ABSTRACT
A selection of wavelets that are commonly used are shown in Figure 2.1; we will consider some of them in more detail as we proceed through the text. As we can see from the figure, they have the form of a small wave, localized on the time axis. There are, in fact, a large number of wavelets to choose from for use in the analysis of our data. The best one for a particular application depends on both the nature of the signal and what we require from the analysis (i.e. what physical phenomena or process we are looking to interrogate, or how we are trying to manipulate the signal). We will begin this chapter by concentrating on a specific wavelet, the Mexican hat, which is very good at illustrating many of the properties of continuous wavelet transform analysis. The Mexican hat wavelet is shown in Figure 2.1b and in more detail in Figure 2.2a. The Mexican hat wavelet is defined as
y( ) ( )t t e t= −1 2 2 2
− (2.1)
The wavelet described by Equation 2.1 is known as the mother wavelet or analyzing wavelet. This is the basic form of the wavelet from which dilated and translated versions are derived and used in the wavelet transform. The Mexican hat is, in fact, the second derivative of the Gaussian distribution function e t−( )
2 2/ , that is, with unit variance but without the usual
FIGURE 2.1 Four wavelets: (a) Gaussian wave (first derivative of a Gaussian). (b) Mexican hat (second derivative of a Gaussian). (c) Haar. (d) Morlet (real part).
