## 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

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 gives

(D.22) |

Therefore,

(D.23) |

or

(D.24) |

Integrating both sides with respect to yields

(D.25) |

In §D.7, we found that , so that, finally, exponentiating gives

(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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Gaussian Integral with Complex Offset