CWT using cwtft and Morlet Wavelet

Question

Lorenzo Bernardini 2020-1-15

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https://ww2.mathworks.cn/matlabcentral/answers/500460-cwt-using-cwtft-and-morlet-wavelet

评论： Sindar 2020-1-16

Hello there,

My name is Lorenzo Bernardini, and currently I'm encountering some issues exploring the world of CWT in MATLAB and I would like to share them with you.

The question is the following: does the cwtft function algorithm work with a complex or a real valued Morlet function, when we specify inside its input arguments 'morl'?

cwtS1 = cwtft({Signal1,dt},'wavelet','morl','scales',scales);

This doubt is due to the fact that:

when using waveinfo('morl') it turns out the following real expression: morl(x) = exp(-x^2/2) * cos(5x)
but when I do the cwtft of a real signal it gives me as outputs complex coefficients. How is this possible? I read from theory that a CWT of a real signal gives real coefficients,therefore how is this possible that I got complex coefficients if this algorithm uses the real valued Morlet wavelet defined as in the previous point?
In addition to this, there's another issue. Using cwtftinfo I can see the Fourier transform of the Morlet wavelet, and it looks exactly like the Fourier transform of a complex Morlet function, found in books. Moreover the central frequency provided by default is 6/(2+pi) which is the one for complex Wavelet function.

Therefore, the question is, which Morlet wavelet does the cftf exploit in its algorithm? A real one? A complex one?

If it is real, how is it possible that the coefficients are complex?

If it is complex (as supposed by the resulting coefficients, the center frequency and the Fourier transform provided by matlab in picture 1, why when I write waveinfo('morl') It turns out to be real valued ( morl(x) = exp(-x^2/2) * cos(5x))?

Thanks for the attention, I would really appreciate your answer.

Lorenzo

P.S: As I asked in previous questions, still unanswered, what's the meaning of analytic and non analytic when referring to Wavelets (picture2)?