Fourier Series & Wavelets

Math 414-502 — Spring 2012

Instructor: Dr. Francis J. Narcowich
Office: 302 Milner Hall
E-mail: fnarc@math.tamu.edu
Phone: 979-845-7554 (Message only.)
URL: /~francis.narcowich/
Office Hours: MWF 1:40-2:40, and by appointment.

Catalogue Description: MATH 414. Fourier Series & Wavelets. Fourier series and wavelets with applications to data compression and signal processing. Prerequisite: MATH 323 or MATH 304 or MATH 311

Required Text: A First Course in Wavelets and Fourier Analysis, 2nd Edition, by Bogess & Narcowich

Time & Place: MWF 12:40 pm - 1:30 pm, ENPH 206

Programming language: Experience with MATLAB would be very helpful.

Grading System & Tests: Your grade will be based on a project, homework, two in-class tests ( February 27 & April 11). The project will count for 25% of your grade, homework for 25%, each in-class test for 25%. Your letter grade will be assigned this way: 90-100%, A; 80-89%, B; 70-79%, C; 60-69%, D; 59% or less, F.

Make-up Policy: I will give make-ups (or satisfactory equivalents) only in cases authorized under TAMU Regulations. In borderline cases, I will decide whether or not the excuse is authorized. Also, if you miss a test, contact me as soon as possible.

Homework and Projects: You may consult with each other on homework problem sets, BUT only submit work which is in your own words AND be sure to cite any sources of help (either texts or people). Late homework will not be accepted. The further information and the due-date for the project will be announced later on.

Academic Integrity

Copying Course Materials:   "All printed hand-outs and web-materials are protected by US Copyright Laws. No multiple copies can be made without written permission by the instructor."

Aggie Honor Code:   "An Aggie does not lie, cheat, or steal or tolerate those who do."

Americans with Disabilities Act Policy Statement: "The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe that you have a disability requiring an accommodation, please contact the Department of Student Life, Services for Students with Disabilities, in Room 126 of the Koldus Building or call 845-1637."

Schedule

Date Section Topic
1/18/12 0.1, 0.2 Inner products on vector spaces
1/20/12 0.3.1 Spaces for discrete and continuous signals
1/23/12 0.3.2 Approximations and convergence
1/25/12 0.4, 0.5.1 Important inequalities; orthogonality
1/27/12 0.5.2, 0.5.3 Orthogonal projections and the Gram-Schmidt process
1/30/12 1.2.1, 1.2.2 Fourier series and how to compute them
2/1/12 1.2.3, 1.2.4 Sine and cosine series; examples of Fourier series
2/3/12 1.2.4, 1.2.5 Complex form of a Fourier series
2/6/12 1.3.1, 1.3.2 Riemann-Lebesgue lemma; pointwise convergence of Fourier series
2/8/12 1.3.2, 1.3.3 Poinwise convergence of Fourier series
2/10/12 1.3.4 Uniform convergence and the Gibbs phenomenon
2/13/12 1.3.5 Parseval's equation and convergence in the mean/energy
2/15/12 2.1 Fourier transform (FT) -- definition and examples
2/17/12 2.2.1 Properties of the FT
2/20/12 2.2.1, 2.2.2, 2.2.4 Parseval's equation, convolutions
2/22/12 2.3.1, 2.3.2 Filters
2/24/12 2.3.2 Catch up, review
2/27/12 §§ 0.1-0.5, 1.2-1.3, 2.1-2.3 Test 1
2/29/12 2.4, 3.1.1 Sampling theorem, discrete Fourier transform (DFT)
3/2/12 3.1.2, 3.1.3 DFT, fast Fourier transform (FFT)
3/5/12 3.1.4 3.1.5 FFT approximation to FT, applications
3/7/12 3.2.1 Discrete signals
3/9/12 3.2.2 Z transform.
3/12 - 3/16 N/A Spring Break
3/19/12 4.1, 4.2.1 Wavelets, Haar scaling function
3/21/12 4.2.1, 4.2.2 Properties of the Haar scaling function
3/23/12 4.2.3 Haar wavelet
3/26/12 4.2.3, 4.3.1 Haar wavelet decomposition
3/28/12 4.3.2, 4.3.3, 4.4 Haar reconstruction, filters
3/30/12 5.1.1 Multiresolution analysis (MRA) -- definition
4/2/12 5.1.2, 5.1.3 Scaling relation, scaling function, wavelet
4/4/12 5.1.3, 5.1.4 Decomposition and reconstruction
4/6/12 N/A Good Friday/Reading day
4/9/12 5.1.4 Catch up, review
4/11/12 §§ 2.4, 3.1.1-3.1.5, 3.2, 4.2-4.4, 5.1 Test 2
4/13/12 5.2.1, 5.2.2 Decomposition and reconstruction algorithms
4/16/12 5.2.2, 5.2.3 Processing a signal
4/18/12 5.3.3, 5.3.4 Fourier transform of the scaling function, iterative construction
4/20/12 6.1 The Daubechies wavelet construction
4/23/12 6.2 Classification of the Daubechies wavelets
4/25/12 6.3, 7.2 Signal extensions, two-dimensional wavelets
4/27/12 7.2 Processing an image
4/30/12 N/A Project presentations
5/1/12 N/A Project presentations

Updated 1/14/2012.