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.

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 T