MorletWellen

MorletWellen, commonly referred to in English as Morlet wavelets, are a family of analytic wavelets used for time–frequency analysis of signals. They are constructed as a complex sinusoid multiplied by a Gaussian envelope, providing a well-localized representation in both time and frequency.

The standard continuous-wavelet form is psi(t) = pi^(-1/4) exp(i w0 t) exp(-t^2/2), where w0 is a non-dimensional central frequency. The real and imaginary parts yield a cosine and sine carrier modulated by the Gaussian window, allowing estimation of instantaneous amplitude and phase. The parameter w0 controls the trade-off between time and frequency localization; larger w0 improves frequency resolution but broadens the wavelet in time.

For many practical purposes, Morlet wavelets are treated as approximately admissible when w0 is sufficiently large (commonly w0 ≥ 5–6), enabling stable inversion of the continuous wavelet transform. At small w0, corrections are applied in software implementations to enforce admissibility.

Applications of MorletWellen span neuroscience, where they analyze EEG and MEG signals; geophysics and seismology for analyzing impulsive signals; acoustics and audio processing; and more generally any domain requiring joint time–frequency localization. They can be used in continuous wavelet transforms and, in extended form, in two dimensions for image analysis.

Historically, the Morlet wavelet is named after Jean Morlet, a French geophysicist who, with co-developer Alex Grossmann, helped develop wavelet-based methods in the 1980s. Since then, Morlet wavelets have become a standard analytic tool in signal processing.