Improved discrete fractional Fourier transform

Abstract

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT's) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.

A Low-Complexity Approach to Computation of the Discrete Fractional Fourier Transform

“…To calculate the output vector y (a) N it is necessary to perform N 2 complex multiplications and N (N − 1) complex additions. However, if we use decomposition (14) when N is an even number or decomposition (18) when N is odd, the number of arithmetical operations required for calculating the discrete fractional Fourier transform can be significantly reduced. We can multiply each component of the sum by the input vector separately, and finally add the results.…”

“…The definition (3) of DFRFT was first introduced by Pei and Yeh [18,19]. They defined the DFRFT in terms of a particular set of eigenvectors, which constitute the discrete counterpart of the set of Hermite-Gaussians functions (these functions are well-known eigenfunctions of FT and the fractional Fourier transform was defined through a spectral expansion in this base [14]).…”

“…Many discrete fractional transforms, such as discrete fractional cosine and sine transforms [22], discrete fractional random transform [12], discrete fractional Hartley transform [23], discrete fractional Hilbert transform [21], discrete fractional Hadamard transform [20], and discrete fractional Fourier transform [18] have been found very useful for signal processing [27], digital watermarking [5], image encryption [7,10,15], image and video processing [9]. Among other transformations, the discrete fractional Fourier transform is perhaps most frequently used.…”

“…In other words, it is not a discrete version of the continuous transform. The third approach is based on the idea of eigenvalue decomposition [3,18,19]. This type of DFRFT possesses all essential properties which are posed as requirements for DFRFT such as unitarity, additivity, reduction to discrete Fourier transform (when the power is equal to unity), approximation of the continuous FRFT.…”

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

“…where denotes the tensor product R θ 1 and R θ 2 , where R θ 1 and R θ 2 are the 1D-DFrFT kernel proposed by Pai and Yes (Pei & Yeh, 1996, 1997. Two parameters θ 1 and θ 2 in the DFrFT indicate the individual fractional orders in two dimensions.…”