# A unified framework for the fractional Fourier transform

@article{Cariolaro1998AUF, title={A unified framework for the fractional Fourier transform}, author={Gianfranco Cariolaro and Tomaso Erseghe and Peter Kraniauskas and Nicola Laurenti}, journal={IEEE Trans. Signal Process.}, year={1998}, volume={46}, pages={3206-3219} }

The paper investigates the possibility for giving a general definition of the fractional Fourier transform (FRT) for all signal classes [one-dimensional (1-D) and multidimensional, continuous and discrete, periodic and aperiodic]. Since the definition is based on the eigenfunctions of the ordinary Fourier transform (FT), the preliminary conditions is that the signal domain/periodicity be the same as the FT domain/periodicity. Within these classes, a general FRT definition is formulated, and the…

## 91 Citations

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A new source for the multiplicity of the weight-type fractional Fourier transform (WFRFT) is proposed, which can generalize the weight coefficients of WFRFT to contain two vector parameters, which provides a unified framework for the FRFT.

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A general FRT definition is generated, based on eigenfunctions and eigenvalues of the ordinary Fourier transform, which allows us to generate all possible definitions and gives explicit relationships between the different FRTs.

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This work applies the language of the unified FT to develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity.

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It is shown that the MP FRFT may provide a novel understanding of sampling process and the proposed sampling theorem generalizes classical Shannon sampling theorem and Fourier series expansion, and provides a full-reconstruction procedure of certain signals that are not bandlimited in the conventional Fourier transform domain.

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