On the stationary phase approximation of chirp spectra

The use of the stationary phase principle is often advocated for evaluating the spectrum of a chirp. This issue is considered here in detail, especially with respect to the quantitative control of the corresponding approximation error. A careful analysis leads to the introduction of a refined criterion, which turns out to be much more complicated than the heuristic conditions which are usually considered in this context. It is moreover evidenced, by means of two counterexamples belonging to the important class of power-law chirps, that-as opposed to a common belief-usual heuristic conditions are by themselves neither necessary nor sufficient for validating a stationary phase approximation.