Fourier Transform of Complex Gaussian

Theorem:

$$\zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} \, e^{-\frac{\omega^2}{4p}},\quad \forall p\in \mathbb{C}: \; \mbox{re}\left\{p\right\}>0}$$ (D.16)

Proof: [202, p. 211] The Fourier transform of $e^{-pt^2}$ is defined as

$$\int_{-\infty}^\infty e^{-pt^2} e^{-j\omega t}dt \eqsp \int_{-\infty}^\infty e^{-(pt^2+j\omega t)} dt.$$ (D.17)

Completing the square of the exponent gives

$$\begin{eqnarray*} pt^2 + j\omega t - \frac{\omega^2}{4p} + \frac{\omega^2}{4p} &=& p\left(t+j\frac{\omega}{2p}\right)^2 + \frac{\omega^2}{4p} \end{eqnarray*}$$

Thus, the Fourier transform can be written as

$$e^{-\frac{\omega^2}{4p}} \int_{-\infty}^\infty e^{-p\left(t+j\frac{\omega}{2p}\right)^2} dt \eqsp \sqrt{\frac{\pi}{p}}\, e^{-\frac{\omega^2}{4p}}$$ (D.18)

using our previous result.