Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 271: Applied and Computational Harmonic Analysis
Approved: 2009-11-01, Naoki Saito and Thomas Strohmer
Units/Lecture:
Winter, alternate years; 4 units; lecture/term paper or discussion
Suggested Textbook: (actual textbook varies by instructor; check your
instructor)
No required textbook. Instructor will supply info about some key articles to read during the course. See below for optional reference books.
Prerequisites:
(MAT 127B or MAT 201C) and (MAT 128B or MAT 167) and (MAT 129 or equivalent), or consent of instructor.
Course Description:
Introduction to mathematical basic building blocks (wavelets, local Fourier basis, and their relatives) useful for diverse fields (signal and image processing, numerical analysis, and statistics). Emphasis on the connection between the continuum and the discrete worlds.
Suggested Schedule:
| Lectures | Sections | Topics/Comments |
|---|---|---|
| Fourier series and transforms, the Shannon sampling theorem | ||
| Discrete Fourier, cosine, and sine transforms | ||
| Karhunen-Loeve transform and principal component analysis | ||
| Frame theory | ||
| Frames and sparse representations | ||
| The uncertainty principle | ||
| Windowed (or short-time) Fourier transform | ||
| Gabor (Weyl-Heisenberg) systems | ||
| Continuous and Discrete wavelet algorithms | ||
| Multiresolution analysis and fast wavelet algorithms | ||
| Applications to signal/image processing and communication |
Additional Notes:
Optional reference books:
- S. Mallat, A Wavelet Tour of Signal Processing, 3rd Ed., Academic Press, 2009.
- I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.
- S. Jaffard, Y. Meyer, and R. Ryan, Wavelets: Tools for Science and Technology, SIAM, 2001.
- K. Groechenig, Foundations of Time-Frequency Analysis, Birkhauser, 2001.
- W. L. Briggs and V. E. Henson, The DFT: An Owner's Manual for the Discrete Fourier Transform, SIAM, 1995.