Fourier Series & Wavelets

Math 414-501 — Spring 2015

Instructor: Dr. Francis J. Narcowich
Office: 611D Blocker
E-mail: fnarc@math.tamu.edu
Phone: 979-845-7554 (Message only.)
URL: http://www.math.tamu.edu/~francis.narcowich/
Office Hours: TR, 11 am-1 pm, 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 Boggess & Narcowich

Time & Place: MWF 12:40-1:30 pm, BLOC 160

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 20 & April 17). and a final exam ( 10:30 am - 12:30 pm, Friday, May 8 ).The project will count for 20% of your grade, homework for 20%, each in-class test for 20%, and the final exam for 20%. 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). Be aware that usually only some of the problems from an assignment will be graded. Late homework will not be accepted. Information concerning projects may be found on at this webpage: Project Information.

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 Disability Services: Cain Hall, Room B118. Phone: 979-845-1637. E-mail: disabilities@tamu.edu "

Schedule

Date Section Topic
1/21/15 0.1, 0.2 Vector spaces and inner products
1/23/15 0.3.1, 0.3.2 Spaces for discrete and continuous signals; approximations
1/26/15 0.3.2, 0.4 Approximations, convergence & important inequalities
1/28/15 0.4, 0.5.1, 0.7.1 Inequalities, orthogonality & least squares
1/30/15 0.5.2, 0.5.3 Orthogonal projections & the Gram-Schmidt process
2/2/15 1.2.1, 1.2.2 Fourier series
2/4/15 1.2.2, 1.2.3 Sine and cosine series
2/6/15 1.2.5, 1.2.4 Complex form of a Fourier series & examples
2/9/15 1.2.4, 1.3.1 Examples, convergence & Riemann-Lebesgue lemma
2/11/15 1.3.2 Poinwise convergence of Fourier series
2/13/15 1,3.2, 1.3.3, 1.3.4 Poinwise convergence, uniform convergence & Gibbs phenomenon
2/16/15 1.3.5 Parseval's equation & convergence in the mean
2/18/15 1.3.2-1.3.5 Catch up, review
2/20/15 §§ 0.1-0.5, 0.7.1, 1.2-1.3 Test 1
2/23/15 2.1, 2.2.1 Fourier transform (FT), properties of the FT, examples
2/25/15 2.2.1, 2.2.2, 2.2.4 Parseval's equation, convolutions
2/27/15 2.3.1, 2.3.2 Filters
3/2/15 2.3.2 Filters
3/4/15 2.4, 3.1.1 Sampling theorem & discrete Fourier transform (DFT)
3/6/15 3.1.2, 3.1.3 DFT, fast Fourier transform (FFT)
3/9/15 3.1.4 3.1.5 FFT approximation to FT & applications
3/11/15 3.2.1, 3.2.2 Discrete signals & the Z transform
3/13/15 3.2.2 Z transform.
3/16 - 3/20/2015 N/A Spring Break
3/23/15 4.1, 4.2.1 Wavelets, Haar scaling function
3/25/15 4.2.1, 4.2.2 Properties of the Haar scaling function
3/27/15 4.2.3 Haar wavelet
3/30/15 4.2.3, 4.3.1 Haar wavelet decomposition
4/1/15 4.3.2, 4.3.3, 4.4 Haar reconstruction, filters
4/3/15 N/A Good Friday/Reading day
4/6/15 5.1.1 Multiresolution analysis (MRA) -- definition
4/8/15 5.1.2 Scaling relation & scaling function
4/10/15 5.1.3 Wavelets & wavelet spaces
4/13/15 5.1.4 Decomposition and reconstruction
4/15/14 5.2.1 Decomposition algorithm & filters
4/17/15 §§ 2.1-2.4, 3.1.1-3.1.5, 3.2, 4.2-4.4, 5.1 Test 2
4/20/15 5.2.2 Reconstruction algorithm & filters
4/22/15 5.3.1, 5.3.3 Fourier transform of the scaling function
4/24/15 5.3.3, 5.3.4 Iterative construction of the scaling function
4/27/15 6.1 The Daubechies wavelet construction
4/29/15 6.2 Moments
5/1/15 6.2, 6.3 Classification of the Daubechies wavelets, singularity detection & signal extensions
5/4/15 6.3 Signal extensions
5/5/15 N/A Catch up, review
5/8/15 N/A Final exam (10:30 am-12:30 pm)

Updated 1/19/2015.