skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Pseudo-differential representation of the metaplectic transform and its application to fast algorithms

Abstract

The metaplectic transform (MT), also known as the linear canonical transform, is a unitary integral mapping that is widely used in signal processing and can be viewed as a generalization of the Fourier transform. For a given function $ψ$ on an $N$-dimensional continuous space q, the MT of $ψ$ is parameterized by a rotation (or more generally, a linear symplectic transformation) of the 2$N$-dimensional phase space (q,p), where p is the wavevector space dual to q. In this study, we derive a pseudo-differential form of the MT. For small-angle rotations, or near-identity transformations of the phase space, it readily yields asymptotic differential representations of the MT, which are easy to compute numerically. Rotations by larger angles are implemented as successive applications of $$K \gg 1$$ small-angle MTs. The algorithm complexity scales as $$O(KN^3N_p)$$ where $$N_p$$ is the number of grid points. Here, we present a numerical implementation of this algorithm and discuss how to mitigate the associated numerical instabilities.

Authors:
ORCiD logo [1];  [2]
+ Show Author Affiliations
  1. Princeton Univ., NJ (United States). Dept. of Astrophysical Sciences
  2. Princeton Univ., NJ (United States). Dept. of Astrophysical Sciences; Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
Publication Date:
Research Org.:
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1573502
Alternate Identifier(s):
OSTI ID: 1570879
Grant/Contract Number:  
AC02-09CH11466
Resource Type:
Accepted Manuscript
Journal Name:
Journal of the Optical Society of America. A, Optics, Image Science, and Vision
Additional Journal Information:
Journal Volume: 36; Journal Issue: 11; Journal ID: ISSN 1084-7529
Publisher:
Optical Society of America (OSA)
Country of Publication:
United States
Language:
English
Subject:
79 ASTRONOMY AND ASTROPHYSICS

Citation Formats

Lopez, N. A., and Dodin, I. Y. Pseudo-differential representation of the metaplectic transform and its application to fast algorithms. United States: N. p., 2019. Web. doi:10.1364/JOSAA.36.001846.
Copy to clipboard
Lopez, N. A., & Dodin, I. Y. Pseudo-differential representation of the metaplectic transform and its application to fast algorithms. United States. doi:10.1364/JOSAA.36.001846.
Copy to clipboard
Lopez, N. A., and Dodin, I. Y. Tue . "Pseudo-differential representation of the metaplectic transform and its application to fast algorithms". United States. doi:10.1364/JOSAA.36.001846. https://www.osti.gov/servlets/purl/1573502.
Copy to clipboard
@article{osti_1573502,
title = {Pseudo-differential representation of the metaplectic transform and its application to fast algorithms},
author = {Lopez, N. A. and Dodin, I. Y.},
abstractNote = {The metaplectic transform (MT), also known as the linear canonical transform, is a unitary integral mapping that is widely used in signal processing and can be viewed as a generalization of the Fourier transform. For a given function $ψ$ on an $N$-dimensional continuous space q, the MT of $ψ$ is parameterized by a rotation (or more generally, a linear symplectic transformation) of the 2$N$-dimensional phase space (q,p), where p is the wavevector space dual to q. In this study, we derive a pseudo-differential form of the MT. For small-angle rotations, or near-identity transformations of the phase space, it readily yields asymptotic differential representations of the MT, which are easy to compute numerically. Rotations by larger angles are implemented as successive applications of $K \gg 1$ small-angle MTs. The algorithm complexity scales as $O(KN^3N_p)$ where $N_p$ is the number of grid points. Here, we present a numerical implementation of this algorithm and discuss how to mitigate the associated numerical instabilities.},
doi = {10.1364/JOSAA.36.001846},
journal = {Journal of the Optical Society of America. A, Optics, Image Science, and Vision},
number = 11,
volume = 36,
place = {United States},
year = {2019},
month = {10}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI:  10.1364/JOSAA.36.001846
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal

Figures / Tables:

FIG. 1 FIG. 1: The Riemann sheet (colored) of the function $f$($z$) ≐ $\sqrt{z}$ illustrates the relationship between the family { $\hat{M}_t$} (more generally, the metaplectic group) and the family of all phase-space rotations (more generally, the symplectic group) for all $t$. As depicted in the figure, $f$ maps a closed curvemore » on the $z$ plane (dashed) to a closed curve on the Riemann sheet only if the winding number is even. Likewise, it takes two rotation periods for $\hat{M}_t$ to return to its original value $\hat{M}_ 0 $= $\hat{1}$. In a more general formulation, the metaplectic group forms a double-cover of the symplectic group. For details, see Ref. [1].« less
All figures and tables (9 total)
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals
journal, January 2010

  • Koç, Aykut; Ozaktas, Haldun M.; Hesselink, Lambertus
  • Journal of the Optical Society of America A, Vol. 27, Issue 6
  • DOI: 10.1364/JOSAA.27.001288

Lectures on nonlinear orbit dynamics
conference, January 1982

Segmented fast linear canonical transform
journal, January 2018

  • Sun, Yan-Nan; Li, Bing-Zhao
  • Journal of the Optical Society of America A, Vol. 35, Issue 8
  • DOI: 10.1364/JOSAA.35.001346

Quasioptical modeling of wave beams with and without mode conversion. I. Basic theory
journal, July 2019

  • Dodin, I. Y.; Ruiz, D. E.; Yanagihara, K.
  • Physics of Plasmas, Vol. 26, Issue 7
  • DOI: 10.1063/1.5095076

Fast numerical algorithm for the linear canonical transform
journal, January 2005

  • Hennelly, Bryan M.; Sheridan, John T.
  • Journal of the Optical Society of America A, Vol. 22, Issue 5
  • DOI: 10.1364/JOSAA.22.000928

Synchrotron radiation representation in phase space
journal, May 2012

A non-negative Wigner-type distribution
journal, January 1976

Digital computation of the fractional Fourier transform
journal, January 1996

  • Ozaktas, H. M.; Arikan, O.; Kutay, M. A.
  • IEEE Transactions on Signal Processing, Vol. 44, Issue 9
  • DOI: 10.1109/78.536672

The Symplectic Group and Classical Mechanics
journal, June 2005

Noninterferometric phase retrieval using a fractional Fourier system
journal, December 2007

  • Gopinathan, Unnikrishnan; Situ, Guohai; Naughton, Thomas J.
  • Journal of the Optical Society of America A, Vol. 25, Issue 1
  • DOI: 10.1364/JOSAA.25.000108

Linear Canonical Transformations and Their Unitary Representations
journal, August 1971

  • Moshinsky, M.; Quesne, C.
  • Journal of Mathematical Physics, Vol. 12, Issue 8
  • DOI: 10.1063/1.1665805

On the Quantum Correction For Thermodynamic Equilibrium
journal, June 1932

Improved implementation algorithms of the two-dimensional nonseparable linear canonical transform
journal, January 2012

  • Ding, Jian-Jiun; Pei, Soo-Chang; Liu, Chun-Lin
  • Journal of the Optical Society of America A, Vol. 29, Issue 8
  • DOI: 10.1364/JOSAA.29.001615

Fast linear canonical transforms
journal, December 2009

  • Healy, John J.; Sheridan, John T.
  • Journal of the Optical Society of America A, Vol. 27, Issue 1
  • DOI: 10.1364/JOSAA.27.000021

Chirp slope keying for underwater communications
conference, May 2005

  • Kaminsky, Edit J.; Simanjuntak, Lastri
  • Defense and Security, SPIE Proceedings
  • DOI: 10.1117/12.605426

Phase-space tomography with a programmable Radon–Wigner display
journal, January 2011

  • Cámara, Alejandro; Alieva, Tatiana; Rodrigo, José A.
  • Optics Letters, Vol. 36, Issue 13
  • DOI: 10.1364/OL.36.002441

Operator methods in physical optics
journal, January 1981

Image rotation, Wigner rotation, and the fractional Fourier transform
journal, January 1993

Metaplectic formulation of linear mode conversion
journal, September 1993

Some properties of a non-negative quantum-mechanical distribution function
journal, September 1981

Lens-System Diffraction Integral Written in Terms of Matrix Optics*
journal, January 1970

The semiclassical evolution of wave packets
journal, May 1986

Two-dimensional nonseparable discrete linear canonical transform based on CM-CC-CM-CC decomposition
journal, January 2016

  • Pei, Soo-Chang; Huang, Shih-Gu
  • Journal of the Optical Society of America A, Vol. 33, Issue 2
  • DOI: 10.1364/JOSAA.33.000214

Local fields for asymptotic matching in multidimensional mode conversion
journal, August 2007

  • Tracy, E. R.; Kaufman, A. N.; Jaun, A.
  • Physics of Plasmas, Vol. 14, Issue 8
  • DOI: 10.1063/1.2748051

Symplectically Invariant WKB Wave Functions
journal, April 1985

Precision measurement beyond the shot-noise limit
journal, July 1987

Smoothing and Differentiation of Data by Simplified Least Squares Procedures.
journal, July 1964

  • Savitzky, Abraham.; Golay, M. J. E.
  • Analytical Chemistry, Vol. 36, Issue 8
  • DOI: 10.1021/ac60214a047
    [ × clear filter / sort ]

    Figures / Tables found in this record:

    FIG. 1 (p. 2)figure
    FIG. 2 (p. 6)figure
    FIG. 3 (p. 7)figure
    FIG. 4 (p. 8)figure
    FIG. 5 (p. 10)figure
    FIG. 6 (p. 11)figure
    FIG. 7 (p. 11)figure
    FIG. 8 (p. 12)figure
    FIG. 9 (p. 13)figure
      [ × clear filter / sort ]
      Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.

      Similar Records in DOE PAGES and OSTI.GOV collections: