A metaplectic perspective of uncertainty principles in the Linear Canonical Transform domain

TL;DR

This paper develops a comprehensive framework for uncertainty principles in the Linear Canonical Transform domain using metaplectic operators, introducing new phase-space distributions and extending classical principles to higher dimensions.

Contribution

It generalizes uncertainty principles to all signals and dimensions via metaplectic operators, and introduces a novel quadratic phase-space distribution with interpretative advantages.

Findings

Derived Heisenberg uncertainty principles for LCTs using metaplectic operators.

Proposed a new quadratic phase-space distribution with non-negative marginals.

Extended Hardy and Paley-Wiener theorems to the LCT and metaplectic context.

Abstract

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-Schr\"odinger 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.