Morlet wavelet

Example

Fix a parameter $\sigma > 0$ and put $\kappa_\sigma = e^{-\sigma^2/2}$. The Morlet wavelet is the complex-valued function $$\psi_\sigma(t) = c_\sigma \, \pi^{-1/4} e^{-t^2/2} \left(e^{i\sigma t} - \kappa_\sigma\right), \qquad t \in \R,$$ where $c_\sigma > 0$ is the constant that makes $\norm{\psi_\sigma}_2 = 1$. It is a plane wave $e^{i\sigma t}$ under a Gaussian envelope, with a small correction term.

Check of the zero mean. Since $\int_{\R} e^{-t^2/2} e^{i\sigma t} \, dt = \sqrt{2\pi} \, e^{-\sigma^2/2}$ and $\int_{\R} e^{-t^2/2} \, dt = \sqrt{2\pi}$, the integral of $\psi_\sigma$ is $c_\sigma \pi^{-1/4} \sqrt{2\pi} \left(e^{-\sigma^2/2} - \kappa_\sigma\right) = 0$. Its Fourier transform is a combination of two Gaussians and vanishes at frequency $0$. Together with the Gaussian decay, this gives the admissibility condition, so $\psi_\sigma$ can be used in the continuous wavelet transform.