Summary

We presented a model for approximations of the FFT of linear frequency chirps, derived from Fresnel integral analysis. We rigorously showed that when the chirp increases or decreases quickly enough within an analysis frame, our current set of Fresnel assumptions are valid, and the FFT will show parabolic phase. We presented results showing that the model is invertible, allowing calculation of the chirp parameter from the FFT. We showed that the inversion techniques may also be used to make linear approximations to nonlinear frequency trajectories, and to conclude that a frequency trajectory is or is not monotonic.

In appendix A below, we discuss other algorithms that may be used specifically when our $\alpha$ value is too small for the above analysis. These algorithms have already been implemented, and will be fully integrated into a later report.