Journal ArticleDOI
Geometry and dynamics in the fractional discrete Fourier transform
TLDR
This work compares the fidelity in reproducing the classical harmonic motion of discrete coherent states of the N x N Fourier matrix with several options considered in the literature.Abstract:
The N×N Fourier matrix is one distinguished element within the group U(N) of all N×N unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N→∞ contraction limit for the integral Fourier transform and its fractional powers. In the finite-N case, several options have been considered in the literature. We compare their fidelity in reproducing the classical harmonic motion of discrete coherent states.read more
Citations
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Journal ArticleDOI
Digital Computation of Linear Canonical Transforms
TL;DR: The algorithms compute LCTs with a performance similar to that of the fast Fourier transform algorithm in computing the Fouriertransform, both in terms of speed and accuracy.
Wigner distribution function for finite systems
TL;DR: In this paper, a Wigner distribution function for finite optical data sets is proposed, which assigns classical c-numbers to the operators of position, momentum, and wave guide mode.
Journal ArticleDOI
Analysis and comparison of discrete fractional fourier transforms
Xinhua Su,Ran Tao,Xuejing Kang +2 more
TL;DR: This paper systematically analyze and compare the main DFRFT types: sampling-type DFR FTs and eigenvector decomposition-typeDFRFTs and discrete counterparts of the linear canonical transform (LCT), simplified FRFT (SFRFT) are summarized and classified.
Journal ArticleDOI
Discrete repulsive oscillator wavefunctions
TL;DR: In this paper, the authors used the three-dimensional Lorentz algebra and group SO(2,1) to model the repulsive oscillator in the form of a hypergeometric function, where the right and left-moving wavefunctions are given by hypergeometrical functions that form a Dirac basis for � 2 (Z).
Journal ArticleDOI
On discrete Gauss-Hermite functions and eigenvectors of the discrete Fourier transform
TL;DR: inspired by concepts from quantum mechanics in finite dimensions, this approach furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the G-H differential operator and in the process furnishes two generators of the group of matrices that commute with the DFT.
References
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Journal ArticleDOI
The Fractional Order Fourier Transform and its Application to Quantum Mechanics
TL;DR: In this article, a generalized operational calculus is developed, paralleling the familiar one for the ordinary transform, which provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians.
Book
The Fractional Fourier Transform: with Applications in Optics and Signal Processing
TL;DR: The fractional Fourier transform (FFT) as discussed by the authors has been used in a variety of applications, such as matching filtering, detection, and pattern recognition, as well as signal recovery.
Journal ArticleDOI
Lens-System Diffraction Integral Written in Terms of Matrix Optics
TL;DR: In this paper, a diffraction integral is derived which relates the electromagnetic fields on the input plane of a lens system to those on its output plane, which indicates a connection between ray optics and diffraction theory.
Journal ArticleDOI
Digital computation of the fractional Fourier transform
TL;DR: An algorithm for efficient and accurate computation of the fractional Fourier transform for signals with time-bandwidth product N, which computes the fractionsal transform in O(NlogN) time.
Book
Classical Orthogonal Polynomials of a Discrete Variable
TL;DR: In this article, the orthogonality relation (2.0.1) is reduced to 2.0, where w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ i at the points x = x i.