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AIP conference proceedings

The Wigner-Ville distribution associated with quadratic-phase Fourier transform

Pratap Bhanu Sharma

Published Jan 1, 2022• 11 citations
Publisher

Abstract

The generalization of the Wignar-Ville distribution to the quadratic-phase Fourier transform so-called the Wigner-Ville distribution associated with quadratic-phase Fourier transform (WVD-QPFT) is defined and its some fundamental properties are investigated. Moreover, reconstruction formula and Moyal's formula are established for the WVD-QPFT.

DOI
10.1063/5.0083549

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The generalization of the Wignar-Ville distribution to the quadratic-phase Fourier transform so-called the Wigner-Ville distribution associated with quadratic-phase Fourier transform (WVD-QPFT) is defined and its some fundamental properties are investigated. Moreover, reconstruction formula and Moyal's formula are established for the WVD-QPFT.
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The main objective of the paper is to generalize and enrich the Weyl transform by introducing the Weyl correspondence in the linear canonical transform (LCT) domain. In this paper, we propose the linear canonical-Wigner transform in harmonic analysis of phase space along with the admissible Wigner-Ville distribution (WVD) and Weyl transform in the LCT domain and discuss some useful results. Further we establish the relationship between the Wigner-Ville distribution and the Weyl transform in the LCT domain.
The Fourier–Wigner transform of two functions is first defined and is given an alternative expression. The Wigner transform is then defined to be the Fourier transform of the Fourier–Wigner transform. The Moyal identity, which is an analog of the Plancherel formula, is then proved for the Wigner transform. The Weyl transform with a given symbol is then defined in terms of the Wigner transform. Another formula for the Weyl transform with a given symbol is then proved and used to establish the fact that the product of two Weyl transforms is also a Weyl transform with symbol given by a twisted convolution of the two given symbols.
The metaplectic operator has shown to be a valid technique for generalizing the notion of cross Wigner distribution to achieve time-frequency superresolution. Inspired by the latest work (Cordero and Rodino, 2022), we revisit the notion of cross Wigner distribution in metaplectic transform domains by introducing two2N×2N2N\times 2N2N×2Nsymplectic matrices rather than integrating them into one4N×4N4N\times 4N4N×4Nsymplectic matrix. We style the derived general formulation as the cross metaplectic Wigner distribution and obtain its basic properties including time translation property, frequency modulation property, time translation and frequency modulation property, Moyal formula, complex conjugate symmetry, time reversal symmetry, and scaling property. We use it to define the so-called short-time metaplectic transform and clarify the equivalence between them. We establish the standard Heisenberg’s uncertainty principles for the cross metaplectic Wigner distribution, i.e., an attainable lower bound for two real-valued functions in cross metaplectic Wigner distribution domains and a sequence of attainable (unattainable) lower bounds for two complex-valued (one real-valued and the other complex-valued) functions in orthogonal, orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative cross metaplectic Wigner distribution domains. We further demonstrate the time-frequency superresolution superiority of the derived results over the conventional one through theoretical analyses and numerical experiments.
In the paper we made a generalization of the Fourier transform in the complex space, applicable to the pair of Husimi and P-quasi distributions, in the representation of nonlinear coherent states. Implicitly, our result is a generalization similar result of Mehta, but which referred only to the canonical coherent states (associated with the one-dimensional harmonic oscillator). Our result is valid for both types of coherent states (defined in the Barut-Girardello, respectively Klauder-Perelomov manner).