Pseudodifferential 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 pseudodifferential form of the MT. For smallangle rotations, or nearidentity 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 $$K \gg 1$$ smallangle 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:

 Princeton Univ., NJ (United States). Dept. of Astrophysical Sciences
 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:
 AC0209CH11466
 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 10847529
 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. Pseudodifferential representation of the metaplectic transform and its application to fast algorithms. United States: N. p., 2019.
Web. doi:10.1364/JOSAA.36.001846.
Lopez, N. A., & Dodin, I. Y. Pseudodifferential representation of the metaplectic transform and its application to fast algorithms. United States. doi:10.1364/JOSAA.36.001846.
Lopez, N. A., and Dodin, I. Y. Tue .
"Pseudodifferential 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.
@article{osti_1573502,
title = {Pseudodifferential 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 pseudodifferential form of the MT. For smallangle rotations, or nearidentity 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 $K \gg 1$ smallangle 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}
}
Figures / Tables:
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