Morlet wavelet

In mathematics, the Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud, Grossmann and Morlet in 1984 as a constant $kappa\_\{sigma\}$ subtracted from a plane wave and then localised by a Gaussian:

$Psi\_\{sigma\}(t)=c\_\{sigma\}pi^\{-frac\{1\}\{4\}\}e^\{-frac\{1\}\{2\}t^\{2\}\}(e^\{isigma\; t\}-kappa\_\{sigma\})$

where $kappa\_\{sigma\}=e^\{-frac\{1\}\{2\}sigma^\{2\}\}$ is defined by the admissibility criterion and the normalisation constant $c\_\{sigma\}$ 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" $omega\_\{Psi\}$ is the position of the global maximum of $hat\{Psi\}\_\{sigma\}(omega)$ which, in this case, is given by the solution of the equation:

$(omega\_\{Psi\}-sigma)^\{2\}-1=(omega\_\{Psi\}^\{2\}-1)e^\{-sigmaomega\_\{Psi\}\}$

The parameter $sigma$ in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction $sigma>5$ is used to avoid problems with the Morlet wavelet at low $sigma$ (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 $sigma$. In this case, $kappa\_\{sigma\}$ becomes very small (e.g. ) and is, therefore, often neglected. Under the restriction $sigma>5$, the frequency of the Morlet wavelet is conventionally taken to be $omega\_\{Psi\}simeqsigma$.

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