Morlet wavelet

Morlet wavelet

In mathematics, the Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud, Grossmann and Morlet in 1984 as a constant κ_σ subtracted from a plane wave and then localised by a Gaussian window:

$\Psi_{\sigma}(t)=c_{\sigma}\pi^{-\frac{1}{4}}e^{-\frac{1}{2}t^{2}}(e^{i\sigma t}-\kappa_{\sigma})$

where $\kappa_{\sigma}=e^{-\frac{1}{2}\sigma^{2}}$ is defined by the admissibility criterion and the normalisation constant c_σ is:

$c_{\sigma}=\left(1+e^{-\sigma^{2}}-2e^{-\frac{3}{4}\sigma^{2}}\right)^{-\frac{1}{2}}$

The Fourier transform of the Morlet wavelet is:

$\hat{\Psi}_{\sigma}(\omega) = c_\sigma \pi^{-\frac{1}{4}} \left( e^{-\frac{1}{2}(\sigma-\omega)^2} - \kappa_\sigma e^{-\frac{1}{2}\omega^{2}} \right)$

The "central frequency" ω_Ψ is the position of the global maximum of $\hat{\Psi}_{\sigma}(\omega)$ which, in this case, is given by the solution of the equation:

$(\omega_{\Psi}-\sigma)^{2}-1=(\omega_{\Psi}^{2}-1)e^{-\sigma\omega_{\Psi}}$

The parameter σ in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction σ > 5 is used to avoid problems with the Morlet wavelet at low σ (high temporal resolution).

For signals containing only slowly varying frequency and amplitude modulations (audio, for example) it is not necessary to use small values of σ. In this case, κ_σ becomes very small (e.g. $\sigma>5 \quad \Rightarrow \quad \kappa_{\sigma}<10^{-5}\,$) and is, therefore, often neglected. Under the restriction σ > 5, the frequency of the Morlet wavelet is conventionally taken to be $\omega_{\Psi}\simeq\sigma$.

References

• P. Goupillaud, A. Grossman, and J. Morlet. Cycle-Octave and Related Transforms in Seismic Signal Analysis. Geoexploration, 23:85-102, 1984

• N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani. Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies. IEEE Trans. Inf. Th., 38:644-664, 1992