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

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