"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

• 1 Introduction.- 1.1 Motivation and Objectives.- 1.2 Core Material and Development.- 1.3 Hybrid Media Components.- 1.4 Signal Processing Perspective.- 2 Mathematical Preliminaries.- 2.1 Basic Symbols and Notation.- 2.2 Basic Concepts.- 2.3 Basic Spaces.- 2.4 Operators.- 2.5 Bases and Completeness in Hilbert Space.- 2.6 Fourier Transforms.- 2.7 Linear Filters.- 2.8 Analog Signals and Discretization.- Problems.- 3 Signal Representation and Frames.- 3.1 Inner Product Representation (Atomic Decomposition).- 3.2 Orthonormal Bases.- 3.3 Riesz Bases.- 3.4 General Frames.- Problems.- 4 Continuous Wavelet and Gabor Transforms.- 4.1 What Is a Wavelet?.- 4.2 Example Wavelets.- 4.3 Continuous Wavelet Transform.- 4.4 Inverse Wavelet Transform.- 4.5 Continuous Gabor Transform.- 4.6 Unified Representation and Groups.- Problems.- 5 Discrete Wavelet Transform.- 5.1 Discretization of the CWT.- 5.2 Multiresolution Analysis.- 5.3 Multiresolution Representation.- 5.4 Orthonormal Wavelet Bases.- 5.5 Compactly Supported (Daubechies) Wavelets.- 5.6 Fast Wavelet Transform Algorithm.- Problems.- 6 Overcomplete Wavelet Transform.- 6.1 Discretization of the CWT Revisited.- 6.2 Filter Bank Implementation.- 6.3 Time-Frequency Localization and Wavelet Design.- 6.4 OCWT Examples.- 6.5 Irregular Sampling and Frames.- Problems.- 7 Wavelet Signal Processing.- 7.1 Noise Suppression.- 7.2 Compression.- 7.3 Digital Communication.- 7.4 Identification.- 7.5 Conclusion.- Problems.- 8 Object-Oriented Wavelet Analysis with MATLAB 5.- 8.1 Wavelet Signal Processing Workstation.- 8.2 MATLAB Coding.- 8.3 The sampled_signal Object.- 8.4 Wavelet Transform Implementation.- 8.5 The wavelet Object.- 8.6 Processing Example.- 8.7 Supporting Functions and Globals.- References.

"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

• 1 Introduction.- 1.1 Motivation and Objectives.- 1.2 Core Material and Development.- 1.3 Hybrid Media Components.- 1.4 Signal Processing Perspective.- 2 Mathematical Preliminaries.- 2.1 Basic Symbols and Notation.- 2.2 Basic Concepts.- 2.3 Basic Spaces.- 2.4 Operators.- 2.5 Bases and Completeness in Hilbert Space.- 2.6 Fourier Transforms.- 2.7 Linear Filters.- 2.8 Analog Signals and Discretization.- Problems.- 3 Signal Representation and Frames.- 3.1 Inner Product Representation (Atomic Decomposition).- 3.2 Orthonormal Bases.- 3.3 Riesz Bases.- 3.4 General Frames.- Problems.- 4 Continuous Wavelet and Gabor Transforms.- 4.1 What Is a Wavelet?.- 4.2 Example Wavelets.- 4.3 Continuous Wavelet Transform.- 4.4 Inverse Wavelet Transform.- 4.5 Continuous Gabor Transform.- 4.6 Unified Representation and Groups.- Problems.- 5 Discrete Wavelet Transform.- 5.1 Discretization of the CWT.- 5.2 Multiresolution Analysis.- 5.3 Multiresolution Representation.- 5.4 Orthonormal Wavelet Bases.- 5.5 Compactly Supported (Daubechies) Wavelets.- 5.6 Fast Wavelet Transform Algorithm.- Problems.- 6 Overcomplete Wavelet Transform.- 6.1 Discretization of the CWT Revisited.- 6.2 Filter Bank Implementation.- 6.3 Time-Frequency Localization and Wavelet Design.- 6.4 OCWT Examples.- 6.5 Irregular Sampling and Frames.- Problems.- 7 Wavelet Signal Processing.- 7.1 Noise Suppression.- 7.2 Compression.- 7.3 Digital Communication.- 7.4 Identification.- 7.5 Conclusion.- Problems.- 8 Object-Oriented Wavelet Analysis with MATLAB 5.- 8.1 Wavelet Signal Processing Workstation.- 8.2 MATLAB Coding.- 8.3 The sampled_signal Object.- 8.4 Wavelet Transform Implementation.- 8.5 The wavelet Object.- 8.6 Processing Example.- 8.7 Supporting Functions and Globals.- References.