A generalization of the Fourier transform and its application to spectral analysis of chirp-like signals

We show that the de Branges theory provides a useful generalization of the Fourier transform (FT). The formulation is quite rich in that by selecting the appropriate parametrization, one can obtain spectral representation for a number of important cases. We demonstrate two such cases in this paper: the finite sum of elementary chirp-like signals, and a decaying chirp using Bessel functions. We show that when defined in the framework of de Branges spaces, these cases admit a representation very much similar to the spectral representation of a finite sum of sinusoids for the usual FT.