# 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.

Monday, Wednesday, Friday:  3:00 -- 3:55
Room Napier 202
```

### About This Course

One of the most important areas of applied mathematics is signal analysis. Mathematically, a signal is just a function, usually a real- or complex-valued function of one or more real variables. In applications, the most typical examples of signals are sounds (considered as functions of time giving the amplitude of the acoustic signal at each moment of time) and images (with the color at each point of an image given as a function of two spacial variables).

The most traditional type of signal analysis is frequency or spectrographic analysis, where a signal is decomposed into a sum of pure waveforms such as sine and cosine functions. This is Fourier Analysis, named after the mathematician Joseph Fourier, who invented the basic ideas in the 1820s. Fourier analysis is very widely used in both pure and applied mathematics. However, it is not appropriate for every task. For example, when analyzing a sound, the original signal represents the sound purely in terms of how the amplitude varies over time, while Fourier analysis represents the sound purely in terms of the frequencies that occur in it. Neither of these representations can tell you that a note of a certain frequency was played at a certain time! Wavelets, on the other hand, can provide a "mixed-mode" analysis that contains information about both frequency and time.

Wavelet and Fourier analysis are big, complicated fields that intersect with several areas of advanced mathematics. We will cover only parts of these fields, and we will do so with only linear algebra and basic calculus as prerequisites. The principal textbook for the course is A Primer on Wavelets and Their Scientific Applications by James S. Walker. This book introduces many of the fundamental ideas about wavelets using very little mathematical background. It also offers a pretty clear perspective on many real-world applications. The plan for the course is that we will work through this book, with digressions to cover some additional mathematical background and additional aspects of the theory of wavelets and Fourier analysis. I have not planned the course in complete detail, but we will begin by reading Chapter 1 from Walker, and then spend some time on complex numbers, inner products on vector spaces, and orthonormal bases. Some of the additional readings for the course will be taken from the book Wavelets Made Easy, by Yves Nievergelt.

### Assignments and Projects

I will assign homework problems from time to time and collect them for grading, generally a week after they are assigned. In addition to this regular homework, there will be two longer projects. For each project, you will do some research and produce a paper or some other type of work to turn in. You will also do a presentation on the project for the class. You will select your project topics in consultation with me. The first project will probably have to do with the history of signal analysis and will be due before Spring break. The second will be on a specific application of wavelets and will be due at the end of the term. More information about the projects will be available later in the course.

### Tests and Grading

There will be two tests. The first test will take place in class on Wednesday, March 1. The second will be during the regularly scheduled final exam period for this course, at 7:00 PM on Sunday, May 7. However, the second test will not be cumulative and will be no longer than the first test. (We can discuss the possibility of moving the second test to the last week of class and using the final exam period for presentation of final projects.)

Your grade for the course will be computed as follows:

```            First Test:     25%
Second Test:    25%
Assignments:    25%
Projects:       25%
```

### Attendance

I expect you to be present and on time for all classes. There might be extraordinary circumstances that force you to miss a few classes. You should discuss any such cases with me at the earliest possible time.

### Office Hours, E-mail, and Web

My office is room 301 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. I will announce my official office hours as soon as I schedule them.

My e-mail address is eck@hws.edu. E-mail is good way to communicate with me, since I usually answer messages the day I receive them.

The Web page for this course is at http://math.hws.edu/eck/math371/. I will post weekly readings and assignments on that page.