# Morlet Wavelet CWT

Hello Everybody,

I am PhD student and I work with regional Earthquakes in which the first signal onset has energy with the bandwith [2 12] Hz. I carry out two test.info : Earthaquakes are non-stationary transient signals

FIRST;
I compute the CWT to build the scalogram with the Morlet Wavelet with a fixed number of cycles w = 6.
–>Good results, when there is NO a burst of energy in the first onset for lower and higher energies simultaneously

SECOND;

I compute the CWT to build the scalogram with the Moret but, now I linearly space the number of cycles [5 10] in the frequency bandwith [2 12] Hz. I carry on this operation to smooth the frequency lost resolution of the CWT for the higher frequency and be able to distinguish the oscillatory behavior for the higher frequencies

My question is;

1. Anyone that can give me some paper reference that uses the sam as the SECOND analysis (e.g. neuroscience paper)

2. Anyone that can give me a well elaborated explanation about, why SECOND procedure is better for my case. Because in principle, with a fix number of cycles the frequency resolution is spread too much over higher frequency (this is up to my concern). However, I am afraid of violating some thermal rule related with de design of the cwt to build the scalogram.

I thank you very much the help of the scientific community.

Hello Francois, thank you very much. I will send a message to @pantazis.

What I’ve also read is that the pseudo-Morelet wavelet fulfills the admissibility condition if X(f) is near 0. for f=0. so just for number of cycles larger than 5.

So from my previous message I think I am not violating any theoretical condition, I am just tuning the width of the wavelet as I INCREASE THE CENTRAL FREQUENCY

Hi rcabdia,

A couple of articles showing the scaling of the morlet wavelets at different frequencies are here:
https://www.sciencedirect.com/science/article/pii/S1364661399012991
https://www.sciencedirect.com/science/article/pii/S1053811917305906

Morlet wavelets have a constant product FWHM_t * FWHM_f = 8*log2 / 2pi, which means that you trade off temporal for spectral resolution (FWHM = Full Width Half Maximum is defined as the width of a Gaussian kernel at half its maximum value).

Unfortunately in the article above I wrote one equation wrong: The wavelets were selected to have a constant ration FWHM_t * f = 3 (and not FWHM_t / f = 3 as I wrote in the article.) This means that the temporal resolution was set to be FWHM_t = 3/f, which is inversely proportional to the frequency of interest. At f = 1Hz, the temporal resolution was 3s; at f = 2Hz, the temporal resolution was 1.5s; etc.

Best,
Dimitrios

Hello Dimitrios,
Thank you very much for your answer, I've understand in terms of time frequency resolution in FWHM units.
**Earthquakes are also a non-stationary transient signal.

I have done an analysis of an earthquake with the morels wavelet,

1. with a numer of cycles set 6
2. with a number cycles set 16
3 with a linear spaced number of cycles with the frequency 6 16

For the 3 option I get a moderate frequency smoothing of the signal while keeping constant the product FWHM_t * FWHM_f for a range of frequencies 2-12 Hz.

So Dimitros or anyone, Do you think that I am doing right?
I've read your references but in that works, they keep the number of cycles constant. Any work you know that implement a linear spaced number of cycles with the frequency?

sorry,

1. with a numer of cycles set 5
2. with a number cycles set 16
3. with a linear spaced number of cycles with the frequency 5 16 for the frequency range [2-12] Hz