	Department of Mathematics & Computer Science

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MA4021 Wavelets and Signal Processing

MC4021 Wavelets and Signal Processing

Credits: 20

Convenor: Dr J. Levesley

Semester: 2

Prerequisites:

essential: MA1151(=MC146), MA1152(=MC147), MA2102(=MC241), MA2101(=MC248)

desirable: MA2121(=MC243)

Assessment:

Computer practicals	20%
Project	20%
Fortnightly worksheets	10%

: Total 50%

Three hour exam: 50%

Lectures:	36	Problem Classes:	5
Tutorials:	none	Private Study:	94
Labs:	10	Seminars:	5
Project:	none	Other:	none
Surgeries:	none	Total:	150

Explanation of Pre-requisites

I will use some results from MA2121 about best approximations. Also since the main space under study will be an inner product space, again material from MA2121 will smooth the student's path to understanding. However, it is not essential to have done MA2121, which deals with abstract inner product spaces, whereas we have a single concrete space in mind.

Course Description

Wavelets are very useful tools for data analysis and compression. Current research into the uses of wavelets for these and other purposes is intense. In this course you will be introduced to the mathematical theory of wavelets. One of the key tools here is harmonic analysis, which is a very useful tool in a variety of settings. You will also learn, in a practical setting, how to use wavelets to analyse and compress digital signals.

Aims

The main aims of this course is to be able to use wavelets for signal processing and compression, and to understand the fundamentals of the mathematical theory of wavelets.

Objectives

To learn basic harmonic analysis.

To understand and apply the idea of a filter bank.

To learn the mathematical and computational theory of wavelets.

To be able to analyse and compress a signal using wavelets.

To improve skills in oral and written communication.

To improve IT skills.

Transferable Skills

Students will make presentations of their projects, both in written and oral form. They shall also improve their IT skills.

Syllabus

Filter banks. The basic space

and the basic facts we need to know about it. Results about the Fourier transform, including Parseval's identity. Brief discussion of Fourier series. Introduction to wavelets, the Haar wavelet as an elementary model example, the two-scale equation, multiresolution analysis, how the multiresolution analysis aids the construction of wavelets, stable bases in

, the decomposition and reconstruction algorithms, the symbol, compactly supported wavelets, the cascade algorithm for constructing wavelets from the two scale equation.

Reading list

Background:

C. K. Chui, An Introduction to Wavelets, Academic Press, 1992.

Details of Assessment

The written examination lasts for three hours, and contains 6 questions. The best 4 answers only will contribute to the final mark.

The coursework for the continual assessment consists of computer practicals, worth 20%, a project worth 20%, and fortnightly exercises worth 10%.