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Wavelets and Filter Banks Course Notes
Copyright ©Dr. W. J. Phillips
January 9, 2003
- 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
- 5.1 Analysis: From Fine Scale to Coarser Scale
- 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
- 7.1 Short Filters
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