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

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