Articoli correlati a Computational Signal Processing with Wavelets
Sinossi
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
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