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Time–Frequency Localization for the Fractional Fourier Transform in Signal Processing and Uncertainty Principles

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Abstract

The fractional Fourier transform (FrFT) is a generalization of the usual Fourier transform. The aim of this paper is to show the compression of sound signal in FrFT domain and to prove the qualitative and quantitative uncertainty principles for the FrFT. The first of these results consists the Hardy’s and an \(L^p-L^q\) version of Miyachi’s theorems for the FrFT, which estimates of decay of two fractional Fourier transforms \(F_{\alpha }(f)\) and \(F_{\gamma } (f)\), with \(\gamma -\alpha \ne n\pi , \forall n\in \mathbb {Z}\). The second result consists an extension of Faris’s local uncertainty principle which states that if a non zero function \(F_{\alpha }(f) \in L^2(\mathbb {R})\) is highly localized near a single point then \(F_{\gamma } (f)\) cannot be concentrated in a set of finite measure with \(\gamma -\alpha \ne n\pi , \forall n\in \mathbb {Z}\). From our results we deduce the usual uncertainty principles for the fractional Fourier transform which states these theorems between a function f and its fractional Fourier transform \(F_{\gamma }(f)\).

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The datasets analysed during the current study are available from the corresponding author on requested. The corresponding author had full access to all the data (Matlab algorithm, results) in the study and takes responsability 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 improved the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science of Tunis, University of Tunis el Manar, Tunis, Tunisia

    Zaineb Aloui

  2. Department of Mathematics, College of Science, University of Bisha, P.O.Box 344, Bisha, 61922, Kingdom of Saudi Arabia

    Kamel Brahim

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  1. Zaineb Aloui
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  2. Kamel Brahim
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Correspondence to Zaineb Aloui.

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Aloui, Z., Brahim, K. Time–Frequency Localization for the Fractional Fourier Transform in Signal Processing and Uncertainty Principles. Circuits Syst Signal Process 40, 4924–4945 (2021). https://doi.org/10.1007/s00034-021-01698-6

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