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Contents
Contents
Wavelets and Filter Banks Course Notes
Copyright ©Dr. W. J. Phillips
January 9, 2003
Contents
1. Analysis and Synthesis of Signals
2. Time-Frequency Analysis
2.1 The Short Time Fourier Transform
2.2 The spectrogram
2.3 An Orthgonal Basis of Functions
3. Time-Scale Analysis
3.1 The Continuous Wavelet Transform
3.2 Comparision with STFT
3.3 The Scalogram
3.4 Examples of Wavelets
3.5 Analysis and Synthesis with Wavelets
3.6 The Haar Wavelet
4. Multiresolution Analysis
4.1 The Scaling Function
4.2 The Discrete Wavelet Transform
5. Filter Banks and the Discrete Wavelet Transform
5.1 Analysis: From Fine Scale to Coarser Scale
5.1.1 Filtering and Downsampling
5.1.2 The One-Stage Analysis Filter Bank
5.1.3 The Analysis Filter Bank
5.2 Synthesis: From Course Scale to Fine Scale
5.2.1 Upsampling and Filtering
5.2.2 The One-Stage Synthesis Filter Bank
5.2.3 Perfect Reconstruction Filter Bank
5.2.4 The Synthesis Filter Bank
5.2.5 Approximations and Details
5.3 Numerical Complexity of the Discrete Wavelet Transform
5.4 Matlab Examples
5.4.1 One-Stage Perfect Reconstruction
5.4.2 Approximations and Details
5.4.3 A Useful Function
5.5 Initialization of the Discrete Wavelet Transform
6. Properties of the Filters, and the Scale and Wavelet Functions
6.1 Double Shift Orthogonality of the Filters
6.2 Frequency Domain Formulas
6.3 Support of the Scale Function
6.4 The Cascade Algorithm
7. Designing Wavelets
7.1 Short Filters
7.1.1 Length 2 Filter
7.1.2 Length 4 Filter
7.1.3 Length 6 Filter
7.2 K-Regular Scaling Filters
7.2.1 The db2 Wavelet
7.2.2 The db3 Wavelet
7.3 Characterizing K-Regular Filters
7.4 The Daubechies Maximally Flat Polynomial
7.4.1 Factoring the Daubechie Maximally Flat Polynomial
7.5 Coiflets
7.5.1 Coif1
7.5.2 Coif2
About this document ...
Dr. W. J. Phillips
2003-04-03
