SÁCH

18. DIGITAL IMAGE PROCESSING
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PREFACE
In January 1978, I began the preface to the first e***ion of Digital Image Processing
with the following statement:
The field of image processing has grown considerably during the past decade
with the increased utilization of imagery in myriad applications coupled with
improvements in the size, speed, and cost effectiveness of digital computers and
related signal processing technologies. Image processing has found a significant role
in scientific, industrial, space, and government applications.
In January 1991, in the preface to the second e***ion, I stated:
Thirteen years later as I write this preface to the second e***ion, I find the quoted
statement still to be valid. The 1980s have been a decade of significant growth and
maturity in this field. At the beginning of that decade, many image processing techniques
were of academic interest only; their execution was too slow and too costly.
Today, thanks to algorithmic and implementation advances, image processing has
become a vital cost-effective technology in a host of applications.
Now, in this beginning of the twenty-first century, image processing has become
a mature engineering discipline. But advances in the theoretical basis of image processing
continue. Some of the reasons for this third e***ion of the book are to correct
defects in the second e***ion, delete content of marginal interest, and add discussion
of new, important topics. Another motivating factor is the inclusion of interactive,
computer display imaging examples to illustrate image processing concepts. Finally,
this third e***ion includes computer programming exercises to bolster its theoretical
content. These exercises can be implemented using the Programmerâ?Ts Imaging Kernel
System (PIKS) application program interface (API). PIKS is an International
Standards Organization (ISO) standard library of image processing operators and
associated utilities. The PIKS Core version is included on a CD affixed to the back
cover of this book.
The book is intended to be an â?oindustrial strengthâ? introduction to digital image
processing to be used as a text for an electrical engineering or computer science
course in the subject. Also, it can be used as a reference manual for scientists who
are engaged in image processing research, developers of image processing hardware
and software systems, and practicing engineers and scientists who use image processing
as a tool in their applications. Mathematical derivations are provided for
most algorithms. The reader is assumed to have a basic background in linear system
theory, vector space algebra, and random processes. Proficiency in C language programming
is necessary for execution of the image processing programming exercises
using PIKS.
The book is divided into six parts. The first three parts cover the basic technologies
that are needed *****pport image processing applications. Part 1 contains three
chapters concerned with the characterization of continuous images. Topics include
the mathematical representation of continuous images, the psychophysical properties
of human vision, and photometry and colorimetry. In Part 2, image sampling
and quantization techniques are explored along with the mathematical representation
of discrete images. Part 3 discusses two-dimensional signal processing techniques,
including general linear operators and unitary transforms such as the
Fourier, Hadamard, and Karhunenâ?"Loeve transforms. The final chapter in Part 3
analyzes and compares linear processing techniques implemented by direct convolution
and Fourier domain filtering.
The next two parts of the book cover the two principal application areas of image
processing. Part 4 presents a discussion of image enhancement and restoration techniques,
including restoration models, point and spatial restoration, and geometrical
image modification. Part 5, entitled â?oImage Analysis,â? concentrates on the extraction
of information from an image. Specific topics include morphological image
processing, edge detection, image feature extraction, image segmentation, object
shape analysis, and object detection.
Part 6 discusses the software implementation of image processing applications.
This part describes the PIKS API and explains its use as a means of implementing
image processing algorithms. Image processing programming exercises are included
in Part 6.
This third e***ion represents a major revision of the second e***ion. In ad***ion to
Part 6, new topics include an expanded description of color spaces, the Hartley and
Daubechies transforms, wavelet filtering, watershed and snake image segmentation,
and Mellin transform matched filtering. Many of the photographic examples in the
book are supplemented by executable programs for which readers can adjust algorithm
parameters and even substitute their own source images.
Although readers should find this book reasonably comprehensive, many important
topics allied to the field of digital image processing have been omitted to limit
the size and cost of the book. Among the most prominent omissions are the topics of
pattern recognition, image reconstruction from projections, image understanding,
image coding, scientific visualization, and computer graphics. References to some
of these topics are provided in the bibliography.
WILLIAM K. PRATT
Los Altos, California
August 2000

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2. Real-time digital signal processing
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3. Independent Component Ana&sis
Preface
Independent component analysis (ICA) is a statistical and computational technique
for revealing hidden factors that underlie sets of random variables, measurements, or
signals. ICA defines a generative model for the observed multivariate data, which is
typically given as a large database of samples. In the model, the data variables are
assumed to be linear or nonlinear mixtures of some unknown latent variables, and
the mixing system is also unknown. The latent variables are assumed nongaussian
and mutually independent, and they are called the independent components of the
observed data. These independent components, also called sources or factors, can be
found by ICA.
ICA can be seen as an extension to principal component analysis and factor
analysis. ICA is a much more powerful technique, however, capable of finding the
underlying factors or sources when these classic methods fail completely.
The data analyzed by ICA could originate from many different kinds of application
fields, including digital images and document databases, as well as economic
indicators and psychometric measurements. In many cases, the measurements are
given as a set of parallel signals or time series; the term blind source separation is used
to characterize this problem. Typical examples are mixtures of simultaneous speech
signals that have been picked up by several microphones, brain waves recorded by
multiple sensors, interfering radio signals arriving at a mobile phone, or parallel time
series obtained from some industrial process.
The technique of ICA is a relatively new invention. It was for the first time introduced
in early 1980s in the context of neural network modeling. In mid-1990s,
some highly successful new algorithms were introduced by several research groups,
together with impressive demonstrations on problems like the ****tail-party effect,
where the individual speech waveforms are found from their mixtures. ICA became
one of the exciting new topics, both in the field of neural networks, especially unsupervised
learning, and more generally in advanced statistics and signal processing.
Reported real-world applications of ICA on biomedical signal processing, audio signal
separation, telecommunications, fault diagnosis, feature extraction, financial time
series analysis, and data mining began to appear.
Many articles on ICA were published during the past 20 years in a large number
of journals and conference proceedings in the fields of signal processing, artificial
neural networks, statistics, information theory, and various application fields. Several
special sessions and workshops on ICA have been arranged recently [70, 348], and
some e***ed collections of articles [315, 173, 150] as well as some monographs on
ICA, blind source separation, and related subjects [105, 267, 149] have appeared.
However, while highly useful for their intended readership, these existing texts typically
concentrate on some selected aspects of the ICA methods only. In the brief
scientific papers and book chapters, mathematical and statistical preliminaries are
usually not included, which makes it very hard for a wider audience to gain full
understanding of this fairly technical topic.
A comprehensive and detailed text book has been missing, which would cover
both the mathematical background and principles, algorithmic solutions, and practical
applications of the present state of the art of ICA. The present book is intended to fill
that gap, serving as a fundamental introduction to ICA.
It is expected that the readership will be from a variety of disciplines, such
as statistics, signal processing, neural networks, applied mathematics, neural and
cognitive sciences, information theory, artificial intelligence, and engineering. Both
researchers, students, and practitioners will be able to use the book. We have made
every effort tomake this book self-contained, so that a reader with a basic background
in college calculus, matrix algebra, probability theory, and statistics will be able to
read it. This book is also suitable for a graduate level university course on ICA,
which is facilitated by the exercise problems and computer assignments given in
many chapters.
Scope and contents of this book
This book provides a comprehensive introduction to ICA as a statistical and computational
technique. The emphasis is on the fundamental mathematical principles and
basic algorithms. Much of the material is based on the original research conducted
in the authorsõ?T own research group, which is naturally reflected in the weighting of
the different topics. We give a wide coverage especially to those algorithms that are
scalable to large problems, that is, work even with a large number of observed variables
and data points. These will be increasingly used in the near future when ICA
is extensively applied in practical real-world problems instead of the toy problems
or small pilot studies that have been predominant until recently. Respectively, some
what less emphasis is given to more specialized signal processing methods involving
convolutivemixtures, delays, and other blind source separation techniques than ICA.
As ICA is a fast growing research area, it is impossible to include every reported
development in a textbook. We have tried to cover the central contributions by other
workers in the field in their appropriate context and present an extensive bibliography
for further reference. We apologize for any omissions of important contributions that
we may have overlooked.
For easier reading, the book is divided into four parts.

Part I gives the mathematical preliminaries. It introduces the general mathematical
concepts needed in the rest of the book. We start with a crash course
on probability theory in Chapter 2. The reader is assumed to be familiar with
most of the basic material in this chapter, but also some concepts more specific
to ICA are introduced, such as higher-order cumulants and multivariate
probability theory. Next, Chapter 3 discusses essential concepts in optimization
theory and gradient methods, which are needed when developing ICA
algorithms. Estimation theory is reviewed in Chapter 4. A complementary
theoretical framework for ICA is information theory, covered in Chapter 5.
Part I is concluded by Chapter 6, which discusses methods related to principal
component analysis, factor analysis, and decorrelation.
More confident readers may prefer to skip some or all of the introductory
chapters in Part I and continue directly to the principles of ICA in Part II.

In Part II, the basic ICA model is covered and solved. This is the linear
instantaneous noise-freemixingmodel that is classic in ICA, and forms the core
of the ICAtheory. The model is introduced and the question of identifiability of
themixingmatrix is treated in Chapter 7. The following chapters treat different
methods of estimating the model. A central principle is nongaussianity,whose
relation to ICA is first discussed in Chapter 8. Next, the principles of maximum
likelihood (Chapter 9) and minimum mutual information (Chapter 10) are
reviewed, and connections between these three fundamental principles are
shown. Material that is less suitable for an introductory course is covered
in Chapter 11, which discusses the algebraic approach using higher-order
cumulant tensors, and Chapter 12, which reviews the early work on ICA based
on nonlinear decorrelations, as well as the nonlinear principal component
approach. Practical algorithms for computing the independent components
and the mixing matrix are discussed in connection with each principle. Next,
some practical considerations, mainly related to preprocessing and dimension
reduction of the data are discussed inChapter 13, including hints to practitioners
on howto really apply ICAto their own problem. An overviewand comparison
of the various ICA methods is presented in Chapter 14, which thus summarizes
Part II.

In Part III, different extensions of the basic ICAmodel are given. This part is by
its nature more speculative than Part II, since most of the extensions have been
introduced very recently, and many open problems remain. In an introductory
course on ICA, only selected chapters from this part may be covered. First,
in Chapter 15, we treat the problem of introducing explicit observational noise
in the ICA model. Then the situation where there are more independent
components than observed mixtures is treated in Chapter 16. In Chapter 17,
the model is widely generalized to the case where the mixing process can be of
a very general nonlinear form. Chapter 18 discusses methods that estimate a
linear mixing model similar to that of ICA, but with quite different assumptions:
the components are not nongaussian but have some time dependencies instead.
Chapter 19 discusses the case where the mixing system includes convolutions.
Further extensions, in particular models where the components are no longer
required to be exactly independent, are given in Chapter 20.

Part IV treats some applications of ICA methods. Feature extraction (Chapter
21) is relevant to both image processing and vision research. Brain imaging
applications (Chapter 22) concentrate on measurements of the electrical and
magnetic activity of the human brain. Telecommunications applications are
treated in Chapter 23. Some econometric and audio signal processing applications,
together with pointers to miscellaneous other applications, are treated in
Chapter 24.
Throughout the book, we have marked with an asterisk some sections that are
rather involved and can be skipped in an introductory course.
Several of the algorithms presented in this book are available as public domain
software through the World Wide Web, both on our own Web pages and those of
other ICA researchers. Also, databases of real-world data can be found there for
testing the methods. We have made a specialWeb page for this book, which contains
appropriate pointers. The address is
www.cis.hut.fi/projects/ica/book
The reader is advised to consult this page for further information.
This book was written in cooperation between the three authors. A. Hyvăarinen
was responsible for the chapters 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 20, 21, and 22;
J. Karhunen was responsible for the chapters 2, 4, 17, 19, and 23; while E. Oja was
responsible for the chapters 3, 6, and 12. The Chapters 1 and 24 were written jointly
by the authors.
Acknowledgments
We are grateful to the many ICA researchers whose original contributions form the
foundations of ICA and who have made this book possible. In particular, we wish to
express our gratitude to the Series E***or Simon Haykin, whose articles and books on
signal processing and neural networks have been an inspiration to us over the years.
Some parts of this text are based on close cooperation with other members of our
research group at the Helsinki University of Technology. Chapter 21 is largely based
on joint work with Patrik Hoyer, who also made all the experiments in that chapter.
Chapter 22 is based on experiments and material by Ricardo Vig´ario. Section 13.2.2
is based on joint work with Jaakko Săarelăa and Ricardo Vig´ario. The experiments in
Section 16.2.3 were provided by Razvan Cristescu. Section 20.3 is based on joint
work with Ella Bingham, Section 14.4 on joint work with Xavier Giannakopoulos,
and Section 20.2.3 on joint work with Patrik Hoyer and Mika Inki. Chapter 19 is
partly based on material provided by Kari Torkkola. Much of Chapter 17 is based
on joint work with Harri Valpola and Petteri Pajunen, and Section 24.1 is joint work
with Kimmo Kiviluoto and Simona Malaroiu.
Over various phases of writing this book, several people have kindly agreed to
read and comment on parts or all of the text. We are grateful for this to Ella Bingham,
Jean-Francáois Cardoso, Adrian Flanagan, Mark Girolami, Antti Honkela, Jarmo
Hurri, Petteri Pajunen, Tapani Ristaniemi, and Kari Torkkola. Leila Koivisto helped
in technical e***ing, while Antti Honkela, Mika Ilmoniemi, Merja Oja, and Tapani
Raiko helped with some of the figures.
Our original research work on ICA as well as writing this book has been mainly
conducted at theNeuralNetworksResearchCentre of theHelsinkiUniversity of Technology,
Finland. The research had been partly financed by the project õ?oBLISSõ? (European
Union) and the project õ?oNew Information Processing Principlesõ? (Academy
of Finland), which are gratefully acknowledged. Also, A. H. wishes to thank Găote
Nyman and Jukka Hăakkinen of the Department of Psychology of the University of
Helsinki who hosted his civilian service there and made part of the writing possible.
AAPO HYVAă RINEN, JUHA KARHUNEN, ERKKI OJA
Espoo, Finland
March 2001
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4) Tracking and Kalman filtering made easy
PREFACE
At last a book that hopefully will take the mystery and drudgery out of the g?"h,

?", g?"h?"k,
?"?" and Kalman filters and makes them a joy. Many books
written in the past on this subject have been either geared to the tracking filter
specialist or difficult to read. This book covers these filters from very simple
physical and geometric approaches. Extensive, simple and useful design
equations, procedures, and curves are presented. These should permit the reader
to very quickly and simply design tracking filters and determine their
performance with even just a pocket calculator. Many examples are presented
to give the reader insight into the design and performance of these filters.
Extensive homework problems and their solutions are given. These problems
form an integral instructional part of the book through extensive numerical
design examples and through the derivation of very key results stated without
proof in the text, such as the derivation of the equations for the estimation of the
accuracies of the various filters [see Note (1) on page 388]. Covered also in
simple terms is the least-squares filtering problem and the orthonormal
transformation procedures for doing least-squares filtering.
The book is intended for those not familiar with tracking at all as well as for
those familiar with certain areas who could benefit from the physical insight
derived from learning how the various filters are related, and for those who are
specialists in one area of filtering but not familiar with other areas covered. For
example, the book covers in extremely simple physical and geometric terms the
Gram?"Schmidt, Givens, and Householder orthonormal transformation procedures
for doing the filtering and least-square estimation problem. How these
procedures reduce sensitivity to computer round-off errors is presented. A
simple explanation of both the classical and modified Gram?"Schmidt procedures
is given. Why the latter is less sensitive to round-off errors is explained in
physical terms. For the first time the discrete-time orthogonal Legendre
polynomial (DOLP) procedure is related to the voltage-processing procedures.
Important real-world issues such as how to cope with clutter returns,
elimination of redundant target detections (observation-merging or clustering),
e***ing for inconsistent data, track-start and track-drop rules, and data
association (e.g., the nearest-neighbor approach and track before detection)
are covered in clear terms. The problem of tracking with the very commonly
used chirp waveform (a linear-frequency-modulated waveform) is explained
simply with useful design curves given. Also explained is the important
moving-target detector (MTD) technique for canceling clutter.
The Appendix gives a comparison of the Kalman filter (1960) with the
Swerling filter (1959). This Appendix is written by Peter Swerling. It is time for
him to receive due cre*** for his contribution to the ?~?~Kalman?"Swerling?T?T filter.
The book is intended for home study by the practicing engineer as well as for
use in a course on the subject. The author has successfully taught such a course
using the notes that led to this book. The book is also intended as a design
reference book on tracking and estimation due to its extensive design curves,
tables, and useful equations.
It is hoped that engineers, scientists, and mathematicians from a broad range
of disciplines will find the book very useful. In ad***ion to covering and relating
the g?"h,
?", g?"h?"k,
?"?", Kalman filters, and the voltage-processing
methods for filtering and least-squares estimation, the use of the voltageprocessing
methods for sidelobe canceling and adaptive-array processing are
explained and shown to be the same mathematically as the tracking and
estimated problems. The massively parallel systolic array sidelobe canceler
processor is explained in simple terms. Those engineers, scientists, and
mathematicians who come from a mathematical background should get a good
feel for how the least-squares estimation techniques apply to practical systems
like radars. Explained to them are matched filtering, chirp waveforms, methods
for dealing with clutter, the issue of data association, and the MTD clutter
rejection technique. Those with an understanding from the radar point of view
should find the explanation of the usually very mathematical Gram?"Schmidt,
Givens, and Householder voltage-processing (also called square-root) techniques
very easy to understand. Introduced to them are the important concepts of
ill-con***ioning and computational accuracy issues. The classical Gram?"
Schmidt and modified Gram?"Schmidt procedures are covered also, as well as
why one gives much more accurate results. Hopefully those engineers,
scientists, and mathematicians who like to read things for their beauty will
find it in the results and relationships given here. The book is primarily intended
to be light reading and to be enjoyed. It is a book for those who need or want to
learn about filtering and estimation but prefer not to plow through difficult
esoteric material and who would rather enjoy the experience. We could have
called it ?~?~The Joy of Filtering.?T?T
The first part of the text develops the g?"h, g?"h?"k,
?",
?"?", and
Kalman filters. Chapter 1 starts with a very easy heuristic development of g?"h
filters for a simple constant-velocity target in ?~?~lineland?T?T (one-dimensional
space, in contrast to the more complicated two-dimensional ?~?~flatland?T?T).
Section 1.2.5 gives the g?"h filter, which minimizes the transient error resulting
from a step change in the target velocity. This is the well-known Benedict?"
Bordner filter. Section 1.2.6 develops the g?"h filter from a completely different,
common-sense, physical point of view, that of least-squares fitting a straight
line to a set of range measurements. This leads to the critically damped (also
called discounted least-squares and fading-memory) filter. Next, several
example designs are given. The author believes that the best way to learn a
subject is through examples, and so numerous examples are given in Section
1.2.7 and in the homework problems at the end of the book.
Section 1.2.9 gives the con***ions (on g and h) for a g?"h filter to be stable
(these con***ions are derived in problem 1.2.9-1). How to initiate tracking with
a g?"h filter is covered in Section 1.2.10. A filter (the g?"h?"k filter) for tracking a
target having a constant acceleration is covered in Section 1.3. Coordinate
selection is covered in Section 1.5.
The Kalman filter is introduced in Chapter 2 and related to the Benedict?"
Bordner filter, whose equations are derived from the Kalman filter in Problem
2.4-1. Reasons for using the Kalman filter are discussed in Section 2.2, while
Section 2.3 gives a physical feel for how the Kalman filter works in an optimum
way on the data to give us a best estimate. The Kalman filter is put in matrix
form in Section 2.4, not to impress, but because in this form the Kalman filter
applies way beyond lineland?"to multidimensional space.
Section 2.6 gives a very simple derivation of the Kalman filter. It requires
differentiation of a matrix equation. But even if you have never done
differentiation of a matrix equation, you will be able to follow this derivation.
In fact, you will learn how to do matrix differentiation in the process! If
you had this derivation back in 1958 and told the world, it would be your
name filter instead of the Kalman filter. You would have gotten the IEEE
Medal of Honor and $20,000 tax-free and the $340,000 Kyoto Prize,
equivalent to the Nobel Prize but also given to engineers. You would be world
famous.
In Section 2.9 the Singer g?"h?"k Kalman filter is explained and derived.
Extremely useful g?"h?"k filter design curves are presented in Section 2.10
together with an example in the text and many more in Problems 2.10-1 through
2.10-17. The issues of the selection of the type of g?"h filter is covered in
Section 2.11.
Chapter 3 covers the real-world problem of tracking in clutter. The use of the
track-before-detect retrospective detector is described (Section 3.1.1). Also
covered is the important MTD clutter suppression technique (Section 3.1.2.1).
Issues of eliminating redundant detections by observation merging or clustering
are covered (Section 3.1.2.2) as well as techniques for e***ing out inconsistent
data (Section 3.1.3), combining clutter suppression with track initiation
(Section 3.1.4), track-start and track-drop rules (Section 3.2), data association
(Section 3.3), and track-while-scan systems (Section 3.4).
In Section 3.5 a tutorial is given on matched filtering and the very commonly
used chirp waveform. This is followed by a discussion of the range bias error
problem associated with using this waveform and how this bias can be used to
advantage by choosing a chirp waveform that predicts the futurê?"a fortunetelling
radar.
The second part of the book covers least-squares filtering, its power and
voltage-processing approaches. Also, the solution of the least-squares filtering
problem via the use of the DOLP technique is covered and related to voltageprocessing
approaches. Another simple derivation of the Kalman filter is
presented and ad***ional properties of the Kalman filter given. Finally, how to
handle nonlinear measurement equations and nonlinear equations of motion are
discussed (the extended Kalman filter).
Chapter 4 starts with a simple formulation of the least-squares estimation
problem and gives its power method solution, which is derived both by simple
differentiation (Section 4.1) and by simple geometry considerations (Section
4.2). This is followed by a very simple explanation of the Gram?"Schmidt
voltage-processing (square-root) method for solving the least-squares problem
(Section 4.3). The voltage-processing approach has the advantage of being
much less sensitive to computer round-off errors, with about half as many bits
being required to achieve the same accuracy. The voltage-processing approach
has the advantage of not requiring a matrix inverse, as does the power method.
In Section 4.4, it is shown that the mathematics for the solution of the
tracking least-squares problem is identical to that for the radar and
communications sidelobe canceling and adaptive nulling problems. Furthermore,
it is shown how the Gram?"Schmidt voltage-processing approach can be
used for the sidelobe canceling and adaptive nulling problem.
Often the accuracy of the measurements of a tracker varies from one time to
another. For this case, in fitting a trajectory to the measurements, one would like
to make the trajectory fit closer to the accurate data. The minimum-variance
least-squares estimate procedure presented in Section 4.5 does this. The more
accurate the measurement, the closer the curve fit is to the measurement.
The fixed-memory polynomial filter is covered in Chapter 5. In Section 5.3
the DOLP approach is applied to the tracking and least-squares problem for
the important cases where the target trajectory or data points (of which there
are a fixed number L þ 1) are approximated by a polynomial fit of some
degree m. This method also has the advantage of not requiring a matrix
inversion (as does the power method of Section 4.1). Also, its solution is
much less sensitive to computer round-off errors, half as many bits being
required by the computer.
The convenient and useful representation of the polynomial fit of degree m in
terms of the target equation motion derivatives (first m derivatives) is given in
Section 5.4. A useful general solution to the DOLP least-squares estimate for a
polynomial fit that is easily solved on a computer is given in Section 5.5.
Sections 5.6 through 5.10 present the variance and bias errors for the leastsquares
solution and discusses how to balance these errors. The important
method of trend removal to lower the variance and bias errors is discussed in
Section 5.11.
In Chapter 5, the least-squares solution is based on the assumption of a fixed
number L þ 1 of measurements. In this case, when a new measurement is made,
the oldest measurement is dropped in order to keep the number measurements
on which the trajectory estimate is based equal to the fixed number L þ 1. In
Chapter 6 we consider the case when a new measurement is made, we no longer
throw away the oldest data. Such a filter is called a growing-memory filter.
Specifically, an mth-degree polynomial is fitted to the data set, which now
grows with time, that is, L increases with time. This filter is shown to lead to the
easy-to-use recursive growing-memory g?"h filter used for track initiation in
Section 1.2.10. The recursive g?"h?"k (m ¼ 2) and g?"h?"k?"l (m ¼ 3) versions of
this filter are also presented. The issues of stability, track initiation, root-meansquare
(rms) error, and bias errors are discussed.
In Chapter 7 the least-squares polynomial fit to the data is given for the case
where the error of the fit is allowed to grow the older the data. In effect, we pay
less and less attention to the data the older it is. This type of filter is called a
fading-memory filter or discounted least-squares filter. This filter is shown to
lead to the useful recursive fading-memory g?"h filter of Section 1.2.6 when the
polynomial being fitted to is degree m ¼ 1. Recursive versions of this filter that
apply to the case when the polynomial being fitted has degree m ¼ 2, 3, 4 are
also given. The issues of stability, rms error, track initiation, and equivalence to
the growing-memory filters are also covered.
In Chapter 8 the polynomial description of the target dynamics is given in
terms of a linear vector differential equation. This equation is shown to be very
useful for obtaining the transition matrix for the target dynamics by either
numerical integration or a power series in terms of the matrix coefficient of the
differential equation.
In Chapter 9 the Bayes filter is derived (Problem 9.4-1) and in turn from it
the Kalman filter is again derived (Problem 9.3-1). In Chapters 10 through 14
the voltage least-squares algorithms are revisited. The issues of sensitivity to
computer round-off error in obtaining the inverse of a matrix are elaborated in
Section 10.1. Section 10.2 explains physically why the voltage least-squares
algorithm (square-root processing) reduces the sensitivity to computer roundoff
errors. Chapter 11 describes the Givens orthonormal transformation voltage
algorithm. The massively parallel systolic array implementation of the Givens
algorithm is detailed in Section 11.3. This implementation makes use of the
CORDIC algorithm used in the Hewlett-Packard hand calculators for
trigonometric computations.
The Householder orthonormal transformation voltage algorithm is described
in Chapter 12. The Gram?"Schmidt orthonormal transformation voltage
algorithm is revisited in Chapter 13, with classical and modified versions
explained in simple terms. These different voltage least-squares algorithms are
compared in Section 14.1 and to QR decomposition in Section 14.2. A recursive
version is developed in Section 14.3. Section 14.4 relates these voltage-
processing orthonormal transformation methods to the DOLP approach used in
Section 5.3 for obtaining a polynomial fit to data. The two methods are shown
to be essentially identical. The square-root Kalman filter, which is less sensitive
to round-off errors, is discussed in Section 14.5.
Up until now the deterministic part of the target model was assumed to be
time invariant. For example, if a polynomial fit of degree m was used for the
target dynamics, the coefficients of this polynomial fit are constant with time.
Chapter 15 treats the case of time-varying target dynamics.
The Kalman and Bayes filters developed up until now depend on the
observation scheme being linear. This is not always the situation. For example,
if we are measuring the target range R and azimuth angle  but keep track of the
target using the east-north x, y coordinates of the target with a Kalman filter,
then errors in the measurement of R and  are not linearly related to the
resulting error in x and y because
x ¼ R cos  ð1z
and
y ¼ R sin  ð2z
where  is the target angle measured relative to the x axis. Section 16.2 shows
how to simply handle this situation. Basically what is done is to linearize
Eqs. (1) and (2) by using the first terms of a Taylor expansion of the inverse
equations to (1) and (2) which are
R ¼ ffixffiffi2ffiffiffiffiffiffiffiffiffiffiffiffi p þ y2 ð3z
 ¼ tan
y
x ð4z
Similarly the equations of motion have to be linear to apply the Kalman?"
Bayes filters. Section 16.3 describes how a nonlinear equation of motion can be
linearized, again by using the first term of a Taylor expansion of the nonlinear
equations of motion. The important example of linearization of the nonlinear
observation equations obtained when observing a target in spherical coordinates
(R, , ) while tracking it in rectangular (x, y, z) coordinates is given. The
example of the linearization of the nonlinear target dynamics equations
obtained when tracking a projectile in the atmosphere is detailed. Atmospheric
drag on the projectile is factored in.
In Chapter 17 the technique for linearizing the nonlinear observation
equations and dynamics target equations in order to apply the recursive Kalman
and Bayes filters is detailed. The application of these linearizations to a
nonlinear problem in order to handle the Kalman filter is called the extended
Kalman filter. It is also the filter Swerling originally developed (without the
target process noise). The Chapter 16 application of the tracking of a ballistic
projectile through the atmosphere is again used as an example.
The form of the Kalman filter given in Kalman?Ts original paper is different
from the forms given up until now. In Chapter 18 the form given until now is
related to the form given by Kalman. In ad***ion, some of the fundamental
results given in Kalman?Ts original paper are summarized here.
ELI BROOKNER
Sudbury, MA
January 1998
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6) Fibre optic communication systems
Preface
Since the publication of the first e***ion of this book in 1992, the state of the art of
fiber-optic communication systems has advanced dramatically despite the relatively
short period of only 10 years between the first and third e***ions. For example, the
highest capacity of commercial fiber-optic links available in 1992 was only 2.5 Gb/s.
A mere 4 years later, the wavelength-division-multiplexed (WDM) systems with the
total capacity of 40 Gb/s became available commercially. By 2001, the capacity of
commercial WDM systems exceeded 1.6 Tb/s, and the prospect of lightwave systems
operating at 3.2 Tb/s or more were in sight. During the last 2 years, the capacity
of transoceanic lightwave systems installed worldwide has exploded. Moreover, several
other undersea networks were in the construction phase in December 2001. A
global network covering 250,000 km with a capacity of 2.56 Tb/s (64 WDM channels
at 10 Gb/s over 4 fiber pairs) is scheduled to be operational in 2002. Several conference
papers presented in 2001 have demonstrated that lightwave systems operating at a bit
rate of more than 10 Tb/s are within reach. Just a few years ago it was unimaginable
that lightwave systems would approach the capacity of even 1 Tb/s by 2001.
The second e***ion of this book appeared in 1997. It has been well received by
the scientific community involved with lightwave technology. Because of the rapid advances
that have occurred over the last 5 years, the publisher and I deemed it necessary
to bring out the third e***ion if the book were to continue to provide a comprehensive
and up-to-date account of fiber-optic communication systems. The result is in your
hands. The primary objective of the book remains the same. Specifically, it should be
able to serve both as a textbook and a reference monograph. For this reason, the emphasis
is on the physical understanding, but the engineering aspects are also discussed
throughout the text.
Because of the large amount of material that needed to be added to provide comprehensive
coverage, the book size has increased considerably compared with the first
e***ion. Although all chapters have been updated, the major changes have occurred in
Chapters 6?"9. I have taken this opportunity to rearrange the material such that it is better
suited for a two-semester course on optical communications. Chapters 1?"5 provide
the basic foundation while Chapters 6?"10 cover the issues related to the design of advanced
lightwave systems. More specifically, after the introduction of the elementary
concepts in Chapter 1, Chapters 2?"4 are devoted to the three primary components of a
fiber-optic communications?"optical fibers, optical transmitters, and optical receivers.
Chapter 5 then focuses on the system design issues. Chapters 6 and 7 are devoted to
the advanced techniques used for the management of fiber losses and chromatic dis-
persion, respectively. Chapter 8 focuses on the use of wavelength- and time-division
multiplexing techniques for optical networks. Code-division multiplexing is also a part
of this chapter. The use of optical solitons for fiber-optic systems is discussed in Chapter
9. Coherent lightwave systems are now covered in the last chapter. More than 30%
of the material in Chapter 6?"9 is new because of the rapid development of the WDM
technology over the last 5 years. The contents of the book reflect the state of the art of
lightwave transmission systems in 2001.
The primary role of this book is as a graduate-level textbook in the field of optical
communications. An attempt is made to include as much recent material as possible
so that students are exposed to the recent advances in this exciting field. The book can
also serve as a reference text for researchers already engaged in or wishing to enter
the field of optical fiber communications. The reference list at the end of each chapter
is more elaborate than what is common for a typical textbook. The listing of recent
research papers should be useful for researchers using this book as a reference. At
the same time, students can benefit from it if they are assigned problems requiring
reading of the original research papers. A set of problems is included at the end of
each chapter to help both the teacher and the student. Although written primarily for
graduate students, the book can also be used for an undergraduate course at the senior
level with an appropriate selection of topics. Parts of the book can be used for several
other related courses. For example, Chapter 2 can be used for a course on optical
waveguides, and Chapter 3 can be useful for a course on optoelectronics.
Many universities in the United States and elsewhere offer a course on optical communications
as a part of their curriculum in electrical engineering, physics, or optics. I
have taught such a course since 1989 to the graduate students of the Institute of Optics,
and this book indeed grew out of my lecture notes. I am aware that it is used as a textbook
by many instructors worldwide?"a fact that gives me immense satisfaction. I am
acutely aware of a problem that is a side effect of an enlarged revised e***ion. How can
a teacher fit all this material in a one-semester course on optical communications? I
have to struggle with the same question. In fact, it is impossible to cover the entire book
in one semester. The best solution is to offer a two-semester course covering Chapters
1 through 5 during the first semester, leaving the remainder for the second semester.
However, not many universities may have the luxury of offering a two-semester course
on optical communications. The book can be used for a one-semester course provided
that the instructor makes a selection of topics. For example, Chapter 3 can be skipped
if the students have taken a laser course previously. If only parts of Chapters 6 through
10 are covered to provide students a glimpse of the recent advances, the material can
fit in a single one-semester course offered either at the senior level for undergraduates
or to graduate students.
This e***ion of the book features a compact disk (CD) on the back cover provided
by the Optiwave Corporation. The CD contains a state-of-the art software package
suitable for designing modern lightwave systems. It also contains ad***ional problems
for each chapter that can be solved by using the software package. Appendix E provides
more details about the software and the problems. It is my hope that the CD will help
to train the students and will prepare them better for an industrial job.
A large number of persons have contributed to this book either directly or indirectly.
It is impossible to mention all of them by name. I thank my graduate students and the
students who took my course on optical communication systems and helped improve
my class notes through their questions and comments. Thanks are due to many instructors
who not only have adopted this book as a textbook for their courses but have also
pointed out the misprints in previous e***ions, and thus have helped me in improving
the book. I am grateful to my colleagues at the Institute of Optics for numerous discussions
and for providing a cordial and productive atmosphere. I appreciated the help
of Karen Rolfe, who typed the first e***ion of this book and made numerous revisions
with a smile. Last, but not least, I thank my wife, Anne, and my daughters, Sipra,
Caroline, and Claire, for understanding why I needed to spend many weekends on the
book instead of spending time with them.
Govind P. Agrawal
Rochester, NY
December 2001
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