Discrete wavelet analysis of signals

International Journal of Electrical Engineering Education, Jul 1999 by Asamoah, F

ABSTRACT A discrete wavelet transform based on Daubechies coefficients is used to decompose a signal into levels. The method discussed is applicable to any signal of length 2^sup N^. Where the length is less than 2^sup N^ it is zero-padded to obtain the required length. A program written in MATLAB is used for the computations.

KEYWORDS decomposition; dilation; levels; reconstruction; wavelets

1 INTRODUCTION

Signal analysis has traditionally been studied by Fourier methods. However, in recent times another concept has emerged which is increasingly being used as an alternative to the Fourier approach; this new method is based on wavelets. A wavelet is a wave of very short duration which has its energy concentrated in time. Wavelet analysis is more efficient particularly where the signal being analyzed has transients or discontinuities. Wavelet analysis is increasingly being applied in diverse fields, including medicine and biology, geophysical signal processing and numerical methods1-3. This paper discusses a simple decomposition and reconstruction of a given discrete signal using Mallat's algorithm4.

The field of wavelets is still being researched in universities, colleges and industry. However, it is increasingly being taught in graduate and undergraduate courses in mathematics digital signal processing and communications. The Electrical and Computer Engineering Department of The University of the West Indies has begun a course with final year undergraduate projects in wavelet-based analysis of electroencephalograms (EEG) and speech waveforms. We hope to start teaching the basics as part of the discrete signal processing course from the next academic year. The paper presents an easy-to-implement program (available on request) for the analysis of any one-dimensional signal.

REFERENCES

[1] Rioul, O. and Vetterli, M., `Wavelets and signal processing, IEEE Spectrum, 10, pp. 14-38 (1991)

[2] Bruce, M., Donoho, D. and Gao, H.-Y., `Wavelet analysis', IEEE Spectrum, 10, pp. 26-35 (1996)

[3] Akay, M., `Wavelet applications in medicine', IEEE Spectrum, 5, pp. 50-56 (1997)

[4] Mallat, S., 'A theory of multiresolution signal decomposition, the wavelet representation', IEEE Trans. Pattern Anal. Machine Intelligence, 11, pp. 674-93 (1989)

[5] Newland, D. E., An Introduction to Random Vibrations, Spectral and Wavelet Analysis, John Wiley, Ch. 17, pp. 295-339 (1994)

[6] Daubechies, I., `Orthonormal bases of compactly supported wavelets', Comm. Pure Appl. Maths, 41, pp. 909-996 (1988)

F. ASAMOAH

Department of Electrical and Computer Engineering, University of the West Indies, Trinidad. E-mail: fasamoah@eng.uwi.tt