Morlet wavelet

In 1946, physicist Dennis Gabor, applying ideas from quantum physics, introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as atoms, and which provide the best trade-off between spatial and frequency resolution.[1] These are used in the Gabor transform, a type of short-time Fourier transform.[2] In 1984, Jean Morlet introduced Gabor's work to the seismology community and, with Goupillaud and Grossmann, modified it to keep the same wavelet shape over equal octave intervals, resulting in the first formalization of the continuous wavelet transform.[4] (See also Wavelet history)

The wavelet is defined as a constant ${\displaystyle \kappa _{\sigma }}$ subtracted from a plane wave and then localised by a Gaussian window:[5]

${\displaystyle \Psi _{\sigma }(t)=c_{\sigma }\pi ^{-{\frac {1}{4}}}e^{-{\frac {1}{2}}t^{2}}(e^{i\sigma t}-\kappa _{\sigma })}$

where ${\displaystyle \kappa _{\sigma }=e^{-{\frac {1}{2}}\sigma ^{2}}}$ is defined by the admissibility criterion, and the normalisation constant ${\displaystyle c_{\sigma }}$ is:

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

The Fourier transform of the Morlet wavelet is:

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

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

${\displaystyle \omega _{\Psi }=\sigma {\frac {1}{1-e^{-\sigma \omega _{\Psi }}}}}$

which can be solved by a fixed-point iteration starting at ${\displaystyle \omega _{\Psi }=\sigma }$ (the fixed point iterations converge to the unique positive solution for any initial ${\displaystyle \omega _{\Psi }>0}$).

The parameter ${\displaystyle \sigma }$ in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction ${\displaystyle \sigma >5}$ is used to avoid problems with the Morlet wavelet at low ${\displaystyle \sigma }$ (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 ${\displaystyle \sigma }$. In this case, ${\displaystyle \kappa _{\sigma }}$ becomes very small (e.g. ${\displaystyle \sigma >5\quad \Rightarrow \quad \kappa _{\sigma }<10^{-5}\,}$) and is, therefore, often neglected. Under the restriction ${\displaystyle \sigma >5}$, the frequency of the Morlet wavelet is conventionally taken to be ${\displaystyle \omega _{\Psi }\simeq \sigma }$.

The wavelet exists as a complex version or a purely real-valued version. Some distinguish between the "real Morlet" vs the "complex Morlet".[6] Others consider the complex version to be the "Gabor wavelet", while the real-valued version is the "Morlet wavelet".[7][8]

• Constant-Q transform
• Gabor wavelet

• 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