On the Fourier Transform of Finite Chirps

In this letter, closed-form expressions for the discrete Fourier transform (DFT) of a finite chirp are derived. It is shown that when the normalized chirp rate is coprime with the chirp length, then the DFT of a finite chirp is again a finite chirp with magnitude, chirp rate, and carrier frequency appropriately scaled. In particular, when the normalized chirp rate is of unit value, then the DFT of a finite chirp is the same chirp, up to a complex scaling factor. Conversely, when the normalized chirp rate has a common factor with the chirp length, then the support of the DFT of a finite chirp is equal to the ratio of chirp length and the common factor. Among other things, results given here complement certain results, obtained by Janssen, on the computation of time-continuous chirps with rational sweep rates