The Fourier transform of a complex Gaussian pulse is derived in
§D.8 of Appendix D:
 |
(11.27) |
This result is valid when

is complex.
Writing

in terms of real variables

and

as
 |
(11.28) |
we have
![$\displaystyle x(t) \eqsp e^{-p t^2} \eqsp e^{-\alpha t^2} e^{j\beta t^2} \eqsp e^{-\alpha t^2} \left[\cos(\beta t^2) + j\sin(\beta t^2)\right].$](http://www.dsprelated.com/josimages_new/sasp2/img1876.png) |
(11.29) |
That is, for

complex,

is a chirplet (Gaussian-windowed
chirp). We see that the chirp oscillation frequency is zero at time

. Therefore, for
signal modeling applications, we typically add
in an arbitrary frequency offset at time 0, as described in the next
section.
Next Section: Modulated Gaussian-Windowed ChirpPrevious Section: Further Reading