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MATLAB has the complex Morlet Wavelet in the following form:

$$\psi(t) = \frac{1}{\sqrt{\pi f_b}}e^{\frac{-t^2}{f_b}}e^{j2\pi f_ct}$$

I arrived at its Fourier transform as shown below (another arrived at the same: Fourier Transform of Morlet wavelet Function?).

$$X(f)=\mathcal F(x)(f)=e^{-\pi^2 f_b(f-f_c)^2}$$

I sought a relationship between $f_b$ and the standard deviation of the Gaussian form of $X(f)$. Is it right? Is there any citable published literature (book or journal article) arriving at this relationship?

Rearranging this into the familiar Gaussian form, $$X(f) = e^{\frac{-(f-f_c)^2}{2\left({\frac{1}{\pi\sqrt{2f_b} })}\right)^2}}$$

The standard deviation $(\sigma_f)$ of this frequency domain Gaussian is then, $$\sigma_f = \frac{1}{\pi\sqrt{2f_b}}$$ Rearranging it, $$f_b = \frac{1}{2{(\pi\sigma_f)}^2}$$

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I might have the ingredients to an answer here and here and here.

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