I am trying to make an algorithm that plots out the Discrete-Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) of the Gaussian filter. The impulsive response and its transfer function (TF) are respectively given by $$ g(t) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{t^2}{2\sigma^2}} $$ and $$ G(f) = \mathscr{F}\big\{g(t)\big\} = \int_{-\infty}^{\infty} g(t)e^{2 \pi f t} dt = e^{- \frac{(2\pi\sigma f)^2}{2}} $$

Where $\mathscr{F}\{\cdot\}$ indicates the Continuous-Time Fourier Transform (CTFT, or just FT). I have studied DSP for a while, but I am still not sure how to proceed in order to derive (and plot) the DTFT and the DFT from this impulsive response. I would like to avoid built-in command that automatically plots it and compute it by myself. Comparing the obtained plot with these built-in results would be interesting, though.