Math 371: Wavelet and Fourier Analysis
First Project (and Homework #3)
You will do two projects for this course. The first project is due on Wednesday, February 22. Each person in the class will choose a different topic for the project and will do a presentation to the class on February 22 or 24, possibly with a handout or other materials for the other students. The content of the presentations and handouts will be included in the material for the first test, which is coming up on Wednesday, March 1.
The homework for the week is to select a topic, in consultation with me, and to prepare a short description of your topic and of several sources that you have found for use in your research. The description of the topic should show that you have already done enough research and planning to have a basic idea of what will be in the project. You should not simply list sources. You should describe the content of each source and how it might be used in the project. It is likely that most of your sources will be on the Web (and they will probably include several pages from wikipedia.org). You should turn in at least one full page of writing. This homework is due next Friday, February 10, and it will be graded.
The rest of this page suggests some possible topics. You are not absolutely required to choose one of the topics that are discussed here. If you have other ideas, you are welcome to discuss them with me. The topic must, however, be related to the history or mathematics of signal processing. Because the final project will be on an application of wavelets, topics in that area are not appropriate for this first project.
History and Background
One possible type of project would be a paper on some aspect of the history of Fourier analysis or signal processing. The paper would probably not include much mathematics, but would discuss how discoveries were made, the people who were involved, or the impact that the discoveries have had. Six pages would be a reasonable length for such a paper. Some possible specific topics:
- Mr. Fourier and the Heat Equation. Discuss the problems that Fourier was trying to solve, how he was led to invent Fourier series, and possibly some of the reactions to his work.
- How the FFT Changed the World. The DFT (Discrete Fourier Transform) is the version of Fourier analysis that applies to finite, discrete signals. The FFT (Fast Fourier Transform) is a very fast computational method for computing the DFT. The invention of a fast transform has enabled digital signal processing techniques that would be impractical without the speed of the FFT. The FFT is implemented in hardware on many digital signal processing chips.
- Prehistory of Wavelets. When wavelets were invented, it was realized that they were a generalization of many older ideas from signal and image processing, such as: quadrature mirror filters, filter banks and sub-band coding, pyramid algorithms and self-similar Gabor functions for image processing. You could look into some of all of these and discuss their relation to wavelets.
- Audio Compression Techniques. Look into some of the signal analysis techniques used in practical audio compression techniques such as MP3 and Ogg Vorbis.
You might want to write a more mathematical paper -- one that includes more equations and perhaps even some proofs. The first three topics listed here are things that we will have to cover at some point in the course, whether by a student or by me.
- The Nyquist/Shannon Sampling Theorem. This tremendously important theorem says that a band-limited signal (one made up of only a finite range of frequencies) can be perfectly reconstructed from discrete samples of the signals, provided that those signals are spaced closely enough together in time. It is probably not possible for you to give the proof of the theorem, but you can discuss what it says, what it means, and how it is used.
- Convolutions and the Convolution Theorem. The convolution product is a mathematical operation that can be performed on two functions. The convolution theorem tells what convolution corresponds to in the frequency domain. You could define convolution and explain the theorem and its importance in digital signal processing and the theory of filters.
- The Fourier Transform. When Fourier analysis is applied to a periodic function, the result is a Fourier series, since a periodic function can only contain a discrete (though infinite) set of frequencies. A non-periodic function, however, can contain all possible frequencies. When Fourier analysis is applied to a non-periodic function, the result is another function, which is called the Fourier transform of the original function. You could discuss the definition and basic properties of the Fourier transform.
- The Discrete Cosine Transform and JPEG Compression. The traditional technique for JPEG image compression is the discrete cosine transform. Discuss the mathematics of the DST and how it is used in JPEG compression.
James Walker, the author of our textbook, has written a program called FAWAV for wavelet and Fourier analysis. FAWAV is only available for Windows. Version 2.0 of the program can be downloaded through a link on his home page (http://www.uwec.edu/walkerjs/) or directly through the link http://www.uwec.edu/walkerjs/media/Fw2_0.zip. A possible project would be to learn some of the things that this program can do and how to use it. For this project, instead of a paper and presentation, you could prepare an overview of the program and a set of exercises that use the program. We would then use a class day for a computer lab in which you would demonstrate the program and the other students in the course would work through the exercises with your help.
David Eck, 2 February 2006