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