### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Text

Strang, and Nguyen. *Wavelets and Filter Banks.* Wellesley-Cambridge Press, 1997.

### Course Structure

The course will consist of lectures, homework assignments and a project on a topic related to the student’s area of interest. We will aim for the right balance of theory and “applications”. The course has no specific prerequisites, although a basic knowledge of Fourier transforms is recommended. We start with time-invariant filters and basic wavelets. The text gives an overall perspective of the field – which has grown with amazing speed. The topics will include

- Analysis of Filter Banks and Wavelets
- Design Methods
- Applications
- Hands-on Experience with Software

These four key areas will be developed in detail.

#### Analysis

- Multirate Signal Processing: Filtering, Decimation, Polyphase, Perfect Reconstruction and Aliasing Removal.
- Matrix Analysis: Toeplitz Matrices and Fast Algorithms.
- Wavelet Transform: Pyramid and Cascade Algorithms, Daubechies Wavelets, Orthogonal and Biorthogonal Wavelets, Smoothness, Approximation, Boundary Filters and Wavelets, Time-Frequency and Time-Scale Analysis, Second-Generation Wavelets.

#### Design Methods

- Spectral Factorization, Cosine-Modulated Filter Banks, Lattice Structure, Ladder Structure (Lifting.)

#### Applications

- Audio and Image Compression, Quantization Effects, Digital Communication and Multicarrier Modulation, Transmultiplexers, Text-Image Compression: Lossy and Lossless, Medical Imaging and Scientific Visualization, Edge Detection and Feature Extraction, Seismic Signal Analysis, Geometric Modeling, Matrix Preconditioning, Multiscale Methods for Partial Differential Equations and Integral Equations.

#### Simulation Software

- MATLAB
^{®}Wavelet Toolbox, Software for Filter Design, Signal Analysis, Image Compression, PDEs, Wavelet Transforms on Complex Geometrical Shapes.

We encourage you to learn about wavelets and their applications.

Multiresolution representation of a complex shape. Courtesy of Igor

Guskov, University of Michigan. Used with permission.