Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Mathematics

The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

Rui Jing, Bei Liu, Rui Li, Rui Liu

Published Oct 1, 2020• 28 citations
Publisher

Abstract

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.

DOI
10.3390/math8101685

Related papers

Uncertainty Principles for the Short-time Free Metaplectic Transform

M. Younus Bhat, Aamir H. Dar

The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short time free metaplectic transform (STFMT). In this paper, we investigate the STFMT. Firstly, we propose the definition of the STFMT, and provide the time frequency analysis of the proposed transform in the FMT domain. Secondly, we investigate the basic properties of the proposed transform including the reconstruction formula, Moyals formula. The emergence of the STFMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. Finally, we extend some different uncertainty principles (UP) from quantum mechanics including Liebs, Pitts UP, Heisenbergs uncertainty principle, Hausdorff-Young, Hardys UP, Beurlings UP, Logarithmic UP, and Nazarovs UP which have already been well studied in the FMT domain

arXiv:2204.07048•Mar 24, 2022•1 citation

A Metaplectic Perspective of Uncertainty Principles in the Linear Canonical Transform Domain

Nuno Costa Dias, Maurice A. de Gosson, João Nuno Prata

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.

arXiv:2405.10651•Journal of Functional Analysis•Jan 1, 2022•15 citations

Anisotropic uncertainty principles for metaplectic operators

Elena Cordero, Gianluca Giacchi, Edoardo Pucci

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$. Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.

math.APCross-listedarXiv:2601.16279•Jan 26, 2026

Generalized Uncertainty Principles for the Offset Linear Canonical Transform

Haiye Huo

The offset linear canonical transform (OLCT) provides a more general framework for a number of well known linear integral transforms in signal processing and optics, such as Fourier transform, fractional Fourier transform, linear canonical transform. In this paper, to characterize simultaneous localization of a signal and its OLCT, we generalize some different uncertainty principles (UPs), including Nazarov's UP, Hardy's UP, Beurling's UP, logarithmic UP and entropic UP, which have already been well studied in the Fourier transform domain over the last few decades, to the OLCT domain in a broader sense.

•Feb 11, 2018•0 citations

Free Metaplectic Wigner Distribution: Definition and Heisenberg’s Uncertainty Principles

Zhichao Zhang, Zhicheng Zhu, Dong Li +1 more

Inspired by a definition of the closed-form instantaneous cross-correlation Wigner distribution (Zhang, 2019), we generalize the notion of Wigner distribution to the so-called free metaplectic Wigner distribution (FMWD) through three free metaplectic transforms, in order to tackle a challenge in high-dimensional complex features information processing. We provide some representative special cases for this general form, including the$N$-dimensional nonseparable affine characteristic Wigner distribution, kernel function Wigner distribution, convolution representation Wigner distribution and instantaneous cross-correlation Wigner distribution. We establish the standard Heisenberg’s uncertainty principles (HUPs) of the real-valued function for the FMWD. We also establish the standard HUPs of the complex-valued function for some specific (i.e., the orthogonal, the orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative) FMWDs. In view of the one-dimensional case of our results, we solve a burning question regarding the limit of time-frequency superresolution triggered by the linear canonical transform free parameters.

IEEE Transactions on Information Theory•Jun 20, 2023•20 citations

N -dimensional Heisenberg’s uncertainty principle for fractional Fourier transform

Zhichao Zhang

A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N -dimensional Heisenberg’s uncertainty principle is obtained, and extended from N -dimensional Fourier transform domain to two N -dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals’ applications in time-frequency analysis and optical system analysis are also given.

arXiv:1906.05451•Mar 9, 2020•1 citation
The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.
The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various application in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short time free metaplectic transform (STFMT). In this paper, we investigate the STFMT. Firstly, we propose the definition of the STFMT, and provide the time frequency analysis of the proposed transform in the FMT domain. Secondly, we investigate the basic properties of the proposed transform including the reconstruction formula, Moyals formula. The emergence of the STFMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. Finally, we extend some different uncertainty principles (UP) from quantum mechanics including Liebs, Pitts UP, Heisenbergs uncertainty principle, Hausdorff-Young, Hardys UP, Beurlings UP, Logarithmic UP, and Nazarovs UP which have already been well studied in the FMT domain
We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.
We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on L2(Rd)L^2(\mathbb{R}^d)L2(Rd), including degenerate cases associated with symplectic matrices whose BBB-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by rank(B)\mathrm{rank}(B)rank(B). First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to ker⁡(B)⊥\ker(B)^\perpker(B)⊥, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of BBB. Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both fff and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.
The offset linear canonical transform (OLCT) provides a more general framework for a number of well known linear integral transforms in signal processing and optics, such as Fourier transform, fractional Fourier transform, linear canonical transform. In this paper, to characterize simultaneous localization of a signal and its OLCT, we generalize some different uncertainty principles (UPs), including Nazarov's UP, Hardy's UP, Beurling's UP, logarithmic UP and entropic UP, which have already been well studied in the Fourier transform domain over the last few decades, to the OLCT domain in a broader sense.
Inspired by a definition of the closed-form instantaneous cross-correlation Wigner distribution (Zhang, 2019), we generalize the notion of Wigner distribution to the so-called free metaplectic Wigner distribution (FMWD) through three free metaplectic transforms, in order to tackle a challenge in high-dimensional complex features information processing. We provide some representative special cases for this general form, including theNNN-dimensional nonseparable affine characteristic Wigner distribution, kernel function Wigner distribution, convolution representation Wigner distribution and instantaneous cross-correlation Wigner distribution. We establish the standard Heisenberg’s uncertainty principles (HUPs) of the real-valued function for the FMWD. We also establish the standard HUPs of the complex-valued function for some specific (i.e., the orthogonal, the orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative) FMWDs. In view of the one-dimensional case of our results, we solve a burning question regarding the limit of time-frequency superresolution triggered by the linear canonical transform free parameters.
A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N -dimensional Heisenberg’s uncertainty principle is obtained, and extended from N -dimensional Fourier transform domain to two N -dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals’ applications in time-frequency analysis and optical system analysis are also given.