The discrete fractional Fourier transform

Abstract

Abstract-We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.

Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform

“…It is known that the discrete FRT approximately maps the samples of a function to the samples of its FRT in the same sense that the ordinary discrete FT does for the ordinary FT [10][11][12][13]. Therefore, the values of the diffracted field at the natural sampling grid points can be well approximated by the discrete FRT of the samples of the input field.…”

Automatic Modulation Classification of LFM and Polyphase-coded Radar Signals

“…Therefore, the optimal value of  is expected to be the same as that of the LFM signal, i. e. equation (9). The numerical search for α opt confirms this hunch.…”

“…The FrFT is a general form of the Fourier transform (FT) that transforms a function into an intermediate domain between time and frequency by rotating the time-frequency plane [8][9]. Compared with FT, the FrFT of optimal angle α opt applied to a LFM signal, maximally concentrates the energy distribution of the signal in the fractional domain This illustrates the use of the FrFT for pulse compression of LFM signals [10].…”

An approach for security of color image data in coordinate, geometric, and frequency domains

“…We have also used the angles of the discrete fractional Fourier transform (FrFT) (Candan, Kutay, & Ozaktas, 2000) as keys for security of color image data for secure transmission in open networks. Almedia (Almedia, 1994) and Namias (Namias, 1980) defined the two-dimensional FrFT (2D-FrFT) kernel of the cryptosystem as follows: …”