Closed-Form Analytical Expression of Fractional Order Differentiation in Fractional Fourier Transform Domain
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.
KeywordsFractional order calculus Fractional order differentiation Fractional Fourier transform Grünwald–Letnikov fractional derivative Kummer confluent hypergeometric function Riemann–Liouville fractional derivative
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