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