Understanding CWT Morlet: Time and frequency resolution

Hello all,

I have been using a continuous 1-D wavelet transform (MATLAB cwt function) to compute and plot wavelet scalograms. For my research, I need to know the time and frequency resolution of the scalograms, but I could not find this information from any of the MATLAB documentations I looked into.

I am using 'amor' for the cwt, and it seems like 'amor' and 'cmor' operate with essentially the same equation (please correct me if I am wrong).

Complex Morlet Wavelet:

Analytic Morlet (Gabor) Wavelet: where

I am not sure if the equation for the analytic Morlet wavelet is correct since it is not documented, but given such equations, the temporal and frequency resolution for the wavelet transform then would be and , respectively, where 'a' is a scale value.

My question is what value or (or width w if ) and is used to define the Morlet wavelet in the 'cwt' function.

EDIT: For 'cmor', it says that and by default. Would these values also apply for 'amor' when using 'cwt'?

Thank you very much.

Answer 1

[Wayne King]

Hi SungJo, the analytic Morlet wavelet used in the cwt() function and cwtfilterbank as 'amor' is defined in the frequency domain as

where the \hat{U}(\omega) is the unit step in frequency. This makes the wavelet purely analytic. If we ignore the unit step part for a moment, the inverse Fourier transform of is

So you can take this as the basic definition of the analytic Morlet wavelet you obtain with 'amor'. Keep in mind a Morlet wavelet is nothing but a modulated Gaussian.

Of course to be technically accurate, we actually have a convolution of the time domain wavelet given above with the inverse Fourier transform of the unit step, so something like

where the \ast denotes convolution.

The above assumed the definition of the Fourier transform and its inverse as

with slight (trivial) differences if a different normalization is used.

Hope that helps,

Wayne

[SungJun Cho]

Hi Wayne,

Thank you so much for your answer. It was immensely helpful to me.

Just to make sure, given that the time domain Morlet wavelet follows a Guassian formulation, would I be correctly understanding the parameters if I take and as and ?

Also, is there any reference I can refer to for more information about the analytic Morlet wavelet?

Thank you very much.

[Wayne King]

Hi SungJo, you can look at the following article:

https://paos.colorado.edu/research/wavelets/bams_79_01_0061.pdf

See equation (1) and Table 1 where the expression for the Morlet wavelet in the Fourier domain is almost exactly what is used in MATLAB's Wavelet Toolbox. In that equations \omega_0 is 6.

The other differences just come down to normalization constants.

[SungJun Cho]

Thank you very much! This was very helpful.

Can I ask lastly why you ignored the unit step part to take the inverse Fourier transform?

[SungJun Cho]

Sorry for the question. I got it now.