This paper proposes a novel closed-form analytical expression of the fractional derivative of a s... more This paper proposes a novel closed-form analytical expression of the fractional derivative of a signal in the Fourier transform (FT) and the fractional Fourier transform (FrFT) domain by utilizing the fundamental principles of the fractional order calculus. The generalization of the differentiation property in the FT and the FrFT domain to the fractional orders has been presented based on the Caputo's definition of the fractional differintegral, thereby achieving the flexibility of different rotation angles in the time-frequency plane with varying fractional order parameter. The closed-form analytical expression is derived in terms of the well-known higher transcendental function known as confluent hypergeometric function. The design examples are demonstrated to show the comparative analysis between the established and the proposed method for causal signals corrupted with high-frequency chirp noise and it is shown that the fractional order differentiating filter based on Caputo's definition is presenting good performance than the established results. An application example of a low-pass finite impulse response fractional order differentiating filter in the FrFT domain based on the definition of Caputo fractional differintegral method has also been investigated taking into account amplitude-modulated signal corrupted with high-frequency chirp noise. Index Terms-Caputo fractional derivative, fractional Fourier transform (FrFT), fractional order calculus (FOC), fractional order derivative, Kummer confluent hypergeometric function.
A new mathematical model for obtaining the fractional Fourier transforms of Dirichlet and General... more A new mathematical model for obtaining the fractional Fourier transforms of Dirichlet and Generalized ''Hamming'' window functions is presented. The different parameters for the window functions are also obtained with the help of simulations. The fractional Fourier transformation contains an adjustable parameter with which the main lobe width and correspondingly, the minimum stop band attenuation of the resulting window function can be controlled.
In this paper, a closed-form analytical expression for fractional order differentiation in the fr... more 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 wellknown 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.
This paper proposes a novel closed-form analytical expression of the fractional derivative of a s... more This paper proposes a novel closed-form analytical expression of the fractional derivative of a signal in the Fourier transform (FT) and the fractional Fourier transform (FrFT) domain by utilizing the fundamental principles of the fractional order calculus. The generalization of the differentiation property in the FT and the FrFT domain to the fractional orders has been presented based on the Caputo's definition of the fractional differintegral, thereby achieving the flexibility of different rotation angles in the time-frequency plane with varying fractional order parameter. The closed-form analytical expression is derived in terms of the well-known higher transcendental function known as confluent hypergeometric function. The design examples are demonstrated to show the comparative analysis between the established and the proposed method for causal signals corrupted with high-frequency chirp noise and it is shown that the fractional order differentiating filter based on Caputo's definition is presenting good performance than the established results. An application example of a low-pass finite impulse response fractional order differentiating filter in the FrFT domain based on the definition of Caputo fractional differintegral method has also been investigated taking into account amplitude-modulated signal corrupted with high-frequency chirp noise. Index Terms-Caputo fractional derivative, fractional Fourier transform (FrFT), fractional order calculus (FOC), fractional order derivative, Kummer confluent hypergeometric function.
A new mathematical model for obtaining the fractional Fourier transforms of Dirichlet and General... more A new mathematical model for obtaining the fractional Fourier transforms of Dirichlet and Generalized ''Hamming'' window functions is presented. The different parameters for the window functions are also obtained with the help of simulations. The fractional Fourier transformation contains an adjustable parameter with which the main lobe width and correspondingly, the minimum stop band attenuation of the resulting window function can be controlled.
In this paper, a closed-form analytical expression for fractional order differentiation in the fr... more 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 wellknown 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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Papers by Sanjay Kumar