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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
• 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