relationship. However, if sound power, or energy flow, was responsible for hearing damage then
you would expect the risk of hearing loss to double for every 3 dB increase in
42
3. Fundamentals of Noise Control
sound pressure (as sound power is related to sound pressure squared). When
you plot points of sound pressure level versus hearing loss for a given exposure time the data should reflect one of these "laws" (risk of hearing loss
doubles with every 3 dB or 6 dB increment). This will provide the answer to
the original question of what is responsible for hearing loss.
Unfortunately, data relating biological damage to some environmental factor
never plots a neat straight line. In the case of noise levels and hearing loss, there
is so much data scatter that drawing a line through the points has some degree of
subjectivity. The present accepted line of thought (generally) is that power damages hearing, and so the line should indicate that the risk of hearing damage
doubles every time the background pressure level increases by 3 dB, not 6 dB
(although this view is being challenged in the academic publications). Note that
this is also the conservative result, which is safer for workers.
What difference does it all make? If the base figure for worker occupational health and safety legislation is, say, 90 dB(A) for 8 hours exposure
every working day, then how long can a worker stay in an area with noise
levels above this? For example, if a worker is situated in an environment of 96
dB(A) (without hearing protection), how long can they work there? If sound
pressure was thought to be responsible for hearing damage, then the exposure
time should be halved for every 6 dB increase in level. The conclusion is that
the worker can remain in the 96 dB(A) environment for 4 hours (8 hours/2).
However, if sound power is taken to be responsible for hearing damage, then
the exposure time should be halved for every 3 dB increase in level. The
conclusion is that the worker can remain in the 96 dB(A) environment for 2
hours (8 hours/2 (to 93 dB(A»/2 (to 96 dB (A» =2 hours). Most countries and!
or organizations have legislation that sits somewhere between the 3 dB and 6
dB rules.
Real and Imaginary Impedance
Thus far we have discussed "impedance," although occasionally the word
"resistance" has crept into the text once or twice. What is the difference
between impedance and resistance? Mathematically, impedance is a complex
number quantity, with both a real and imaginary component. The real part,
often labeled resistance relates to acoustic power flow (or "real" acoustic
power). This is the quantity that governs how much sound energy moves
away from the source, travels up the street, and bothers you and me. Resistance, or "active impedance," is usually the quantity of most interest in noise
control.
The imaginary part of impedance, or "reactive impedance," relates to sound
which does not travel away from the source. This sound naturally reduces in
level as you move away from the source, and generally does not bother people
who aren't sitting immediately next to the source. In this text, we will not be
terribly interested in reactive impedance.
What Is Passive Noise Control?
43
What Is Noise Control?
Put simply, noise control is the reduction in amplitude, or the attenuation, of
unwanted sound. In terms of the previous discussion, we can divide noise
control approaches into two general categories: reduction in the total sound
power flowing away from a sound source, and redirection of the flow of acoustic
energy such that the pressure in the direction of humans is reduced. The
former approach will often provide "global" results, where noise levels are
reduced everywhere. The latter approach will often yield an increase in noise
levels in one location in response to a decrease in noise levels elsewhere.
This is because the total energy in the form of sound is unchanged, and you
are simply channeling the energy flow in another direction.
What Is Passive Noise Control?
Passive noise control refers to noise control which results from modification
of the environment in which the sound source operates. Passive noise control
probably accounts for 99.9999% (or more) of all noise control currently practiced. It is often cheap, simple, and has a mix of common sense and the laws of
physics.
While there are countless specific techniques for passive noise control,
they can all be lumped into the two categories mentioned above: those which
aim to reduce energy flow in the form of sound, and those which simply
redirect the path of sound energy flow away from humans. The former category can be further divided into several subgroups. The first subgroup includes techniques which aim to "absorb" sound energy and turn it into a very
small amount of heat. The use of acoustic insulation ("fluff') to muffle sound
provides a common example of this. A second subgroup aims to reduce the
volume velocity of the noise source, often by attenuating vibration. Common
examples here include the undercoating of cars and the use of rubber isolators
under motors. The third subgroup of energy-based passive noise control techniques explicitly targets a change in acoustic impedance as a way to reduce
acoustic power output. A common example here is a car muffler.
Techniques which aim to redirect acoustic energy flow generally involve
the erection of some sort of barrier or wall. While this may incorporate some
form of energy absorption, such as a surface treatment (fluff) on the wall, the
main effect is to force the acoustic energy flow elsewhere. A common example
can be found in the erection of walls along busy highways to shield residents
who live close-by.
It is interesting to consider what happens when you erect such a wall.
Suppose, for example, you have a swimming pool, and the pump and filter
system is near the boundary of your property and your neighbors. Your neighbor complains about the noise, and so being a socially responsible person
you decide to erect a large brick wall behind the pump to shield your neigh-
44
3. Fundamentals of Noise Control
bor from the noise. What have you done? You have redirected the acoustic
energy flow away from your neighbor's house. Where does the energy go? A
small amount of it will be absorbed when the acoustic waves hit the wall, but
in the absence of an explicit surface treatment on the wall most of the energy
will be reflected back in your direction. The end result: twice as much energy
is flowing onto your side. You have just doubled the sound pressure levels in
your yard! The price of luxury.
What Is Active Noise Control?
Active noise control is a noise control technique that aims to reduce sound
levels by "canceling" the unwanted acoustic waves with a second set of electronically generated acoustic waves of equal amplitude but opposite phase.
Cancellation works because the acoustic environment is "linear," and so the
principle of superposition holds (in plain English, 1 + (-1) = 0 when adding
two acoustic waves).
Active noise control accounts for a minute proportion of noise control
applications. There are several reasons for this. Traditionally, the major stumbling blocks in system implementation have related to the physical requirements of the technology (the quality of loudspeakers, the capabilities of the
electronics, etc.) and the associated cost. Combined with this is the inaccessibility of the technology. While the general idea of two waves canceling each
other out is a simple one, actually implementing a control system to do this,
in a way which is robust enough to call "commercial," is extremely difficult.
Active control has also suffered from the "universal panacea" problem. Commercial product developers (and their bosses) have had unrealistic expectations about what the technology is capable of, and view the whole area as
slightly "fraudulent" when their attempts at implementation fail (usually for
sound physical reasons). Slowly, these problems are being overcome. A cynical person would add they are being replaced by new problems, which are
entirely legal and patent-related.
As with passive noise control, active noise control applications can be
split into two groups: those which aim to reduce acoustic energy flow, and
those which aim for local sound attenuation without any overall reduction in
energy flow. The majority of applications which are viewed as "successful"
are in the former category, where the energy flow is reduced and so attenuation is global. Local sound attenuation, which in the case of active noise
control means local sound cancellation rather than energy redirection, usually leads to disappointment. The reasons for this, and the mechanisms employed by active control to reduce the flow of acoustic energy, will be
discussed in the next chapter.
Having broadly outlined what active and passive noise control are, and
now being armed with a knowledge of acoustic power and impedance, we are
in a position to examine a few common noise control targets, or categories of
What Is Active Noise Control?
45
problems. The discussion will in no way constitute a thorough investigation
of techniques. Rather, the aim is to describe broad physical principles. The
categories of problems are free space radiation (Chapter 4), sound in enclosed
spaces (Chapter 5), and sound propagation in ducts (Chapter 6).
4
Free Space Noise Control
The first noise control category of interest here is acoustic radiation into free
space. What is acoustic radiation, and what is free space? Acoustic radiation
is simply the generation of sound waves by a source. Acoustic radiation is a
term commonly used amongst noise control practitioners. The term "free space,"
or "free field," refers to the environment in which the sound source is operating. Free space means that there is nothing to reflect the sound back; it can
travel away forever. This is as opposed to acoustic radiation into an enclosed
space, like a room. The free space environment is also referred to as "anechoic"
(literally, no echo). If you are working in the field of noise control, then you
may further subdivide free space into spaces where unreflected acoustic radiation is possible in only some directions, like a "half-space" (for example,
if the sound source is sitting on the floor, it can only radiate up, not down).
This might also be referred to as "semianechoic." However, for our purposes
of qualitative results examination, we will not be particularly fussed. We will
simply assume that the sound can travel away without interference.
Having defined free space, there is a variety of common acoustic radiation
problems of this type which immediately come to mind: electrical transformer
substations, swimming pool pumps, the Rolling Stones playing at an outdoor
stadium (remember, noise is in the ear of the beholder), traffic on a busy road
next to a residential area, etc. What can be done to fix these problems, to
reduce the noise levels experienced by some human observer?
Passive Noise Control Approaches
Put simply, with passive noise control there are two approaches to solving the
problem: put a wall between you and the noise source, or else put a box over
the source. Consider the first of these options, the wall (noise control practitioners tend to call these "barriers," which sounds much more technical than
a wall; we will adopt this terminology here, too). The aim of building a barrier
is to redirect the acoustic power flow away from whatever is behind the barrier. To be effective at this, the barrier must be constructed from "heavy"
material. Technically, this means material with a high surface density. This
point will be discussed further in the next chapter.
46
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
Passive Noise Control Approaches
47
Side Story. I was once doing some consulting work for an automobile manufacturer, looking at problems of noise in the Press Shop. The Press Shop is
basically a huge warehouse-like building filled with machines that punch
components out of rolls of sheet metal, up to 6 mm (quarter inch) thick. The
average noise level in walking around the plant was of the order of 110
dB(A), which is loud. Near a machine, however, it could peak at around 130
dB(A), which is really loud. What I remember most about the job was the staff
coffee area, which was a large fenced-off area in the middle of the shop. The
fence was the common mesh-stuff that is seen everywhere, around schools,
playgrounds, etc. To reduce the noise in the coffee area, someone had decided
to turn the fence into an acoustic barrier. The staff apparently rallied around
the idea, and decided to do this conversion themselves (I imagine there was
no money available from management, although I never asked). In looking
around for a material which they could obtain cheaply, with a contribution
from everyone, they decided on: egg cartons. I was dumbfounded at the sight:
thousands of egg cartons, each lovingly tied to the fence. I was told that the
staff were very proud of this - and that the coffee break was much quieter now.
Unfortunately, I fear that the construction would have been useless insofar
as an acoustic barrier goes. The underlying assumption of an acoustic barrier
is that sound cannot pass through the barrier; it can only go around. Aside
from the fence being too low, the egg cartons would not have been able to
impede the low-frequency noise that dominated the environment (discussed
in a moment). I didn't have the heart to tell them that, apart from being a
source of morale improvement for the staff, they had erected art, not function.
Upon later consideration, the comment of a "quieter coffee period" puzzled
me. Knowing that the plant did not possess acoustic measuring equipment
(that's why I was there), the result must have been subjective. Knowing how
much attenuation is required before a difference is really noticeable (discussed in the previous chapter), I am left with several possible conclusions:
1. Mind over matter; the importance of worker happiness in any environ-
ment can never be underestimated.
2. The staff were all deaf.
3. All of the machines were turned off during coffee-break time.
4. The egg cartons were in fact made of lead, and so my hypothesis about
them not being able to impede the sound was incorrect.
Assuming that the barrier is of sufficient surface density, the sound field on
the "shadow" side of the barrier (the side without the noise source) will be
due entirely to sound field diffraction over the top, and possibly around the
edges if the wall is short. Diffraction refers to the deviation of a wave which
occurs when the wave encounters an object. The most important parameter in
determining the sound levels from the diffracted wave is (obviously) the
height of the wall relative to the height of the sound source and the wavelength of the sound. Technically, this is normally quantified in noise control
work by a Fresnel number, which relates the difference between the length of
48
4. Free Space Noise Control
a direct path from source to observer to the length of the path over the wall
relative to the wavelength of sound. Other important factors include the ability ofthe ground to absorb sound (as sound can travel from the source, reflect
off the ground, hit the top of the wall, and diffract around), the ability of the
wall to absorb sound, the coherence of the sound, and the thickness of the
barrier. While calculation of the exact attenuation levels on the shadow side
is complicated and will vary from installation to installation, a reasonable
sized wall usually provides something of the order of 10 dB of attenuation.
Attenuation levels of 20 dB or more are almost impossible with a simple
barrier.
Consider now the technique of building an enclosure around a sound
source. As mentioned, this is a technique that aims to provide global sound
attenuation by reducing the flow of energy into the acoustic field. The first
question to ask is, if a sound source is enclosure by an air-tight structure, and
therefore the sound waves cannot escape, how does the sound get out? The
answer is vibration: the sound field shakes the walls of the enclosure, and the
vibrating walls reradiate the acoustic field. While this might be taken for
granted, it is interesting to ponder this point for a while. The amount of
energy flowing out of a sound source and into a sound field is incredibly
small, often less than the energy consumed by a single light bulb. And yet,
this sound field can shake massively thick walls and generate noise on the
other side.
Given that it is the vibration of the enclosure walls that is responsible for
sound generation, it is intuitive that "how difficult" a given wall is to shake,
quantified by its mechanical input impedance, will play an important role in
determining how effective the enclosure is at reducing noise. Generally speaking, a mechanical "thing," be it a wall, structure, whatever, is easier to shake
at low frequencies than at high frequencies. This leads to a general conclusion: enclosures do a better job at providing attenuation at high frequencies
than at low frequencies. This is a point that most people would already be
aware of, based upon their personal experience. However, we will state this
explicitly as it will be important when we come to compare active and passive noise control.
Just stating that "enclosures work better at high frequencies" tends to
overly simplify the situation. Shown in Figure 4.1 is the typical shape of a
"transmission loss" (T.L.) curve for a panel as used in the wall of an enclosure.
Transmission loss is the negative of the fraction of the energy in the incident
acoustic field which is reradiated by the panel. The greater the transmission
loss, the less the fraction of energy that is transmitted. In terms of decibels,
transmission loss =noise inside - noise outside.
There are two important points to note regarding the plot shown in Figure 4.1.
The first is the large dip in the curve at the first resonance frequency of the panel.
What is the resonance frequency, and why should the curve dip there? A resonance frequency is a frequency at which the mechanical component wants to
Passive Noise Control Approaches
en
~
TL controlled
by panel stiffness
49
TL controlled
by panel damping
c:
o
·iii
en
·E
en
c:
(1j
t=
Coincidence frequency
TL controlled
by panel mass
First panel
resonance frequency
Frequency (Hz)
FIGURE 4.1.
Typical plot of panel transmission loss as a function of frequency.
vibrate. It is the frequency where the energy contained within the structure is
most easily exchanged between potential and kinetic. Most people will have
experienced resonant behavior at some time. A common example is a plucked
guitar string, which vibrates predominantly at its first resonance frequency.
At resonance, the input impedance of the structure is very small; the structure almost "wants" to move. Therefore, there is a lot of movement for little
energy input. A lot of movement means a lot of sound being generated on the
other side. That is why there is a dip in the curve at this point.
Side Story. I was once asked by a white goods manufacturer to look at a
domestic refrigerator, to see why it was so noisy and if it could be made
quieter. The manufacturer was convinced that the noise was coming from the
compressor. These units were imported from South America, and the manufacturer wanted to return the entire shipment. However, after some investigation
it turned out that the problem was the small plastic fan used to stop the
buildup of frost in the freezer compartment (noisy, but frost free!). This fan
emitted a very small acoustic signal at a frequency of the order of 30 Hz. The
problem was, the side panel on the opposite end of the refrigerator had a
resonance at 30 Hz, and so for little input energy the refrigerator produced a
relatively large amplitude low-frequency humming noise. The refrigerator
was essentially a poorly designed enclosure for the fan noise source.
The other point to note on the transmission loss curve of Figure 4.1 is the
coincidence frequency. This is the frequency where the wavelength of waves in
the structure is the same as the wavelength of acoustic waves. At this point the
"sizes match," and the impedance is slightly reduced. The coincidence frequency
50
4. Free Space Noise Control
TABLE 4.1. Transmission loss of some typical materials.
Octave band (Hz)
Material
63
125
250
500
lk
3mmlead
24
30
31
27
38
sheet
22 g steel
14
20
23
3
8
sheet
21
16 g steel
9
14
27
32
sheet
¥linch
24
22
27
28
plywood
Plasterboard
15
20
24
29
9
on wood frame
37
40
46
Brick and
30
36
plaster wall
Single ',4 inch
32
17
11
24
28
glass window
Typical
12
14
16
13
interior door
2k
44
4k
33
8k
38
26
27
35
37
43
42
25
27
32
35
38
54
57
59
27
35
39
18
24
26
can vary markedly between materials. For a concrete wall, it may be a few tens of
Hertz, while for a lead curtain it may be tens of thousands of Hertz.
Given this variation amongst materials, is there any common factor that
produces a panel with a high transmission loss? There is: surface density. The
greater the surface density, the better the material. Lead, for example, is an
excellent material to make enclosures from (acoustically, that is; there are
other obvious health problems which can arise from constructing a "lead
wall"). A common commercial material is vinyl impregnated with minerals to
increase the surface density. The worst enclosure materials are those which
are light and stiff. Table 4.1 lists the measured transmission loss values in dB
for several materials.
Side Story. I was once investigating whether active noise control could be
applied to a small medical appliance. This appliance had an air compressor
inside, which produced a sound field with a fundamental frequency of 25 Hz
and every harmonic imaginable. The sound used to drive the patients crazy
after several days. The manufacturer had gone to great lengths to mechanically isolate the compressor from the case with rather expensive silicon mounts.
He had also called in an "expert" to advise on materials to put on the walls of
the cabinet enclosure, to improve the transmission loss. The expert recommended a custom foam product, which, by coincidence, he manufactured.
What was the problem? The foam had negligible surface density; it was essentially a % inch thick sponge. As such, it was completely useless in providing
low-frequency attenuation of the noise. The unit could have stood a few other
minor improvements. However, the overwhelming problem was the poor selection of material for providing transmission loss at low frequencies.
Active Control Approaches
51
The design of high-quality acoustic enclosures involves much more than
simply picking a good material. A description of design techniques is beyond
the scope of this book. The most important points, for our purposes here, are
(1) enclosures work best at high frequencies and worst at low frequencies, and
(2) a high-quality enclosure using means construction with materials of highsurface density, which usually means "heavy."
Active Control Approaches
Passive noise control techniques aim to reduce acoustic levels by altering the
"physical" environment in which the sound source operates, by adding enclosures or barriers in the case of free space radiation. Active noise control
techniques aim to reduce acoustic levels by altering the "acoustic" environment in which the sound source operates, by adding an additional acoustic
field which can (hopefully) cancel the output of the sound source. When we
examine the active control of free field acoustic radiation, our interest is in
variables which relate to the ability to provide cancellation, and provide it in
such a way that the sound reduction is global. 1 Typical variables of interest
include:
1. control source (loudspeaker) placement;
2. characteristics of the sound field (harmonic versus random noise);
3. group delays;
4. error microphone placement (if the active control system is adaptive, or
self-tuning) .
We will examine these variables by answering a series of questions.
Question 1: How does active noise control "physically" work?
Up to now, we have been content with the idea that active noise control works
through sound field cancellation. If we have an unwanted sound field, and we
are able to produce a second sound field with loudspeakers that is equal in
amplitude but opposite in phase, then the two will add together to be "zero,"
or quiet. This is a perfectly legitimate way to think of the technique, both
mathematically and intuitively.
How can we now fit this line of thought into our general framework of
noise control, which says we are either reducing or redirecting acoustic energy flow? The redirecting option, which provides local attenuation at the
1 Global is normally used to mean that the sound field at all locations is reduced.
However, we will use a slightly more lenient definition: global will be taken to mean
that, on average (over all locations), the sound level is reduced. This definition means
that it is possible to have some locations which actually experience an increase in sound
level when the active control system is switched on, although on average the sound
levels go down.
52
4. Free Space Noise Control
expense of increasing noise levels elsewhere, is straightforward to visualize.
Consider two sound sources operating in free space and producing a series of
sound waves, as illustrated in Figure 4.2. At some points in space the waves
cancel (50% shading in the sketch), and at some points in space they add
(very dark/very light portions of the sketch). Local areas of attenuation (where
the waves cancel) are provided at the expense of other areas of increased
sound level (where the waves add). However, unlike the passive control barrier, the addition of active noise control in the form of a second sound source
has meant that acoustic power in the environment has actually increased, and
in fact doubled if the sound sources are separated by a great enough distance.
We now have two sound sources pumping out acoustic energy. This implies
that, on average, the sound levels have increased, not decreased.
How is it then that we can ever end up with global sound attenuation when
active noise control techniques are employed? The only possible way this
can happen is if the total energy flow is reduced. For this to occur, the introduction of a second loudspeaker must do one of the following:
1. cause a reduction in the acoustic power output of both sound sources,
such that the total is less than the power output of the original source
operating alone; or
2. one of the sound sources must "absorb" power (energy flow into the
loudspeaker, not out of the loudspeaker) while the energy flow from the
other sound source stays roughly the same or reduces.
We will leave the second option for the moment, and concentrate on the first.
How is it that two sound sources operating in the same environment can output less acoustic power than either single sound source operating on its own? The
key to understanding how this is possible is to keep in mind that power flow is a
FIGURE 4.2. Interference pattern between two coherent sound sources.
Active Control Approaches
53
function of two parameters from the potential difference/impedance/flow tripartite (because of the relationship between these parameters, only two can ever be
adjusted independently with the third defined by the amplitudes of the other
two). If we can fiddle even one of these parameters, we can reduce the power flow.
Recall again the electrical analogy: the electrical socket on the wall always has
the potential for producing energy flow, but will only do so to a degree which is
mediated by the electrical impedance.
Consider the simple example of two loudspeakers, one producing unwanted
noise and the other introduced into the environment with the aim of reducing
the unwanted noise. Loudspeakers can be viewed, to a first approximation, as
constant volume velocity sources. That is, for a given electrical signal input
the loudspeaker cones will move a constant amount, regardless of the acoustic environment which is in front of the cone. In doing so, they will displace
a constant volume of air. This displaced volume will not change if, for example, you alter the acoustic environment by standing in front of the loudspeaker. The loudspeaker cone will continue to move with the same amplitude,
and so the volume velocity, or the volume displaced per second, is constant.
Volume velocity is the flow parameter in the set. To calculate how this flow
is turned into acoustic power which will move away from the loudspeaker to
bother humans remote from it requires consideration of a second parameter,
either impedance or pressure. Let us consider pressure for a minute. If the
loudspeaker is operating on its own, then as the loudspeaker cone moves
outward (positive volume displacement of the air) it will compress the air in
front of it (pressure increase). Similarly, as the loudspeaker cone moves back
in (negative volume displacement of the air) the air will be rarefied (pressure
decrease). Noting that acoustic power is a function ofthe product (pressure) x
(volume velocity), we can conclude that a positive amount of acoustic energy
will flow into the environment with the scenario. We are producing noise.
Suppose now we were able to make the acoustic pressure in front of the
loudspeaker cone equal to 0 at all times. If this could be accomplished then
even though the loudspeaker continued to displace a volume of air, there
would be no energy flow into the acoustic environment (power = volume
velocity x pressure = 0). If you were to stick your hand in front of the loudspeaker you would feel a breeze, but you would hear no sound. Is this possible? The answer is yes, using active noise control.
If we introduce a second loudspeaker into the picture in close proximity to
the first loudspeaker, and appropriately tune the amplitude and phase of the
second loudspeaker, we can arrange it so that the acoustic pressure of the
second loudspeaker either partially or completely cancels the acoustic pressure directly in front of the first loudspeaker. The real bonus is, if the acoustic
pressure of the second loudspeaker cancels the acoustic pressure of the first,
then to some degree the acoustic pressure from the first loudspeaker will end
up canceling the acoustic pressure immediately in front of the second loudspeaker. The result is neither loudspeaker has energy flowing from it into the
acoustic environment. The world is quiet!
54
4. Free Space Noise Control
This is very much a strange-but-true phenomenon. We can end up with two
loudspeakers essentially pushing a volume of air between them, but producing little or no sound. This is all because the flow of energy requires two
parameters. Simply displacing a volume of air is not enough.
Note here that as acoustic impedance is equal to (pressure)/(flow), we have
essentially reduced the impedance that the sound sources are driving into.
In light of this physical picture, we can immediately postulate a few important factors:
1. The separation distance between the sound sources must be small.
When radiating into free space, the acoustic pressure wave does not travel as
a big "lump" directly from one loudspeaker to another. The waves spread out,
just as waves spread out from the point of impact when you drop a rock in a
pond. From an energy point of view, this means that a little bit of the displaced volume of air will "sneak" out the sides, with a resulting generation of
acoustic power. The further apart the sound sources are, the more is lost. The
conclusion is that sound sources must be close to each other for energy flow
to be significantly reduced. We will quantify this notion of "close" shortly.
2. The two sound sources must be coherent.
For the flow of acoustic energy to be reduced on average, the pressure must
be canceled in front of the loudspeaker cone on average. This means that one
loudspeaker will have to produce a sound field which is a mirror image of that
produced by the other. Every time one has an increase in pressure, the other
must have a similar-sized decrease in pressure, and vice versa. This means
that the two sound sources must be related to some "master" driving signal, so
that one loudspeaker "knows" what the other is going to do and can act
accordingly. Technically, we say that the sound sources must be coherent.
3. The sound sources must be of roughly the same size.
For the reduction in energy flow to be significant, the acoustic pressure must
be canceled/reduced at the large majority of points in front of the source.
Extrapolating from this, there is little to be gained from using a small loudspeaker to control sound from some huge source. The small loudspeaker simply won't be able to cancel the pressure over a large enough region. This
doesn't necessarily mean that if the source of unwanted noise is large, such as
a large electrical transformer, a loudspeaker the size of a truck will be required. It does, however, mean that multiple loudspeakers distributed over
the front of the source of unwanted noise may be required.
It was mentioned earlier that it might be possible to use one sound source to
absorb the energy flow from another (the acoustic black hole!). Two questions
which then arise are: (1) How is this possible? and (2) Where does the energy go?
The first of these is straightforward to answer. Noting that acoustic power is
Active Control Approaches
55
defined by the product of volume velocity and pressure, if the (total) pressure in
front of one sound source can be made negative with respect to the displacement
of air, then the acoustic power output will be negative and the sound source will
be absorbing. This requires that we have compression (positive pressure) when
the loudspeaker cone is moving in (negative volume displacement), and rarefaction (negative pressure) when the loudspeaker cone is moving out (positive volume displacement). When does the energy go? Recall that the actual amount of
energy associated with an acoustic wave is very small, typically, a fraction of a
watt. Therefore, the amount of energy being absorbed is also very small. Physically, the absorbed energy ends up helping to actually move the cone against the
mechanical impedance associated with the diaphragm and suspension, the magnetic stiffness and damping, etc.
In general, for free-space acoustic radiation problems the absorption idea
is a poor one. If you follow the analysis through you find that in the process
of absorbing power the control source often induces more energy flow out of
the source of unwanted noise than it absorbs. The end result is often an
increase in energy flow, not a decrease.
Question 2: What limits the amount of cancellation which can be achieved?
Based upon our earlier discussion of the mechanisms that are responsible for
global sound attenuation with active control, we can construct a list of variables which will influence the performance of an active noise control system.
This list can be arranged in a hierarchical fashion, with the entries generalized into four categories. Doing this, the performance of an active noise control system is dependent upon:
1.
2.
3.
4.
Control source (loudspeaker) arrangement; this sets an upper limit on
how much global sound attenuation can be achieved.
Error sensor (microphone) placement; this determines how close to the
upper limit (set by the control source arrangement) the given system can
possibly come.
Reference signal quality; the coherence between the reference signal and
the error signal, that is inherently the coherence between the canceling
sound field and the unwanted sound field, sets a limit upon the performance of the "electronic" part of the control system. The coherence must
be very high for high levels of sound cancellation.
Quality of the controller software; this determines how much cancellation at the error microphones actually occurs, given the constraints placed
by signal coherence.
Question 3: How close do the sound sources have to be?
Based on an intuitive argument, it was stated above that the cancellation, or
control, sound source should be placed in close proximity to the source of
56
4. Free Space Noise Control
unwanted noise for a reduction in total energy flow into the acoustic environment to be possible. Reducing the total acoustic power output is desirable, as
it is the only way to achieve global sound attenuation.
The question which immediately follows is, how close? Before we can quantify this requirement we must decide upon the units to measure distance. Intuitively, the control source must be able to duplicate the shape of the sound field
generated by the primary noise source, as defined by the amplitude of the sound
pressure at all points. In this way, the phase can simply be inverted and cancellation will be possible at most points in space. Referring to Figure 4.3, the sound
field patterns of two sources quickly become "different" as the noise sources are
separated. Note that the rings in the sketch in Figure 4.3 are meant to indicate
successive wavefronts as they leave the sound source, with the rings representing
crests and troughs. Based upon this, it is more correct to say that the shapes of the
sound field patterns begin to differ as the sources separate relative to the length
between crests and troughs. Inherently, we are quantifying the effect of separation distance relative to the wavelength of the sound, which was defined in
Chapter 2 as the distance between successive sound wave crests. Wavelength is a
quantity that changes with frequency, or is frequency dependent: for a given
frequency of sound, the wavelength is approximately:
wavelength (in meters) = 343/frequency (in Hz).
Plotted in Figure 4.4 is the maximum level of sound power reduction which is
possible when using one small sound source to control acoustic radiation
from another small sound source into free space. This equates to the quality of
the global result. Note that as the distance between the two sources increases
the maximum possible reduction decreases rapidly, until at one-half wavelength separation there is no reduction to be had. A useful result to remember
from this curve is that the separation distance should be less than one-tenth
of the wavelength to achieve 10 dB power attenuation.
Small
separation
One wavelength
separation
FIGURE 4.3. The sound fields of two sources quickly differ as the separation distance
moves from "small" (less than one-fifth of a wavelength) to one full wavelength.
Active Control Approaches
a:l
57
20
"0
c:
0
ca 15
:;:::;
::J
c:
Q)
:=
....ca 10
~
a.
u
:;:::;
5
(/)
::J
0
«
U
0
0.00
0.25
0.50
0.75
Separation distance (wavelengths)
1.00
FIGURE 4.4. Maximum possible acoustic power attenuation for two small sound sources,
plotted as a function of separation distance between them.
It is interesting to consider what the above result means for a practical
problem. Consider the arrangement shown in Figure 4.5, where a loudspeaker
is being used to cancel the noise being emitted from a car exhaust. Exhaust
noise comes from the periodic firing of the cylinders in the engine, and as
such contains a huge number of harmonics (practitioners might say the signal
is harmonically rich, to sound more technical). For arguments sake, let's say
that the fundamental frequency is 50 Hz, with the second harmonic at 100 Hz,
the third at 150 Hz, etc. Let's also say that the center of the exhaust and the
center of the loudspeaker are separated by 343 millimeters, which is a little
over 1 foot. At 50 Hz, this distance is equal to 0.05 wavelength; at 100 Hz,
this distance is 0.1 wavelength, etc. With reference to Figure 4.5, this means
that we will achieve pretty good attenuation for the first couple of harmonics,
~
1 foot (343 mm)
Separation
FIGURE 4.5. Possible arrangement for active control of exhaust noise.
58
4. Free Space Noise Control
moderate attenuation for the next couple of harmonics, a little attenuation for
the next couple, and then no attenuation for the high harmonics. In other
words, active noise control is useful at low frequencies, and useless at higher
frequencies.
The above result is a good example of why active and passive noise control are complementary: passive noise control works best at high frequencies,
while active noise control works best at low frequencies. An astute vehicle
manufacturer, particularly a heavy vehicle manufacturer whose exhaust noise
spectrum contains a lot of low-frequency harmonics, might attempt to combine active and passive noise control for the best result at the best size and
best price. Remember, low-frequency passive noise control equates to mass,
size, and bulk, and so if the responsibilities of low-frequency sound attenuation can be removed from the passive techniques there is tremendous potential for cost savings.
Question 4: What is the relationship between the primary noise source and
control source output?
We have mentioned previously that the primary noise source and the control
source must be coherent for cancellation to occur. Every primary sourcegenerated wave peak must be paired to a control source-generated wave trough,
etc. However, it is worthwhile having a closer look at the relationship between the primary source and control source outputs.
In our discussion about how active noise control can bring about an overall reduction in energy flow into the acoustic field, we mentioned that the
acoustic pressure at the primary source could be canceled by the control
source output, and vice versa. However, completely canceling the sound pressure at both sources simultaneously is not really possible in the free space
environment. This is due to the "spherical spreading" phenomena of waves as
they move away from a source. Sound waves spread out in a circle as the move
away from a source, in the same way waves in a pond move away from the
point of impact when a stone is thrown in. It is useful to think of each wavefront
as having a set amount of energy, which must be distributed over the entire
wavefront. As the wavefront expands in diameter when moving away from the
source, the energy, and hence displacement amplitude, at any given point
will decrease. As the distance from the source increases (the radius of the
circle), the length of the wavefront (the circumference of the circle) will increase proportionally. Therefore, we might expect the amplitude of the wave
to decrease proportional to the radius; this is exactly the case. For example,
the sound pressure amplitude at 1 meter from a sound source is twice (6 dB
more than) the amplitude at 2 meters from the source.
Consider now the case where there are two sound sources operating in free
space, where the aim is to adjust the sources so as to minimize the radiated
acoustic power. If the acoustic pressure in front of source 1 is x dB, then for
this pressure to be canceled by source 2 the acoustic pressure in front of
Active Control Approaches
59
source 2 will have to be greater than x dB. This is because as the sound wave
travels from source 2 to source 1, its amplitude will decrease by an amount
proportional to the distance separating the two sources. However, if source 2
is adjusted to this level, then the sound pressure amplitude at source 2 from
the source 1 acoustic wave will not be sufficient to cancel completely the
source 2 sound pressure. Unless the two sources are on top of each other, it
will only ever be possible to cancel completely the sound pressure at one
source. The other source pressure will only be canceled partially.
There is a general result which states that the best global result,2 resulting
from the maximum reduction in total sound power, will occur when the control source amplitude is adjusted such that the sound pressure in front of the
control source is completely canceled. In this way, the control source acoustic power output will be zero, and the output from the primary (unwanted
noise) source will be reduced to some degree. Shown in Figure 4.6 is the
relationship between the primary and control source outputs for the twosource active control arrangement considered previously in Figure 4.4. Note
that the amplitude of the control source output drops as the separation distance increases. This is in response to the drop in the primary source-generated sound pressure amplitude arriving at the control source location (which
must be canceled) which accompanies an increase in separation distance.
You may ask, what happens if the primary and control source amplitudes are
equal? Will the result be much different? The answer is yes. As before, as the
separation distance increases the acoustic power reduction reduces. However,
'5 '[ 0.8
"S"S
00
(J)~
.
~ .... 04
::;,::;,
o
oCl)
CI)
e co
-~
-E
§ ._ 0.0 f---------~-------"7i
00:
-0.4
L -_ _---L-_ _-----L_ _ _--'-----_ _----'
0.00
0.25
0.50
0.75
Separation distance (wavelengths)
1.00
FIGURE 4.6. Ratio of optimum control source output (volume velocity) to the primary
source output (volume velocity) for total acoustic power reduction.
2
Technically, this is true only if reciprocity in the acoustic environment exists.
60
4. Free Space Noise Control
this time, as the separation distance increases, the reduction is not limited at 0
dB. As the separation distance approaches one-half wavelength and beyond, the
total acoustic power output increases over the original case. The two sources
look just like that: two sources, operating independently in space. Needless to
say, this is a poor result to obtain in practice.
Question 5: What is the effect of error microphone location?
The results presented previously are based upon the notion of minimizing the
total acoustic power radiated into space. The problem is, in practice, we are
not usually in a position to measure total radiated acoustic power and provide this quantity to our adaptive control system with the aim of minimizing
it (we would like to, if it was possible). Instead, active noise control systems
are most often implemented with microphones as sensors, and microphones
measure pressure, not power. Typically, we will place a microphone somewhere in space, and ask our adaptive control system (politely, of course) to
adjust the output of the control loudspeaker until the sound pressure measured at the microphone position is minimized. Does this "approximation" to
what we would really like have any influence upon system performance?
The answer is yes. Illustrated in Figure 4.7 is a plot of the reduction in total
radiated sound power for the case of one small sound source being used to
control the output of another small source radiating in free space, with the
two sources separated by one-tenth of a wavelength. From the previously
discussed acoustic power considerations, the maximum possible amount of
acoustic power attenuation that can be achieved by this arrangement of sources
is approximately 10 dB. From the plot, note that at some microphone loca-
(i)
~ 0.50
]j
~
~ 0.00
i
~
-0.50
-1.00 '----LL..--"--'-'--.w....1......l.....l......J...>...-l...~
1.00
-1.00
-0.50
0.00
0.50
x-position (wavelengths)
-0.1 0 '--""c......:.....<.l.<~"-'-.........---'-->..J.-"-_"""
0.05
0.10
-0.10
-0.05
0.00
x-position (wavelengths)
FIGURE 4.7. Acoustic power attenuation as a function of error sensor placement, for two
small sound sources separated by one-tenth wavelength (maximum possible acoustic
power attenuation = 10 dB).
Active Control Approaches
61
tions (near a line running between the sources) this maximum result is possible. However, in some locations, the result is actually an increase in total
power output. If the microphone is located close to the primary noise source,
this can be a significant increase.
This result is somewhat disheartening, as we often do not know where the
best microphone location is a priori. The practitioner can perform some analytical modeling only, but this is normally beyond the scope of the "suck it
and see" projects that represent a relatively high proportion of active noise
control trials. It is easier to simply move the microphone around the place and
see what happens. Alternatively, additional microphones can be added if the
scope of the electronics permits.
A few rules of thumb which can be used to guide microphone placement
include:
1. Never locate the microphone too close to the primary noise source. This
inevitably leads to a control source output that is too large, and often
controller saturation in response to trying to satisfy the requirement of
cancellation at the microphone location.
2. Do not locate the microphone too close to the control source if global
sound attenuation is desired. If the microphone is too close to the control
source, then the control source output will be too low to provide cancellation away from the microphone position.
3. If multiple error microphones are used, avoid too much symmetry in placement. Often a random placement will work best.
4. The sensitivity of microphone placement tends to reduce as the control
and primary source separation distance reduces.
Question 6: What is the effectiveness of local sound cancellation?
Our previous discussion has indicated that as the separation distance between the sound sources increases toward one-half wavelength and beyond,
the amount of global sound attenuation that is possible becomes zero. However, local sound cancellation is still possible. This may be of interest, for
example, if the noise source is a large machine and the aim is to cancel the
noise only at the machine operator's location. The question which follows is,
if local cancellation is targeted, over how large an area (around the error
microphone) will the effect prevail?
The answer to this question is dependent upon a number of variables,
including the characteristics of the radiation pattern of the primary noise
source which are often not known a priori. However, as a conservative limit,
consider the case where the sound field is diffuse, which means that the sound
waves can be coming from any angle. While this characteristic is not often
associated with a free space radiation problem, the result does provide us with
a worse-case result for tonal noise radiation. For this case, cancellation will
be limited to a small sphere around the error microphone, with a radius that is
62
4. Free Space Noise Control
a small fraction of a wavelength. Some of the early NASA trials on active
noise control described a region of local cancellation, or a "zone of silence,"
that was the size of a small grapefruit. As the distance from the error microphone increases toward one-half wavelength and beyond, the reduction becomes zero. It is possible to improve upon the result using multiple
microphones arranged in sophisticated patterns, but the improvement is marginal when viewed relative to, say, the area occupied by the operator of a
large machine (the zone of silence might increase in size from a "grapefruit"
to a "honeydew melon"!).
The assumption of a diffuse sound field is most often associated with a
"nonlow" frequency sound in an enclosed space, where "nonlow" is quantified in terms of the size ratio of the wavelength of sound and the typical
dimensions of the enclosure. "Nonlow" is where the dimensions of the enclosure are, say, five or more times the wavelength of the sound of interest for an
enclosure with dimensions that are of a similar order-of-magnitude on all
sides. There are a number of possible applications of local-cancellation active noise control which are "spoiled" by the physics described above. Two
of these include:
Spoiled application 1: Cancellation of siren noise inside emergency vehicles.
Inside many emergency vehicles, such as ambulances, it is not possible to
hear the radio with the siren on. It would therefore be useful to be able to
cancel out the siren noise inside the cab. The acoustic wave from a siren is
essentially a square wave, which is a combination of the fundamental harmonic and all odd harmonics (first, third, fifth) up to the point where the siren
and horn no longer respond. The fundamental harmonic will typically vary in
frequency as the siren ramps up and down between several hundred Hertz and
one to two thousand Hertz. In terms of wavelength, the fundamental wavelength is varying from something like one-half meter (20 inches) to 150 millimeters (6 inches). If an active noise control system is placed into the vehicle
cab, we could expect the area of usable sound reduction to be limited to, say,
a region around the microphone that is 25% of the wavelength of sound. This
is a few inches, or a few tens of millimeters, at best. This is not a particularly
useful result.
Spoiled application 2: Cancellation of noise inside aircraft.
Noise levels inside aircraft, even modern jet aircraft, can lead to passenger discomfort after a period of time. Active noise control systems in many different
forms are being developed and implemented as a way of reducing this noise. One
suggested implementation is to include active noise control in the passenger
headrests. If, for argument's sake, it is decided that the zone of quiet was to have
a minimum radius around the noise source of 100 millimeter (four inches), then
the limit on frequency where this active noise control system would be of use is
Active Control Approaches
63
around the 500 Hz mark. In practice, this can be expanded somewhat using
multiple microphones, but the upper frequency limit is still under 1000 Hz.
Fortunately, for an aircraft (unlike an ambulance), this is still a useful result. The
difference between the ambulance and the aircraft is that passengers in an aircraft
sit still, and so placing a small zone of silence around the ear is a possibility; the
same is not true for an emergency worker.
Question 7: What about causality?
Adaptive feedforward active noise control systems calculate a canceling signal on the basis of a reference signal provided to the controller as a measure of
the impending disturbance. If the unwanted noise is periodic, such as is generated by a piece of machinery with rotating parts, then the reference signal
can be taken from a measure of the machine rotation. As shown in Figure 4.8,
it is important to note that if the disturbance is periodic then the reference
signal can be used to predict the characteristics of the sound field in I millisecond, 10 milliseconds, or even 1 minute. This is because the sound field
continually repeats itself (the definition of periodic sound). This greatly simplifies the control system requirements, as it is not necessary to consider the
timing of the reference signal relative to the unwanted sound field; it will
always be a good predictor of the sound field. This type of control arrangement, where the reference signal and sound field do not have to be perfectly
matched, is referred to as noncausal. The reference signal does not have to be
the precise "cause" of the section of sound field that is canceled out based
upon its acquisition by the controller.
Consider now the case where the sound field is not periodic. In this case,
every section of the sound field can be directly related to a section in the
reference signal. Further, as shown in Figure 4.9, this matching of the sound
point in the reference signal
can be used to predict the sound
field at any point in time.
FIGURE 4.8. A given spot on the reference signal for a periodic noise source can be used
to predict the sound field at practically any point in time. Therefore, the control system
can be noncausal, where a given point in the reference signal can be used to calculate the
canceling signal for any point in the sound field . Precise matching in time is not
required.
64
4. Free Space Noise Control
FIGURE 4.9. A given spot on the reference signal for a nonperiodic noise source can only
be used to predict the sound field at one point in time. Therefore, the control system can
be causal, with each point in the reference signal used to calculate the canceling signal
for the matching point in the sound field.
field and reference signal is exclusive: only one section of the reference signal is responsible for, or correlated with, any given section of the sound field.
Therefore, for active noise control to cancel out a given section of a sound
field, the corresponding section of the reference signal must be measured and
processed, and the canceling signal fed out at precisely the right time. The
pairing of reference signal and unwanted noise is critical. Such a control
arrangement is termed causal. The reference signal must be the precise cause
of the section of sound field that is canceled out based upon its acquisition
by the controller.
The major problem with implementing a causal active noise control system for free space radiation problems is time delays. It takes a finite period of
time for a signal to pass through a digital control system. There are delays
associated with input and output filtering, sampling, and calculation of results, as discussed in Chapters 7 and 8. There are also delays associated with
driving a loudspeaker, as it takes a finite amount of time for the loudspeaker
to produce sound once an electrical signal is fed to the loudspeaker. These
delays are dependent upon a number of physical variables, such as loudspeaker size and filter cutoff frequency, and are frequency dependent (different frequency components take different amounts of time to pass through the
control system). However, typical values of delay are of the order of several
milliseconds or more. On the surface, this sort of time delay may sound trivial.
However, recall that sound waves travel through space at 343 meters per
second. During the delay of several milliseconds, the sound wave would have
traveled 1 meter, 2 meters, or even more. What does this mean physically? To
implement a causal active noise control system, a reference signal measurement must be taken from the target noise disturbance several milliseconds
before it arrives, in order to accommodate the delays in the electronics and
Active Control Approaches
65
loudspeakers (note again that a reference signal provides a measure of the
impending noise disturbance, a measure which is used by the system to derive
a corresponding canceling signal). This means the reference microphone must
be located upstream of the loudspeaker by I meter, 2 meters, or more, so that
the unwanted noise disturbance is arriving at the loudspeaker just as the
canceling sound wave is being generated.
To see what impact this has upon the potential of active noise control for
free space radiation problems, recall that for global noise control to be possible the control source and primary source must be situated in close proximity, preferably less than one-tenth of a wavelength. If, for argument's sake, we
determine that the delays in the loudspeaker and electronics are of the order
of 6 milliseconds, during which time the unwanted sound wave will travel
(0.006 seconds x 343 meters per second =) approximately 2 meters, then for a
causal control system the sound sources must be separated by at least 2 meters.
So for good global control we will want to target frequency components
where one-tenth wavelength is of the order of 2 meters, or where the wavelength is of the order of 20 meters. As the wavelength of sound equals 3431
frequency in Hertz, this means we are looking at controlling sound fields with
frequencies of the order of a few tens of Hertz at most (a frequency of 17 Hz
has a wavelength of 20 meters). There are not a whole lot of nonperiodic noise
problems with frequency components limited to under 20-30 Hz which are of
interest!
The end conclusion is that active noise control of free space acoustic radiation is almost entirely limited to periodic noise problems. Fortunately,
there are a lot of these: electrical transformers, commercial garden vacuums,
motors, and practically any other machine which has rotating parts.
Side Story. One nonperiodic free space acoustic radiation problem that is
possibly amenable to solution through the use of active noise control is found
in outdoor rock concerts. Several years ago, I was contacted by the local
organizer for the Australian leg of a Rolling Stones world tour. The Rolling
Stones were in the process of booking large outdoor concert venues, such as
football stadiums, and many city councils were placing restrictions on noise
levels. In particular, councils were concerned about the bass noise. At previous concerts of this nature, residents several kilometers away had complained
about this. The organizers' question was this: would it be possible to cancel
the bass noise (say, frequencies under 50 Hz) which is propagating away from
the stadium only? Of course, it wouldn't be acceptable to globally cancel
bass noise, including inside the stadium. As it turns out, it is possible to get a
reference signal from the bass several milliseconds before it comes out of the
loudspeaker, by tapping into the input to the sound mixer. The problem then
becomes one of making canceling sound sources which are directional, throwing sound away from the stadium and not in. This would potentially cancel
the noise outside the stadium only. This too is possible in theory, by properly
designing trumpet-like horns on the front of the speakers and using a little
66
4. Free Space Noise Control
electronic filtering. However, I decided to turn the job down (despite my
thoughts of contract requirements like personally escorting the Rolling Stones
around the country, purely for "technical reasons," of course).
The reason comes back to the requirement that the general dimensions of
the sound sources should be the same. After considering the size of the PA
system being used by the Rolling Stones, the thought of building a "canceling" PA system of similar size, with all of the associated cost and without
actually knowing it would work, left me cold. Most contracts have a performance clause, and if the system didn't perform I would be left with the world's
largest home stereo loudspeaker system!
Still, it seemed technically possible. Now, what other bands would I like
free tickets to?
5
Enclosed Space Noise Control
The second group of noise control problems that we will briefly discuss here
concerns the control of sound in enclosed spaces. Examples in this group
include noise in rooms, noise in vehicle cabins, and noise inside aircraft. If
the concept of control is taken in the most general way, it can also mean
preferential modification of the acoustic environment in places like concert
halls and video conferencing rooms.
Before discussing the control of sound in an enclosed space, it will be
worthwhile discussing a few physical phenomena associated with enclosed
sound fields.
Where Does the Sound Come From?
On the surface, this appears a somewhat trivial question. However, from the
standpoint of looking at our two noise control mechanisms (reduction or
redirection of energy flow) later in this section, it is worth putting an answer
on paper.
Noise can come from sources inside or outside the enclosure. For noise
sources outside the enclosure, there are two ways in which the unwanted
noise can enter the enclosed space. The first is via sound waves traveling
through openings in the enclosure, referred to as "airborne" disturbances. It is
amazing how small the opening in an enclosure actually has to be in order to
get a significant amount of noise entering. To see this, try putting a noise
source outside of a room and opening the door, even a few millimeters. The
other way in which unwanted sound from an external source can enter an
enclosed space is via vibration of part or all of the enclosure boundaries,
referred to as a "structure-borne" disturbance. In some instances the path
between the noise source and the enclosure wall vibration can be very complicated, and not immediately visible.
For a noise control practitioner working on an enclosed space noise problem, it is very important to assess the relative importance of structure-borne
and airborne disturbances. This assessment will guide the strategy used for
correcting the noise problem. However, assessment is not always a straightforward exercise. A common trick employed by practitioners is to tape record
67
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
68
5. Enclosed Space Noise Control
the noise source (say, a motor, for argument's sake), and then turn off the
source and replay the tape through a loudspeaker in the same location and at
the same volume as the sound source. Assuming that the loudspeaker cabinet
is not moving wildly, the noise can now only propagate from the source to the
enclosed space via an airborne path; the vibration option has been eliminated by turning off the original noise source. The noise which now exists in
the enclosed space is the airborne component, and the difference between
these acoustic levels and those which exist when the real source is operating
are due to the structure-borne path.
How Does the Sound Get Out Again?
This is an important question to ask, because from a noise control point of
view we would like to "help" the noise get out by enhancing the physical
mechanism that is responsible.
Consider the case where you are driving in a noisy car. If you open the
window, does the noise escape? Probably not. More noise will likely enter
from outside. In fact, once noise has entered an enclosed space it doesn't
really "escape." Rather, to remove the noise, the energy in the sound field
must be absorbed and converted to some other form. This is generally done on
the surfaces of the enclosure. When the sound wave strikes a surface, the
surface moves slightly. Typically, the sliding of the particles against one
another generates a small amount of heat, the energy of which has been taken
(absorbed) from the sound field. Remember that the amounts of energy in a
sound field are usually small fractions of a single watt, and so the sound field
is unlikely to cause the surface to get hot! This conversion of energy from the
sound wave to heat is referred to as damping. Technically, we should say
acoustic damping in order to differentiate it from other forms.
Based upon the description above, we can postulate a few phenomena. First,
for sound absorption to be possible the surface must be moveable. Technically,
the surface density should be low, which is exactly the opposite of the case for
stopping sound transmission. Second, the acoustic wave should be able to penetrate into the absorbing material, to increase the portion of the material that is
actually absorbing. This ability to penetrate is often quantified by aflow resistivity, which specialist acoustics material manufacturers will cite for their products.
Third, the surface area of the absorbing material should be large enough to provide more area for the wave to strike and be absorbed over. Fourth, the movement
between particles should be inefficient in that it generates a lot of heat. This
requires a high degree of friction as opposed to a low degree of friction, which is
quantified by viscosity for fluids.
In a vehicle cabin, sound absorption is usually carried out by the interior
trim, the foam in the seats, and even the bodies of the occupants. In professional acoustics environments, special treatments are often applied to the
walls to enhance absorption. One of the most notable cases occurs in an
How Does the Sound Field Arrange Itself?
69
acoustics testing facility called an anechoic room. The idea in these rooms is
to absorb the entire incident acoustic wave. To do this, large wedges of foam
or other absorbing material are mounted on the walls. The resulting high-tech
appearance makes these rooms common candidates for background settings
in loudspeaker brochures. Amongst other things, the wedge shape provides a
greatly increased surface area for the sound wave to be absorbed over versus
a flat wall, which in turn greatly enhances the absorption.
How Does the Sound Field Arrange Itself?
When a sound source is radiating into free space, the sound waves can travel
unimpeded in all directions. The result of this is a sound field that is characterized by the spherical spreading phenomena, where the sound waves spread
out in a manner resembling ripples in the water after a stone is thrown in.
If a sound source is radiating in an enclosed space, the sound waves will
initially move away from the source in the same way. However, when the
waves reach the walls of the enclosed space they will be reflected back into
the room. Usually there is a slight reduction in amplitude and change in
phase, which is dependent upon the absorption and vibration properties of
the wall. If the path of the waves is traced as they bounce from wall to wall it
will be found that there are some paths of travel that tend to repeat upon
themselves. These are the paths which seem to fit properly in the given enclosed space. At certain frequencies it would most likely be found that, as the
waves travel along the preferred path, they actually build upon one another,
employing constructive interference to increase in amplitude. The frequencies where this happens most strongly are the resonance frequencies of the
space, and the dominant pattern of wave travel at those frequencies will be
the associated mode shape. 1
Resonance frequencies and modes are extremely important quantities in
acoustics and noise control, and so it is worthwhile deriving some more qualitative information to describe the notion of a "preferred path" and a "good
fit" for a sound wave. Consider the enclosed space shown in Figure 5.1, where
it is assumed that the walls are perfectly reflecting. Suppose we wish to produce a sound wave which travels from end to end, building upon itself: what
kind of sound wave would we want? One sound wave which would do this is
one that has a distance from peak to trough to peak to trough (one and onehalf wavelengths) that is exactly the same as the end-to-end dimension of the
room. In this way, the peak would build upon the peak as the wave is reflected, and the trough would build upon the trough. The resulting sound
pressure amplitude pattern across the room would look something like that
1 A note on terminology: the frequency at which the resonance phenomena occurs is
called the resonance frequency (not resonant frequency); at this frequency, the room is
said to be at resonance, or resonant.
70
5. Enclosed Space Noise Control
1.0
0.5.
~
O·
-----,.- - 0. 5 ;
- 1.0 i
25 20
Enclosed space
.
15 10
.
5
FIGURE 5.1. Sound field in an enclosed space, showing a standing wave pattern (dark
areas are locations of high-pressure amplitude, light areas are locations of low-pressure
amplitude). Note that there are three one-half wavelengths displayed in the above mode.
which is shown in Figure 5.1. This is the pattern of a standing wave, a wave
pattern that does not propagate away but "stands" in one place. This is also
one of the mode shapes of the room.
There are two comments which must be made concerning this idea.
1. This perfect fit of the sound wave into the enclosure to form the specific
mode shape shown will only occur at one frequency. For the case of an
enclosure that appears rectangular from end to end, this frequency is
where the wall-to-wall length is equal to one and one-half acoustic wavelengths. This frequency is one of the resonance frequencies of the enclosed space. Aresonance frequency is also called a natural frequency, as
it is a frequency which naturally fits with the enclosed space.
2. The pattern shown in the figure is not the only one which perfectly fits
into the enclosure. If the end-to-end distance of the enclosure is the same
as any integer number of half-wavelengths the pattern will fit perfectly.
The frequencies which correspond to these wavelengths are all resonance
frequencies of the enclosure.
If we examine the standing wave pattern we will see areas where the pressure peaks, and other areas where the pressure is essentially zero. The areas of
peak pressure are referred to as antinodes, while the areas of zero pressure are
referred to as nodes. In a standing wave, these areas of peak and zero pressure
will remain in the same position in the room for all time. This means that if
you walk around in the room while it is resonating at one of its natural frequencies you will find areas where it is very loud (the antinodes), and areas
where it is dead quiet (the nodes).
Side Story. If you want to experience a mode first-hand, then next time you
are in a toilet cubicle, shower enclosure, or some other small enclosed space
that is reasonably free of soft hanging materials, try humming a single tone.
Start humming at as Iowa frequency as you can manage and gradually increase the frequency. At some point you should notice the sound level increase dramatically-you have just hit a modal resonance frequency.
How Does the Sound Field Arrange Itself?
71
Sometimes you can strike these frequencies by accident. Two incidents
which immediately come to mind are:
1. My "favorite" toilet cubicle at work has a resonance frequency that is
exactly the same as the tone emitted by the large-leaf vacuum used by our
gardening staff. If I time my trek poorly, I can be in the cubicle when they
turn this thing on and end up deafened for the entire morning! The funny
thing is, it is only loud in the cubicle and not in the surrounding toilet
area. This is because the cubicle has a resonance at that frequency.
2. I used to live near a train line that was on the main route between Adelaide
and Melbourne, Australia. During the day the rail traffic was dominated
by small local trains, but at night the large diesel goods trains would use
the line. On a clear night, these trains could be heard miles away, coming
toward my house. Because of the Doppler effect, the sound of the trains
increased in frequency as the trains came closer. As the trains approached
my house, the effect of this was to essentially run through the frequency
response of my bedroom, slowly and painfully, over a 5 minute period. As
the train got closer, I could hear the sound pass through resonance frequency after resonance frequency, with the amplitude of the noise increasing then decreasing, increasing then decreasing, until the train
finally passed. Fortunately it passed through a tunnel shortly after passing, so I wasn't subject to an encore performance as the frequency decreased as the train moved away.
Up to now we have shown arrangements of the standing wave pattern to
be determined, by areas of peak pressure at the bounding walls. All other
antinodes and associated nodes are suitably placed relative to these end
conditions. The question then arises, does this have to be so? Why can't the
standing waves simply arrange themselves as they chase, maybe with a
node at the boundaries? The answer is yes, it must be this way for an enclosure with rigid walls; pressure antinodes must exist at the walls. The precise
mathematical reasons for this are beyond the scope of this book, but the
logic can be understood by considering the garden hose analogy first outlined in Chapter 3. We have said throughout this book that a proper description of sound requires description of two of the variables in the pressure/
velocity/impedance set (the third is a function of the other two). Consider
now a simple relationship between pressure and velocity. If the garden tap
is turned off, and so the velocity of each minute particle of water traveling
through the tap is zero, then the pressure on the mains side of the tap will be
at its maximum. As soon as the tap is opened, water begins to flow and so
each minute particle of water has a nonzero velocity. Associated with this,
the pressure will drop. We have all encountered this effect: when you first
turn on the hose the water gushes out with great pressure, which is the
pressure associated with zero velocity. However, after this transient condition has subsided, the water pressure will drop to the value associated with
the given water velocity. For a given amount of energy, there is an inverse
72
5. Enclosed Space Noise Control
relationship between pressure and velocity. 2 If there is a large velocity then
there is low pressure, with the converse also true.
The same pressure/velocity phenomena will happen in the enclosed space.
When the sound wave strikes a rigid wall, the wall doesn't move. This forces
the velocity of the particles currently under the influence of the wave to
become zero; they have "hit the wall." At this point, then, the associated
sound pressure must have a peak amplitude, as all of the energy contained in
the sound field will be associated with pressure. This is why there must be an
antinode, or pressure peak, at the boundary of a rigid walled enclosure.
Following on, what happens if the walls are not perfectly rigid? The particle velocity will not be zero, and so the pressure antinode will not be perfectly placed at the wall. The result will be a shift in the placement of the
nodes and antinodes in the standing wave.
We have so far discussed only modes which have a pressure variation in
one direction. These are referred to as axial modes in the field of room acoustics. However, this neglects the three-dimensional nature of the enclosed space.
Visualizing a rectangular box, there are three types of modes that can exist:
axial modes, that exist between two parallel walls (one-dimensional modes);
"tangential modes," that involve reflections from two sets of walls at one time
(two-dimensional modes, analogous to a drumskin where the wave reflections are off all boundaries in a given plane); and "oblique" modes, that
involve reflections off all walls (three-dimensional modes, similar to a balloon blowing up and shrinking down). Each of these modes has an associated
resonance frequency, that is a frequency where the wave fits into the dimensions and geometry of the enclosed space.
The word "geometry" in the preceding line is important. An enclosure does
not have to be rectangular to exhibit modal response characteristics. Any
geometry, regular or irregular, will have modes and resonance frequencies.
They may just be somewhat more difficult to calculate mathematically.
Having stated that "rooms respond modally," let us now consider the relationship between different resonance frequencies for a given mode type. Consider axial modes in a rigid walled enclosure: a resonance will occur at every
frequency that yields a sound wave whose one-half wavelength fits an integer
number of times from end-to-end in the enclosed space. Suppose our enclosure measures I meter from end-to-end, and we assume that the speed of sound
is 343 meters per second. For one half-wavelength to fit in to the 1 meter
space (that is, to have a wavelength of 2 meters), the frequency would be
(343/2) = 171.5 Hz. This is the first axial mode resonance frequency for our
enclosure. The second resonance frequency occurs when two half-wavelengths
fit in to the 1 meter space (that is, the wavelength is 1 meter): this is (343/1) =
Technically, the total amount of energy in the system is the sum of two components:
potential energy, associated with pressure, and kinetic energy, associated with velocity. As
one decreases the other increases, such that the total remains the same. Interestingly, this is
what vibration is, a swap of the total energy between the potential and kinetic states.
2
How Does the Sound Field Arrange Itself?
73
343 Hz. The third resonance frequency occurs when three half-wavelengths
fit into the 1 meter space (that is, the wavelength is 0.667 meters): this is
(343/0.667) = 514.5 Hz. Each subsequent axial mode resonance frequency
will be found at an increment of 171.5 Hz. Limiting consideration to axial
modes, if the frequency response of the enclosed space were plotted (say,
pressure amplitude at one of the walls versus frequency), we would see an
even distribution of pressure peaks at each resonance frequency. Physically,
if you were to stand at the wall of the enclosed space, generate a sine wave in
the space and vary the frequency, the amplitude would appear to get louder as
the frequency approached one of the resonance frequencies. The sound pressure would peak right at resonance and then get quieter as the frequency
increased beyond the resonance frequency. This amplitude up/amplitude
down phenomenon would repeat itself as each resonance frequency was approached. For the example above, this is every 171.5 Hz.
The relationship between axial mode resonance frequencies is straightforward. Life becomes more complicated when considering tangential, or two-dimensional, modes. Using the idea of a wave fitting into the space, in a
rectangular enclosure the dimension "seen" by the wave that defines a tangential mode resonance frequency is really the length of the two sides considered in parallel. This is somewhat hard to visualize, but relatively straightforward to derive mathematically. The antinode/node pattern of a tangential
mode will divide the two sides into sections, like a patchwork tablecloth.
This will produce a modal pattern which appears as a regular arrangement of
peaks and troughs in the enclosed space, as shown in Figure 5.2. There will be
1.0
0.5
0
-0.5
-1.0
25
25
FIGURE 5.2. Typical tangential mode pressure distribution (in an enclosed space); the
displayed mode forms a 3 x 2 pattern of nodal "areas."
74
5. Enclosed Space Noise Control
a separate resonance frequency for each combination of divisions (1 x 1, 1 x
2,2 x 37, etc.).
It is not terribly important for the discussion here to derive an expression
for the resonance frequencies of the tangential modes. However, what is important is that the "fit" of the wave is dependent upon the dimensions of each
"patch." As the number of integer divisions of each side increases, the average
dimensions of the sides of the patch, from one case to the next, increment
only slightly. The result is that as the number of divisions of the sides increases, the resonance frequencies increase only slightly. Stated another way,
for a given frequency span (say, 50 Hz), there will be many more tangential
mode resonances over the span at high frequencies (say, 1000-1050 Hz) than
at low frequencies (say, 50-100 Hz). This is quite different for the axial mode
case, where the resonance frequencies were always evenly spaced.
Oblique modes are even more complicated than tangential modes, with all
three dimensions divided into pieces. The resonance frequencies are also
more closely packed together at high frequencies.
If we were to draw some conclusion about frequency response from the
previous few paragraphs, they would be that, given a room:
1. the frequency response would exhibit a few, discrete peaks at low frequencies corresponding to the low frequency resonances; and
2. the density of the peaks would increase as the frequency increases, until
discrete peaks can no longer be distinguished.
At low frequencies, where individual resonance peaks are distinguishable,
the room response is referred to as modal. At higher frequencies, the sound
field in the room is referred to as diffuse, meaning that all sound waves have
approximately the same amplitude and are traveling in all directions.
It is important to note that the amplitude of the resonance peaks and the
frequency where the modal response ends are both dependent upon the damping in the enclosed space. In fact, at the height of the peaks at resonance are
only limited by the damping in the enclosed space. If there was no damping
they would, theoretically, be infinite. If the damping is very high, then it may
not be possible to find a "modal" response at all. This is because there will be
no distinguishable resonance peaks in the spectrum.
Passive Noise Control Approaches
With passive noise control there are basically two approaches to solving
problems of noise in an enclosed space: stop the noise from getting there and/
or absorb the noise once it is inside. Both of these approaches are directed at
the reduction rather than redirection of the acoustic energy. Given that waves
reflect off the walls of an enclosed space and so eventually fill the space, it is
difficult to actually redirect the flow of energy away from an object. The
sound will eventually find a way there.
Passive Noise Control Approaches
75
Consider the first of the two approaches. Methods that are employed to
"stop the noise from getting there" vary depending upon whether the noise
disturbance is structure-borne or airborne. If the disturbance is structure-borne,
then the simplest approach is to isolate mechanically the source using some
form of elastic isolator (such as a piece of rubber). Most machinery is mounted
using isolators; look at how a car engine or refrigerator compressor is mounted
for example. It is also important to isolate the lines, wires, etc., going to a
machine, as they are also sources of vibration.
Side Story. Vibration isolation is beyond the scope of this book. However, it
is worth commenting on the last sentence in the above paragraph. The number of noise problems that have been created by overlooking the isolation of
lines and wires, going to and coming from a machine, is incredible. Two
examples which immediately come to mind are:
1.
2.
A name-brand refrigerator manufacturer had the compressor beautifully
isolated, and then simply bolted the heat exchanging coils coming from
the compressor to the back of the refrigerator. As a result, the refrigerator
cabinet shook and radiated more noise than the compressor.
A medical equipment manufacturer spent a small fortune on custom silicon isolators for a small compressor inside a piece of equipment, and
then tied all of the lines to and from the compressor to the outer case. As
a result, the case shook and made a lot of noise.
Of course, there are still a large number of manufacturers who will not spend
the extra 2 cents on rubber grommets or a piece of foam to isolate moving
components from the equipment structure. I would estimate that 95% of the
products we have examined for the possible installation of active noise control, which usually involves noise from a motor or other rotating machine,
have had their problems solved by a piece of rubber! This is the original 2
cent solution (plus my consulting fee, of course).
If the disturbance is airborne, then the first port of call is to make sure that
all cracks and openings are sealed. Even a small gap can lead to a large noise
problem.
If all openings are sealed and all machinery is isolated but the enclosed
space is still noisy, then the remaining approach to keeping the noise out is
to reduce the vibration of the walls of the enclosure. The most common remedial approach here is to cover the walls with some form of vibration-damping material, something which is soft, heavy, and has a reasonably high
viscosity. The bitumen-based undercoating material in automobiles is a good
example of this. This material is applied to reduce vibration, which inherently reduces noise. The classic example of vibration-damping material not
being there is an aircraft. In a propeller aircraft, all of the openings to the
outside are certainly sealed. Still, it can be very noisy inside. However, it is
not practical to smother the plane in tons of undercoating, for obvious reasons of weight.
76
5. Enclosed Space Noise Control
The best approach is to design the enclosed space by maximizing sound
transmission loss in the first place, rather than relying on remedial action
afterward. Proper selection of materials (preferably something heavy, or floppy
with a high surface density) is important, as is the use of correct building
techniques. For example, even simple steps, like "staggering studs" when
building a partition wall, will help. With this, partition coverings are nailed
to every other stud, with one side nailed to the first, third, etc., and the other
to the second, fourth, etc., to reduce the efficiency of the transmission path
from one side to the other. Most building material manufacturers provide
advice, as do most engineering noise control texts.
Consider now the second approach to passive noise control, that of absorbing the sound once it is in the enclosure. In general, for a material to
be good at absorbing sound it must be thick and porous; curtains, foam,
and insulation are common examples of sound-absorbing materials. Thick
is a relative term, quantified by relating the actual material thickness to the
wavelength of sound. Even simple steps such as making a set of curtains
look thicker, and increasing their surface area by hanging them with folds
will improve their sound absorption properties. Typically, absorbing materials work well at mid- and high-frequencies and poorly at low frequencies.
It is worth mentioning where the sound absorbing material should be hung
in order to obtain the best result. It is intuitive that you would like the maximum pressure impinging upon the absorber in order to get the best effect.
Based upon our previous discussion of modes in enclosed spaces, this is
generally the walls of an enclosed space (where the antinodes exist). If there
is a particular mode which is of interest, for example, an axial mode, then
placing the absorbing material on the walls which form the end boundaries of
the mode will have a much greater effect than placing the absorbing material
on the side waveguide walls.
Active Noise Control Approaches
The use of active control for noise problems in enclosed spaces has been one
of the driving forces of the rapid expansion of research and development in
the area over the past 15 years. The particular problem that has been the
greatest impetus (that is, where the greatest amount of research funding has
been available) has been the active control of noise in aircraft, particularly
hi-tech propeller-driven aircraft. This has been closely followed by problems
of noise in ground-based vehicles. There has also been a great deal of interest
in controlling noise in prefabricated dwellings, particularly in Japan (although
this has waned to some degree by a lack of positive outcomes, for physical
reasons that will be described shortly).
To be in a position to assess the potential of each of these applications, we
must develop a set of qualitative performance indicators for active control in
Active Noise Control Approaches
77
enclosed spaces. The most important indicator relates to the modal (or otherwise) response of the enclosure at the frequencies of interest.
We have discussed already that the physical mechanism which provides a
reduction in energy flow in an active noise control implementation is a change
in at least one of the parameters in the set of pressure/volume velocity/impedance. When this happens, and global sound attenuation is achieved, a byproduct result is that the sound field produced by the canceling sources has
the same amplitude as, and opposite phase to, the original unwanted sound
field. For the previously considered case of active noise control in free space,
we found that this required the two sound sources (primary and control) to be
in close proximity.
Let us apply this line of thought of duplicating sound field shapes to the
enclosed space noise control problem. At low frequencies, the response of the
enclosed space is dominated by the response at one or more resonance frequencies. We have seen that these are very discrete entities at the low-end of
the frequency response spectrum of the enclosed space. Each of these resonance frequencies is associated with a mode of the enclosed space, as we have
also discussed. The important fact about this modal type of response is that it
is essentially independent of the position of the source of noise in the enclosed space. Unless you happen to be extremely unlucky and have the noise
source located exactly at a node of the mode, you can drive the mode with a
sound source at almost any location. This is extremely advantageous for
active noise control. It means that if the source of unwanted noise is eliciting
a modal response from the enclosed space, then the canceling source can
elicit the same form of response from a large variety of locations in the enclosure. That is, when the response is modal the canceling source does not have
to be located close to the source of unwanted noise. This is quite different
from the free space radiation case.
Consider now what happens at high frequencies. If the response of the
enclosed space to a noise source is a diffuse sound field, then essentially
there is no structure to the sound field. A given point in space is equally
likely to be struck by a sound wave from any given direction. In other words,
the response is more or less random in nature; sound waves are running amuck
in the enclosed space! How can the canceling sound source possibly reproduce this form of sound field? The only way is if the source of unwanted noise
is small and the canceling sound source is placed next to it. This is more or
less the same as the free space radiation criterion, where "close" means a
small fraction of a wavelength. We have already decided that such a criterion
is impractical at mid- and high-frequencies.
We can therefore conclude that active noise control in enclosed spaces has
the potential to provide global sound attenuation at low frequencies, where
the response is dominated by a few modes. There is almost no potential for
global sound attenuation with active methods at mid- and high-frequencies.
What is a low frequency? Based upon our previous discussion of modes, and
78
5. Enclosed Space Noise Control
how a sound wave must fit into the enclosure for a mode to be resonant, we
can say that a low frequency is one where the size of the wavelength of sound
is of the order of the largest dimension of the enclosure. If one dimension is
disproportionately large, as is the case with a tube, then this criterion can be
relaxed to some degree.
Summarizing, in an enclosed space the location of the control source(s) is
secondary in importance to the form of the response of the enclosed space (modal
versus diffuse) at the frequency of interest insofar as determining the potential of
active noise control. The frequency must be low and the response modal for
active noise control to be effective at providing global sound attenuation.
Side Story. The requirement of a modal response for active noise control to be
effective in enclosed space problems is one of the unfortunate limiting factors in the field. There are a number of potential applications which are not
physically (practically) possible because of it, including:
1. Control of sound inside domestic dwellings situated in noisy
environments
Noise in rooms next to roads, train lines, airports, factories, etc., is a major
problem in many areas. If the noise could be reduced using active control the
benefits would be immense. Consider that an average room dimension is of
the order of 4 meters square, meaning that the first modes of the room are
resonant at a few tens of Hertz. Unfortunately, by the time 100 Hz is reached,
most rooms will no longer be exhibiting modal response characteristics and
so active noise control is no longer practical. Owing to the preferential nature
of human hearing, if an active noise control system is implemented to simply
remove the (very) low-frequency component of a noise spectrum, the result
will subjectively be poor.
2. Control of noise inside passenger vehicles
It is a dream of many automobile manufacturers to include active noise control in their cars, essentially reusing the stereo system to reduce noise inside
the vehicle. At the time of writing this text, Nissan was actually offering
active noise control as an option in one line of their vehicles. The average car
has inside dimensions of the order of 1.5-2 meters (wagons a bit longer),
meaning that we can expect modal response characteristics up to a few hundred Hertz. This appears to be a worthwhile range for implementing an active
noise control system. There are, however, many problems with active noise
control inside cars. While not all of these are "physical," one impediment is
the actual high level of acoustic damping inside modern cars. This arises from
both deliberate design and the plush seats and trim. Because of the high
levels of damping, the modal response ends at a lower frequency than would
be expected by dimensions alone, and with it ends the potential for active
noise control to provide useful levels of global sound attenuation.
Active Noise Control Approaches
79
The active noise control system offered by Nissan specifically targets a
phenomenon referred to as "booming noise." This is where one of the first
resonance frequencies is excited. In the case of the Nissan system, they are
looking at excitation by engine noise, rather than, say, excitation from having a window slightly open. The cynic would say that this resonance, which
is a few tens of Hertz in frequency, is too low in frequency and too low in
amplitude in a luxury automobile to be a problem in the first place. The
system is only there for "gimmick value." Attempts to extend the frequency
range of the system, and to target noise generated by the road rather than just
by the engine, have had little success. Noise reduction has been found to be
local rather than global, and the level is only 5-6 dB at best.
3. Control of noise in jet aircraft
It was mentioned earlier that one of the problems which has driven the field of
active noise control is noise inside propeller-driver aircraft. While there are numerous technical difficulties related to implementation, the concept appears, at
least physically possible as the fundamental frequency of excitation by the propellers is low and so the response inside the target planes can still be classified as
modal. Further, if the fundamental bladepass frequency was reduced in level it
would make a huge difference in many propeller-driver aircraft.
It would also be nice to apply active control technology to problems of
noise inside jet aircraft, as these make up the bulk of the passenger-carrying
fleets. One of the physical problems with this, however, is that the frequency
content of the noise is higher than that of propeller aircraft, and so the response inside the plane can not be classified as modal. The noise is also
largely nontonal, arising from the flow of air around the fuselage. Without a
modal response, it is practically impossible for active noise control to produce a global result.
As an alternative, at least one manufacturer is now putting active noise
control into passenger headrests (for first class passengers, of course). This
active control implementation specifically targets local, rather than global,
attenuation. Basically, it is working within the physical constraints placed on
active control by the acoustic response of the aircraft interior.
Side Story. When an enclosed space is sealed, such as the case of the aircraft
interior, the noise must enter the space via vibration of the enclosure walls.
Therefore, rather than attack the noise problem once it enters the space, it is
sometimes possible to actively cancel out the vibration on the walls (essentially by shaking the walls out-of-phase). For low-frequency problems, especially those where the vibration travels along compact structural components
such as struts, this can be quite an effective approach.
The last point to mention in this brief discussion on active noise control in
enclosed spaces relates to a question which has been given a lot of attention
by researchers in recent years: how many canceling sources and error micro-
80
5. Enclosed Space Noise Control
phones are required to provide good sound attenuation? To some degree the
answer to this question is application specific. However, an upper limit on the
minimum number of sources required for global attenuation is equal to the
number of modes which are excited in the target frequency band. If there are
two modes resonant then two canceling sources are required, etc. For error
microphones, it is often the case that the more error microphones there are,
the better the result. To a point. If an extremely large number of error microphones are used then there are problems with the control system implementation, as will be discussed in later chapters.
6
Control of Sound Propagation
in Ducts
The final group of noise control problems which we will briefly discuss
here concerns the control of sound propagation in ducts. Ducts can be
viewed as enclosures where one dimension is very long, often terminating
into open space. Common examples of a duct include the airways used in
central heating and cooling systems, and any piping system (including
vehicle exhaust systems). Another example of a duct which may not be so
obvious is a long hallway connecting two adjacent rooms or halls. The
essential acoustic ingredient for a duct is that sound waves be constrained
in two dimensions while being allowed to travel more-or-less freely in the
third. For this reason, ducts are often referred to in acoustics literature as
waveguides, where the constraining walls guide the travel of sound waves
in the third dimension.
Once again, before we examine passive and active noise control approaches
to attenuating sound propagation in ducts, we will examine important characteristics of the sound field in ducts. Once a set of characteristics is established it is straightforward to understand and optimize the physical
mechanisms behind the noise control approaches.
Sound Fields in Ducts
The idea that a duct is simply an enclosed space with the boundary removed
on one side provides us with clues about the structure of a sound field in a
duct. It is intuitive that in describing the sound field we should essentially
be considering the bounded sides and the open sides separately.
The bounded sides can be expected to have a modal response, similar to
that of a fully enclosed space. When a sound wave fits in the cross section
the response will peak. The frequency where this occurs is a resonance frequency. If, for example, the duct cross section is rectangular, then we will
have resonances for what are essentially axial and tangential enclosure modes.
The open side can be expected to have a response similar to that associated with free space sound radiation. In the absence of any boundaries, the
81
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
82
6. Control of Sound Propagation in Ducts
sound waves will simply travel away and there will be no resonance response associated with this side of the duct.
This intuitive model of sound fields in ducts is essentially correct. There
are, however, some details which must be added to provide a complete
description.
Modes in Ducts
Ducts do have a modal form of response. However, unlike modes in an enclosed space, duct modes are restricted to being one- or two-dimensional
(associated with the cross section). Duct modes also propagate, or move
down the duct.
The fundamental, or lowest frequency, mode in a duct is the plane wave
mode. Referring to Figure 6.1, a plane wave has a uniform sound pressure
distribution in the duct cross section, and has the same waveform of pressure
distribution down the duct as an acoustic wave in free space. A plane wave
does not have a "true" resonance response associated with cross section. In
theory, the plane wave mode has a resonance frequency of 0 Hz, being an
axial mode with an infinite length in one direction.
All modes in the duct other than the plane wave mode are referred to as
higher-order modes. These modes have an enclosure-like modal pressure
distribution in the cross section.
It was mentioned that duct modes will travel, or propagate, down a duct.
We can view the acoustic energy that is flowing down a duct to be divided
up amongst the traveling modes; each mode carries a bit of the total energy.
One of the most important results in duct acoustics is that a duct mode can
only travel if the frequency of sound is greater than or equal to what we have
been thinking of as the resonance frequency of the mode. Because of this
result, ducts modes are usually referred to as having a cut-on frequency,
rather than a resonance frequency. At all frequencies above this one, the
duct will be "cut-on" and physically allowed to carry acoustic energy down
Sound source
FIGURE 6.1. Plane wave sound propagation in a duct.
Note that the pressure distribution is uniform in all duct cross sections, and that the pressure peaks and troughs
travel down the duct at the speed of sound.
Impedance in Ducts
83
the duct. If the frequency is below a cut-on of a given duct mode the mode is
said to be "cut-off," or evanescent.
The fact that duct modes can only travel and carry energy at frequencies
above their cut-on frequency has significant implications for noise control
approaches. For example, if the duct cross section is small compared to the
frequency of interest, and so none of the higher-order modes are cut-on,
then essentially all of the acoustic energy flowing down the duct will be
carried by the plane wave mode. Therefore, any noise control approach we
may wish to adopt must specifically target the plane wave mode. Higherorder modes do not explicitly have to be targeted for attenuation.
The process of cutting-on a mode is not binary. It is not the case at 1 Hz
below the cut-on that the mode is doing nothing, and at 1 Hz above the cut-on it
is carrying a large amount of acoustic energy. As mentioned, when the frequency
is below the cut-on of a mode the response of the mode will decay with increasing distance from the source. As the frequency approaches cut-on, the rate of
decay decreases. To illustrate the mode is like a bit of chewing gum stuck to the
bottom of your shoe. While it will stay attached to the location of the noise
source, it can be "stretched" in the same way chewing gum will stretch when
you try to pull it off your shoe. The closer the noise source frequency is to cuton, the longer it will stretch. At cut-on, a bit of the mode is finally allowed to
break away, just like a bit of chewing gum coming off your shoe. The entire
mode does not suddenly propagate down the duct, just a bit of it. As the frequency increases beyond the cut-on, more and more bits of the mode can propagate, and so the acoustic energy carried by the mode increases as the frequency
increases beyond the cut-on frequency. Therefore, when assessing how important a duct mode is in carrying the portion of the total acoustic energy, and so
assessing how much attention should be paid to it in the noise control process, it
is not enough to know simply whether the mode is cut-on. When it cut on is also
important.
Impedance in Ducts
When a sound waves travels in free space, it sees only one impedance. Technically, this is the specific acoustic impedance, defined by the ratio of (pressure)/(particle velocity), where particle velocity is the velocity of a small
bit of the continuous medium in which the acoustic wave flows. Specific
acoustic impedance for a given medium is defined by the product: (speed of
sound in the medium) x (density of the medium). If a duct is infinitely long,
and so the acoustic wave never reached the end, then the impedance the
1 A cynical person might ask, how many infinitely long ducts are there in the world?
None, I suppose. However, if an acoustic wave is completely absorbed before it
reaches the end of the duct, then the duct appears to be infinitely long to the wave as
it travels along; the end is never reached. There are many examples of this.
84
6. Control of Sound Propagation in Ducts
wave would see in the duct would be the same as the free space impedance. 1
However, most real ducts are very much finite in length. When the wave
hits the end of the duct, it will usually hit an impedance change in moving
from the duct to the open environment into which the duct exhausts. This
impedance change will cause some of the traveling wave to be reflected and
move back upstream. This scenario has obvious implications for acoustic
energy flow and noise control approaches.
Calculation of the actual impedance seen by an acoustic wave as it looks
down a duct with some specific end conditions is complicated, and beyond
the scope of this book. However, a few general comments can be made.
1. Impedance is a complex number quantity. The real part of the impedance,
which is the resistance, is associated with acoustic energy flow. The imaginary part of the impedance, which is the reactance, is not associated with
acoustic energy flow. Therefore, to reduce acoustic energy flow, it is necessary to reduce the real part of the impedance only. If the end conditions of
the duct were such that there was no real part to the impedance, then the
acoustic power output of the duct would be zero. Achieving this is often the
aim of vehicle muffler design.
2. For a duct exhausting into free space, the resistance is dependent upon a
number of quantities. These include: duct cross-sectional area, duct perimeter, duct length, characteristics of the medium (such as air), frequency of sound, and the velocity of air flowing down the tube.
3. It is possible to have a given end arrangement such that the impedance
of the duct is the same as that of the free space environment. In this
case, there will be no reflection of the sound waves back upstream. Wave
reflection requires an impedance change.
Passive Noise Control Approaches
The noise control practitioner has a wide variety of techniques available for
tackling the problem of sound propagation inside of, and eventually out of,
ducts this. While these methods can again be divided into two groups, acoustic
energy flow redirection and acoustic energy flow reduction, the redirection
options are rather trivial from a technical standpoint. They basically consist
of pointing the duct exhaust in another direction, such as straight up, or
erecting a wall in front of the exhaust to direct the energy flow elsewhere.
The acoustic energy flow reduction techniques for the duct noise problem
can be further divided into two categories: those which specifically aim to
alter the duct impedance, referred to as reactive techniques; and those which
specifically aim to absorb the acoustic energy as the wave propagates down
the duct, referred to as dissipative techniques. In general, reactive techniques
are best applied to low-frequency noise problems as they tend to provide
more compact solutions than dissipative techniques. Dissipative techniques
are best applied to mid- and high-frequency noise problems as they tend to
Side-branch Resonator
85
work over a wider frequency range and are cheaper and simpler to build.
We will first consider reactive techniques for passive noise control, and restrict ourselves to three common devices: the side-branch resonator (which includes the Helmholtz resonator), the expansion chamber, and the Helmholtz
filter.
Side-branch Resonator
A side-branch resonator is a useful device for attenuating pure tone sound
propagation in a duct, such as might arise from a fan. A side-branch resonator is basically a tuned piping arrangement placed off the main piping run,
as shown in Figure 6.2. The tuned arrangement is commonly a sealed section of pipe with a length equal to one-quarter the wavelength of the target
frequency (a "quarter-wave stub"), or a "volume" section connected to the
pipe via a smaller orifice section (a "Helmholtz resonator," which is shown
in the diagrams of Figure 6.2). The aim of the side-branch resonator is to
essentially offer the sound wave a parallel alternative to the main piping
run, a parallel that is designed to have negligible impedance at the target
frequency. This is analogous to running through the jungle and being offered an alternative, parallel path of an open field (which would you take?).
The state of negligible impedance occurs at the resonance of the side-piping
section, which is why it is called a resonator. If the resonator is perfect and
so the impedance is zero, then all of the acoustic energy will flow into the
resonator and none will continue down the duct.
For the Helmholtz resonator sketched in the figure, the resonance frequency is defined by
Sound source
f
)
Direction of sound propagation
FIGURE 6.2. A side-branch resonator.
86
6. Control of Sound Propagation in Ducts
where c is the speed of sound (343 meters per second, nominally), A is the
cross-sectional area of the pipe connecting the volume to the pipe, L is the
length of the connecting pipe, and V is the volume of the enclosed space
behind. It should be noted that it is difficult to manufacture a "perfect" sidebranch resonator, one which will resonate at some specified frequency when
it is placed in the duct. It is common to include some way of adjusting the
system for in situ tuning.
In order for the resonator to be most effective in providing sound attenuation, it should be placed at a point in the duct where the acoustic pressure is
maximum. Intuitively, the pressure will then be most sensitive to an impedance change. This is commonly an odd multiple of one-quarter wavelengths
downstream from the noise source (fan, etc.).
Side Story. Side-branch resonators appear in many places. One clever application was in a recent model station wagon. When the wagon was first built,
the designers found that there was an axial acoustic mode running across the
back seat which had a resonance frequency that was the same as the frequency emitted when air entered the intake manifold (part of the engine)
when the throttle was partially open. The result was that if the car was driven
under certain conditions the occupants in the back seat were subjected to
unacceptably high levels of noise, as the axial mode resonance frequency
"latched on" to the noise of air entering the engine. To fix the problem, the
designers included a Helmholtz resonator section in the plastic ductwork
which guides air into the air filter. The resonator essentially removed the
unwanted frequency as the acoustic waves traveled along the air inlet dueting, and so the back seat resonance noise problem was solved!
Side Story. Side-branch resonators have an equivalent in the field of vibration control: the vibration absorber. Basically, if a structure is vibrating at a
single frequency, and the vibration levels are unacceptable, then a "mass" is
attached to the vibrating structure via a spring. The mass and spring pair
have a certain frequency at which they want to vibrate, the resonance frequency of the mass/spring pair. If the mass and spring are chosen such that
the resonance frequency is the same as the frequency of unwanted vibration,
then all of the vibration energy will flow into the mass/spring subsystem and
the structure will stop vibrating! This is a good way to stop a piping system
from vibrating-hang some weights off the vibrating pipe with a spring!
Expansion Chamber
Any reader who has experience with a two-stroke motor cycle will know
immediately what an expansion chamber is. An expansion chamber is basically a relatively large opening in the piping system, as shown in the dia-
Helmholtz Filter
Sound source
87
Expansion chamber
FIGURE 6.3. An expansion chamber in an exhaust system.
gram of Figure 6.3. As inferred by the first sentence, they are perhaps the
most common form of muffler fitted on two-stroke engines.
Expansion chambers are able to provide sound attenuation over a wider
range of frequencies than the previously discussed side-branch resonator.
Referring to the figure, the ability of the expansion chamber to provide sound
attenuation at low frequencies is limited by the resonance of the volume/
exhaust tube combination. This resonance frequency can be calculated using the same equation as given for calculating the resonance frequency of
the Helmholtz resonator, where the speed of sound c is often higher as the
medium is hot, high-speed exhaust gas. At the resonance frequency, the expansion chamber actually amplifies the noise, rather than attenuating it. Above
the chamber/tailpipe resonance frequency, attenuation will be provided up
to the point where resonances of the actual piping system come into play.
Side Story. As mentioned, expansion chambers are commonly used as mufflers on two-stroke engines, such as on motocross motorcycles. When designing the muffler, it is important to consider the effect of which
backpressure has upon engine performance. Backpressure is the increase in
pressure at the outlet of the engine which results from bolting on the muffler. Research has shown that there is an optimal tailpipe length for a given
expansion chamber design which will give the best engine performance. If,
for example, the motorcycle manufacturer has designed the expansion chamber muffler to include this optimum length, then fitting an additional aftermarket tailpipe to the system, no matter how chromed, will result in a
performance reduction. This is usually the opposite of the intended effect.
Helmholtz Filter
An extension of the expansion chamber idea is the low-pass, or Helmholtz,
filter shown in the diagram of Figure 6.4. This device is commonly used to
suppress pressure fluctuations in flowing gas. Qualitatively, the performance
of the system can be viewed as a magnification of the expansion chamber
88
6. Control of Sound Propagation in Ducts
Sound source
FIGURE 6.4. Helmholtz, or low pass, filter.
performance. Low-frequency attenuation is limited by the resonance frequencies of the chambers, as at resonance the noise will be amplified. Attenuation is provided at frequencies above this. The precise high-frequency
performance is dependent upon the tailpipe conditions (long, short, etc.).
In general, the Helmholtz filter attenuates higher frequencies while having little effect upon low frequencies, where low is defined by the chamber
resonance frequencies (these can be very low). For this reason, it has the
form of a "low-pass filter." Low-frequency components of the sound field
(only) are allowed to pass.
Dissipative Passive Control Techniques
Dissipative approaches to passive noise control in ducts aim to absorb the
sound field as it travels along, converting the energy in the acoustic field to
a minute amount of heat. Dissipative passive noise control most commonly
takes the form of a porous lining material (sometimes referred to as "fluff")
placed on the walls of the duct. In industrial settings, the fluff is often protected by a thin, limp piece of plastic, similar in many ways to kitchen plastic wrap, and/or a perforated metal panel. In the perforated panel case,
provided that the open area provided by the holes is approximately 25% or
more of the total panel area, the effect on the acoustic performance of the
liner is negligible (provided that the holes themselves do not "whistle").
Proper design of a dissipative duct liner requires balancing many parameters. Some general trends are as follows:
1. The three most important parameters for determining the level of sound
attenuation for a given duct are: (i) the flow resistivity of the lining
material; (ii) the thickness of the lining material relative to the diameter
of the open (unlined) portion of the duct; and (iii) the area of duct over
which the liner is installed. The area is defined by both the length of
lined section and how much of the duct perimeter is lined (all sides, two
sides, etc.).
2. The level of sound attenuation provided varies with frequency. When
plotted as a function of frequency, the attenuation properties of a given
Active Noise Control Approaches
89
lining arrangement generally peak at some middle frequency and fall
off at higher and lower frequencies. The division of high, low, and middle
frequencies is based upon the ratio of the radius of the open area in the
center of the duct to the wavelength of sound. Performance is often best
when these are approximately equal. In general, the greater the peak
attenuation (in the middle), the worse the performance at frequencies
away from the peak.
3. Generally, a thin layer of lining (say, one-fifth of the radius of the open
areaof the duct) will have a greater peak performance than a thick layer
of lining. However, the performance away from the peak will be significantly worse.
4. Dissipative techniques work best when several higher-order modes in the
duct are cut-on. Conversely, performance is generally the poorest for the
plane wave mode. This makes sense intuitively, as the higher-order modes
impinge upon the sides of the duct whereas the plane wave mode does not.
5. Generally, to achieve any sort of meaningful attenuation, the duct lining
should be placed over a length which is ten times or more the diameter
of the open section of duct.
Active Noise Control Approaches
The application of active noise control to problems of sound propagation in
ducts was one of the originally envisaged uses when the technology was
patented over 60 years ago. Even today, it is the most popular application of
the technology. While there is a variety of possible ways to implement an
active noise control system in a duct, we will concentrate our discussion
here on the feedforward systems. These are by far the most popular, and
arguably most useful, arrangements for the duct noise problem.
Noise
source
+
Direction of sound propagation
Reference
microphone
Control
source
Error
microphone
Control
system
FIGURE 6.5. Feedforward active noise control system for attenuating sound propagation in a duct.
90
6. Control of Sound Propagation in Ducts
Shown in Figure 6.5 is a diagram of the main components of a feedforward
active noise control system in a duct. Ignoring the controller part of the
system for the moment (for a later chapter), the most important items, insofar as determinants of performance of the duct active noise control system,
are in no particular order:
1. the quality of the reference signal, in particular, the ability of the reference
sensor to provide a good measurement of the propagating sound field without a large amount of corruption from air flow past the microphone;
2. the quality of the error signal measurement (similar to (1));
3. the separation distance between the reference sensor and control source;
4. the location of the control source in the duct and
5. the characteristics of the duct response, in particular, what modes are
cut-on or close to cut-on, for the target frequency range.
Note for future reference. While referring to the figure, it is worth noting
that another important consideration is the amount of feedback that occurs
between the control source and reference microphone. This feedback can
lead to the same howling effect as occurs when a microphone is moved toward a loudspeaker in a public address system. Small amounts of feedback
can be compensated for in the controller design. Discussion of this design
consideration will be left for the last chapter.
We will discuss each of the important criteria outlined above separately.
Reference and Error Signal Quality
The signal from the reference sensor (the reference signal) provides the controller with an indication of the impending disturbance. Given this signal, it
is the job of the controller to calculate a suitable canceling signal. It is obvious, then, that the reference sensor must actually be measuring the sound
field propagating down the duct. For active noise control implementations
targeting sound fields in ducts, this is not always as straightforward as it
sounds. If, for example, you want to measure the sound field in an air-conditioning duct and you simply insert a microphone into the duct, chances are
that you will only measure the wind passing over the microphone. This is
true even for slow air speeds. The actual sound field measurement will be
buried somewhere in the air flow noise.
It is common to see wind screens, which resemble foam balls, placed on
microphones in public address systems. These are useful for small amounts of
wind noise, but are usually useless in a duct active noise control implementation. The air flow noise is simply too much of a problem for common wind
screens. Most commercial-grade active noise control systems in ducts use highquality "antiturbulence" microphone systems for both reference and error signal acquisition. These systems were originally developed in the 1960s for
Reference Sensor/Control Source Separation Distance
91
Direction of sound propagation
I
Tube
)
Slit
·
hone I~
MICrop
FIGURE 6.6. Typical arrangement for an antiturbulence microphone probe.
measuring sound fields in wind tunnels, and if designed correctly can yield a
reasonable measurement of the sound field, even in high-speed air flow environments.
Antiturbulence microphone probes commonly consist of a long tube
plugged at the upstream end and having the microphone mounted at the downstream end, as shown in Figure 6.6. In the tube is a slit, that is covered with
a porous cloth material. The idea behind the arrangement makes use of the
fact that the sound waves and the wind responsible for the unwanted wind
noise travel at greatly different speeds. Sound entering the microphone tube
through the slit will tend to amplify itself via additions of more sound from
the outside as it travels along the tube. The wind noise, however, will not.
The wind outside the tube will not travel as fast as the wind noise inside the
tube, and so the wind noise entering at one point in the tube will not be
related to the wind noise entering at another point in the tube. The end result
is that the sound field component of the measurement increases in proportion to the wind noise point. Technically, the signal-to-noise ratio improves.
A similar situation holds for the error microphone signal. The controller
uses the error microphone signal to tune its internal calculation process.
Basically, the controller is looking for any residual component in the sound
field measurement which is related to the information it was provided with
by the reference signal. Technically, it is looking for any correlation between the reference signal and error signal. If the wind noise dominates the
error signal, then when the controller "compares" the reference and error
signals it will think that all of the referenced sound field has been removed.
It cannot find any trace of it in the measurement, as it has been swamped by
the wind noise. Therefore, for the control system to properly tune itself, the
error signal measurement must provide information about the sound field
with as little corruption from wind noise as possible.
Reference Sensor/Control Source Separation Distance
It was previously mentioned that it takes a finite amount of time for the
control system to receive the reference sensor measurement, calculate a suit-
92
6. Control of Sound Propagation in Ducts
able canceling output, and actually turn this into sound coming from the
loudspeaker, or control source. This time period is often referred to as the
system's group delay. In a digital system, this time period can be dominated
by the components used to actually get the signals into and out of the microprocessor (the analog to digital and digital to analog conversion process,
which will be discussed in the next chapter). It will also take a period of
time for the loudspeaker to generate sound once it has received an electrical
signal. All together, the required time period between receiving a reference
signal and outputting a canceling signal is usually a period of milliseconds:
as low as a couple of milliseconds for an active noise control system targeting moderate frequencies (say, up to 1000 Hz) up to as long as several tens
of milliseconds for very low-frequency systems. Typically, the time period
will be between 5-10 milliseconds for industrial duct noise problems.
For active noise control systems targeting the entire sound field spectrum, and not simply the harmonic components associated with, say, the fan
rotation at the end of the duct, the control system must be causal. This means
that the separation between control source and reference signal must be long
enough to give the control system the time it needs to output a signal before
the (referenced) sound field arrives. Sound travels roughly 1 meter every 3
milliseconds, so if the group delay is 9 milliseconds the control source and
reference sensor must be separated by at least 3 meters, or approximately 10
feet. A safety margin should be added to this, so a better separation distance
would be 4 meters.
Control Source Position in the Duct
It was mentioned in the passive noise control section that a side-branch reso-
nator is most effective if placed in a location where the sound pressure amplitude is at its maximum. This is often an odd number of one-quarter
wavelength intervals from the sound source. The same idea holds for active
noise control systems: they work best if the control source is placed at a
location of peak sound pressure amplitude.
While this notion is simple to state, it is not always easy to follow. In a
ducting system with changing air flow and possibly changing temperature,
the locations of maximum pressure will change location over time. This can
have a marked impact upon overall system performance, particularly if there
is a large amount of harmonic content in the sound pressure field (from, say,
fan rotation).
Duct Response Characteristics
In the discussion on noise control in enclosed spaces, it was mentioned that
active control is most effective when there is only one, or at most a few,
Duct Response Characteristics
93
discrete modes responsible for the majority of the unwanted noise. This same
idea holds true for noise control in ducts: active control is most effective
when only one mode (the plane wave mode) is cut-on. In fact, it can be said
that active control will be largely ineffective at frequencies where there are
multiple higher-order modes cut-on. This result is opposite to that of passive
noise control, where passive control is much more effective at controlling
high-order mode sound propagation than plane wave mode sound propagation. The two techniques can therefore be used together to get the best result: active control for low frequencies (where the plane wave mode
dominates), and passive control for mid and high frequencies.
Suppose, then, that you are interested in controlling noise inside an industrial exhaust stack which is several meters in diameter, and that you are
interested in frequencies up to 200 Hz. Your preliminary calculations show
that higher-order modes will start cutting-on at around 50 Hz. What can you
do? The common solution to this problem is to put a splitter in the duct. A
splitter consists of one or more partitions placed in the duct to effectively
turn it into several smaller ducts. The partitions must be long enough to fit
the reference sensor, control source, and error sensor within the section, as
well as provide a few meters at the front to accommodate the sound field
transition from the higher-order mode to plane wave sound propagation. By
splitting the duct, it is possible to force the sound to propagate in the plane
wave mode in the smaller duct sections and to achieve good control using
active methods.
Side Story. Duct splitters can be used in a clever way to provide active
control-like sound cancellation with electronics. Consider the curved duct
section shown in Figure 6.7. If the duct section is split in half, then the
effective length of the top section is longer than the effective length of the
__- - - - - Splitter
tFIOW
FIGURE 6.7. Flow splitter in a curved duct section, that presents a propagating sound
field with two different paths of travel (with two different path lengths).
94
6. Control of Sound Propagation in Ducts
bottom section. Suppose now that a pure tone sound field is propagating
down the duct. When the sound field enters the duct section, the sound waves
are in phase. However, because the top section is longer than the bottom, the
sound waves will no longer be in phase at the exit of the pipe. If the duct
work is designed correctly, the exiting waves will be exactly out of phase
and cancellation will occur-active noise control without electronics! This
technique has been used effectively for problems such as exhaust ducts for
cement kilns, where the disturbance is tonal (from the exhaust fans) but the
environment inside the duct is so terrible that even the most robust loudspeakers and microphones would have a short life span. Metal partitions last
a lot longer!
7
Active Noise Controller Overview
Active noise control systems provide sound attenuation by introducing a
second, electronically generated, sound field into the acoustic environment.
If the amplitude of the second sound field is the same as the first, but the
phase inverted, then the two will cancel. The end result is quiet. As we have
discussed in the preceding chapters, the potential for active control to provide a satisfactory result is largely a function of the physical acoustics of the
problem: the characteristics of the unwanted sound field, the response of the
acoustic system, the location of the loudspeakers, etc. We can conclude that if
the physical acoustics "aren't right," then even the biggest, best, and most
expensive active noise control system will not be able to provide the desired
levels of sound attenuation. Active noise control is not a universal panacea;
the problem must be amenable to the recommended solution.
Suppose now that we have conducted the necessary preliminary investigation and calculations, and have decided that active noise control might be the
answer to our problems. Having selected appropriate positions for loudspeakers and microphones, what do we do next? How do we implement a system?
The missing component is the controller.
In active noise control work, the controller is basically the "electronics"
which are responsible for receiving signals from the microphones, calculating a suitable canceling signal, and then sending the canceling signal to the
loudspeaker system. There is a wide variety of controller software and hardware configurations that find use in this task. It is beyond the scope of this
text to consider most of them in even a superficial way, or any of them in
depth. What we will be concerned with here is one specific type of controller,
which is the most popular amongst researchers and developers: the adaptive
feedforward controller.
The purpose of this chapter is to provide a brief "heuristic" discussion of
the most important parameters in an adaptive feedforward control system.
The discussion is useful if, for example, you are simply contemplating active
noise control for the first time, or you are using a controller that someone else
has developed, or you are interested in interpreting someone else's results.
The discussion in this chapter will not go into any depth at all in regard to
any single component, and will not provide enough information to enable
you to build a controller. However, if you are simply looking for some hints
95
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
96
7. Active Noise Controller Overview
on how to tune a controller, or just want to know what a controller is, then this
chapter may be all that you need to read. Chapter 8 will consider the controller in more depth.
Some Important Facts
Before beginning our heuristic discussion, there are three important facts regarding the adaptive feedforward active noise controller that need to be stated:
• Practically all adaptive feedforward active noise controllers are digital.
Therefore, the fundamental properties associated with digital systems in
general will have an influence on the performance of the active noise
controller.
• The overall task of the adaptive feedforward active noise controller is to
receive the reference signal and calculate an appropriate canceling signal.
As the canceling signal must be the mirror image of the unwanted noise
(sample amplitude, inverted phase), the controller must effectively mimic
the response of the target acoustic system. Consider that if the controller
was a perfect replica of the target acoustic system, then we could simply
reverse the leads on the loudspeaker input to get cancellation.
• In most target acoustic systems, the characteristics of the response of the system will change with time in response to changing temperature, air flow, number of people inside the enclosed space, etc. Also, the response of the
loudspeaker system generating the canceling sound field will change with
time as the components age, fungus grows on them, etc. Therefore, in order to
provide a high level of sound cancellation over an extended period of time,
the controller must be able to adapt to a changing environment.
The design and tuning of all adaptive feedforward active noise controllers
involve an optimization of parameters that are encompassed in these three
statements. These parameters (which have yet to be named) often influence
more than one "statement," and so controller tuning effectively becomes a
negotiating process, an attempt to achieve the best compromise and thus best
performance in the given circumstances.
The discussion in this chapter will be organized to roughly address the
above three statements independently.
Digital System Requirements
Digital systems are used by practically everyone in some way. Your computer,
modem, CD player, even telephone answering machine (maybe) are all digital.
What makes a system "digital"? The most important characteristic is that a
digital system works with discrete numbers: 23, - 47.21, etc. This is as opposed to an "analog" system, which works with continuous signals.
Digital System Requirements
97
To contrast an analog and digital system, consider the answering machine
example outlined in Figure 7.1. An analog answering machine receives a voice as
a voltage signal, the same sort of signal that could be fed to a loudspeaker to
produce sound. The analog answering machine essentially takes this voltage
signal from the telephone, turns it into a magnetic signal, and stores the image of
the magnetic signal onto magnetic tape. When the tape is played back, the magnetic signal which was recorded on the tape is read, converted back to a voltage
signal, and played out from a loudspeaker. The message can then be heard.
A digital system works slightly differently. When the digital system receives the voltage signal from the telephone, it turns it into a set of numbers.
What are these numbers? Referring to Figure 7.2, the numbers represent the
amplitude of the voltage signal at discrete instances in time. For example, at
10.1 milliseconds into the message the voltage amplitude was 0.92 volts, at
10.2 milliseconds the amplitude was 0.03 volts, at 10.3 milliseconds the
amplitude was -0.4 volts. These numbers are stored in memory on the answering machine. When you want to play back a recorded message, the answering
machine will output the recorded amplitude at the appropriate period of time:
0.92 volts at 10.1 milliseconds, 0.03 volts at 10.2 milliseconds, -0.4 volts at
10.3 milliseconds, etc .; see Figure 7.3. With a bit of "smoothing" (described
in the next chapter), the voltage signal will be reconstructed and then sent to
a loudspeaker for the listener to hear.
Analog
Storage
~
~
Magnetic
Electrical
~
~
Numbers ~
Digital
Storage
Computer
Chip
FIGURE 7.1 . Analog answering machines utilize a continuous magnetic signal in their
storage system, whereas digital answering machines store discrete numbers on a computer chip.
98
7. Active Noise Controller Overview
Voltage Signal
~t*-(~i
10
11
12
;>.time
(milliseconds)
Time
10.1
10.2
10.3
Computer
chip
FIGURE 7.2. The numbers stored in a digital system correspond to the amplitude of the
voltage input signal at discrete, evenly spaced, moments in time.
So what is important in a digital system? From the above example, it is
intuitive that two very important parameters are:
• How often the digital signal samples the voltage signal. In the above example, a new voltage measurement was taken every 0.1 millisecond.
• The accuracy with which the voltage measurement is taken. For example,
is the voltage 0.6 volts, or 0.601 volts, or 0.6008137 volts?
Parameter 1, how often the digital system samples the voltage signal, is usually quantified by a system sample rate. The sample rate of a system is the
Computer
chip
0.92
Voltage signal
-D.4
~
Hold and
smooth
FIGURE 7.3. The digital (computer chip) output is turned into discrete voltages at discrete
moments in time. These voltage levels are held constant over the sample period, and the
resulting signal smoothed before output to the loudspeaker.
Digital System Requirements
99
number of samples taken per second (evenly distributed over the second).
Sample rate may be given the units of Hertz for convenience, even though
there are no "cycles" involved. For example, if the sample rate is labeled 1000
Hz then 1000 samples are taken every second, corresponding to a sample
being taken every 0.001 second, or 1 millisecond.
What should the sample rate be for a given implementation? In active
noise control, the answer to this is entirely dependent upon your target frequency. Suppose, for example, the unwanted noise is a tone (sine wave) at 50
Hz. Theoretically, it is possible to measure and control this frequency with a
sample rate which is twice the target frequency, or 100 Hz. However, this is
not an advisable choice, for a number of reasons. Basically, this is the theoretically extreme case, which is not always practically achievable. A much
better choice is ten times the target frequency, which in this example is 500
Hz or 500 samples per second.
While experience has shown that a sample rate of ten times the target
frequency is about optimal, an adaptive feedforward active noise control
system will work satisfactorily with sample rates that are something like 3-50
times the target frequency. The precise range is dependent upon a number of
hardware and software factors, but 3-50 is a good starting point. For the 50 Hz
tonal problem in the example above, this means we could expect reasonable
performance from a system with a sample rate somewhere in the range of 150
Hz-2500 Hz. If it is any slower or faster it will begin to have a telling effect
upon both sound attenuation and controller speed. Note that if there are a
number of discrete target frequencies, or perhaps a frequency range, then
ideally the frequencies should fit within the range defined by one-third to
one-fiftieth of the sample rate.
The second important digital parameter, the accuracy of the sample, needs
to be considered in two parts: the accuracy with which the sample is actually
measured, and the accuracy with which it is recorded and manipulated in the
calculation process by the microprocessor.
The first part, the accuracy with which the measurement is taken, is set by the
analog to digital converter (ADC). This is either a specific piece of hardware on
the circuit board (a microchip dedicated to the process of measuring the input
voltage signal) or possibly a specific hardware section of a single special-purpose microprocessor. The accuracy of the measurement is normally quantified in
terms of a number of "bits." The question that follows is, what is a "bit"?
Normally, a digital number has a binary representation, expressed as a
number of bits. Each bit has two possible values: 0 or 1. A number of bits are
used together to store a range of numbers. If, for example, the digital number
is "3 bits" wide, then there are three bits available for number representation,
each with a possible value of 1 or O. The representation of a number by bits is
referred to as a binary representation. The binary form of a number is written
as X 2X l X o for a three-bit system, where Xl is the value (0 or 1) of bit 2, Xl is the
value of bit 1, and Xo is the value of bit O. One common way of storing numbers in a three-bit format would be as shown in Table 7.1.
100
7. Active Noise Controller Overview
TABLE 7.1. Binary numbers (three-bit)
and their decimal equivalents.
Binary number
Decimal equivalent
o
1
10
11
100
101
110
111
o
1
2
3
4
5
6
7
There are a number of points to be made concerning this method of counting:
• There are only eight possible numbers that can be stored by the three-bit
digital system. Generalizing, the numbers which can be stored by a digital
system are 2n , where n is the number of bits (2 3 8 in the example above).
• When multiple bits are used to store numerical values, each bit represents a
different part of the total value. If a given bit is "set," or equal to 1, then the
value that is represented by the bit must be added to the total which defines
the number in binary notation. For the ith bit, the value represented is 2;,
starting with i =0 for the rightmost bit. To read the binary number the values
of each set bit must be added together: if the right/most bit is set, then 1 must
be added to the total; if the second right/most bit is set, then 2 must be added
to the total; if the third right/most bit is set, then 4 must be added to the total.
So, for example, the number that is represented by the binary notation "011,"
where bit 1 and bit 0 are set, is equal to (2 1 + 2° = 2 + 1 =) 3. For identification
purposes, the rightmost bit which represents the smallest number is referred to
as the least significant bit, while the leftmost bit that represents the largest
number is referred to as the most significant bit.
• There are no negative numbers shown in the previous list. How, then, can a
digital system storage a negative voltage measurement corresponding to
the rarefaction component of a sound wave? It is common for digital systems to use the most significant bit to represent the sign of the number.
Using this idea, the numbers available in our three-bit example could be as
shown in Table 7.2. This way of "counting" in a digital system is technically referred to as twos-complement, as opposed to ones-complement.
From the standpoint of being a digital system user, it is not terribly
important to know what form of counting is used internally in the processor. It can, however, be important to know whether the system can handle
negative voltage inputs or whether all voltages must be positive. In the
former case the inputs will be referred to as bi-polar, while in the latter case
the inputs will be referred to as un i-polar.
• We may ask, why does a digital system have to work this way? Why do
numbers have to be represented by bits? Basically, your wonderful digital
=
Digital System Requirements
101
TABLE 7.2. Twos-complement binary numbers
and their decimal equivalents.
Binary number
Decimal equivalent
o
o
11
100
101
110
111
-4
-3
-2
1
10
1
2
3
-1
system is just a glorified box of switches, each with two states: on (1), or
off (0). There is no underlying intelligence; this is provided by the programmer and/or designer who decides what to do with the switches.
Analog to digital converters are classified by the number of bits they use to
represent a quantity. For example, a 12-bit ADC converter will represent a sampled
physical system variable as a set of 12 bits, each with a value of 0 or 1. This
means that there are 2 12, or 4096, possible numbers that can be used to represent the amplitude of the voltage signal. Common accuracies of ADC used in
active noise control are (at present) 12 bit, 14 bit, and 16 bit. In general, the
more bits, the better the system.
As a digital system user, it is important to recognize that it is not enough to
simply have the bits available to you; you have to use them. The 2" numbers
that are available to you are spread evenly over the input range of the system.
This range must be adjusted to match the actual voltage range you are interested in (or vice versa). Suppose, for example, that the bipolar input voltage
range of the digital system is -10 volts to + 10 volts, and you are feeding it
with a microphone signal that has a voltage range of -50 millivolts to +50
millivolts (not an uncommon output voltage range for small lapel microphones). You will only ever be using a small subset of the digital numbers that
are available to you, as the higher numbered bits that represent larger voltage
values will always be equal to O. This will greatly diminish the performance
of your system. As illustrated in Figure 7.4, the solution is to supply some
gain, or amplification, to your input signal so that it takes up as much of the
-10/+ 10 voltage range of the hardware as possible.
As mentioned, the accuracy of the measurement by the ADC is only one
aspect of the system accuracy. The other is the accuracy of the data storage
and manipulation, or the internal accuracy, of the microprocessor. When assessing the internal accuracy of the microprocessor there are two aspects that
must be considered: the number of bits used, and the format of the number
representation (fixed or floating point). The "number of bits used" aspect
follows the same line of thought as outlined in the ADC discussion. The
microprocessors used in active noise control, at present, are typically l6-bit,
102
7. Active Noise Controller Overview
-32, 45, 145, ...
)
I....L._~-----' To microprocessor
FIGURE 7.4. Gain (amplification) should be applied to input signals to ensure that the
entire ADC input voltage range is used .
32-bit, or 64-bit devices. In general, the more bits, the better (and the more
expensive, of course).
The number format aspect of accuracy refers to how the concept of a "decimal point" is handled by the microprocessor. Effectively, afixed-point microprocessor assumes that all numbers have the same (fixed) decimal point
location. When combined with the previous discussion of accuracy (being
limited by the number of bits) this means, for example, that the number 0.12345
might be stored as 0.12345, but the number 0.000012345 might be stored as
0.00001. The extra information is lost to the system. Referring to Figure 7.5,
this is referred to as quantization error.
Ajloating-point microprocessor represents each number by two pieces of
information: the actual data, without all of the preceding zeros, and the location of the decimal point. Therefore, with a floating point system the numbers
Fixed accuracy
of digital system
(
)I
Actual amplitude at sample
time = 0.432129856432 volts
Mathematical operation
0.43216 x 0 2
. 1734
t
I~ Lost data
0.43212 ~ (quantization error)
I
)
Number used by
microprocessor
Fixed accuracy
of digital system
L -_ _ __ _ _ _ _ _-?)
Actual result = 0.0939256544
)
E>
I( 0.09392): 56544
I
Lost data
(quantization error)
Number used by
microprocessor
)
FIGURE 7.5. The finite accuracy of digital systems means that the measurements and
calculations are truncated, with some data lost. The lost data is referred to as the quantization error.
Digital System Requirements
103
above might be represented as follows: 0.12345: data = 0.12345, decimal
point = 0; 0.000012345: data = 0.12345, decimal point = -4. With the latter
number, the information that was lost in the fixed-point case is retained in the
floating-point case.
In general, a system implemented on a floating-point microprocessor is
easier for a nonexpert to tune, as it is far more forgiving on the selection of
parameters which impact upon the range of numbers used in the calculation
process. The cost of using a floating-point processor is just that: floatingpoint processors are more expensive than fixed-point processors. This is reflected in the general areas of use. Items with low-cost electronics, such as
microwave ovens and PC sound cards, use fixed-point microprocessors. Highend electronic systems such as the latest video game controllers use floatingpoint microprocessors.
You might be wondering what impact the finite accuracy of microprocessors actually has upon the performance of active noise control systems. Surely
if the input ranging is correct, then the information that is lost due to quantization errors is incredibly small. This is true, to a degree. However, if an
active noise controller is operating over an extended period of time, the quantization errors begin to build up in the calculation process. Mathematically,
the quantization errors have the form of additional energy with no place to
go. It is therefore common for commercial systems to include mathematical
tools in the algorithms such as "leakage" (described later) to remove the
negative effects of quantization errors.
There are two final points that should be mentioned here on the topic of
digital system consideration. The first is the type of microprocessor used in
the calculation process. There is a huge range of microprocessors currently on
the market, each with some distinguishing feature: low-cost, ease of programming, speed, etc. In active noise control work, the two operations that the
controller must perform most often are multiplication and addition. This is
simply a result of the mathematical nature of the process of calculating a
canceling sound field. Therefore, the microprocessors which are most suitable for use in adaptive feedforward active noise control systems are those
with specialized features for multiplying quickly and accurately. There is a
class of microprocessor, called a Digital Signal Processor or DSP, that is
specifically designed for this purpose; these are the chips most commonly
used in adaptive feedforward active noise control.
The second point to make concerns the output process. Once a canceling
sound field has been calculated in the digital domain of the microprocessor, it
must be sent to the loudspeaker system. This involves sending a set of numbers to a digital to analog converter (DAC), which in turn is connected to the
amplifiers and loudspeakers. The DAC operation is essentially the inverse of
the ADC operation, with a similar set of considerations: number of bits, operating range, etc. The sample rate of the ADC and DAC are usually tied together, so for every new sample coming in via the ADC a sample is output via
theDAC.
lO4
7. Active Noise Controller Overview
Controller Output (Digital Control Filter) Requirements
Overall, the job of the controller is to calculate a signal to send to the loudspeaker system in order to generate a suitable canceling sound field. As previously discussed, this calculation is based on a reference signal which is in
some way related to the impending disturbance.
As illustrated in Figure 7.6, the inherent job of the controller is to model
the acoustic system between the reference signal and the control source. That
is, the calculation process in the controller must be able to produce largeamplitude outputs at frequencies where the acoustic system is loud, and smallamplitude outputs at frequencies where the acoustic system is quieter. The
amplitudes of the unwanted sound field and electronically generated sound
field must match, and the phases for the two inverted, for cancellation to take
place.
The important questions here are, what is involved in the controller output
calculation process and what parameters are important for influencing the
performance of the system?
The adaptive feedforward active noise controller output is usually calculated by digital filtering the reference signal, as shown in Figure 7.7. Filtering refers to the preferential amplification and attenuation of frequency
components in a signal (the reference signal in this case). Digital filtering
simply means that the filtering process is carried out in the digital domain.
Digital filtering basically involves collecting a number of input, and possibly output, signal samples, multiplying the samples by a set offUter coeffi-
Acoustic (primary) system
Residual
noise
~
Control system
FIGURE 7.6. The control system must do the "opposite" of the acoustic (primary) system
in order to generate a canceling sound field. In other words, the control system must
model the acoustic system and invert the phase of the output.
Controller Output (Digital Control Filter) Requirements
105
Acoustic (primary) system
Residual
noise
~
Reference
signal
Digital filter J -- - - '
~ does this
FIGURE 7.7. The generation of the appropriate canceling sound field from the reference
signal input is performed by a digitalfilter.
cients, and adding the products to produce a new output sample.! This idea is
illustrated in Figure 7.8. The filter coefficients are simply a set of numbers,
and so this multiplication and addition process is the same as an elementary
school mathematics problem. From this description, it is intuitive that there
are three parameters that are important in determining the performance of the
digital filtering process:
1. the values of the filter coefficients;
2. the "form" of the digital filter; and
3. the length of (number of samples and coefficients in) the digital filter.
The tuning of the first of these parameters, the values of the filter coefficients,
is largely the responsibility of an adaptive algorithm. This is discussed later
in this chapter.
To explain the impact that the second of these parameters, the "form," or
type, of digital filter, has upon performance, consider the following: The
calculation process which is running on the microprocessor is simply a set of
mathematical operations (multiplications and additions) that must mirror what
happens in the real world. For example, if in the real world (the acoustic
system) the reference signal is altered in amplitude by 3.2 dB, then the numbers that are output from the mathematical operations must also have altered
in value by a factor of 3.2 dB. In the real world, a sound pressure wave may
travel forward forever, as is the case for radiation into free space. Alterna1
It should be noted that digital filtering is discussed in more detail in the next chapter.
106
7. Active Noise Controller Overview
Acoustic (primary) system
Residual
noise
~
F--+~ X weight 1
\--- +> x weight 2
~--+~ x weight 3
x weight 4
~ x weight 5
> x weight 6
Sum of
products
,r---r-> X weight 7
f---~ X weight 8
~-~ x weight 9
Digital filter
FIGURE 7.8. The digital filter calculation involves multiplying samples of the input
reference signal (and possibly the output signal, in the case of an IIR filter) by a set of
weights, or filter coefficient, and outputting the sum of the products.
tively, the sound pressure wave may reflect off boundaries and/or take the
form of a modal resonance. Each of these characteristics must be mirrored in
the mathematical operations undertaken by the digital filter.
How can the mathematics model sound wave reflections? By including
past values of calculated outputs as well as signal inputs in the calculation
process. This type of arrangement can be referred to as a "feedback loop" in
the digital filter, where outputs are fed back into the calculation process.
Digital filters that include a feedback loop are called Infinite Impulse Response (UR) filters. Digital filters that do not include a feedback loop are
called Finite Impulse Response (FIR) filters.
It is intuitively sensible that if there are reflections of the sound pressure
waves in the target acoustic environment, then an IIR filter is a more appropriate choice for the controller output calculation mathematics than an FIR
filter. Reflections can be obvious, as in systems that exhibit modal response
characteristics. Reflections can also be subtle, such as when there is feedback
from the control source to the reference microphone. To the controller, this
latter case has the same appearance as a wave measured by the reference
microphone being reflected back off a wall. If there are no reflections, then an
FIR filter is an appropriate choice.
There is much more to consider in the selection of an FIR or IIR filter than
just "reflections." However, it is a good intuitive starting point. More detailed discussion can be found in the next chapter.
Adaptive Algorithm Requirements
107
The final parameter which influences the performance of the digital filtering process is the filter length. Filter length refers to the number of samples,
and hence weight coefficients, used in the calculation process. 2 The longer
the filter, the more accurate the calculation process. To a point. If the filters
are too long, the adaptive algorithm (described in the next section) is likely
to be slow in tuning the weighs, or not will not converge at all, meaning that
the weight coefficient values will be (very) suboptimal.
What is a suitable filter length? This is dependent upon the frequency
characteristics of the reference signal, and hence the active noise control
target. If the target is a single tone, then an FIR filter of 4-20 taps will usually
work well. If there are multiple tones/harmonics, then 4-20 taps per tone/
harmonic is a good starting point. The actual number required for a given
level of performance is greatly influence by the system sample rate. The more
appropriate the choice of sample rate, the less the required number of weights.
If there are resonances in the target frequency band, then 4-20 taps in both
the feedforward and feedback paths of an IIR filter is a good starting point. In
general, an IIR filter will require less taps for a given level of performance
than an FIR filter.
There is one more point which should be made here. It must be recognized
that the digital filtering process takes a finite amount of time to produce an
output. When this is added to the finite amount of time that is required for the
analog-to-digital and digital-to-analog conversion process, as well as the
finite time that is required for a loudspeaker to produce sound after receiving
an electrical input, it can take anywhere from a few to a few tens of milliseconds between sampling the reference signal and outputting a canceling sound
wave. In this amount oftime, a sound wave can travel between 1-10 meters.
The location of the reference microphone relative to the control loudspeakers
must accommodate this delay if the system is to be causal.
Adaptive Algorithm Requirements
As mentioned, the values of the weight coefficients in the digital control filters
are directly responsible for calculating the signal sent to the loudspeaker system
to generate the canceling sound field. How does one determine these coefficient
values? In theory, it is possible to make a number of measurements in both the
target acoustic system and the active control system prior to operation and to
calculate the weight coefficient values. The problem is, the performance resulting from such a process would not be very good due to nonlinearities in the
system components (for example, the response of the loudspeaker system will
change slightly when there is a flow of air across it). Further, the performance will
deteriorate over time as components age, as environmental conditions change,
2 The weights are sometimes referred to as taps, as outlined in the next chapter, and so
it is common to talk about "the number of taps" in a filter.
108
7. Active Noise Controller Overview
etc. Each of these changes is reflected by a change in optimum values of the
digital filter weight coefficients.
What most feedforward active control systems implement to overcome this
problem is an adaptive algorithm (hence the title adaptive feedforward active
noise control). The job of the adaptive algorithm is to calculate the optimum
digital filter weights while the control system is operating, continually adjusting
and tuning the weights to mirror changes in operating conditions. For the algorithm, the optimum values of filter weight coefficients are those that minimize
the amplitude of the sound field measurement provided by the error microphone.
The questions of interest here are, how does the adaptive algorithm perform this
tuning operation and what parameters are important for it to function well?
There is a multitude of possible adaptive algorithms that will work in an
active noise controller, each based upon some different strategy. The cynic
might refer to many of these as "Jaguar juice" algorithms: sounding nice, and
perhaps good for something, but not universal panaceas for the problem. The
most widely used algorithms are what is referred to as gradient descent. While
this type of algorithm is described in more detail in the next chapter, it will be
worthwhile here painting a heuristic picture of how a gradient descent algorithm operates, to lead to some conclusions about what is important.
Implemented in an active noise control system, the operation of a gradient
descent algorithm is somewhat analogous to "running down a hill while looking through a periscope," as depicted in Figure 7.9. At the bottom of the hill
§
Wind
Optimum filter weights
FIGURE 7.9. The operation of the adaptive algorithm is analogous to running down a hill
while looking through a periscope. The optimum filter weights are at the bottom of the
hill; this is where the algorithm wants to go.
Adaptive Algorithm Requirements
109
is the optimum set of digital filter weight values, the "holy grail" of the
algorithm. As you move up the hill and away from the optimum values your
performance will decrease.
The parameters and quantities used by the gradient descent algorithm have
a number of effects, analogous to:
1. setting a pace at which you run down the algorithm;
2. blocking the wind that is trying to blow you back up the hill;
3. determining whether you have enough energy to actually get down the
hill, and/or whether you are so "hyper" that you are unable to stop at the
bottom; and
4. determining the direction which your viewing periscope is pointing (down
the hill as desired, or completely in the wrong direction, back up the
hill).
The parameters that influence effects 1-3 can be lumped together, while
the parameters that influence effect (4) should be discussed separately. Consider effect 1, the pace at which you run down the hill. If you try to run too fast
you will become "unstable" and fall over. If you proceed down the hill too
slowly you take an exorbitant amount of time to reach the optimum values. If
the bottom of the hill "moves," which is what happens to the active control
gradient descent algorithm when environmental conditions change, you never
seem to catch up.
The algorithm parameter that is chiefly responsible for the speed of running down the hill is the convergence coefficient, also referred to as the algorithm step size. This is arguably the single most important parameter in the
algorithm, and usually requires some form of manual adjustment. If the value
is too small the progress down the hill is too slow. If the value is too large the
progress is too fast and the algorithm becomes unstable. 3
At this point you may ask, why not simply make the convergence coefficient small? Does it really matter if the adaptive algorithm takes a few seconds to converge? There is a problem associated with making the convergence
coefficient too small, a problem that arises from the finite-precision digital
environment in which the algorithm is implemented. If the convergence coefficient is too small then the calculation process will actually stop prematurely, as the finite precision of the digital environment will essentially treat
small numbers in the calculation of the slope of the hill as '0.' The algorithm
will think it has reached the bottom. This relates to effect 3 in the list above;
the algorithm "runs out of energy" before it reaches the bottom.
You may ask, how do you know if the algorithm becomes unstable? The answer is,
"you will know." When the algorithm becomes unstable the digital filter weight coefficients typically become very large, too large for the finite bounds placed upon the
calculation process by the digital environment. This is referred to as calculation, or filter
weight, saturation. The end result is that the controller output sounds something like a
jet engine exhaust at 10 feet away!
3
110
7. Active Noise Controller Overview
This effect may sound funny, but it is quite real. If you take a stable adaptive algorithm implemented in an active noise control system and increase
the convergence coefficient, you can often hear a marked improvement in the
performance. If you then decrease the convergence coefficient the increased
performance will go away.
The converse effect happens when the convergence coefficient is too large
(but still stable). The adaptive algorithm will have so much energy that it is
unable to stop at the bottom, rather running down and up, down and up, in a
steady and stable fashion. The end result here is that the performance of the
control system is never quite as good as it could be.
Putting together the above factors, the effect that the convergence coefficient value has upon algorithm performance takes a shape that looks something like that which is shown in Figure 7.10. In this figure, performance is
quantified by the final mean square error value, which is basically the average amplitude of the squared value of the error signal (the measured sound
pressure in active noise control).
You might be wondering, what is the optimum value of the convergence
coefficient and how can it be calculated? The optimum value is unfortunately
application specific, being dependent upon a number of factors: signal powers, loudspeaker characteristics, characteristics of the response of the acoustic system, etc . This is why some manual adjustment of the convergence
coefficient is usually required.
It is worth mentioning here that in an active noise control implementation
there is one specific factor that greatly limits the stability of the calculation
process and hence the maximum usable value of the convergence coefficient:
the time delay between calculating a new set of weight values and seeing the
effects of the new weight values registered in the error signal. This time delay
comes about from the finite time it takes for a signal to travel out of the
......
e......
Q)
~
C1l
::l
0-
en
c
C1l
Q)
E
Optimum convergence
' / coefficient value
(U
c
i..L
L -________________
~----~
Convergence coefficient
FIGURE 7.10. The value of the convergence coefficient used by the adaptive algorithm
has a significant impact upon the final performance (cancellation) of the system.
Adaptive Algorithm Requirements
III
controller, be turned into a canceling sound field by the loudspeaker system,
travel to the error microphone location, be measured by the error microphone,
and get through the analog-to-digital conversion process to be received by
the microprocessor. While this may only be a matter of milliseconds, the time
delay is very significant to a system which is sampling new data every fraction of I millisecond. It is often the case that the stability and performance of
the adaptive algorithm can be improved by "slowing it down," by not calculating new weight values every time a new piece of data is taken (maybe
instead calculating new weights after every five data samples). Thus, the
adaptation rate of the algorithm may ideally be slower than the sample rate
of the system. This is analogous to putting stairs down the hill, where you
make a step down the hill with a new weight calculation and then wait for
several samples to make another.
Consider now the analogous condition effect 2 of "blocking the wind."
What is the wind, and where does it come from? The wind is basically an
unwanted impediment to progress down the hill, an impediment which
"niggles" in the beginning, but through continual application will eventually lead to instability (in maybe an hour, or 6 hours, or even I day). The wind
in this analogy comes from the calculation quantization process inherent in
the digital environment. All of the "lost numbers," the small bits and pieces
that were dropped off the end of the calculations, will actually lead to algorithm instability. The simple fix is to add "leakage" to the algorithm. This
continually removes a small percentage of the weight values with every new
calculation, that counteracts the quantization process. Leakage has other stabilizing effects on the adaptive algorithm, such as limiting the effort which is
expended in trying to minimize the error signal. Leakage is essential in any
adaptive system that is to run for an extended period of time.
Let us turn our attention now to the analogous condition effect 4, the
"periscope" part. First, you may be wondering, why is the operation of a
gradient descent algorithm in an active noise control system like "looking
through a periscope" while running down a hill? The answer to this is tied in
with how the "hill," and more precisely the slope of the hill, is calculated
mathematically by the algorithm. 4 As part of the calculation process the algorithm must be "told" what will happen to the controller output signal between leaving the digital filter and arriving in the error signal. The in-between
part, which sees the signal pass through the output digital-to-analog conversion process, then through an amplifier, out of the loudspeaker, through the
acoustic system, enter into the error microphone, pass through the error signal
analog-to-digital conversion process, and reappear in the microprocessor, is
referred to as the cancellation path, or error path. The cancellation path must
The adaptive algorithm uses the slope, or "gradient," of the "hill" to assess the direction in which it should travel (down the hill, hence the name "gradient descent algorithm"). When the bottom is reached, the slope should become "0," and ideally the
algorithm will stop.
4
112
7. Active Noise Controller Overview
be identified, or modeled, by the controller. It is often the case that the cancellation path must be continually modeled, as it will change with changing
temperature, air flow, and even sound field frequency content. This modeling
process often involves inserting a small amount of random noise into the
canceling signal, as will be outlined in the next chapter.
The quality of the model of the cancellation path determines the "direction of the periscope." If the phase estimate of the model is completely wrong,
the periscope will be pointing up the hill instead of down. As a result, the
algorithm will go the wrong way, leading the weight value saturation and
instability. If the model is perfect, the periscope will point directly down the
hill and the algorithm will run in the intended direction. Fortunately, the
model does not have to be perfect for the system to function. It simply has to
point more-or-Iess down the hill (in theory, simply below the horizontal line
which splits "down" from "up"; in practice, a little bit better than this).
8
Controller Fundamentals
Active noise control is a technique that aims to attenuate unwanted noise by
introducing an additional, electronically generated, "canceling" sound/ield.
This is a simple enough concept to state, and with some basic knowledge of
various aspects of acoustics (summarized in the previous chapters of this
book) it would seem entirely possible. However, while the basic idea of active
noise control can be traced back to original patents over 60 years ago, truly
practical implementation has only recently become feasible. The main reason
is that precise calculation and output of the appropriate canceling sound
field is a complicated task, a task which is now practically possible for reasonable cost and effort as a result of advances in microprocessor and acoustic
transducer technology.
At the heart of any active noise control system is a piece of electronics, the
job of which is to calculate the signal that drives the control source (loudspeaker) to produce the canceling sound field. This piece of electronics is
generically referred to as the controller. We might refer to it as the
electronic control system, as opposed to the physical control system that
comprises the loudspeakers, microphones, etc. This idea of differentiating
the electronics from the physical environment is shown in Figure 8.1. The
majority of the preceding chapters in this book have been aimed at developing an understanding of what is important on the physical side. This chapter
will concentrate on what is important on the electronic side.
As might be guessed, there is a wide variety of possible electronic setups
that can be used in active noise control. This chapter will concentrate on the
most popular: an adaptive feedforward active control system. Two of the
advantages of this type of active control system are its attenuation performance (that is normally superior to most other controller arrangements) and
its simplicity of design (the physical control system can essentially be designed independently of the electronic control system). The main disadvantage of feedforward active control systems is the requirement of a reference
signal, a measurement which will predict with good fidelity the impending
unwanted disturbance which is to be canceled. Other controller arrangements
do not have this requirement. However, for many targets of active control,
such as sources of tonal noise (for example, fans and motors) and noise propagating in waveguides (such as air-handling ducts), obtaining a reference
113
S. D. Snyder, Active Noise Control Primer
© Springer Science+Business Media New York 2000
114
8. Controller Fundamentals
Reference
microphone
Control
Error
source.
microphone
sical control sy tern
Ele tronic control ystern
Control
system
FIGURE 8.1. The "physical control system" consists of the microphones and loudspeakers that actually produce and measure the sound field. The "electronic control system"
performs the calculations required to generate the canceling sound field.
signal is relatively straightforward and so a feedforward control system is the
best option.
Before discussing the various components of the controller in an adaptive
feedforward active control system, we will need to outline some basics, including common controller arrangements, common terminology, and some
basics of digital systems.
General Control System Outlines and Definitions
We will begin with the definition of a few terms that which are commonly
used in active noise control literature, and will be used throughout this chapter. A system is defined as a set of individual components acting together as a
whole. The systems of interest in active control work in general (where noise
andlor vibration is targeted for attenuation) fall into three broad categories;
acoustic, structural, and structural/acoustic. Any of these systems, when acted
upon by some form of excitation, will exhibit a certain response. 1 The excitations of interest in active control work are classified as disturbance (which
are also referred to as primary excitation or primary disturbance in the active
control literature) and control. Disturbance inputs are responsible for unwanted excitation of the system (for example, noise), and control inputs are
purposely introduced into the system with the aim of obtaining some desired
response (for example, attenuation of noise). An example of an acoustic system is the acoustic environment in an air handling duct, in which the disturbance input is responsible for unwanted sound propagation. An example of a
structural system is a beam, where the disturbance input is responsible for
1 For example, your bicycle is a system, and your foot can provide excitation, when you
excite your bike by kicking it, it will respond by falling over.
General Control System Outlines and Definitions
115
unwanted vibration. An example of a structural/acoustic system is an aircraft
interior, where the disturbance input is responsible for the unwanted acoustic
field in the cabin that is generated by the vibration of the fuselage .
A control system is a device that is responsible for generating a control
input. The concept of "control" is a very general and common one. The most
basic form of control system is one in which the system output has no effect
upon the control input. This is known as an open loop control system. A
sketch of a typical open loop control system is shown in Figure 8.2, where a
desired output/outcome is fed to the controller to produce a control input to
the target system. Such an arrangement can be found, for example, in a toaster.
Here the "darkness" setting provides the desired output and the control input
(signal to the toaster ejection mechanism) determines when the toaster will
"pop up."
Desired
ou teo me ... Control
system
Control
input .... Target
system
Output
...,
FIGURE 8.2. Open loop control system.
The diagram of the system shown in Figure 8.2 is know in engineering as a
block diagram. A block diagram displays the functional relationship between
system components using arrows for data flow and blocks for major system
components or sections, rather than mathematical equations.
What happens when our open loop toaster control system is confronted
with the problem of toasting frozen bread, or maybe a muffin, or wholewheat
instead of white bread? If the actual level of toast darkness is unsatisfactory,
due to fluctuations in bread quality or initial bread conditions, there is no
way for the open loop toaster to automatically alter the length of time heat is
applied. The output of the system ("ghost toast"/burnt toast) has no influence
upon the control input.
An arrangement that has the potential for improvement upon the open
loop control system is one in which the system output does have an influence
upon the control input, referred to as a closed loop, or feedback, control
system. A block diagram of a typical closed loop control system is shown in
Figure 8.3. 2 Here some output quantity is measured and compared to a desired
value, and the resulting error is used to correct the system's output. For
In the block diagram of Figure 8.3, the circle with the Greek letter sigma (L) inside
represents a summation of signals (physically, this is the same as two sound fields
"adding"). If a negative sign appears next to one of the inputs to the summing process,
it means that the input is subtracted from the total, rather than added.
2
116
8. Controller Fundamentals
I
"Error" =
desired output actual ::::01
Control
input
r-------, Outp ut
system
Target
system
FIGURE 8.3. Closed loop control system.
example, if our toaster were fitted with a closed loop control system to ensure
correct darkness, the measured darkness of the toast would be compared to
the desired darkness and the control system action (to pop up or not to pop
up) would be based upon the results of this comparison.
In active noise control we are not, in general, interested in toasting an
object, or for that matter moving or altering the equilibrium state of a system.
We are principally interested in disturbance attenuation. For us, then, the
measured system output is an acoustical disturbance (an unwanted sound
field), and the desired system output is normally zero (quiet). Therefore, the
typical feedback control structure used in active control is as shown in Figure
8.4, where the system output is used to derive the control input. The system
output may, for example, be the system response as measured by a microphone. This output is sometimes referred to as an "error signal," as it represents the "error" between the desired response (a response of magnitude zero)
and the actual system response.
In the implementation of active control systems it will, in many instances, be
possible to obtain some a priori measure of the impending disturbance input,
referred to in active control literature as a reference signal. An example of this
occurs when the disturbance propagates along a duct, where it is possible to
obtain an upstream measurement. A second example is where the source of the
disturbance (the primary source) is rotating machinery, the disturbance is peri-
... Control
system
C on t ro I
input
,
"Error" =
residual output from
system (such as noise)
Target
system
Output
I
,
Figure 8.4. Closed loop control system as often constructed in active noise control
implementations.
General Control System Outlines and Definitions
117
odic, and a tachometer signal is available that is related to the disturbance. In
these instances it is possible to "feedforward" a measure of the disturbance to
provide attenuation, producing afeedforward control system as shown in Figure
8.5. Feedforward control systems, when they can be implemented, often offer the
potential for greater disturbance attenuation than feedback control systems. Heuristically, the feedforward control system can be viewed as offering prevention of
the disturbance, producing an output to counteract (cancel) the disturbance upon
its arrival. Feedback control systems must wait until the disturbance has occurred, and been measured, at the system output before they can act to attenuate
the lasting effects the disturbance has upon the system. Feedforward and feedback control systems can be implemented together to produce a control system
that will both effectively attenuate the referenced disturbance to the degree maximally possible, and also provide some attenuation of the unreferenced component of the disturbance.
The feedforward control system shown in Figure 8.5 is an open loop control system. This was the form of control system originally envisaged for
active control by Paul Lueg in his patent of 1933, where the control system
was set to produce a control input which is 180 0 out of phase with the primary
disturbance at the point of application. However, such a control strategy is
unable to cope with changes in the system characteristics (such as due to
temperature and air flow changes), and attenuation would be greatly reduced
after some period of time. The form of feedforward control system currently
implemented in active control systems is an "adaptive" strategy, such as is
shown in Figure 8.6. Here a measure of the system output is used to adjust the
control system to provide maximum attenuation. Hence, an adaptive
feedforward control system is effectively a closed loop implementation of a
feedforward control strategy, where an adaptive algorithm is used to adjust
the system in order to minimize the error signal.
It is adaptive feedforward control systems that are of interest in this chapter.
The active control system depicted in Figure 8.7 is an adaptive feedforward
arrangement. A reference signal measurement of the impending disturbance
(downstream in the duct) is provided by a microphone placed upstream of the
Disturb ance
input
Refere nce
signal
Target
,..
system
, Control
Output
. L
,
"Error" =
) residual signal from
disturbance input
Control
input
system
FIGURE 8.5. Feedforward control system arrangement.
118
8. Controller Fundamentals
Disturbance
input
Target
system
Reference
signal
Output
"Error" =
I-"""T-~ residual signal from
disturbance input
Control
input
Control
system
FIGURE 8.6. Adaptive feedforward control system arrangement.
control system (a "reference microphone"). This arrangement works because
the noise at some upstream point in the duct will be largely the same as the
noise at some point downstream after a finite period of time, this time being a
function of the speed of sound.
Provided with a reference signal, the job of the electronic portion of the
active control system is quite straightforward: derive an acoustic signal that
is equal in amplitude but opposite in phase to the impending disturbance,
and introduce this signal via a control source (loudspeaker) in the duct when
the unwanted noise arrives. As acoustic systems are linear, the control signal
will cancel the unwanted noise. As a result, the acoustic levels downstream of
the control system will be reduced.
While it may be straightforward to state the objectives of the control system, it is much more difficult to realize them. The change in the disturbance
during propagation from the reference microphone to the control source must
be accounted for, as must the change in the control signal as it progresses
through filters, amplifiers, and loudspeakers. The characteristics of these
changes will alter significantly over time , with changing environmental con-
f ./\V/\V/\
Fan noise
source
Cancellation
~~wanted
",olse
Reference
microphone
Residual
noise
~
Control
source
Error
microphone
Control
system
FIGURE 8.7. Adaptive feedforward active noise control system arrangement.
Physical System Limitations
119
ditions and transducer wear. Based upon these factors, it becomes apparent
that the control system must be adaptive. That is, it must continually tune
itself to provide the optimal result. To facilitate this, a measure of the residual
sound field, an error signal, is provided to the control system via an "error
microphone." An adaptive algorithm is normally implemented as part of the
control system that continually alters the characteristics of the controller so
as to minimize the disturbance at this sensor.
Summarizing, a standard feedforward active noise control system has four
basic components: a reference sensor (microphone) to provide a measure of
the impending disturbance; a control source (loudspeaker) to introduce the
controlling or canceling disturbance into the acoustic environment; an error
sensor (microphone) to provide a measure of the residual acoustic field after
the introduction of the controlling disturbance; and an electronic controller,
which uses the reference signal to derive a control signal that will minimize
the acoustic field at the error sensor.
Side Note. As mentioned, adaptive feedforward control systems are most commonly used for a number of types of active noise control system implementation. One area where they are not generally used, however, is in active hearing
protection (active headsets). In these implementations, the problem of group
delays in digital electronic control systems (discussed next) make adaptive
feedforward implementation essentially impossible. Instead, a simple analog
(not digital) feedback control system is used. These are cheap, low power, and
have virtually no group delay. They cannot, however, adapt themselves to a
changing environment. Fortunately, the characteristics of a low-frequency
sound field inside a headset cup do not change very much once a good seal
between cup and head is established.
Physical System Limitations
While the four basic components described above are required for an adaptive feedforward active control system to function, simply having a system
that attenuates the acoustic disturbance at an error sensor does not guarantee
that the sound field at all locations is reduced. Suitable design of the physical
part of the active control system, the number and location of sensors and
loudspeakers, is required for this.
Many of the basic considerations that come into play in designing and
implementing the physical part of an active noise control system (where to
put loudspeakers and microphones, the viability of applying active noise
control given the response of the system in the target frequency band, etc.)
have been discussed in previous chapters. However, it is worth reemphasizing
several points here.
Controlling sound radiation into an open space is more difficult than controlling sound radiation into a confined space (such as a duct). The control source(s)
120
8. Controller Fundamentals
must be able to produce an acoustic field with the same spatial characteristics as
the unwanted sound, a criterion that generally dictates that the loudspeakers be
placed in close proximity to the source of the unwanted noise. For example, if the
source of the unwanted disturbance radiates the same acoustic field in all directions, and the control source is also uni-directional, then, to achieve a reduction
of 10 dB in radiated power, the two acoustic sources must be no greater than onetenth of an acoustic wavelength apart. Noting that the acoustic wavelength for a
given frequency is approximately equal to 343 (meters per second) divided by
the frequency (in Hz), then to control 100 Hz sound (wavelength approximately
3.4 meters) the sources cannot be separated by more than 340 millimeters. For
sound at 400 Hz, this allowable separation distance is in the order of 80 millimeters. Again, this is provided that the control source can mirror the radiation pattern of the primary noise source, a feat not easily achieved when the radiation is
from a complicated piece of machinery. These physical limitations are usually
the critical determinant when assessing what noise problems are amenable to
active control.
The disturbance most amenable to active control is a periodic one, where
the sound is characterized by discrete harmonically related tones. Periodic
sound fields are usually straightforward to "predict" for feedforward control
implementation, as there will be a constant relationship between the sound at
one point in the acoustic environment and the sound at some other point. As
an alterative to a microphone, a tachometer could be used to provide a reference signal for a periodic disturbance, as there will also be a constant relationship between the shaft rotation and the acoustic field.
Periodic disturbance problems provide other benefits when implementing
the electronic control system, one of which is that the control system does not
have to be causal. Basically, this means that it is not necessary to get the
"timing" between the reference signal and the control input correct for the
system to function effectively. The characteristics of periodic sound do not
change with time, and so a measurement of the disturbance at some instant in
time can be used to predict the impending disturbance in 1 millisecond, or in
1 second, or even in 1 minute without any problems. If the disturbance is
random noise, however, the system must be causal. This means that the timing
between the reference signal and control input must be correct. The measurement of the disturbance at some instant in time will only predict the impending disturbance at one other instant in time. For example, a measurement of
random noise at an upstream point in an air-handling duct may be used to
predict the noise at some downstream point in 3 milliseconds time if the two
points are separated by approximately 1 meter. However, the measurement
cannot be used to predict the noise at the downstream point in, say, 10 milliseconds time.
Combining the results of the last two paragraphs, the most viable targets
for active noise control are arguably those where the disturbance is periodic
and of low frequency. This is important to bear in mind in subsequent
discussions.
Background
121
Interfacing a Digital System
The majority of (maybe even all) adaptive feedforward active noise control
systems are implemented using digital electronics. Digital control systems
require some additional componentry to be added to the previous control
system sketches (block diagrams).
Background
What distinguishes digital systems from their counterparts (analog systems)
is how digital systems work with quantities. For a digital system to do some
task, it must be presented with a set of discrete parameters, in response to
which it will produce one or more discrete outputs. This is similar to the way
in which human beings perform mathematical tasks. If, for example, you are
asked to do some addition, you will expect to receive a set of discrete numbers (23, 37, 12, etc.) which are to be added; this is your input. In response,
you will produce a discrete output (the answer is "72").
The alternative to a digital, or discrete, approach is an analog, or continuous, approach. Returning to the addition example, consider the piping arrangement shown in Figure 8.8. Here two smaller pipes are feeding into one
larger pipe. This can be viewed as an arrangement for performing addition:
the flow through the larger pipe is the sum of the flows from the two smaller
pipes. However, the pipe does not work with discrete quantities: it does not
take one discrete "chunk" of liquid from pipe A, one discrete chunk of liquid
from pipe B, add then together and deliver one discrete "result" chunk to
main pipe C. Rather, the addition process is continuously happening. An
engineer might say the process is continuous in time.
At the physical level, active noise control is a continuous process: two
sound fields are continuously adding together in space to provide cancellation. However, digital systems cannot perform continuous operations; they
can only work with discrete numbers (2 x 3 = 6, 1 + 2 = 3, etc.). How can we
integrate these two different modes of operation? The answer is: sampling.
Flow into
pipe A
vvt
Output flow == A+B
Flow into
pipe B
FIGURE 8.8. Example of a continuous (analog) addition: the flow from two smaller pipes
entering a larger pipe.
122
8. Controller Fundamentals
Consider the problem shown in Figure 8.9, where we want to add two waves
together using a digital (discrete) approach. To perform this task in an approximate way, we could sample the waves at certain points in time, obtaining discrete
values of the amplitude of the waves at those precise instants in times. At any
point we could add together these sampled values, and produce an output which
describes the sum of the waves at that one instant. Taken together, the discrete
results yield a skeleton of the continuous (desired) result.
If we wanted to move from the skeletal result shown in Figure 8.9 to one
that more closely resembles the desired continuous outcome, what would we
need to do? Two steps are required: (1) somehow the discrete results must be
joined together in time, and (2) the resulting edges must be smoothed off.
This is, in fact, what occurs in a digital control system implementation.
Consider step 1 first. Perhaps the most ideal way to connect the skeletal results
from the digital process would be to "connect the dots," to draw a straight line
between subsequent results. There is a problem with implementing this idea,
though. If you are at one instant in time and calculate a result, you cannot draw a
straight line to the next result until you calculate it, which is at some future
instant in time. That is, to connect the dots as shown, you would have to be able
to predict the future; not even a digital system can do that!
A compromise position is to hold the current result as an output until a new
result is calculated, at which time it replaces the previous one as the output.
The result is a stepped' output, as shown in Figure 8.10.
The stepped output shown in Figure 8.10 resembles the continuous output
more closely than did the skeletal result. We can now further improve things
by implementing step 2: smooth off the edges.
How is it possible to smooth off the edges? To get an answer to this, we
must go back to our friend Fourier. We know from Fourier that any waveform
can be considered as the sum of a number of sine waves (frequency compo-
Discrete samples
of waveform A
+
I
;>
+
Discrete samples
of waveform B
=
Discrete result:
A+B
FIGURE 8.9. Discrete (digital) addition of two waves.
Background
123
Discrete result:
Discrete result
with data held
constant between
samples:
FIGURE 8.10. Discrete (digital) addition of two waves from Figure 8.9, with the result
held constant between samples.
nents). To construct perfectly an "edged" signal requires a large number of
high-frequency components. If these components are missing the edged signal becomes more approximate. But this is actually what we want: an approximation to the edged signal, which is the continuous result (a bit of
lateral thinking here!) . So what we actually need to do is remove the highfrequency components of the output signal. We can do this by filtering.
Filtering is a common concept: it refers to the preferential sorting of a set
of things into those we want and those we do not (you may filter your lunch
into tasty and nontasty bits). It is possible, and in fact relatively straightforward, to construct (analog) filters which "sort" a signal into different frequency components, allowing some frequency components to pass through
the filter while others are stopped. In order to smooth off the edges from our
digitally generated waveform result, we need to put the signal through a low
pass filter. This is a filter that allows low-frequency components to pass,
while blocking high frequency components. The output, as shown in Figure
8.11, is a smoothed-out result.
The remaining question is, what constitutes a high frequency that must be
removed? This will be answered in a future discussion. At the moment we are
more interested in the qualitative result.
FIGURE 8.11. A low pass filter will remove the edges from the stepped (discrete value)
waveform.
124
8. Controller Fundamentals
So, to summarize, we can use a discrete number system to add the two
continuous waveforms using the following methodology:
1. sample that input data;
2. perform the discrete operation;
3. output and hold the result; and
4. low pass filter the result to smooth the edges.
These are exactly the same steps that are taken when using a digital electronic
system to produce some continuous result, with one addition:
Step 0: In a "real" implementation, there is an additional step that must top
the list above. This is low pass filtering of the input data, for reasons of
aliasing to be discussed later in this chapter. Further, in a real system the
input sampling process often involves some form of comparison operation
between the measured signal and a set of reference voltage levels to assign a
value to the sample (described later). During this time the input must remain
constant. For this reason, step 1 is usually augmented with a "sample and
hold" operation, which is responsible for sampling the continuous data and
holding the value constant while electronic hardware figures out what the
numerical value is.
Note that in some instances the output side of the procedure, steps 3 and 4,
might not be required. This is commonly the case in monitoring systems,
where the aim is to collect samples of continuous data only (for example, the
temperature of a liquid in a chemical process). Also, "step 2: perform the
operation," may be long-winded and distributed. For example, in digital communications, the input speech may be sampled on one end of the world,
relayed to another part of the world as a set of discrete numbers, and then
output on a loudspeaker at the receiver's end.
Step I on the list above, which entails sampling a continuous time signal
and deriving a discrete value, is usually referred to as analog-to-digital conversion. Similarly, step 4 on the list above, the output of a discrete result, is
usually referred to as digital-to-analog conversion. In an electronic system,
these processes are usually handled by dedicated microchips: an analog-todigital converter, or ADC, for the input and a digital-to-analog converter, or
DAC, for the output.
Required Additions for Digital Control
The adaptive feedforward control system of Figure 8.7 is shown modified for
digital implementation in Figure 8.12. The digital implementation involves
the addition of an antialiasing (low pass) filter, sample and hold circuitry, and
an ADC on the inputs to the controller, and the addition of a DAC, sample and
hold circuitry (to hold the output constant between subsequent samples), a
reconstruction filter on the output of the controller, and the addition of a
clock to synchronize events such as sampling.
Required Additions for Digital Control
125
Fan noise
source
+f\J\NOise "><XX
Unwanted
Cancellation
~--------------------~..
Reference
Control
source
microphone
~
m~----__~-Error
microphone
Smoothing
filter
Anti-alias
filter
Residual
noise
....--.. %. . .- -.
Anti-alias
filter
Microprocessor
FIGURE 8.12. Adaptive feedforward active noise control system with the required digital
system components shown in block form.
As mentioned, ADCs and DACs provide an interface between the real (continuous) world and the world of a digital system. ADCs take some physical variable,
usually an electrical voltage, and convert it to numbers that are sent to the digital
system. Referring to Figure 8.13, these numbers usually arrive at intervals of some
fixed time period, called the sample period. The numbers arriving from the ADC
are usually representative of the value of the signal at the start of the sample period,
as the data input to the ADC is normally sampled and then held constant during
the conversion process to enable an accurate conversion (discussed further shortly).
Commonly, the sampling period Ts is implicitly referred to by a sample rate1" which
is the number of samples taken in 1 second; Ts = 1/j s' Thus, the ADC provides
discrete time samples of a physical variable which is continuous in time. The entire
system, consisting of both continuous and discrete time signals, is referred to as
a sampled data system.
The digital signal coming from the ADC is quantized in level. This simply
means that the stream of numbers sent to the digital control system has some
finite number of digits, hence finite accuracy. For example, the continuous
signal may have a value of 23.00012735, but the digital system is only accurate to four decimal points, so it sees the value as 23.0001. Referring to Figure
8.13, it can be seen that, as a result, while the value of the analog signal fed
into the ADC is increased in a continuous nature, the output is increased by
126
8. Controller Fundamentals
Value
FIGURE 8.13. Discrete representation of a continuous signal, as provided to the microprocessor by the ADC.
discrete increments given by the quantum size. Normally, the digital signal
has a binary representation, expressed as a number of bits, each with a state of
oor 1 (binary signal representation was discussed in more depth in the previous chapter). This leads to ADCs being classified by the number of bits they
use to represent a quantity. For example, a 16-bit ADC will represent a sampled
physical system variable as a set of 16 bits, each with a value of 0 or 1. It
follows that the accuracy of the digital representation of the analog (continuous) value is limited by the quantum size, given by
.
full scale range
quantum Size =-=--------=2n
where n is the number of bits. For example, if the full scale range of the ADC is
±10 volts, the quantum size of the 16 bit digital representation is (20 volts)/(2 16)
=0.305 millivolts. The difference between the actual analog value and its digital
representation is referred to as the quantization error. The dynamic range of the
ADC is also determined by the number of bits used to digitally represent the
analog value, and is usually expressed in decibels, or dB. For example, an ideal
16 bit ADC has a dynamic range of (20 log(216)), or 96.3, dB.
A signal is said to be digital if it is both discrete in time and quantized in
level. The values of both the sample rate and quantum size, that define the
digital system, have a significant influence upon its performance. This will
be outlined later in this chapter.
Overview of the Controller
127
The "dual" of the ADC, the DAC, works in an opposite fashion to the ADC
in that it provides a continuous output signal in response to an input stream
of numbers. This continuous output is achieved using the sample and hold
circuit, normally incorporated "on-chip." This circuit is designed to progressively extrapolate the output signal between successive samples in some
prescribed manner, most often simply holding the output voltage constant
between successive samples. This type of extrapolation process is referred to
as a zero-order hold. By incorporating a zero-order hold circuit, the output of
the DAC is continuous in time but cantiest in level. To smooth out this pattern, a low pass smoothing filter, or reconstruction filter, is placed at the
output of the DAC and sample and hold circuitry as shown in Figure 8.12.
Overview of the Controller
The electronic controller portion of an adaptive feedforward active noise
control system is basically a special-purpose digital control system. A basic
outline of an adaptive feedforward active noise controller is shown in Figure
8.14, where the signal lines (reference, error, and control) are assumed to be
digital (the ADCs and DACs are not shown). The controller has three basic
components: a digital filter, an adaptive algorithm, and a cancellation path
transfer function modeler (or system identification component). These components are interrelated, and each must be designed correctly if the system is
to function properly.
The digital filter component of the feedforward controller (sometimes referred to as the control filter) is responsible for calculating the control signal.
This in turn will be fed to a control source (such as a loudspeaker) which will
C on t ro I I\,
Referen ce ~
input
r
output
Digital
~ filter
I
Cancellation
path modeler
~
Adaptive
algorithm I'
I Error
~ in put
~
I
Adaptive feedforward controller
FIGURE 8.14. Major components of an adaptive feedforward controller.
128
8. Controller Fundamentals
actually input the canceling disturbance into the acoustic system. The control signal is derived in response to the reference signal via a digital filtering
operation. As will be described shortly, the digital filter takes discrete samples
of current and past reference inputs and possibly filter outputs, multiplies
them by a set of coefficients or weights, and adds the products to produce an
output sample. The values of the filter weights determine the relationship
between the reference signal and control signal. For an active noise control
system this means that, given some reference signal, the derived control signal is a function of the digital filter weights.
The adaptive algorithm component of the controller is responsible for tuning
the digital filter weights such that the derived control signal provides the optimum level of disturbance attenuation. To do this, the adaptive algorithm requires
a measurement of three items. The fIrst of these is one or more error signals, that
are measurements of the residual sound field that exists after the introduction of
the canceling sound fIeld. These measurements are taken at locations where the
unwanted noise is to be minimized. In an active noise control system, these
measurements are typically provided by one or more microphones.
The second quantity the algorithm must have is a measurement of the signals
upon which the control signal calculation has been based. This at least entails
taking copies of the reference signal samples, and sometimes copies of the actual
control output samples.
The third quantity required by the adaptive algorithm is not actually a
signal, but rather an effect. The adaptive algorithm requires a knowledge of
what will happen to the control signal between its calculation in the digital
filtering operation and its appearance in the error signal measurement. Such a
knowledge is quantified technically as a transfer function, which defines in
numbers, as a function of frequency, the relationship between the control
signal and the error signal. In active noise control work, the transfer function
between the control signal and error signal is often referred to as the cancellation path transfer function.
The third component of the adaptive feedforward controller shown in Figure
8.14 is the cancellation path transfer function modeler. As might be guessed, this
component is responsible for obtaining a measurement of the cancellation path
transfer function for use by the adaptive algorithm component of the system.
A more detailed discussion of these three adaptive feedforward control
system components follows.
Controller Component 1: The Digital Filter
What Is a Digital Filter?
Consider the sketch shown in Figure 8.15. Given a reference signal that is in some
way related to the impending (noise) disturbance, the active noise controller
must calculate an appropriate signal to send to the loudspeaker such that a can-
Controller Component l: The Digital Filter
129
Structural/acoustic system
transfer function
Primary
disturbance
Error
Frequency
Control
input
~1~J
Reference
signal
n~
Frequency
Ideal control source
transfer function
FIGURE 8.15. The control system transfer function should have the same amplitude, but
inverted phase, as the target structural/acoustic system.
celing sound field is generated. In essence, the response of the controller must be
a mirror image of the response of the acoustic system to the reference signal: the
amplitude of the response must be the same, but the phase inverted. So, for example, if the reference signal input to the acoustic system produces an output at
the loudspeaker location of "23," then the reference signal input to the controller
must produce an output at the loudspeaker location of "-23" for cancellation to
occur. In engineering, the relationship between the signal going into a system
and the signal coming out of a system is referred to as a transfer function (specifically, a transfer function is the ratio of (output)/(input), which usually varies with
changing frequency of the signal). So, we can say that the controller calculation
process must mirror the transfer function of the acoustic system (to the reference
signal). "Component I" in our discussion of the controller is responsible for this
calculation procedure.
So, a system transfer function defines the relationship between the signal
coming into a system and the signal coming out of the system. This relationship is usually frequency dependent. Consider, for example, what happens
when an input voltage is sent to a small loudspeaker. If a low-frequency
signal (say, 30 Hz) is fed into the loudspeaker, the output sound field will be
very small in amplitude. If a high-frequency signal (say, 3000 Hz) is fed into
the loudspeaker, the output sound field amplitude will be much larger. These
130
8. Controller Fundamentals
characteristics for the entire range of frequencies are the frequency response
characteristics of the loudspeaker, that are quantified by a set of numbers in a
transfer function.
Calculation of the control canceling signal in an adaptive feedforward
active noise control system is accomplished by digitally filtering the reference signal input. A digital filter is a mathematical structure, or series of
mathematical operations (specifically, multiplications and additions), that
can mimic some desired transfer function. In active noise control, the desired
transfer function would be that which transforms the reference signal into a
control signal that provides the maximum level of disturbance attenuation.
That is, the desired transfer function is the mirror image of the acoustic transfer function of the system targeted for active noise control.
The transfer function model provided by a digital filter is referred to as a
discrete transfer function, as it is calculated using a series of multiplications
and additions with discrete samples of the reference signal (it is implemented
digitally). This is as opposed to the continuous transfer function of the acoustic system it is mimicking.
The transfer function of a system reflects its inner workings. For example, if
our system was an amplifier which increased the size of the signal by a factor of
10, then the transfer function of the amplifier at all frequencies would be "10";
this result reflects the inner workings of the system. For physical systems in
general (responding to heat transfer, fluid flow, vibration, etc.), and acoustic
systems in particular, the inner workings are described by calculus, using differential equations. These are relatively complex mathematical expressions, expressions that should be well known to all engineering students.
The discrete transfer functions that digital filters implement have their
roots in finite difference equation approximations of differential equations.
Finite difference equations provide a simple way to solve approximately a
differential equation, using simple multiplication and addition operations
performed on a number of discrete signal samples. As such, the basic building
blocks of digital filters are the same as those of finite difference equations:
multiplication and addition operations performed on discrete samples of the
input and output signals.
Hint. In engineering, the symbol Z-l is used to denote a unit time delay (delay of
one sample period). Representation of the unit time delay is shown in Figure
8.16, where the input x(k) is the value of some sampled signal at sample time k,
and the output x(k-l) is the value of the signal at the previous sample time (k-l).
New samPI">1 Z -1
I
Previous:amPle
FIGURE 8.16. Representation of a unit time delay in a digital system.
Controller Component 1: The Digital Filter
131
In its most general form, the current output value y(k) of a digital filter is
equal to the weighted sum of present and past inputs and past outputs, defined by the expression:
y(k) = box(k) + b,x(k-l) + .. . +bnx(k-n) +
a,y(k-l) + a2 y(k-2) + ... + amY(k-m).
In this expression, the a and b terms that multiply the signal samples are the filter
coefficients or weights. Derivation of the filter output is therefore constructed
from a series of multiplications (signal sample values times filter weights) and
additions (of the products). In engineering, this is referred to as a convolution
operation, or a multiply/accumulate (MAC) operation. Figure 8.17 contains a
"sketch" of the above equation. This is, in fact, a sketch of a "direct form" digital
filter. While there is a variety of ways to structure the mathematical operations
that define a digital filter, the direct form filter is the simplest and most common
as it directly reflects the underlying mathematical expression.
The discrete transfer function associated with the above filter output equation can be summarized as
where z-x relates the filter coefficient to what it multiplies: the signal samples
that were taken x times ago.
In the diagram of Figure 8.17, as each new input sample x(k) arrives, the
previous input samples are shifted by one position. The pipeline which contains the data samples is sometimes referred to as a delay line or delay chain.
Once a new input sample has been received, and the old input and output
samples have been shifted one position in the delay line, the filter output is
Sampled ....--_ _ _ _ __ _ _ _ _ __ _ _ _ __ _ _ _ _ _--,
input
Output
FIGURE 8.17. A direct-form digital filter.
132
8. Controller Fundamentals
Sam pled
in put
Feedforward
transfer function
bo+bl Z-1+b2Z-2+ ...
Out put
~
---
Feedback
transfer function ~
a 1z - 1+ a 2 z - 2+ ...
FIGURE 8.18. Direct-form digital filter split into feedforward and feedback components.
derived by multiplying the values at each position in the delay lines by a
weight assigned to that position, and by adding the products.
It is useful to think of the digital filter in Figure 8.17 as being comprised of
two components: an input, or feedforward, section, and an output, or feedback, section. This split is illustrated in Figure 8.18. The structures in the
feedforward and feedback sections of the filter are identical: a delay line, a set
of weights, and an accumulator to add the products. This structure is sometimes referred to as a transversal filter, or a tapped delay line. Using the latter
nomenclature, the number of "stages" in the filter, or the number of positions
in the delay line, is sometimes referred to as the number of filter "taps."
There are two different digital filter types commonly found in active noise
controllers: infinite impulse response (IIR) filters, and finite impulse response
(FIR) filters. The basic difference between these two is that IIR filters have a
feedback section as shown in Figure 8.18, and FIR filters do not (they are purely
feedforward devices). The filter names arise from characteristics associated with the
feedback part of the filter: if a feedback filter section exists, then a unit impulse
input signal (an input value of 1.0 for one sampled, followed by inputs of O's for
all other samples) will result in an infinite length output signal, as the output
signals continue to circulate in the feedback loop. If the feedback filter section does
not exist, then the unit impulse will simply propagate through the feedforward
delay line and disappear. The output signal lasts for afinite period of time.
Side Note. It is worth noting that FIR filters are also referred to in the engineering literature as nonrecursive filters, all-zero filters (as there are no poles,
or denominator terms, in the transfer function, but there are zeros, or numerator terms), moving average (MA) filters, or simply tapped delay lines or transversal filters. Similarly, IIR filters are also referred to as recursive filters ,
pole-zero filters, and autorecursive moving average (ARM A) filters.
Specifying the "Appropriate" Digital Filter
One question that often arises in active noise control work is, which type of
filter should be used for which type of application? Unfortunately, the answer
Specifying the "Appropriate" Digital Filter
133
to this question is far from straightforward. Before providing some general
guidelines, there are a few pertinent characteristics of the two filter types
which need to be outlined.
The first characteristic to note is that FIR filters are inherently stable
architectures, while IIR filters are not. By stable, we mean that the mathematics will not "blow up," or yield an infinite number output in response to a
finite number input. This lack of inherent stability in the IIR filter is a result
of the presence of the feedback section of the filter: if the amplification in the
feedback loop becomes too great the system will become unstable. This is
similar to what happens in the physical world if you insert a microphone into
a loudspeaker in a public address system: the loudspeaker output becomes
incredibly large, or "blows up." Remember that digital filters are mathematical models of real-world transfer functions, and so anything that can happen
in the real world can, in theory (given certain mathematical constraints), also
happen in the digital filter calculation process.
In an FIR filter, the gain, or input signal amplification, can certainly become extremely large. However, without a feedback loop the filter output
cannot reinforce itself to drive the system into an unstable mode of operation.
The second characteristic to note is somewhat obvious: as FIR filters do
not contain a feedback loop, they are not particularly good at mimicking
systems which do have feedback, or long reverberation times. Hence they are
not particularly good at controlling such systems. If, for example, attenuation
of the response of a system with several resonances in the target frequency
band is being considered for active control, then the control system itself
must have a multiple resonance-like response. In previous discussions in this
book we have noted that a resonance occurs when waves bounce back and
forth between boundaries with a good fit. In other words, the wave "feeds
back" to its point of origin, or circulates through the system, just like that
which happens in the digital filter with a feedback loop. The obvious choice
of digital filter in this instance is an IIR filter, that most accurately mimics the
characteristics of the target physical system.
The third characteristic to note it that for realizing a given transfer function
with some desired degree of accuracy with the minimum number of total filter
weights, an IIR filter is often more efficient than an FIR filter. The discrete transfer
function, with terms in the numerator and denominator, can be viewed as a division, the result of which will be a purely feedforward series (a function of past
inputs only, that corresponds to a single value of 1.0 in the denominator). Such a
transfer function could be implemented "exactly" with an FIR filter. However, the
feedforward series required to do this is often of infinite length, and so, in practice, an extremely long FIR filter must often be used to obtain satisfactory results.
The computational load associated with an extremely long filter (how many
multiplications and additions the microprocessor must do) can provide problems
for even the fastest digital signal processing chips.
The fourth characteristic is one which arises from including or excluding a
feedback loop in the digital filter (that is, using an IIR or FIR filter), but is by no
l34
8. Controller Fundamentals
means obvious. Jumping ahead slightly, it was mentioned in the previous chapter
that the adaptive algorithms used to tune the digital filters in an active noise
controller are gradient descent algorithms. These algorithms rely on the characteristics of the error criterion to achieve satisfactory results: if there is a single
(global) optimum set of digital filter weight values then gradient descent algorithms work well. However, if there is a number oflocally optimum sets of weight
values the algorithm can become trapped in a local optimum, and the weights
will not converge to the globally optimum values. When FIR filters are used,
there is always a single (global) optimum set of weight values. When IIR filters
are used, there can be several (local) optimum sets of weight values. Therefore,
gradient descent adaptive algorithms do not always provide the best possible
result when used with IIR filters.
Given the above outlined characteristics, it is possible to put forward a few
guidelines for selection of the correct filter for a given problem. First, it
should be stated that, where possible, use of an FIR filter is arguably a better
option than use of an IIR filter. This is due to the inherent stability and
algorithm behavior associated with FIR filters. FIR filters are ideally suited to
tonal noise problems, where the reference signal is one or more sinusoids
(probably the most common reference signal in active noise control work),
and implementations where the control signal does not in any way corrupt the
reference signal.
IIR filters are better suited to broadband work, where the target is a wide
range of frequency values. This is especially true where the target system has
resonances in the referenced frequency band or where the phase speed is not
constant (such as higher-order modes propagating in air-handling ducts). A
second situation where IIR filters are the preferred options are in systems
where there is feedback from the control source to the reference sensor. This
can occur, for example, when implementing active noise control in air-handling ducts where the reference signal is provided by a microphone in the
duct. Once again, if there is feedback in the physical system, the mathematic
model (the digital filter) should also include it. IIR filters are the best option
for this.
Specifying the Digital Filter Length
Having chosen the type of filter for a given application, the next problem is
to decide on the filter length. Again, by filter length we are referring to the
number of weights or taps in the filter, corresponding to the number of multiplications and additions that are necessary to calculate the filter output. As
with filter type, there has no straightforward selection criterion. There are,
however, a number of guidelines.
When the reference signal is a pure sine wave, in theory it is possible to
obtain an arbitrary gain and phase change with only two taps (weights) in an
FIR filter. However, in practice, the use of only two taps can lead to require-
Controller Component 2: The Adaptive Algorithm
135
ments of very large weight values. This is especially true if the sampling rate
is significantly greater (say, more than 20 times) than the reference sine wave
frequency. It is better to use an FIR filter with 4-20 taps for a sine wave
reference signal to avoid these very large weight values (move toward the
higher number of taps as the target disturbance becomes lower in frequency
relative to the sample rate). If multiple sine waves are present in the reference
signal, then 4-20 taps per tonal component is a useful starting point. If the
result is unsatisfactory then usually the number of taps should be increased.
For broadband reference signals, where the system is targeting a wide frequency range, the question of tap numbers is more complex. If an IIR filter is
being used in a system designed to attenuate a resonant response, then a good
starting point is 4-10 taps per resonance peak in both the feedforward and
feedback weight banks. However, the number of taps can increase to several
hundred or more for applications such as broadband control of noise in an air
duct. Unfortunately, selecting the number of taps to use is largely a matter of
experience and trial and error.
Controller Component 2: The Adaptive Algorithm
Background: Making Use of Adaptive Signal Processing
The second component of the adaptive feedforward active noise controlier is
the adaptive algorithm. The purpose of the adaptive algorithm is to tune the
digital filter, to adjust the values of its weights so that the residual sound field
after cancellation, as measured by the error microphone(s), is minimized. More
precisely, the algorithm will attempt to remove any correlation between the
reference signal and error signal, and so will effectively be blind to
unreferenced frequency components.
The adaptive algorithms used in the most active noise control systems are
variants of algorithms used in the broader field of adaptive signal processing
(there is no point in reinventing the wheel!). Adaptive digital signal processing is a field born out of the requirements of modern telecommunication
systems. In these systems, the need often arises to filter a signal, so that it can
be extracted from contaminating noise (recall that by filtering, we mean sorting out signal components into those we want, that are allowed to pass through
the filter, and those we do not, that are stopped by the filter). Conventional
signal processing systems employed to do this operate in an open loop fashion, using a filter with permanently fixed characteristics. The underlying
assumptions accompanying the use of fixed filters are that a description of
the input signal is known, and that the system disturbance and response characteristics are time invariant. If this is the case, a satisfactory fixed filter may
be designed. It is often the case, however, that the characteristics of the input
signal and system response are unknown, or may be slowly changing with
time. In these instances the use of a filter with fixed characteristics may not
136
8. Controller Fundamentals
give satisfactory performance.
To overcome this problem, a large number of adaptive algorithms has been
developed for modifying the filter characteristics through a change in weight
values to optimize system performance. Indeed, adaptive digital signal processing has become a firmly established field, encompassing a wide range of
applications (one of which is active noise control). The adaptive algorithms
used in active noise control systems are essentially generalizations of the
adaptive algorithms used in systems such as telephone echo cancelers, equipment to measure foetal heart beats (where the mother's heartbeat must be
removed from the signal), and antenna sidelobe interference chancellors.
Gradient Descent Algorithms
The adaptive algorithm part of the feedforward control system is responsible
for modifying the weights of the digital filter such that attenuation of the
unwanted acoustic or vibration disturbance is maximized. To do this, one or
more error signals are provided to the control system. It is these signals, that
provide a measure of the unwanted disturbance, that the active noise controller will attempt to minimize.
There is a variety of adaptive algorithms, with varying degrees of practicality, that have been put forward for modifying the weights of the digital
filters used in active noise control systems. The majority of these algorithms,
and in fact the majority of algorithms in the wider field of adaptive signal
processing, are gradient descent algorithms. Gradient descent algorithms
operate by adding to the current value of the filter weights a small percentage
of the negative gradient of the error surface to calculate an improved set of
filter weights. Note that the error surface is the error criterion plotted as a
function of the filter weights, a plot which for active noise control describes
the residual noise for a particular combination of digital filter weight values;
see Figure 8.19. In active noise control work, the error criterion is the sum of
the squared values of the error signals. Minimization of this error criterion
will lead to minimization of the unwanted acoustic or vibration disturbance
at the error sensing locations.
Side Note. A squared error criterion, such as the squared value of the acoustic
pressure at the error microphone location, is used because if minimization of
the unsquared error signal was the control object, a very large negative error
signal would result. This is clearly undesirable.
To understand how a gradient descent algorithm works, consider the error
surface shown in Figure 8.19. This is the typical shape ofthe plot of the mean
square value of a single error signal as a function of filter weights for a twotap FIR filter. The error surface shape has the appearance of a "bowl", and is
technically a hyper-paraboloid. There is a single combination of weight values that will minimize the error criterion; these values are located at the
bottom of the bowl. The task set for the adaptive algorithm is to modify the
Controller Component 2: The Adaptive Algorithm
137
Mean square error
FIGURE 8.19. Typical error surface ("bowl"), that is a plot of the mean square value of the
error input as a function of two digital filter weights.
filter weight values to arrive at this optimum set, thereby minimizing the
error criterion.
To obtain an intuitive derivation of a gradient descent algorithm for calculating the optimum weight coefficients of the FIR filter, consider what would happen if the error criterion bowl was constructed and a ball was placed at some point
on its edge, as shown in Figure 8.20. When released, the ball would roll down the
sides of the bowl, eventually coming to rest (after some oscillation) at the bottom.
This is exactly what we would like our algorithm to do to find the optimum set of
filter weights. When first released, the ball will roll in the direction of maximum
negative change in the slope, or gradient, of the error surface. If we examine the
position of the ball at discrete moments in time as it descends, we would find that
FIGURE 8.20.
Ball and bowl arrangement.
l38
8. Controller Fundamentals
its new position is equal to its old position (one discrete moment ago) plus some
distance down the negative gradient of the bowl.
As with the digital filter, what we want to do is put together a mathematic
expression that can be implemented on a microprocessor to mimic the real world.
The characteristics of the "ball and bowl" are somewhat formalized in a
gradient descent algorithm. This type of algorithm attempts to arrive at a
calculation of the optimum set of filter weights (at the bottom of the bowl) by
adding to the present estimate of the optimum weight coefficient vector a
portion of the negative gradient of the error surface at the location defined by
this estimate. In this way, the current value of the mean square error descends
down the sides of the error bowl, eventually arriving at the bottom. This is the
location corresponding to the optimum weight coefficients.
Mathematically, this notion, that defines a generic gradient descent algorithm, can be expressed as
w(k)= w(k) - 11 Aw(k)
where Aw is the gradient of the error surface at the location given by the
current weight coefficient vector and 11 is a positive number that defines the
portion of the negative gradient to be added, referred to as the convergence
coefficient.
The question now arises, how is the gradient of the error surface at the
location of the current weight values calculated? It is worthwhile doing some
simple mathematics to answer this question. If consideration is limited to a
single error sensor system for simplicity, then at any instant in time k the error
signal e(k) is a function of two components: a component p(k) due to the
unwanted disturbance and a component s(k) generated by the active noise or
vibration control system. As acoustic and structural systems are linear in the
normal operating range, the error signal is the sum of these two components
e(k) = p(k) + s(k).
Note that these two components are actually the signals which are output
from the error sensor measurement system when either the unwanted noise
source or the active control system are operating alone.
As mentioned, the error criterion is the squared value of the error signal. Mathematically, the gradient is calculated by differentiating this error criterion with
respect to the filter weights. For a single error sensor system, noting that the
unwanted disturbance component p(k) of the error signal is not a function of the
digital filter weights, this differentiation produces the following expression:
oe 2 (k)
oe(k)
os(k)
AW(k)=-(-) =2e(k)-(-)=2e(k)-().
owk
owk
owk
Controller Component 2: The Adapti ve Algorithm
139
Evaluating the Gradient
The equation above shows that the gradient of the error surface at the location
of the current filter weight values is equal to twice the product of the current
error signal sample e(k) and the partial derivative ds(k)/dW(k) of the control
source component of the error signal with respect to the filter weights.
While it is straightforward to obtain the error signal component of the
gradient estimate (by sampling the error signal), obtaining the partial derivative component requires additional work. In this section a descriptive account of how to obtain a measure of the partial derivative will be given.
A description of how to obtain the partial derivative component of the
gradient can be given with respect to Figure 8.21, which depicts the cancellation path of the single channel active noise control system being used in this
section for illustrative purposes. The cancellation path is the path the control
signal must take from the filter output to the error signal input of the adaptive
algorithm. Along this path the control signal must pass through a number of
"obstacles," including: the various filters associated with converting the digital control signal to an analog signal, the control source (loudspeaker) amplifier, the control source itself, the acoustic path between the control source
and error sensor (microphone), the amplifying and conditioning circuitry
associated with the error measurement system, and the filters associated with
converting the analog error signal into digital format. Each of these "obstacles" has associated with it a transfer function, which describes the relationship between its input signal and output signal. Because the systems that
are of interest here are linear, the various transfer functions contained within
the cancellation path can be lumped together into a single cancellation path
transfer function. This transfer function quantifies what happens to the control signal between its output from the digital filter and its appearance in the
error signal.
The cancellation path transfer function is important when evaluating the partial differential component of the gradient. The change in error signal component
that accompanies a change in the digital filter weights (the partial differential
Cancellation path
Reference
microphone
Control
source
~--~-T~--~~
Error
microphone
Control
system
FIGURE 8.21. Cancellation path in an adaptive feedforward active noise control system.
140
8. Controller Fundamentals
term in the gradient) is equal to the change in the control signal (filter output)
that accompanies a change in filter weights modified by the cancellation path
transfer function. As a simple illustrative example, suppose that the cancellation
path was an amplification by a factor of 2. Any change in the control signal would
be seen as a similar change, with twice the amplitude, in the control source
component of the error signal. Therefore, for a control signal y(k), the partial
derivative as(k)/aw(k) would be equal to (2 X ay(k)law(k)).
In practice, the cancellation path transfer function is unlikely to be simply
a gain. Rather, the change in amplitude and phase that it describes is usually
frequency dependent, and can vary dramatically over the operating frequency
range of the system. This is especially true if the target structural and/or
acoustic system has resonances in the operating frequency range.
Calculation of the partial derivative ay(k)/aw(k) is relatively straightforward. From the previous description of the digital filter, the filter output y(k)
is the product of two components: the filter weights w(k) and the signal samples
in the filter delay lines. Therefore, the partial derivative ay(k)/aw(k) is simply
equal to the values of the signal samples in the delay lines (the partial derivative for each individual filter weight is equal to the value of the signal sample
at the point in the delay line where the weight is situated). It follows that the
partial derivative as(k)/aw(k) used in the gradient calculation is equal to the
signal samples in the filter delay chain modified by the cancellation path
transfer function. This modification can be viewed as a filtering operation,
where the signal samples are filtered by (a model of) the cancellation path
transfer function to produce the partial derivative used in the gradient calculation. This filtered set of signals is then multiplied by the error signal to
produce the gradient estimate used to modify the current weight values in
such a way that the levels of disturbance attenuation are improved.
This filtering of the signal samples in the process of deriving the gradient
estimate is what differentiates the active noise control implementation of
adaptive filtering from the more common implementations, such as those
used in telephone echo cancellation. In the common implementations there is
no cancellation path, and so the raw signal samples in the digital filter are
used in the gradient calculation. This need to filter the signal samples, to
derive the gradient in the active noise and vibration control implementation,
has led to adaptive algorithm names such as the "filtered-x LMS algorithm,"
that is the active noise and vibration control version of the standard "LMS
algorithm."
The Convergence Coefficient
Having calculated the error surface gradient using the filtered signal samples
and the error signal, a portion of the negative gradient is added to the current
weight values to produce a new and improved set of filter weights. The portion of the negative gradient added to the current weight values is set by the
convergence coefficient Il. Selection of a suitable value of convergence coef-
Controller Component 2: The Adaptive Algorithm
141
ficient is extremely important, as it impacts upon both the speed of adaptation and the stability of the adaptive algorithm. If the value of the convergence coefficient is too small the weights will adapt slowly, and possibly stop
adapting before the optimum values are reached. If the value of the convergence coefficient is too large the weights will fail to stay close to the optimum values, and either change constantly or diverge completely. In the last
scenario the adaptive system becomes completely unstable.
Unfortunately, selection of an appropriately sized convergence coefficient
is often a trial-and-error procedure. There are a number of influencing factors,
notably: the number of control sources and error sensors used (increasing the
number of sources and sensors requires the use of a smaller convergence coefficient); the characteristics of the cancellation path transfer function (an increase in gain requires a decrease in convergence coefficient value); the time
delay present in the cancellation path, such as the time taken for sound to
propagate from the control source to the error sensor (as the time delay increases, the value of the convergence coefficient must be reduced, sometimes
dramatically to avoid algorithm instability); digital filter length (a longer
digital filter requires the use of a smaller convergence coefficient; and the
characteristics of the reference signal (a low-frequency tone which is heavily
oversampled (digital sampling frequency 100 times or more the frequency of
the reference signal) requires the use of a much smaller reference signal than
a higher-frequency reference signal or random noise).
In most common adaptive filtering systems, there is a number of ways in
which a suitable value of convergence coefficient can be derived automatically by the system. These are typically based upon the signal power of the
reference input, that can be shown to be the critical determinant in convergence coefficient selection in standard arrangements. However, these automatic selection strategies are not suitable for direct implementation in active
noise control systems. This is because in active noise control systems the
critical determinant is usually the time delay component of the cancellation
path transfer function, and not input signal power. The relationship between
the time delay and the bounds placed upon the convergence coefficient for
stable operation is extremely complex, and not easily implemented in an
automatic selection strategy. Therefore, it is often most straightforward to
select a convergence coefficient value manually, using a trial and error process. There is an upper bound placed on the convergence coefficient for stable
operation of the gradient descent adaptive algorithms. This value is a function of, amongst other items, control source and error sensor numbers, the
various system gains and attenuations, and the time delay a signal experiences in traveling from a given control source to a given error sensor. With
large values of convergence coefficient, the algorithm will "wander" in the
vicinity of the optimum set of weight coefficient values, rather than remain
stationary at this desired location on the error surface. However, large values
of convergence coefficient also increase the speed of algorithm adaptation to
the optimum set of values.
142
8. Controller Fundamentals
As adaptive algorithm stability is usually the most important factor, and
analytical evaluation of the maximum value of convergence coefficient that
will facilitate stable algorithm operation is difficult, it may appear that the
best choice of convergence coefficient is a very small one. This will both be
stable and minimize the unwanted sound field. Although this will mean that
the speed of convergence is reduced, this may be viewed as not to be terribly
detrimental in many cases (it may be of the order of a few seconds). This is
because the time scale of active noise and vibration control systems is constrained anyway to be longer than the time scale in most other digital filtering applications, owing to significant signal propagation times between
control sources and error sensors.
These properties, however, are based upon the analog characteristics of the
adaptive algorithm. In an analog, or infinite precision, implementation of the
adaptive algorithm, reducing the convergence coefficient will reduce the residual value of error criterion ad infinitum. In fact, for this case a good balance between speed and accuracy can be attained by continuously decreasing
the convergence coefficient during the adaptation process. For the digital
implementation of the algorithm, however, smaller is not always better. In
fact, if the convergence coefficient is chosen to be too small the final value of
the error criterion will be increased. This comes about due to the quantization
inherent in digital systems. If the portion of the gradient estimate used in the
adaptive algorithm is equal to less than half the value of the least significant
bit of the digital control system, convergence will stop. Although this may
seem an obvious point, it is one for which the implications cannot be overlooked; making the convergence coefficient too small will stop adaptation
too soon. Increasing the convergence coefficient value will rectify this. It
may be surprising to note that if the algorithm is initially adapted using a
given value of convergence coefficient, and when steady state is reached the
value is reduced, the result may actually be an increase in the value of the
error criterion.
Combining the analog characteristics associated with large values of convergence coefficient and the digital characteristics associated with small values of convergence coefficient, it can be concluded that neither too large nor
too small a convergence coefficient is a good choice.
The question to be asked now is, how can the optimum value of convergence
coefficient be chosen? While it is a poor answer, the best way for an active control
system is by trial and error. Following are a number of guidelines that are useful
for aiding the choice of the convergence coefficient value:
1. Start small-it is better to start with a convergence coefficient value that
is too small and gradually increase it, than have the active control system
go unstable as soon as it is switched on.
2. Different types of reference signals lead to different optimum and stable
values of convergence coefficient. In general, the maximum stable value
of the convergence coefficient is larger when the reference signal is ran-
Controller Component 2: The Adaptive Algorithm
143
dom noise, than it is when the reference signal is sinusoidal. Also, when
the reference signal consists of one or more sinusoids, the maximum stable
value of the convergence coefficient decreases markedly if the sample
rate is many times greater than the frequencies of the sinusoids (say, more
than 50 times faster). In other words, systems with low-frequency reference signals require smaller values of convergence coefficient than similar implementations with higher-frequency reference signals.
3. If any gains in the system are increased the convergence coefficient value
should be decreased. If the system gains are reduced the convergence
coefficient can be increased.
4. If more control sources or error sensors are added to the system, the value
of the convergence coefficient should be decreased. Conversely, if the
number of control sources or error sensors is reduced, the convergence
coefficient can be increased.
5. If the separation distance between the control source( s) and error sensor( s)
is increased, so that the time required for a signal to propagate from
source to sensor is increased, the value of convergence coefficient should
be decreased. Conversely, if the distance between source(s) and sensor(s)
is decreased, a larger value of convergence coefficient can be used.
6. If the size of the digital filter(s) used to generate the control signal(s) is
increased in size, then the value of the convergence coefficient should be
reduced. Conversely, if the size of the filter is reduced, the convergence
coefficient can be increased.
7. The maximum stable value of the convergence coefficient for FIR digital
filter implementations is often greater than the maximum stable value of
the convergence coefficient for IIR digital filter implementations.
Quantization Errors and Leakage
While the gradient descent algorithm is useful for deriving a set of filter weights
that will lead to the minimization of an acoustic disturbance at one or more
sensing locations, long-term operation of the algorithm can lead to instability.
This instability arises because the adaptive algorithm is implemented digitally,
and so is susceptible to bias arising from quantization errors.
In the implementation of the adaptive digital filter in an active noise control system, there are two sources of quantization error: the quantization error
that occurs in the analog-to-digital signal conversion, and the truncation
error that occurs when multiplying two numbers in a system employing arithmetic of finite precision. It may be tempting to ignore these errors in the
implementation of the adaptive algorithm, as they would appear to be random in sign and of an order less than the least significant bit of the system.
However, such assumptions can prove disastrous.
Heuristically, the effect of quantization errors can be viewed as one of adding
additional energy to each weight coefficient in the adaptive digital filter. An
144
8. Controller Fundamentals
explanation of the effect that this has upon the long-term operation can be made
with reference to Figure 8.22. Illustrated in this figure is a typical two-dimensional plot of the mean square value of the error criterion (mean square error, or
MSE) as a function of two weights in an FIR filter, that is essentially how the bowl
in Figure 8.19 would appear if the viewer was on top looking down. There is a
number of combinations of weight values that will result in the same mean square
value of the error criterion. These combinations form a set of concentric contours
centered around the optimum combination of filter weights. During long-term
operation of the adaptive algorithm, quantization errors cause the values of the
filter weights to increase in magnitUde, analogous to a build-up of energy. However, this increase in value is not evident to the outside viewer, as the mean square
value of the error signal is unchanged. Rather, the combination of weights moves
along a contour of a constant mean square value of the error criterion. Eventually
the values of the weights become larger than the maximum value allowed by the
digital system, and the filter calculations begin to overflow. Only at this point
does the outside viewer realize that something has gone terribly wrong.
Fortunately, it is relatively simple to fix this problem using what is referred
to as tap leakage. Tap leakage works by removing a small portion of the
current weight values with each new weight calculation. In this way the buildup of energy that arises from quantization errors is avoided and long-term
stable operation of the adaptive algorithm is possible. When tap leakage is
implemented, the gradient descent algorithm is modified to
Area of weight saturation
-
Contours of
constant MSE
..c
OJ
£
convergence
th
Weight
Wo
FIGURE 8.22. Quantization errors will add "energy" to the adaptive algorithm, eventually
driving it unstable (saturate the filter weights) if some precautions are not taken .
Controller Component 2: The Adaptive Algorithm
w(k + 1)= w(k )(1-2)
145
L1w(k)
where (X is some small positive number, referred to as a leakage coefficient. It
is this version of a gradient descent algorithm that is commonly implemented
in active noise control.
For those readers who are familiar with optimal control, it should be noted
that the tap leakage algorithm is effectively a version that includes control
effort weighting in the error criterion (the same as in optimal control). Having
this weighting has been shown to be beneficial in many active control implementations.
Slowing Down the Algorithm to Improve Performance
Earlier in this section, it was mentioned that the time delay that occurs between, when the adaptive algorithm changes the digital filter weights and
when the change is reflected in the error signal, is the major limiting factor for
the convergence coefficient size in an adaptive feedforward active noise control system. Intuitively, the algorithm can be thought of as running blindly
for a number of iterations after changing the digital filter weight values, and
so must proceed slowly. Note that it is algorithm iterations, or the number of
times the adaptive algorithm blindly updates the weights during the time
delay, that is important, and not the actual time delay in seconds. It is not
uncommon to have a situation where the limit placed upon the convergence
coefficient size by the time delay precludes the use of a convergence coefficient value that would provide the "best" result in terms of sound attenuation. If somehow the adaptive algorithm could be stabilized so that a larger
value of the convergence coefficient could be used, then the attenuation
produced by the active control system would be increased.
In the situation described above, as unusual and unintuitive as it sounds, it
is the case that slowing down the adaptive algorithm will improve its performance. By "slowing down," what is meant is performing the adaptive algorithm calculations after every two, three, etc., samples, rather than after every
new input sample. This has the effect of stabilizing the adaptive algorithm, as
it does not have to wait as long (in terms of samples, which is what is important in the digital domain) to receive the results of its past efforts. With the
algorithm stabilized, the user can increase the convergence coefficient value
and so increase sound attenuation.
An experimental example of this behavior is shown in Figure 8.23. In the
figure, the value of the sound attenuation is plotted against the convergence
coefficient value for three different adaptation rates: one that is the same as the
sample rate, one that is one-third of the sample rate, and one that is one-fifth of
the sample rate. Observe that while increasing the convergence coefficient will
increase the level of sound attenuation (as expected), the "faster" algorithm implementations fail to reach the optimum value. The system becomes unstable due to
146
8. Controller Fundamentals
50.---------------------------,
1"'..j(-----
~/'
40
Case 2 instability
Case 1 instability
c:
-co 20
o 30
a:l
:l
c:
Q)
Case 1:
Adaptation every cycle
Case 2:
Adaptation every 3 cycles
Case 3:
Adaptation every 5 cycles
"C
c:
:l
o
C/)
10
O~----~
o
100
__~____~____~____
-J
200
300
400
500
Convergence coefficient value
FIGURE 8.23. A plot of the sound attenuation versus the convergence coefficient for a
typical active noise control implementation. Note that the "faster" algorithm implementations are not always amenable to the "optimum" convergence coefficient value, due to
premature instability arising from long time delays in the cancellation path.
the time delay in the cancellation path. By effectively reducing this time delay
(as measured in numbers of weight updates that occur in the time delay period),
the algorithm is stabilized and the convergence coefficient can be increased.
One additional point that is of interest here is that slowing down the adaptive algorithm calculation rate need not slow down the actual time it takes
for the algorithm to converge to a set of weight values that provide a given
level of sound attenuation. If a larger convergence coefficient is used in the
stabilized implementations, it will counteract the effect of not updating the
weights as frequently as is physically possible. An example of this is shown in
Figure 8.24, where the error signal response after a restart is shown for two
different adaptation rates and convergence coefficient values (the best result
for each case is shown). Note that each algorithm converges at more-or-Iess
the same actual rate (in seconds).
Controller Component 3: Cancellation Path Modeler
The third component of the adaptive feedforward active noise controller is
the cancellation path transfer function modeler. As was outlined in the previ-
Controller Component 3: Cancellation Path Modeler
...ro
..........
Adapt every cycle
Convergence coefficient = 200
0.4
Q)
c
147
-::::
---~ 0.0
c
C)
e...
(f)
-0.4
UJ
....
..........
ro
Q) 0.4
c
o
1
2
Time (seconds)
3
Adapt every 10 cycles
Convergence coefficient = 1000
-::::.-
---- 0.0
cti
c
C)
e
(f)
....
w
-0.4
Time (seconds)
FIGURE 8.24. Initial convergence of the adaptive feedforward active noise control system
for different adaptation rates and convergence coefficient values.
ous section on the adaptive algorithm, knowledge of how the control signal
is altered between its output from the digital filter and appearance in the error
signal is required to calculate the gradient used in the adaptive algorithm.
This knowledge takes the form of a model of the transfer function, the derivation of which is the job of this part of the control system.
Before progressing further, it should be noted that the general operation of
formulating a mathematical model of a physical transfer function is referred
to as system identification. This terminology will be used frequently in the
subsequent discussion.
The gradient descent adaptive algorithm used to adjust the weights of the
digital filter model of the cancellation path transfer function is slightly different than the gradient descent algorithm described previously for adjusting
the weights of the control signal-generating digital filter. Referring to Figure
8.25, the difference occurs because there is no transfer function in the cancellation path with the system identification arrangement. The control signal
148
8. Controller Fundamentals
Cancellation
ouct
Path
rvv
~~ Speaker
Microphone
"-
Modeling
s ignal
(noise, etc.)
+
- L
Model
')
~
LMS
Algorithm
error
FIGURE 8.25. Arrangement for modeling the cancellation path in an active noise control
implementation.
generated by the digital filter model is used directly in the calculation of the
error signal, without having to first pass through loudspeakers, microphones,
filters, etc. This means that the signal samples in the digital filter can be used
directly in the gradient calculation, rather than having to be filtered through
a model of the cancellation path transfer function. That is, the algorithm
implementation has the standard adaptive filtering form, as found in applications such as telephone echo cancellation.
Before discussing the system identification procedure further, it is worthwhile considering an important question: how accurate does the model of the
cancellation path transfer function have to be for the control filter adaptive
algorithm (described in the previous section) to function correctly? Fortunately, it does not have to be exact. Errors in the estimate of the amplitude of
the transfer function have the simple effect of reducing the maximum stable
value of the control filter adaptive algorithm convergence coefficient in an
inverse proportional manner (if the gain estimate is too high, the maximum
stable value of the convergence coefficient is reduced; if the gain estimate is
too low, the maximum stable value of the convergence coefficient is increased).
However, the most important parameter is the phase of the transfer function.
Theoretically, for a single control source, single error sensor system, it is
possible to have stable operation of the control filter adaptive algorithm
Controller Component 3: Cancellation Path Modeler
149
provided that the phase response of the model is within ±90° of the actual
phase response for the frequency components being targeted for active noise
or vibration control. In practice, it is better to restrict errors in the estimate of
the phase response to be less than half (45°) of this value. For systems with
multiple control sources and multiple error sensors, the robustness of the
system increases. In this case, it is possible to have stable operation of the
control filter adaptive algorithm provided that the total error in the phase
response of the models between a given control source and each error sensor (the
sum of the errors in each model) is less than ±N x 90°, where N is the number of
error sensors in the system (this is actually an approximate value; the actual value
is dependent upon the characteristics of the transfer functions themselves).
The characteristics of the cancellation path transfer function are usually
not constant, but rather are slowly time varying. In some instances this variation can be extremely slow, such as due to the change in loudspeaker response due to mold growth! In other instances, the variation can be more
rapid, such as due to the change in flow rate in an air-handling duct. In either
case it is necessary to update the model of the cancellation path transfer
function to account for these changes. In most practical active noise control
systems, modeling of the cancellation path transfer function is done in parallel to the adaptation of the control filter weights. To do this, the signal used in
the modeling procedure, injected into both the cancellation path and the
transfer function model as shown in Figure 8.25, must be included in the
output from the digital control filter.
The most common way of conducting this "on-line" modeling, in both
active noise control and other (general) adaptive control implementations. is
to inject random noise into both the system to be modeled and the model
itself. The advantage of random noise is that it is uncorrelated with the other
disturbances in the system to be modeled, which in this case includes the
unwanted noise and control disturbances. This reduces the chance of bias in
the model (although some bias can still occur, as will be discussed shortly).
While this effectively means adding an additional, uncontrollable disturbance into the system targeted for active noise or vibration control, the amplitude of the modeling disturbance can usually be quite small and still
produce an adequate model (say, 30 dB below the peak signal levels in the
external environment).
There is a second approach to modeling that can be useful, and is sometimes used, in some circumstances. This approach uses the actual control
signal as the modeling disturbance, injecting it into the model of the cancellation path transfer function (it is already being injected into the cancellation
path itself). Intuitively, this would appear to be a very risky way of conducting system identification, as the control signal is correlated with disturbances
in the external environment. This can potentially lead to such high degrees of
signal bias that the control filter adaptive algorithm becomes unstable. However, there are means of accounting for the correlated environment which
make this approach useful for tonal disturbances.
150
8. Controller Fundamentals
Note. The procedure used for identification of the cancellation path transfer
function is similar to the procedure used for tuning the control filter weights.
In both instances, an adaptive algorithm is used to modify the weights of a
digital filter so as to minimize some error criterion. When adjusting the control filter weights, this error criterion is minimization of the sum of the squared
error signal amplitudes. When modeling the cancellation path transfer function, the error criterion is minimization of the squared value of the prediction
error, which is the difference between the model output and the system output. Because of the similarities between the two procedures, and the similarity of parameters and effects, many of the parameter details discussed in the
previous section are mirrored here and so will not be discussed.
Side Note. Modeling of the cancellation path transfer function entails directing the output of a modeling signal into both the cancellation path and its
model, and the use of an adaptive algorithm to minimize the difference between the system (cancellation path) and model outputs. If the modeling
disturbance is injected into an environment which has "auto-correlated" noise,
noise with strong periodic characteristics, such as systems with tonal disturbances, or systems with strong resonances, the model can become biased or
even completely wrong. This can result in system instability. One way to get
around this problem is to use extended system identification procedures.
Consult a more "in-depth" active noise control text for more information.
Selecting the Sample Rate
One of the, if not the, most important parameters in the design and implementation of the active noise controller is the digital system sample rate.
While it is fine to say "the digital filter will do this," and "the adaptive
algorithm will do that," how well it all actually works is largely determined
by the sample rate. Now armed with some basic facts about the desired workings of the controller, we are in a position to assess the influence of this
extremely important parameter.
Selecting the correct sample rate for a given application is often not easy,
requiring some judgment to balance a number of competing influences. In
this section, we will look at the effect of high and low sample rates with the
aim of arriving at a suitable compromise.
An absolute limit on the lower (minimum) value of sample rates is set by
the phenomena of aliasing. Aliasing refers to the fact that in sampled systems,
images ofthe true value of the sampled spectrum repeat themselves at infinite
numbers of multiples of the sample frequency f.. Practically, the phenomena
of aliasing means that it is impossible to tell the difference between two or
more sinusoids based upon the sampled signal. This effect is illustrated in
Figure 8.26, where two sinusoids have exactly the same sampled values and
can therefore not be distinguished from one another based upon the sampled
Controller Component 3: Cancellation Path Modeler
151
Sinusoid 2 ~
FIGURE 8.26. An example of two signals aliasing.
data. Aliasing can have a significant detrimental effect upon control system
performance if there are substantial levels of high-frequency data, f> f s/2,
that are allowed to alias onto the low-frequency data, f < f,l2. To combat this
problem, antialiasing filters are placed in front of the analog-to-digital converter (ADC). These are low-pass filters that remove frequency components
greater than half the sample frequency, f > f,l2, from the input spectrum.
In theory, then, the lower bound on the sample rate for a given problem is
twice the highest frequency of interest. However, actually implementing a
system with this sample rate is not advisable. First, while it is theoretically
possible to reconstruct a harmonic signal sampled at twice its frequency, the
filter required to do so is of infinite length, and not bounded input, bounded
output (BIBO) stable. By this, we mean that for a finite input there is not
necessarily a finite output. Second, there is no margin for error in the upper
frequency limit. Any slight change in the upper frequency results in aliasing.
Third, practical antialiasing filters do not have perfectly square pass/stop
characteristics, but rather have some finite transition band from pass (frequencies below half the sample rate that are allowed through the filter) to
stop (frequency components above half the sample rate that are filtered out).
Therefore, expecting the antialiasing filter to pass all frequencies up to half
the sample rate, while stopping all frequencies above half the sample rate, is
optimistic.
To paint a qualitative picture of what happens if the sample rate is too low,
consider the problem of sampling the "step response" of a system that has a
resonance at 1 Hz (that is, we are looking at the response of a system that is
dominated by a frequency of 1 Hz). As shown in Figure 8.27, if the step
response is sampled at 2 Hz the characteristics are indistinguishable to the
viewer. Sampled at 5 Hz, the characteristics begin to appear. At 10 Hz, the step
is apparent. In fact, if the samples are connected by straight lines, the reconstruction of the step is in error by less than 4%. Intuitively, then, we can
postulate that the filtering exercise, which is analogous to the reconstruction
of a signal, becomes "easier: as the sample rate increases.
There is, however, a limit to this process for an adaptive control system. At
high sample rates, tens or even hundreds of times the disturbance frequency,
152
8. Controller Fundamentals
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Sample rate = 2 Hz
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Sample rate = 5 Hz
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FIGURE 8.27. Sampling the response of a system with a dominant 1 Hz frequency
component at three different sample rates: 2 Hz, 5 Hz, and 10 Hz. Note that the x's
indicate the sampled values.
there are problems with numerical accuracy in the digital environment. Perhaps more seriously, there are problems in the convergence behavior of the
adaptive algorithm.
The upper bound on sample rate selection for a given problem is usually
determined by adaptive algorithm performance characteristics. When discussing
adaptive algorithm performance, the focus is usually on two competing factors:
algorithm stability and speed of adaptation. The key factor that influences these
characteristics is the adaptive algorithm convergence coefficient. Algorithm sta-
Controller Component 3: Cancellation Path Modeler
153
bility is enhanced by reducing the size of the convergence coefficient, while
speed of adaptation is often enhanced by increasing the size of the convergence
coefficient. In practice, there must be a balance: converge as fast as possible
without becoming unstable. When items such as sample rate are assessed for their
influence upon adaptive algorithm performance, what is normally examined is
their influence upon this balance point via qualification of how they affect the
upper bound place on the convergence coefficient size for stable operation. If the
item of interest reduces the maximum stable value of convergence coefficient it is
regarded as retarding algorithm performance. Conversely, if the item of interest
increases the maximum stable value of the convergence coefficient, it is regarded
as improving algorithm performance.
While the effect that items such as sample rate have upon the maximum
stable value of the convergence coefficient can be mathematically quantified, it will be more instructive here to paint a qualitative picture. This can be
done with reference to the error surface bowl. As outlined previously, this
bowl describes the shape of the plot of the algorithm error criterion (the mean
square value of the measured error signal) as a function of the weights of the
digital control filter. The control filter weight adaptation process can be
viewed as one of searching this surface for the optimum weights which produce the minimum error criterion value; in other words, a process of "looking
for the bottom of the bowl." It is therefore intuitive that the shape of the bowl,
such as the slope of the sides and symmetry, will have an influence upon the
performance of the algorithm in its search for the bottom. Related to the
discussion here, the degree to which the bowl is "squashed" is largely determined by the choice of sample rate.
The steepness of the slope of the sides of the error surface bowl (mathematically characterized by the eigenvalues of the error surface) influences
both the speed of convergence and stability of the adaptive algorithm. If an
error surface has steep sides (mathematically, large eigenvalues), then the
adaptive algorithm is less stable and the convergence coefficient must be
small. If the error surface has shallow sides, then the adaptive algorithm is
more stable, and the convergence coefficient can be increased. This phenomenon is evident in Figure 8.28, which illustrates the convergence, then divergence, path of an adaptive algorithm, plotted on contours of mean square
error. Note that the algorithm aligns itself along the axis of maximum slope,
rocks, then goes unstable; the algorithm is least stable along the steepest
axis. With such an error surface, the bounding of the maximum stable value of
convergence coefficient is set by the steep sides.
With regard to adaptation speed, for a given stable value of the convergence coefficient, convergence is always faster along the steep sides. This is
because the gradient is larger, and as gradient descent algorithms operate by
adding a portion of the gradient to the current weight values to adapt, larger
gradients mean faster adaptation.
From this discussion, it can be surmised that the worst type of error surface
for algorithm convergence is one that is highly squashed (that is, some very
154
8. Controller Fundamentals
2
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Weight Wo
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FIGURE 8.28. Example of adaptive algorithm divergence.
steep sides and some very shallow sides). With such an error surface, the
convergence coefficient must be very small to maintain stability due to the
steep sides. However, with such a small convergence coefficient the speed of
adaptation will be very poor due to the shallow sides. Conversely, the best
error surface for algorithm convergence is one that is perfectly symmetric.
Here the steep and shallow sides, and hence speed and stability, are perfectly
in balance.
The reason for this discussion is that sample rate largely determines the
degree to which the error surface bowl is squashed. In particular, as the sample
rate increases relative to the target frequency of excitation, the steep and
shallow sides of the error surface become more and more disparate. If the
target frequency is heavily oversampled (say, 50 or more samples per excitation frequency cycle), then the error surface will have some very steep sides
and some very shallow sides. In this case, the adaptive algorithm performance
will be very poor. The convergence coefficient will have to be very small, and
the speed of convergence will be very slow. If the degree of oversampling is
several hundred or more, attempts at control are largely futile.
So What Is the Optimum Sample Rate?
For tonal excitation, if the sampling frequency is synchronized to be exactly
four times the excitation frequency, then for a two-tap finite impulse response
filter the error surface bowl is perfectly uniform, which means that the competing effects of stability and convergence speed are perfectly balanced. This,
however, is not recommended as optimum for a number of reasons. First, in
general, we cannot expect to be able to synchronize the sample rate of the
controller with the unwanted disturbance, even if it is harmonic. Second, the
use of only two taps in an FIR filter would not be recommended in practice
Controller Component 3: Cancellation Path Modeler
155
(see the control filter section of this chapter for a discussion of recommended
tap numbers). Third, as could be deduced from the step response example,
having only four samples per cycle may lead to accuracy problems.
The optimum sample rate is a compromise between fast and slow. Both of
these extremes lead to problems with adaptive algorithm convergence and
stability, and to problems with numerical accuracy. The "optimum" sample
rate compromise is often cited as ten times the frequency of interest. In
practice, this sample rate provides for rapid convergence of the adaptive algorithm and reasonable levels of stability. In implementing active control systems we often find that for a given sample rate f. the system will work
reasonably well from frequencies approaching f/100 to frequencies up to f/
3. On the low end of the scale, adaptation of the controller with frequencies
below 0)/100 is often (extremely) slow, and not particularly stable. While this
is sometimes improved by increasing the length of the digital control filter,
the only real solution is a reduction in sample rate. On the high end of the
scale, the adaptive algorithm appears ineffective with excitation frequencies
above 0)s/3.
Index
A-weighting, 30
values, 31
acoustic pressure, 9
active noise control, 44
definition, 3
ducts, 89
enclosed spaces, 76
feedforward systems, 89
free space, 51
mechanisms, 51
adaptation rate, Ill, 145
adaptive algorithm, 105, 108
adaptive feedforward active noise
control
definition, 4
adaptive feedforward control
system arrangement, 117
aliasing, 150
analog to digital converter (ADC), 99,
124
anechoic, 46
anti -aliasing filters, 151
anti-nodes, 70
axial modes, 72
model, 140, 147
required model accuracy, 148
causality, 63
closed loop control system, 116
constant pressure sound source, 38
constant volume velocity source, 38
control filter, 104
description, 128
filter types, 131
selection guidelines, 132
control inputs, 114
control source, 5
control system, 5
controller, 95
convergence coefficient, 109
description, 138
barrier, 46
beating, 27
binary numbers, 99
decibels, 9, 40
digital filtering, 104
Digital Signal Processor, 103
digital system components, 121,
124
digital system requirements, 96, 124
digital systems, 96-99
accuracy, 99
digital to analog converter, 103, 124
disturbance inputs, Il4
ducts, 81
cancellation path transfer function, 111
description, 111, 139
effect upon gradient descent
algorithm, 140
identification, 146
enclosed space noise control, 67
error microphone, 5
error microphone location, 60
error signal, 5, 89, 116
expansion chamber, 86
157
158
Index
Fast Fourier Transform, 21
feedback control
system arrangement, 116
feedforward control, 117, 118
feedforward systems, 89
feedforward taps, 132, 135
filter coefficients, 104
filtered adaptive algorithms, 139
filtering, 104
FIR filters, 106
advantages and disadvantages, 133
fixed point microprocessor, 102
floating point microprocessor, 102
Fourier, 19
Fourier analysis, 19, 20
Fourier transform, 21
free field, 46
free space, 46
frequency, 14, 15
frequency analysis, 14, 20
frequency content, 14
fundamental harmonic, 21, 22
noise control, 8, 36, 43
oblique modes, 74
octave band, 29
one-third octave band, 29
open loop control system, 115
Pascal
definition, 8
passive noise control, 43
dissipative techniques, 84, 88
ducts, 84
enclosed spaces, 74
free space, 46
reactive devices, 84
physical system limitations,
119
potential difference, 37
power, 39
pressure
definition, 8
quantization error, 102
gradient descent algorithm, 136,
138
harmonics, 21
hearing loss, 41
Helmholtz filter, 87
Helmholtz resonator, 85
human hearing
frequency range, 27, 28
perception, 27
range of pressure levels, 9
IIR filters, 106
advantages and disadvantages, 133
impedance, 37, 38,42, 83
input range, 101
inverse Fourier transform, 21
local sound cancellation, 61
modes, 67, 70, 82
nodes, 70
noise
definition, 8
reconstruction filter, 127
reference microphone, 4
reference signal, 4, 90
resonance frequencies, 69
RMS
definition, 17
sample period
definition, 125
sample rate, 98
definition, 125
effect upon system performance,
153, 154
lower bound, 151
optimum, 154
upper bound, 152, 154
sidebranch resonator, 85
sine wave, 15
amplitude, 17, 18
phase, 17, 19
sound
definition, 7
sound field, 7
ducts, 81
Index
sound power, 39
magnitude, 40
sound power reduction, 56
sound pressure
acceptable levels, 30
definition, 8
subjective assessment, 9,
10
typical levels, 9
sound source size, 54
sound waves, 10
generation, 10, 11
type, 12
source separation distance, 54
spectral analysis, 20
speed of sound, 12, 13
standing wave, 70
tangential modes, 74
tap leakage
description, 144
tonal, 16
transmission loss, 48, 50
waveform, 11
periodic, 21
159