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Understanding CWT Morlet: Time and frequency resolution

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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.

Accepted Answer

Wayne King
Wayne King on 8 May 2021
Edited: Wayne King on 8 May 2021
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,
SungJun Cho
SungJun Cho on 11 May 2021
Edited: SungJun Cho on 11 May 2021
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?

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