The Fourier transform of a complex Gaussian pulse is derived in
§D.8 of Appendix D:
![$\displaystyle \zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-\frac{\omega^2}{4p}},\quad \forall p\in {\bf C}: \; \mbox{re}\left\{p\right\}>0} \protect$](http://www.dsprelated.com/josimages_new/sasp2/img1874.png) |
(11.27) |
This result is valid when
![$ p$](http://www.dsprelated.com/josimages_new/sasp2/img1009.png)
is complex.
Writing
![$ p$](http://www.dsprelated.com/josimages_new/sasp2/img1009.png)
in terms of real variables
![$ \alpha $](http://www.dsprelated.com/josimages_new/sasp2/img4.png)
and
![$ \beta $](http://www.dsprelated.com/josimages_new/sasp2/img6.png)
as
![$\displaystyle p \eqsp \alpha - j\beta,$](http://www.dsprelated.com/josimages_new/sasp2/img1875.png) |
(11.28) |
we have
![$\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].$](http://www.dsprelated.com/josimages_new/sasp2/img1876.png) |
(11.29) |
That is, for
![$ p$](http://www.dsprelated.com/josimages_new/sasp2/img1009.png)
complex,
![$ x(t)$](http://www.dsprelated.com/josimages_new/sasp2/img109.png)
is a chirplet (Gaussian-windowed
chirp). We see that the chirp oscillation frequency is zero at time
![$ t=0$](http://www.dsprelated.com/josimages_new/sasp2/img1877.png)
. Therefore, for
signal modeling applications, we typically add
in an arbitrary frequency offset at time 0, as described in the next
section.
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