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# 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:

- 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:
- 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.
```

```
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.
```

```
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.
```

```
@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}

}

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