Development of a wavelet-based algorithm to detect and determine ...

Chapter 6
Wavelets
6.1 Introduction
The wavelet transform is a new and developing branch of mathematics. Recently it
has become one of the hot points in the research of engineering applications. This is
neither a novelty nor a fashion. The wavelet transform leads to the signal processing
techniques with special characteristics that have been expected for a long time. [6]
The term "wavelet" literally mens little wave because a wavelet decays quickly (lit-
tle) with oscillations (wave) (see figure 6.1). Although countless functions may be little
waves, the item wavelet is reserved for those little waves that are associated with a par-
ticular choice: narrow-band in frequency. Simply, wavelets are window functions not
only in time domain but also in frequency domain. The wavelet transform represents a
signal through wavelets. It is a linear transformation much like the Fourier transforma-
tion but with important differences: the wavelet transform can localize simultaneously
in time and in frequency, adjust the window widths according to frequency automati-
cally and allow the flexible selection of wavelets to match different applications.
The wavelet transform as a new item was proposed by Morlet for geophysics signal
processing in 1982 [8], but there was a long time preparation for its development. The
wavelet transform was developed as a new theory in mathematics by Meyer, Gross-
mann and Daubechies during 1984-1988. In 1989, Mallat proposed the multiresolution
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