Computation of the fractional Fourier transform

  title={Computation of the fractional Fourier transform},
  author={Adhemar Bultheel and Hector Martinez Sulbaran},
  journal={Applied and Computational Harmonic Analysis},
A New Formulation of the Fast Fractional Fourier Transform
This work derives a Gaussian-like quadrature of the continuous fractional Fourier transform from a bilinear form of eigenvectors of the matrix associated to the recurrence equation of the Hermite polynomials, which becomes a more accurate version of the FFT and can be used for nonperiodic functions.
Review of Computing Algorithms for Discrete Fractional Fourier Transform
A comparative analysis of the most famous algorithms for the computation of Discrete Fractional Fourier Transform is presented, to portray the major advantages and disadvantages of the previously proposed algorithms so that appropriate algorithm may be selected as per requirements.
Riesz transform associated with the fractional Fourier transform and applications
The fractional Riesz transform associated with fractional Fourier transform is introduced, in which the chirp function is the key factor and the technical barriers to be overcome, and the physical and geometric interpretation of the high-dimensional fractional multiplier theorem is given.
Research progress on discretization of fractional Fourier transform
A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented and it is hoped to offer a doorstep for the readers who are interested in the fractionsal Fouriers transform.
Research progress of the fractional Fourier transform in signal processing
The fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view and a course from the definition to the applications is provided, especially as a reference and an introduction for researchers and interested readers.
Image and video processing using discrete fractional transforms
Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractionsal cosine transform is better in encryption of image and video.
The fractional fourier transform and its application to digital watermarking
  • M. T. Taba
  • Computer Science
    2013 8th International Workshop on Systems, Signal Processing and their Applications (WoSSPA)
  • 2013
This paper describes the implementation of a watermark embedding technique for images using the discrete fractional Fourier transform to recognize the watermark if there is a strong correlation with the embedded watermark.
Spectrum Estimation of Pseudo-random Nonuniformly Sampled Signals in the Fractional Fourier Transform Domain
  • Xu Huifa, Liu Feng
  • Computer Science
    2010 WASE International Conference on Information Engineering
  • 2010
First, the digital spectrum of nonuniformly sampled signals in the Fractional Fourier transform domain is introduced and the result shows that these theories are also effective in the fractional Fouriers transform domain.
Optimal Step Size for the Adaptive Least-Mean Squares Algorithm Applied in the Fractional Fourier Transform Domain for Efficient Signal Estimation in Interference and Noise
© 2015 IEEE. Reprinted, with Permission, from 14 th Canadian Workshop on Information Theory Abstract—The Fractional Fourier Transform (FrFT) has wide applications in communications and signal


Discrete fractional Fourier transform based on orthogonal projections
The proposed DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT and will provide similar transform and rotational properties as those of continuous fractional Fourier transforms.
The fractional Fourier transform and time-frequency representations
The authors briefly introduce the functional Fourier transform and a number of its properties and present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time- frequencies such as the Wigner distribution, the ambiguity function, the short-time Fouriertransform and the spectrogram.
The Fractional Order Fourier Transform and its Application to Quantum Mechanics
We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct
The discrete fractional Fourier transform
This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions and supports confidence that it will be accepted as the definitive definition of this transform.
Digital Computation of Fractional Fourier Transform
  • Zhou Min
  • Engineering, Computer Science
  • 2002
Simulation results indicate that the energy of LFM signal will be collected effectively when the fractional order is matching with its modulation slope and in weak signals detection of underwater acoustic domain, the authors can get high anti-Doppler performance using the Fractional fourier transform algorithm.
A multi-input-multi-output system approach for the computation of discrete fractional Fourier transform
Multiplicity of fractional Fourier transforms and their relationships
A general FRT definition is generated, based on eigenfunctions and eigenvalues of the ordinary Fourier transform, which allows us to generate all possible definitions and gives explicit relationships between the different FRTs.
Improved discrete fractional Fourier transform.
An improved DFRFT is proposed that provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.
Understanding discrete rotations
  • M. Richman, T. Parks
  • Computer Science
    1997 IEEE International Conference on Acoustics, Speech, and Signal Processing
  • 1997
By studying a 90 degree rotation, an algorithm to compute a prime-length discrete Fourier transform (DFT) based on convolutions and multiplications of discrete, periodic chirps is formulated, providing a further connection between the DFT and the discrete Wigner distribution based on group theory.