Gaussian Windowed Chirps

The Fourier transform of a complex Gaussian pulse is derived in §C.7 of Appendix C:

$$\zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-... ...ga^2}{4p}},\quad \forall p\in {\bf C}: \; \mbox{re}\left\{p\right\}>0} \protect$$ (10.9)

This result is valid when $p$ is complex. Writing $p$ in terms of real variables $\alpha$ and $\beta$ as

$$p = \alpha - j\beta,$$

we have

$$x(t) = e^{-p t^2} = e^{-\alpha t^2} e^{j\beta t^2} = e^{-\alpha t^2} \left[\cos(\beta t^2) + j\sin(\beta t^2)\right]$$

That is, for $p$ complex, $x(t)$ is a Gaussian-windowed chirp. We see that the chirp oscillation frequency is zero at time $t=0$ . Therefore, for signal modeling applications, we typically add in an arbitrary frequency offset at time 0, as described in the next section.