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Fractional Fourier Transform, Signal Processing and Uncertainty Principles

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Abstract

The fractional Fourier transform (FrFT) is one of the generalizations of the Fourier transform (FT). This paper is centered on the compression of different forms of signal in FrFT domain in order to extract some properties of each one with a comparison between the FrFT and the usual FT. Also, our focus here will be on two qualitative uncertainty principles for the fractional Fourier transform: The Cowling–Price’s theorem and the \(L^p-L^q\) version of Morgan’s theorem for the FrFT. These two results estimate the decay of two fractional Fourier transforms \(F_{\alpha }(f)\) and \(F_{\gamma } (f)\), with \(\gamma -\alpha \ne n\pi , \forall n\in \mathbb {Z}\), which allows us to deduce the usual uncertainty principles between a function f and its fractional Fourier transform \(F_{\gamma }(f)\).

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Data Availability

The datasets analyzed during the current study are available on request from the corresponding author. The corresponding author has had full access to all the data (MATLAB algorithm and results) in the study and total responsibility for the integrity of the data and the accuracy of the data analysis.

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Acknowledgements

The authors would like to thank the referee for their valuable comments and suggestions which improve the presentation of our paper.

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Correspondence to Zaineb Aloui.

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Aloui, Z., Brahim, K. Fractional Fourier Transform, Signal Processing and Uncertainty Principles. Circuits Syst Signal Process 42, 892–912 (2023). https://doi.org/10.1007/s00034-022-02138-9

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