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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6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 37<br />
• The function is oscilla<strong>to</strong>ry.<br />
• Average value is zero<br />
The first characteristic makes a function "little" whereas the second one makes it "wavy"<br />
<strong>and</strong> hence a <strong>wavelet</strong>. These two characteristics must be simultaneously satisfied for a<br />
function <strong>to</strong> be a <strong>wavelet</strong>. A sinusoid never decays (i.e. wavy but not little) so that can<br />
not be a <strong>wavelet</strong>. A decay function (little but not wavy, e.g. Gauss function) also can<br />
not be a <strong>wavelet</strong>. However, the product <strong>of</strong> a wavy <strong>and</strong> a decay function can lead a<br />
<strong>wavelet</strong>. Figure 6.1 shows, as an example, how a <strong>wavelet</strong> is created: the product <strong>of</strong><br />
a sinusoid (6.1(a)) <strong>and</strong> a Gauss function (6.1(b)) makes a <strong>wavelet</strong> (6.1(c)) - the famous<br />
Morlet <strong>wavelet</strong> [8].<br />
Figure 6.1: Wavelet<br />
In mathematics, any function ψ(t) can be a <strong>wavelet</strong> if it satisfies the following condi-<br />
tion.