Morlet's Wavelet

The wavelet defined by Morlet is [16]:

$$\hat{g}(\omega) = e^{ -2 \pi^2(\nu - \nu_0)^2}$$ (14.7)

it is a complex wavelet which can be decomposed in two parts, one for the real part, and the other for the imaginary part.

$$\begin{eqnarray*}g_r(x) & = & \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} \cos(2\p... ... = & \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} \sin(2\pi\nu_0 x) \end{eqnarray*}$$

where $\nu_0$ is a constant. The admissibility condition is verified only if $\nu_0 > 0.8$. Figure 14.1 shows these two functions.

  

Figure 14.1: Morlet's wavelet: real part at left and imaginary part at right.