The course described on this page ended May 7, 2006

## Math 371 (Topics in Mathematics):

Wavelet and Fourier Analysis

Department of Mathematics and Computer Science Hobart and William Smith Colleges Spring 2006. Instructor: David J. Eck (eck@hws.edu) Monday, Wednesday, Friday: 3:00--3:55 PM. Room Napier 202 Course Handout: http://math.hws.edu/eck/courses/math371_s06.html Scheduled Office Hours (Room Lansing 301): Monday: 12:30 -- 1:30 Tuesday: 1:30 -- 3:00 Wednesday: 11:00 -- 12:00 Friday: 10:00 -- 11:00

Assignments and other information for Math 371

will be posted on this page as the course is taught

during Spring term, 2006.

## End of Term: April 17 through May 7

We have decided on the following schedule for the end of the semester:

- Wednesday, April 26: The second test. A review sheet is available.
- Friday, April 28: Three final project presentations.
- Monday, May 1: Two final project presentations.
- Sunday, May 7: Final project paper due at the scheduled final exam period (7:00 PM).
For the time leading up to the test on April 26, I will be talking about some of the mathematics of the continuous wavelet transform and its relation to the discrete transform that we have been downplaying up until now. We will probably also look at some of the applications that are discussed briefly in Chapter 4 of Walker.

## Twelfth Week: April 10, 12, and 14

This week, we will continue talking about the DFT and its applications. The reading is sections 3.1, 3.2, 3.4, and 3.5. We have already pretty much covered sections 1 and 2 already. We will not be covering the

z-transform (pages 101--102) immediately, but we should get back to it next week. We will probably not cover Section 3.3 at all.The sixth homework assignment is due in class next Wednesday, April 19. This will be the last homework assignment of the term.

## Eleventh Week: April 3, 5, and 7

We will be finishing up Chapter 2 of

A Primer on Waveletsthis week. You should read the remaining sections in that chapter. This will finish our coverage of two-dimensional wavelet transforms. We will then move on to Chapter 3. We have already done some of the material on the Discrete Fourier Transform, but we will look at the two-dimensional DFT and the Fast Fourier Transform. Chapter 3 is largely about applications of this material to the theory of wavelets.

## Tenth Week: March 27, 29, and 31

You should choose a final project topic by the end of the week. For more information on the project, click here.

We will spend this week and probably much of next week talking about two-dimensional wavelets and applications to image processing. The reading for this material is Sections 2.7 to 2.11.

The next homework assignment is due next Monday.

## Ninth Week: March 20, 24, and 26

The reading for the week starts with a handout from

The World According to Wavelets. This handout talks about applications of wavelets. This reading is supposed to help you choose a topic for your final project, which will be on some aspect of applications of wavelets. You should read the handout before Friday, so that we can discuss it then, along with some ideas for final projects.In addition to the handout, we will be picking up where we left off back in

A Primer on Wavelets. You should look through Section 2.3, just to get the basic idea of the families of wavelets that are discussed there. You should read Sections 2.4 and 2.5 in more detail. These sections include more details about compression and denoising of one-dimensional signals. We will also cover the Root Mean Square (RMS) Error, page 25, which I skipped the first time through.

## Eighth Week: March 6 and 8

There is a test on Monday. On Wednesday, we will go over the test and talk about what we will do next in the course. There is no class on Friday because of Spring break. Classes resume on Monday, March 20.

## Seventh Week: February 27; March 1 and 3

There is a test next week, on Monday, March 6. A review sheet is available. Note that a copy of this review sheet will be available to you during the test.

This week, we will have Andrei Romanov's presentation (postponed from last Friday when we ran out of time). After that, we will finish up some material on the Fourier transform and convolution and review for the test.

## Sixth Week: February 20, 22, 24

Projects are due on Friday. On Monday and Wednesday, I will continue presenting some background material on complex Fourier analysis. Here is the handout on this material:

Outline of Complex Fourier Analysis

## Fifth Week: February 13, 15, and 17

This week, everyone picked a final project. We decided that they would be due on Friday, February 24, and everyone will give a presentation on their project on that day. The project topics are:

Ian Cook: Audio compression (mp3 and ogg vorbis). Phil Fiero: Convolution and the convolution theorem. Chris Hagar: Joseph Fourier and the heat equation. Rebecca Gutwin: JPEG image compression with the discrete cosine transform. Andrei Romanov: The sampling theorem.I started a series of lectures on backbround material on Fourier analysis that I thought would be needed before the presentations next Friday.

## Fourth Week: February 6, 8, and 10

We will move on to Daubechies wavelets, which are discussed in Chapter 2 of

A Primer on Wavelets. The reading for the week is Sections 2.1, 2.2, and 2.3.This week's homework is to begin work on the first of the two projects that you will do for the course. You need to choose a topic in consultation with me. A one-page write-up about your project is due on Friday. The project itself will be due on Wednesday, February 22. Complete details can be found at the following link:

Midterm Project (and Homework #3)

For writing a paper that includes some mathematical formulas, you might be interested in learning to use LaTeX, the document preparation system that is used for the mathematical typesetting. A short introduction to LaTeX can be found at this link:

http://math.hws.edu/eck/about_tex/

## Third Week: January 30; February 1 and 3

We will finish Chapter 1 from

A Primer on Wavelets. The remaining sections in this chapter cover application of the Haar Wavelet Transform to compressing audio signals and to removing noise from audio signals. We will discuss in class whether we should move on immediately to Chapter 2, or whether we should first look more deeply into Fourier series.The second homework assignment is due in class on Friday of this week. The homework sheet, which is available through the following link, is also a handout on Orthogonal and Orthonormal Bases:

Part of the homework is based on a Java application that illustrates the discrete Haar wavelet transform. The application is contained in the following executable .jar file. Brief instructions for using it can be found in the Homework Assignment #2.

January 31: I have added another demo program that deals with signal compression using the Haar Transform. The executable .jar file for the appliation is HaarCompressionDemo.jar, However, I have also re-worked both programs as applets to make them easier to use. You can find the applet versions of the two programs by following this link:

## Second Week: January 23, 25, and 27

We will be continuing with Sections 1.3 and 1.4 from

A Primer on Wavelets, and we will try to relate this material back to the continuous wavelet transform. The next reading will be a handout on normed vector spaces and orthonormal bases.Homework #1 is due in class on Friday. The assignment can be found in the following PDF file: hw1.pdf.

## First Week: January 16, 18, and 20

We will begin the course with a general introduction to signal analysis, with a very short overview of Fourier Analysis and Wavelets. We will then start working through

A Primer on Wavelets. The reading for the week in Chapter 1, Sections 1.1 through 1.4. This chapter covers Haar wavelets, a simple type of wavelet that has been know since the early 20-th century. In practice, Haar are not suitable for many applications, but most of the techniques that are used with Haar wavelets extend to the more useful wavelets that are discussed in Chapter 2.