### Chirplet Fourier Transform

The Fourier transform of a complex Gaussian pulse is derived in §D.8 of Appendix D:

This result is valid when is complex. Writing in terms of real variables and as

(11.28) |

we have

(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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