Fourier Transform of the full Morlet wavelet

Question

In 2014 someone asked here the Fourier transform of the Morlet wavelet; link below:

Fourier Transform of Morlet wavelet Function?

However, it was the approximated Morlet wavelet not written with the canonical Gaussian function.

Can somebody help to find the Fourier Transform of the full scaled Morlet wavelet below:

$$\psi_{\mu}(\tfrac{t}{\mu}) = \frac{1}{\mu} \times \frac{1}{\sqrt{2\pi} \, \sigma} \, e^{\frac{ -\left(\frac{t}{\mu} \right)^2}{2\sigma^2}} \big[e^{j\omega_c \frac{t}{\mu}}-e^{-\frac{1}{2} \omega_c^2}\big]$$

where $\omega_c = 2\pi f_c$ and $f_c$ is the centre or carrier frequency of the wavelet (the frequency to which the wavelet oscillates in the temporal domain). Also, $\mu$ is the scaling factor defined as $\mu = \frac{\omega_c}{\omega_{\mu}}$, where $\omega_{\mu}$ is the analysing frequency. Indeed, when $\omega_{\mu} = \omega_{c} \Rightarrow \mu = 1$ we have the mother Morlet wavelet.

In addition, the second term in the brackets is the correction term necessary to enforce zero mean to the Morlet wavelet.

I'd much appreciate a detailed step-by-step demonstration if possible to check where I'm getting stuck in my own demonstration. It could be done in a paper if quicker and a photo sent to me; I'll then publish here the full demonstration here.

Thanks in advance for any help.

Answer 1

if you wanna do this with ordinary frequency $f$ (rather than angular frequency $\omega$) youcan start with

$$ \mathscr{F}\Big\{ x(t) \Big\} \triangleq \int\limits_{-\infty}^{\infty} x(t) \, e^{-j 2 \pi f t} \, dt $$

$$ \mathscr{F}\left\{ e^{-\pi t^2} \right\} = e^{-\pi f^2} $$

$$ \mathscr{F}\left\{ e^{-\pi \alpha t^2} \right\} = \frac{1}{\sqrt{\alpha}} e^{-\frac{\pi}{\alpha} f^2} \qquad \alpha > 0 $$

$$ \mathscr{F}\left\{ e^{-\pi \alpha (t-\tau)^2} \right\} = \frac{1}{\sqrt{\alpha}} e^{-\frac{\pi}{\alpha} f^2} e^{-j 2 \pi f \tau} $$