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