The Poisson sum formulae associated with the fractional Fourier transform

Authors:
Bing-Zhao Li;Ran Tao;Tian-Zhou Xu;Yue Wang
Affiliations:
Department of Mathematics, Beijing Institute of Technology, Beijing 10081, China;Department of Electronic Engineering, Beijing Institute of Technology, Beijing 10081, China;Department of Mathematics, Beijing Institute of Technology, Beijing 10081, China;Department of Electronic Engineering, Beijing Institute of Technology, Beijing 10081, China
Venue:
Signal Processing
Year:
2009

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Abstract

The theorem of sampling formulae has been deduced for band-limited or time-limited signals in the fractional Fourier domain by different authors. Even though the properties and applications of these formulae have been studied extensively in the literature, none of the research papers throw light on the Poisson sum formula and non-band-limited signals associated with the fractional Fourier transform (FrFT). This paper investigates the generalized pattern of Poisson sum formula from the FrFT point of view and derived several novel sum formulae associated with the FrFT. Firstly, the generalized Poisson sum formula is obtained based on the relationship of the FrFT and the Fourier transform; then some new results associated with this novel sum formula have been derived; the potential applications of these new results in estimating the bandwidth and the fractional spectrum shape of a signal in the fractional Fourier domain are also proposed. In addition, the results can be seen as the generalization of the classical results in the Fourier domain.