The book gives an easy read introduction to the discrete wavelet transform and some of its applications. It is based on a novel approach to discrete wavelets called lifting. The first part is an elementary introduction to the subject, and the prerequisites are knowledge of basic calculus and linear algebra. The second part requires some knowledge of Fourier series and digital signal analysis. It is well suited for presenting the wavelet theory to undergraduate students in mathematics and electrical engineering and engineers in industry.

It is the result of parts of the activities in a group concerned with advanced digital signal processing. It is founded via grant from "Det Tekniske Uddannelsesråd (TUR)".

Target group

The notes preceding the book were aimed at students at the engineering colleges in Denmark. This means that the necessary prerequisites are as limited as possible. The first five chapters can be read with a very little mathematical background (high school level), while the rest requires the same background as would be necessary to comprehend basic digital signal processing; this is linear algebra, some Fourier theory, in particular the Fourier transform, the z-transform, and a little sampling theory.

Ripples and lifting

It is a very recent development in the theory of wavelets that has made the this book possible. The central idea is the concept of lifting, which was introduced by Wim Sweldens in 1995. It was also Sweldens, and colaboration with Daubechies, who in the fall 1996 showed, that all known wavelet transforms (based on FIR filters) can be constructed through lifting. The major advantages of lifting over the classical approach is that it can be interpreted solely in the time domain.

Table of contents

(A complete table of contents is available here)

1. Introduction
2. A First Example
3. The Discrete Wavelet Transform via Lifting
4. Analysis of Synthetic Signals
5. Interpretation
6. Two Dimensional Transforms
7. Lifting and Filters I
8. Wavelet Packets
9. The Time-Frequency Plane
10. Finite Signals
11. Implementation
12. Lifting and Filters II
13. Wavelets in Matlab
14. Applications and Outlook

Ripples in Mathematics - The Discrete Wavelet Transform. Springer Verlag Berlin Heidelberg 2001
Web master: Anders la Cour-Harbo, Aalborg University, Denmark.
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