A Generalization of the Fourier Transform and its Application to Spectral Analysis of Chirp-like Signals

Mamadou Mboup (1, 2) , Tülay Adali (3)

1 NON-A - Non-Asymptotic estimation for online systems
2 CRESTIC - Centre de Recherche en Sciences et Technologies de l'Information et de la Communication - EA 3804
3 MLSP lab - Machine Learning for Signal Processing Laboratory

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 para\-me\-trization, 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.

Mots clés: de Branges spaces; Fourier transform; Chirps; Reproducing kernel

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Mamadou Mboup, Tülay Adali. A Generalization of the Fourier Transform and its Application to Spectral Analysis of Chirp-like Signals. Applied and Computational Harmonic Analysis, 2011, ⟨10.1016/j.acha.2011.11.002⟩. ⟨hal-00640508v2⟩