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
Goals: This is a mathematics course. One of the goals is for you to learn to be able to prove theorems concerning Fourier series, properties of Fourier transforms, filters, discrete Fourier transforms and wavelets. Another is for you to learn how to find and work with these tools. Finally, by means of a project, you will learn how wavelets are applied. To achieve these goals, you must diligently do the homework assignments and read the appropriate sections of the book, as well as additional notes.
Required Text: A First Course in Wavelets and Fourier Analysis, 2nd Edition, by Boggess & Narcowich
Time & Place: MWF 11:30-12:20, BLOC 117
Programming language: Experience with MATLAB would be very helpful.
Grading System & Tests: Your grade will be based on a project, homework, and three in-class tests (February 16, March 23 and April 27).The project will count for 20% of your grade, homework for 20%, and each in-class test 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.
Covid information The latest information about covid policies may be found at the Covid-19 informaiuon link on the TAMU homepage.
||1.1.1-1.1.2, 1.2.1-1.2.2||Fourier series (FS): motivation, calculation, examples|
||1.2.1-1.2.3||FS examples, function extensions, symmetry, Fourier cosine/sine series (FCS/FSS), examples|
||1.2.4-1.2.5||Convergence of FS; Complex form of FS; examples|
||1.3.1-1.3.3||Partial sums, Dirichlet (Fourier) kernel, Riemann-Lebesgue Lemma; proof of pointwise convergence (See the Notes on Pointwise Convergence) for a simplified version); uniform convergence|
|| Test 1 (2/16/22)
|Review, Test 1 (covers 1.2.1-1.2.5, 1.3.1-1.3.3, (Notes on Pointwise Convergence), uniform convergence|
||1.3.4, 0.2-0.5, 1.3,5||inner products, signal spaces ($L^2,\ell^2$), types of convergence, orthogonal bases, Parseval's equation, FS examples|
||2.1, 2.2||Fourier Transform & properties; convolution theorem; Plancherel (Parseval) Theorem|
|2.3, 2.4||Time-invariant filters, causal filters, sampling theorem|
|| Test 2 (3/23/22),
|Review, catch up, Test 2 (covers 1.3.2-1.3.5, 2.1-2.4), discrete Fourier transform|
||3.1.1-3.1.4, 3.2.1||Discrete Fourier transform, fast Fourier transform (FFT), applications, discrete signals & filters|
|4.1, 4.2, 4.3||Haar wavelets, decomposition and reconstruction algorithms, filter representation|
|| 5.1.1, 5.1.2
Good Friday, reading day
|Multiresolution analysis, scaling relation|
||5.1.3, 5.3.3 Theorem 5.2.3, 6.2-6.3 ,||Decomposition and reconstruction algorithms, connection with FT, and existence criteria for wavelets, Daubechies wavelets|
|| Test 3 (4/27/22)|
|Review, catch up, Test 3 (covers Chapter 4, 5.1, 5.2, 5.3.3-5.3.4), Daubechies wavelets|
||6.2, 6.3||Finish Daubechies wavelets, computational issues|
|5/9/22, 8-10||N/A||Brief project presentations|
|Tuesday, 5/10/22||N/A||Written form of projects due at 4 pm|