This unique resource examines the conceptual, computational, and practical aspects of applied signal processing using wavelets. With this book, readers will understand and be able to use the power and utility of new wavelet methods in science and engineering problems and analysis. The text is written in a clear, accessible style avoiding unnecessary abstractions and details. From a computational perspective, wavelet signal processing algorithms are presented and applied to signal compression, noise suppression, and signal identification. Numerical illustrations of these computational techniques are further provided with interactive software (MATLAB code) that is available on the world wide web. Topics and Features: * Continuous wavelet and Gabor transforms * Frame-based theory of discretization and reconstruction of analog signals is developed * New and efficient "overcomplete" wavelet transform is introduced and applied * Numerical illustrations with an object-oriented computational perspective using the Wavelet Signal Processing Workstation (MATLAB code) available This book is an excellent resource for information and computational tools needed to use wavelets in many types of signal processing problems. Graduates, professionals, and practitioners in engineering, computer science, geophysics, and applied mathematics will benefit from using the book and software tools.
"This book provides an expository treatment of wavelets from a signal processing perspective. The focus is on the expansion of signals in overcomplete wavelet systems. All illustrations of the theory are generated in the framework of the Matlab toolbox wavelet signal processing workstation (WSPW) made publicly available by the author.... The last chapter is a manual for WSPW, and the whole book serves as an extended manual." --Mathematical Reviews "This book provides a bridge between theory and practice of wavelet-based signal processing and is written for both students and professionals. A solid mathematical foundation is given in the beginning chapters [1-6].... Several applications of wavelet-based signal processing including noise suppression, signal compression, signal identification and digital communication are presented in Chapter 7. Chapter 8 gives numerical illustrations and examples of wavelet methods using MATLAB 5. The accomanying MATLAB-based software is available on the world wide web. Every chapter of the book contains a collection of exercises." --Zentralblatt MATH "A self-contained text that is theoretically rigorous while maintaining contact with interesting applications. A particularly noteworthy topic...is a class of 'overcomplete wavelets'. These functions are not orthonormal and they lead to many useful results." --Journal of Mathematical Psychology
Overview For over a decade now, wavelets have been and continue to be an evolving subject of intense interest. Their allure in signal processing is due to many factors, not the least of which is that they offer an intuitively satisfying view of signals as being composed of little pieces of wa'ues. Making this concept mathematically precise has resulted in a deep and sophisticated wavelet theory that has seemingly limitless applications. This book and its supplementary hands-on electronic: component are meant to appeal to both students and professionals. Mathematics and en- gineering students at the undergraduate and graduate levels will benefit greatly from the introductory treatment of the subject. Professionals and advanced students will find the overcomplete approach to signal represen- tation and processing of great value. In all cases the electronic component of the proposed work greatly enhances its appeal by providing interactive numerical illustrations. A main goal is to provide a bridge between the theory and practice of wavelet-based signal processing. Intended to give the reader a balanced look at the subject, this book emphasizes both theoretical and practical issues of wavelet processing. A great deal of exposition is given in the beginning chapters and is meant to give the reader a firm understanding of the basics of the discrete and continuous wavelet transforms and their relationship. Later chapters promote the idea that overcomplete systems of wavelets are a rich and largely unexplored area that have demonstrable benefits to offer in many applications.
Introduction * Mathematical Preliminaries * Signal Representation and Frames * Continuous Wavelet and Gabor Transforms * Discrete Wavelet Transform * Overcomplete Wavelet Transform * Wavelet Signal Processing * Object-Oriented Wavelet Analysis with MATLAB 5 * References * Index
1 Introduction
1.1 Motivation and Objectives
1.2 Core Material and Development
1.3 Hybrid Media Components
1.4 Signal Processing Perspective
1.4.1 Analog Signals
1.4.2 Digital Processing of Analog Signals
1.4.3 Time-Frequency Limitedness
2 Mathematical Preliminaries
2.1 Basic Symbols and Notation
2.2 Basic Concepts
2.2.1 Norm
2.2.2 Inner Product
2.2.3 Convergence
2.2.4 Hilbert Spaces
2.3 Basic Spaces
2.3.1 Bounded Functions
2.3.2 Absolutely Integrable Functions
2.3.3 Finite Energy Functions
2.3.4 Finite Energy Periodic Functions
2.3.5 Time-Frequency Concentrated Functions
2.3.6 Finite Energy Sequences
2.3.7 Bandlimited Functions
2.3.8 Hardy Spaces
2.4 Operators
2.4.1 Bounded Linear Operators
2.4.2 Properties
2.4.3 Useful Unitary Operators
2.5 Bases and Completeness in Hilbert Space
2.6 Fourier Transforms
2.6.1 Continuous Time Fourier Transform
2.6.2 Continuous Time-Periodic Fourier Transform
2.6.3 Discrete Time Fourier Transform
2.6.4 Discrete Fourier Transform
2.6.5 Fourier Dual Spaces
2.7 Linear Filters
2.7.1 Continuous Filters and Fourier Transforms
2.7.2 Discrete Filters and Z-Transforms
2.8 Analog Signals and Discretization
2.8.1 Classical Sampling Theorem
2.8.2 What Can Be Computed Exactly?
Problems
3 Signal Representation and Frames
3.1 Inner Product Representation (Atomic Decomposition)
3.2 Orthonormal Bases
3.2.1 Parseval and Plancherel
3.2.2 Reconstruction
3.2.3 Examples
3.3 Riesz Bases
3.3.1 Reconstruction
3.3.2 Examples
3.4 General Frames
3.4.1 Basic Frame Theory
3.4.2 Frame Representation
3.4.3 Frame Correlation and Pseudo-Inverse
3.4.4 Pseudo-Inverse
3.4.5 Best Frame Bounds
3.4.6 Duality
3.4.7 Iterative Reconstruction
Problems
4 Continuous Wavelet and Gabor Transforms
4.1 What is a Wavelet?
4.2 Example Wavelets
4.2.1 Haar Wavelet
4.2.2 Shannon Wavelet
4.2.3 Frequency B-spline Wavelets
4.2.4 Morlet Wavelet
4.2.5 Time-Frequency Tradeoffs
4.3 Continuous Wavelet Transform
4.3.1 Definition
4.3.2 Properties
4.4 Inverse Wavelet Transform
4.4.1 The Idea Behind the Inverse
4.4.2 Derivation for L 2 ( R )
4.4.3 Analytic Signals
4.4.4 Admissibility
4.5 Continuous Gabor Transform
4.5.1 Definition
4.5.2 Inverse Gabor Transform
4.6 Unified Representation and Groups
4.6.1 Groups
4.6.2 Weighted Spaces
4.6.3 Representation
4.6.4 Reproducing Kernel
4.6.5 Group Representation Transform
Problems
5 Discrete Wavelet Transform
5.1 Discretization of the CWT
5.2 Multiresolution Analysis
5.2.1 Multiresolution Design
5.2.2 Resolution and Dilation Invariance
5.2.3 Definition
5.3 Multiresolution Representation
5.3.1 Projection
5.3.2 Fourier Transforms
5.3.3 Between Scale Relations
5.3.4 Haar MRA
5.4 Orthonormal Wavelet Bases
5.4.1 Characterizing W 0
5.4.2 Wavelet Construction
5.4.3 The Scaling Function
5.5 Compactly Supported (Daubechies) Wavelets
5.5.1 Main Idea
5.5.2 Trigonometric Half-Band Filters
5.5.3 Examples
5.6 Fast Wavelet Transform Algorithm
5.6.1 Filter Bank Decomposition of the Identity
5.6.2 Down- and Up-Sampling
5.6.3 Examples
Problems
6 Overcomplete Wavelet Transform
6.1 Discretization of the CWT Revisited
6.1.1 Definition
6.1.2 Semilog Regular Time-Scale Sampling
6.1.3 Invertibility
6.1.4 Overcompleteness and Redundancy
6.2 Filter Bank Implementation
6.2.1 Analysis
6.2.2 Synthesis
6.3 Time-Frequency Localization and Wavelet Design
6.3.1 The Uncertainty Principle
6.3.2 Parametric Bandlimited Wavelets
6.4 OCWT Examples
6.5 Irregular Sampling and Frames
6.5.1 Paley-Wiener Frames
6.5.2 Wavelet Frames
6.5.3 Irregular Sampling in the Time-Scale Domain
Problems
7 Wavelet Signal Processing
7.1 Noise Suppression
7.1.1 Noise Domains
7.1.2 Problem
7.1.3 Approach
7.1.4 Coherence
7.1.5 Frame Localization and Coherence
7.1.6 Reconstruction Error from Threshold OCWTs
7.1.7 Numerical Experiment
7.1.8 Coefficient Noise and Overcompleteness
7.1.9 Measuring Noise (SNR)
7.2 Compression
7.2.1 Problem
7.2.2 Performance Measures
7.2.3 Approach
7.2.4 Numerical Experiment
7.2.5 Remarks
7.3 Digital Communication
7.3.1 Problem
7.3.2 Objectives
7.3.3 Approach
7.3.4 Performance Measures
7.3.5 Computer Implementation
7.3.6 Wavelet Communication Testbed
7.3.7 Numerical Experiments
7.3.8 Remarks
7.4 Identification
7.4.1 Approach
7.4.2 Performance Indicators
7.4.3 Numerical Experiment
7.4.4 Remarks
7.5 Conclusion
Problems
8 Object-Oriented Wavelet Analysis with MATLAB 5
8.1 Wavelet Signal Processing Workstation
8.2 MATLAB Coding
8.2.1 Object-Oriented Programming
8.2.2 Designed Classes
8.3 The sampled_signal Methods
8.3.1 Class Construction
8.3.2 sampled_signal Methods
8.4 Wavelet Transform Implementation
8.4.1 Overcomplete Wavelet Transform (OCWT)
8.4.2 GUI for Filter Bank Specification
8.5 The wavelet Object
8.5.1 Class Reconstruction
8.5.2 wavelet Methods
8.6 Processing Example
8.6.1 Forward OCWT
8.6.2 Inverse OCWT
8.7 Supporting Functions and Globals
8.7.1 Frequency and Time Units
8.7.2 Graphical
8.7.3 Wavelet
References
Index