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 ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 38<br />
�<br />
+∞<br />
−∞<br />
Equation 6.3 is called the admissible condition.<br />
ψ(t)dt = 0 (6.3)<br />
The admissible condition is quite lenient so that there are a lot <strong>of</strong> <strong>wavelet</strong>s, which can<br />
be discrete or continuous, real or complex functions. Figure 6.2 shows three different<br />
<strong>wavelet</strong>s as examples. The Haar <strong>wavelet</strong> (the simplest <strong>wavelet</strong>) <strong>and</strong> the Db2 <strong>wavelet</strong><br />
(one <strong>of</strong> Daubechies <strong>wavelet</strong>s) are discrete while the Mexican Hat <strong>wavelet</strong> is continuous.<br />
The continuous <strong>wavelet</strong>s are described with formulas, but the discrete <strong>wavelet</strong>s nor-<br />
mally have no direct formulas <strong>and</strong> are given through discrete filters.<br />
Figure 6.2: Examples <strong>of</strong> <strong>wavelet</strong>s<br />
Mother <strong>wavelet</strong>, <strong>wavelet</strong> family <strong>and</strong> <strong>wavelet</strong> domain: Any <strong>wavelet</strong> can be taken<br />
as a mother <strong>wavelet</strong>. A mother <strong>wavelet</strong> generates a <strong>wavelet</strong> family by scaling <strong>and</strong><br />
translating. All members in one <strong>wavelet</strong> family have the same energy <strong>and</strong> the similar<br />
waveform as the mother <strong>wavelet</strong>.<br />
Let ψ(t) denote a mother <strong>wavelet</strong>, the corresponding <strong>wavelet</strong> family ψab(t) is defined<br />
as Equation 6.4:<br />
ψab(t) = a −1/2 � �<br />
t − b<br />
· ψ<br />
a<br />
(6.4)<br />
Where a represents "scale" <strong>and</strong> b represents "translation". Figure 6.3 displays the re-<br />
lation between a mother <strong>wavelet</strong> (Morlet <strong>wavelet</strong>) <strong>and</strong> it’s family. It can be seen that