This work provides a guide to wavelet-based signal processing with fractals. Coverage includes wavelet theory and transformations, statistically self-similar signals, deterministically self-similar signals, fractal modulation and linear self-similar systems.
FREE Two-Day Shipping for students on millions of items. Learn more |
| |||||||||||||||
Book Description
Editorial Reviews
From the Inside Flap
SIGNAL PROCESSING WITH FRACTALS: A WAVELET-BASED APPROACH
PREFACE
In recent years, the mathematics of fractal geometry has generated much excitement within the engineering community among those seeking broad new and more realistic classes of models for wide-ranging applications. This optimism has arisen out of the perspective that many natural and man-made phenomena around which we engineer our world are much better described through an inherently irregular "fractal" geometry than by the traditional regular Euclidean geometry. However, despite the apparent promise of fractal geometry in providing useful solutions to important engineering problems, progress in applying these models in such applications has been slower than expected. To a large degree, this has been due to the lack of an adequate set of convenient and efficient engineering-oriented mathematical tools for exploiting these ideas.
From this perspective, the recent emergence of powerful multiscale signal representations in general and wavelet basis representations in particular has been particularly timely. Indeed, out of this theory arise highly natural and extremely useful representations for a variety of important fractal phenomena. This book presents both the development of these new techniques as well as their application to a number of fundamental problems of interest to signal processing and communications engineers.
In particular, this book develops a unified, wavelet-based framework for efficiently synthesizing, analyzing, and processing several broad classes of fractal signals. For example, efficient and practical algorithms for solving some important problems of optimal estimation, detection and classification involving fractals are developed using this framework. As another example, novel and practical signal processing techniques for exploiting fractal signals as information-bearing waveforms in efficient communication systems are also developed.
In preparing this monograph, there has been an effort to make this material as accessible as possible to graduate students and practicing professionals alike. In particular, no prior familiarity with either fractal geometry or wavelets is assumed of the reader. In fact, Chapter 2 is a fully self-contained primer on the relevant wavelet theory. However, to keep the treatment compact, the reader is assumed to possess a basic familiarity with continuous- and discrete-time signals and systems, with stochastic processes, detection and estimation theory, and with the geometry of linear vector spaces and linear algebra. Wherever possible, concepts and intuition are emphasized over mathematical rigor. Nevertheless, unavoidably some measure of that ill-defined but important quantity "mathematical sophistication" is also assumed of the reader.
As a final remark, it should be emphasized that no attempt has been made to describe many of the exciting parallel developments taking place within this active field of research. While I have tried whenever possible to point out those developments most closely related to the material covered, this monograph should not be interpreted as a comprehensive treatise. In fact, rather than a retrospective on a mature topic, it is hoped that this book will serve as a catalyst, stimulating further development in both the theory and applications of the exciting, powerful, and increasingly important ideas in this area.
There are many people who contributed to this book becoming a reality, and to them I am extremely grateful. Alan Oppenheim contributed generously to the development of the ideas in the original work, and strongly encouraged and supported the book. Alan Willsky and William Siebert also provided valuable technical input and suggestions during the development of this material. Henrique Malvar, Jelena Kovacevic, and Jerome Shapiro all read the complete manuscript very carefully and provided a great deal of helpful and detailed feedback. Warren Lam and Haralabos Papadopoulos also did an exceptional job proofreading and helping to debug the manuscript. Jon Sjogren and his program at the Air Force Office of Scientific Research played a critical role in supporting and encouraging the research out of which this book grew. And finally, Karen Gettman at Prentice-Hall was instrumental in steering the project to fruition.
Gregory W. Wornell
Cambridge, Massachusetts
PREFACE
In recent years, the mathematics of fractal geometry has generated much excitement within the engineering community among those seeking broad new and more realistic classes of models for wide-ranging applications. This optimism has arisen out of the perspective that many natural and man-made phenomena around which we engineer our world are much better described through an inherently irregular "fractal" geometry than by the traditional regular Euclidean geometry. However, despite the apparent promise of fractal geometry in providing useful solutions to important engineering problems, progress in applying these models in such applications has been slower than expected. To a large degree, this has been due to the lack of an adequate set of convenient and efficient engineering-oriented mathematical tools for exploiting these ideas.
From this perspective, the recent emergence of powerful multiscale signal representations in general and wavelet basis representations in particular has been particularly timely. Indeed, out of this theory arise highly natural and extremely useful representations for a variety of important fractal phenomena. This book presents both the development of these new techniques as well as their application to a number of fundamental problems of interest to signal processing and communications engineers.
In particular, this book develops a unified, wavelet-based framework for efficiently synthesizing, analyzing, and processing several broad classes of fractal signals. For example, efficient and practical algorithms for solving some important problems of optimal estimation, detection and classification involving fractals are developed using this framework. As another example, novel and practical signal processing techniques for exploiting fractal signals as information-bearing waveforms in efficient communication systems are also developed.
In preparing this monograph, there has been an effort to make this material as accessible as possible to graduate students and practicing professionals alike. In particular, no prior familiarity with either fractal geometry or wavelets is assumed of the reader. In fact, Chapter 2 is a fully self-contained primer on the relevant wavelet theory. However, to keep the treatment compact, the reader is assumed to possess a basic familiarity with continuous- and discrete-time signals and systems, with stochastic processes, detection and estimation theory, and with the geometry of linear vector spaces and linear algebra. Wherever possible, concepts and intuition are emphasized over mathematical rigor. Nevertheless, unavoidably some measure of that ill-defined but important quantity "mathematical sophistication" is also assumed of the reader.
As a final remark, it should be emphasized that no attempt has been made to describe many of the exciting parallel developments taking place within this active field of research. While I have tried whenever possible to point out those developments most closely related to the material covered, this monograph should not be interpreted as a comprehensive treatise. In fact, rather than a retrospective on a mature topic, it is hoped that this book will serve as a catalyst, stimulating further development in both the theory and applications of the exciting, powerful, and increasingly important ideas in this area.
There are many people who contributed to this book becoming a reality, and to them I am extremely grateful. Alan Oppenheim contributed generously to the development of the ideas in the original work, and strongly encouraged and supported the book. Alan Willsky and William Siebert also provided valuable technical input and suggestions during the development of this material. Henrique Malvar, Jelena Kovacevic, and Jerome Shapiro all read the complete manuscript very carefully and provided a great deal of helpful and detailed feedback. Warren Lam and Haralabos Papadopoulos also did an exceptional job proofreading and helping to debug the manuscript. Jon Sjogren and his program at the Air Force Office of Scientific Research played a critical role in supporting and encouraging the research out of which this book grew. And finally, Karen Gettman at Prentice-Hall was instrumental in steering the project to fruition.
Gregory W. Wornell
Cambridge, Massachusetts
From the Back Cover
Fractal signals, derived from wavelet theory, are ideally suited for use in many engineering applications, ranging from communications to remote sensing. This book provides an introduction to wavelet theory from a signal processing perspective, and details fractal signals and a collection of practical wavelet-based techniques for representing and manipulating fractal signals in various applications. Covers wavelet theory and transformations, statistically self-similar signals, detection and estimation with 1/f processes, deterministically self-similar signals, fractal modulation, and linear self-similar systems. For industrial research scientists and practicing engineers within the signal processing and communications communities.
Product Details
Would you like to update product info or give feedback on images?
|
More About the Author
Discover books, learn about writers, read author blogs, and more.
Customer Reviews
|
Share your thoughts with other customers:
|
||||||||||||||||||||||
|
Most Helpful Customer Reviews
2 of 2 people found the following review helpful:
5.0 out of 5 stars
Graduate Student,
By A Customer
This review is from: Signal Processing with Fractals: A Wavelet Based Approach (Textbook Binding)
This is a very good book that deals fractal self similar stocastic processes,linear scale invariant systems and wavelet theory. This book covers synthesizing,estimation methods and modeling of self similar processes and their broad field of applications like communication and signal processing. These areas are extensively covered and the book is organized in a text book manner.Overall I found this book very useful and this is one of the best books I have read.
5.0 out of 5 stars
signals/math/engineering,
By Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews (VINE VOICE) (REAL NAME)
This review is from: Signal Processing with Fractals: A Wavelet Based Approach (Textbook Binding)
Observations of signals, time series in telecommunication, or pictures in medical imaging reveal selfsimilarity, i.e., the signal/picture looks the same as the scale varies. Hence, the name "fractal" ! Pictures in the Intro make this scale-similarity visually apparent. But it is made precise in mathematical statistics, and the book further makes the connection to the engineering of signal/image processing. That's a main point of the book!Selfsimilar processes are stochastic processes that are invariant in distribution under suitable scaling of time and/or space. Fractional Brownian motion or Brownian sheets are the best known of these. They were found by Kolmogorov long ago, but made popular by Mandelbrot and Ness in 1968. We now speak of 1/f processes. More recent use of wavelet bases in telecommunication and in stochastic integration has revived interest. Other even more recent applications include finance. While the underlying idea behind all of this is quite simple, and can be traced back to Kolmogorov in the 1930ties, it is only recently, with the advent of wavelet methods, that the *computational* power has been better appreciated. The idea is analogous to that of random Fourier series: Instead of treating the Fourier coefficients as random variables, it is now wavelet coefficients that are analyzed statistically. Since wavelets have computational advantages, it is not surprising that the engineering applications abound. This little book is well written, and should be attractive both to members of the math community and to engineers. Mathematicians will be pleased to note that wavelet analysis now brings *Hilbert space theoretic features* of the subject back to the fore. Amusingly, this was in fact a dominant feature which motivated both A. N. Kolmogorov and Norbert Wiener in the early days; e.g., curves in Hilbert space. The author is an authority in the field, and his book brings out beautifully the highpoints of the subject. I further expect that the book will go over well in the classroom; nice summaries at the end of each chapter! (Exercises would have helped though!) The book will help bridge mathematical analysis, probability, and engineering. Engineers may like that proofs are relegated to later in the book. Mathematicians will be pleased with having the proofs, and with the clarity of their presentation. Reviewed by Palle Jorgensen, July, 2005.
Share your thoughts with other customers: Create your own review
|
|
Tags Customers Associate with This Product(What's this?)Click on a tag to find related items, discussions, and people.
|
Sell a Digital Version of This Book in the Kindle Store
If you are a publisher or author and hold the digital rights to a book, you can sell a digital version of it in our Kindle Store.
Learn more
Customer Discussions
|
This product's forum
Active discussions in related forums
Search Customer Discussions
|
Related forums
|
