### Modulated Gaussian-Windowed Chirp

By the *modulation theorem* for Fourier transforms,

(11.30) |

This is proved in §B.6 as the dual of the shift-theorem. It is also evident from inspection of the Fourier transform:

(11.31) |

Applying the modulation theorem to the Gaussian transform pair above yields

(11.32) |

Thus, we frequency-shift a Gaussian chirp in the same way we frequency-shift any signal--by complex modulation (multiplication by a complex sinusoid at the shift-frequency).

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Chirplet Fourier Transform