Chirplet Fourier Transform

The Fourier transform of a complex Gaussian pulse is derived in §D.8 of Appendix D:

$$\zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-\frac{\omega^2}{4p}},\quad \forall p\in \mathbb{C}: \; \mbox{re}\left\{p\right\}>0} \protect$$ (11.27)

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

$$p \eqsp \alpha - j\beta,$$ (11.28)

we have

$$x(t) \eqsp e^{-p t^2} \eqsp e^{-\alpha t^2} e^{j\beta t^2} \eqsp e^{-\alpha t^2} \left[\cos(\beta t^2) + j\sin(\beta t^2)\right].$$ (11.29)

That is, for $p$ complex, $x(t)$ is a chirplet (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.