Morlet transform of modulated oscillations

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Morlet transform of modulated oscillations

The modulated waves of section 14 are processed using the Morlet wavelet, and their local norms are shown below.

For the signal, the Morlet wavelet shows the continuous shift toward high frequencies (Fig. 17) (Note: the ordinates should be labeled `Period (s)'. I will redo these plots at some point). The most striking difference with the Mexican-hat transform in section 14 is the shift in emphasis from the individual maxima and minima to the local periodicity.

Figure 17: Morlet-wavelet transform of the signal.

In the case of the amplitude/frequency modulated wave (Fig. 18), the results are less clear than in the Mexican hat case, because no frequency has time to get established and `resonate' with this particular wavelet.

Figure 18: Morlet-wavelet transform of the modulated oscillation.

Finally, for the intermittent data, the Morlet wavelet (Fig. 19) establishes a clear distinction between the random fluctuations and the periodic patches. It is noteworthy that the period of the sine waves is determined much more accurately than with the Mexican hat: the compromise between spatial and spectral localization is clearly demonstrated: Morlet favors spectral accuracy while Mexican hat favors individual event localization.

Figure 19: Morlet-wavelet transform of the intermittent signal.

Thus, the selection of the wavelet will be quite important, and practice will guide the user to some standard wavelets. In all instances, it will be important to interprete the results by keeping in mind what different wavelets can and cannot do.

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Jacques Lewalle
Mon Nov 13 10:51:25 EST 1995