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The Fourier transform of a complex Gaussian pulse is derived in
§D.8 of Appendix D:
![$\displaystyle \zbox {e^{-pt^2} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-\frac{\omega^2}{4p}},\quad \forall p\in \mathbb{C}: \; \mbox{re}\left\{p\right\}>0} \protect$](img1866.png) |
(11.27) |
This result is valid when
is complex.
Writing
in terms of real variables
and
as
![$\displaystyle p \eqsp \alpha - j\beta,$](img1867.png) |
(11.28) |
we have
![$\displaystyle x(t) \eqsp e^{-p t^2} \eqsp e^{-\alpha t^2} e^{j\beta t^2} \eqsp e^{-\alpha t^2} \left[\cos(\beta t^2) + j\sin(\beta t^2)\right].$](img1868.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.
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