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Closed-Form Analytical Expression of Fractional Order Differentiation in Fractional Fourier Transform Domain

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Abstract

In this paper, a closed-form analytical expression for fractional order differentiation in the fractional Fourier transform (FrFT) domain is derived by utilizing the basic principles of fractional order calculus. The reported work is a generalization of the differentiation property to fractional (noninteger or real) orders in the FrFT domain. The proposed closed-form analytical expression is derived in terms of the well-known confluent hypergeometric function. An efficient computation method has also been derived for the proposed algorithm in the discrete-time domain, utilizing the principles of the discrete fractional Fourier transform algorithm. An application example of a low-pass finite impulse response (FIR) fractional order differentiator in the FrFT domain has also been investigated to show the practicality of the proposed method in signal processing applications.

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Acknowledgements

The authors express their sincere thanks to the Editor-in-Chief and to the learned reviewers for their valuable comments and suggestions in developing this article into its present form.

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Authors and Affiliations

  1. Department of Electronics and Communication Engineering, Thapar University, Patiala, Punjab, India

    Sanjay Kumar & Kulbir Singh

  2. Department of Electronics and Communication Engineering, Jaypee University of Engineering and Technology, Raghogarh, Guna, Madhya Pradesh, India

    Rajiv Saxena

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  1. Sanjay Kumar
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  2. Kulbir Singh
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  3. Rajiv Saxena
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Correspondence to Sanjay Kumar.

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Kumar, S., Singh, K. & Saxena, R. Closed-Form Analytical Expression of Fractional Order Differentiation in Fractional Fourier Transform Domain. Circuits Syst Signal Process 32, 1875–1889 (2013). https://doi.org/10.1007/s00034-012-9548-1

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  • DOI: https://doi.org/10.1007/s00034-012-9548-1

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