Fourier Transform of Complex Gaussian
Theorem:
![]() |
(D.16) |
Proof: [202, p. 211]
The Fourier transform of
is defined as
![]() |
(D.17) |
Completing the square of the exponent gives
![\begin{eqnarray*}
pt^2 + j\omega t - \frac{\omega^2}{4p} + \frac{\omega^2}{4p}
&=& p\left(t+j\frac{\omega}{2p}\right)^2 + \frac{\omega^2}{4p}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2775.png)
Thus, the Fourier transform can be written as
![]() |
(D.18) |
using our previous result.
Alternate Proof
The Fourier transform of a complex Gaussian can also be derived using the differentiation theorem and its dual (§B.2).D.1
Proof: Let
![]() |
(D.19) |
Then by the differentiation theorem (§B.2),
![]() |
(D.20) |
By the differentiation theorem dual (§B.3),
![]() |
(D.21) |
Differentiating
![$ g(t)$](http://www.dsprelated.com/josimages_new/sasp2/img565.png)
![]() |
(D.22) |
Therefore,
![]() |
(D.23) |
or
![]() |
(D.24) |
Integrating both sides with respect to
![$ \omega$](http://www.dsprelated.com/josimages_new/sasp2/img89.png)
![]() |
(D.25) |
In §D.7, we found that
![$ G(0)=\sqrt{\pi/p}$](http://www.dsprelated.com/josimages_new/sasp2/img2784.png)
![]() |
(D.26) |
as expected.
The Fourier transform of complex Gaussians (``chirplets'') is used in §10.6 to analyze Gaussian-windowed ``chirps'' in the frequency domain.
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Why Gaussian?
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Gaussian Integral with Complex Offset