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Royal Society Open Science

Uncertainty principles for coupled fractional Wigner–Ville distribution

Andi Tenri Ajeng Nur, Mawardi Bahri, Nasrullah Bachtiar, Amran Rahim

Published May 1, 2024• 3 citations
Publisher

Abstract

The coupled fractional Wigner-Ville distribution is a more general version of the fractional Wigner-Ville distribution. Main properties including boundedness, Moyal's formula and inversion formula are studied in detail for the transformation. Additionally, the relation of the coupled fractional Wigner-Ville distribution with the two-dimensional Fourier transform is studied. We also present the relationship between the coupled fractional Wigner-Ville distribution with the two-dimensional Wigner-Ville distribution. We show how the properties and relations allow us to derive several versions of the uncertainty inequalities related to the coupled fractional Wigner-Ville distribution.

DOI
10.1098/rsos.231579

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The coupled fractional Wigner-Ville distribution is a more general version of the fractional Wigner-Ville distribution. Main properties including boundedness, Moyal's formula and inversion formula are studied in detail for the transformation. Additionally, the relation of the coupled fractional Wigner-Ville distribution with the two-dimensional Fourier transform is studied. We also present the relationship between the coupled fractional Wigner-Ville distribution with the two-dimensional Wigner-Ville distribution. We show how the properties and relations allow us to derive several versions of the uncertainty inequalities related to the coupled fractional Wigner-Ville distribution.
The coupled fractional Fourier transform is known to be a generalized version of the fractional Fourier transform. This research paper aims to propose two inequalities related to the coupled fractional Fourier transform. The results are generalizations of uncertainty principles concerning the Fourier transform, emphasizing that the essential principles have not yet been published in the academic literature. The results show that the uncertainty principles were obtained using the properties of the coupled fractional Fourier transform and its relation to the classical Fourier transform. Keywords: Coupled Fractional Fourier Transform, Fourier Transform, Uncertainty Principle DOI： https://doi.org/10.35741/issn.0258-2724.58.2.32
The focus of this article is to propose two different versions of the Heisenberg uncertainty principle related to the fractional Fourier transform (FrFT). The first version is directly derived using the basic connection between the traditional Fourier transform (FT) and the FrFT, and the second is derived by exploiting time differentiation property and Parseval’s formula for the FrFT. In addition, a weighted uncertainty inequality related to this transformation is investigated. As a simple illustration of the use of the obtained results, we design the fractional characteristic function. An inequality describing the relation between the probability density function and its fractional characteristic function is presented.
The displacement operator is related to the displaced parity operator through a two dimensional \nFourier transform. Both operators are important operators in phase space \nand the trace of both with respect to the density operator gives the Wigner functions \n(displaced parity operator) and Weyl functions (displacement operator). The generalisation \nof the parity-displacement operator relationship considered here is called \nthe bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional \ndisplacement operators lead to the novel concept of bi-fractional coherent states. \nThe generalisation from Fourier transform to fractional Fourier transform can be \napplied to other phase space functions. The case of the Wigner-Weyl function is considered \nand a generalisation is given, which is called the bi-fractional Wigner functions, \nH(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to \ngive the bi-fractional Q−functions and bi-fractional P−functions respectively. The \ngeneralisation is likewise applied to the Moyal star product and Berezin formalism for \nproducts of non-commutating operators. These are called the bi-fractional Moyal star \nproduct and bi-fractional Berezin formalism. \nFinally, analysis, applications and implications of these bi-fractional transforms \nto the Heisenberg uncertainty principle, photon statistics and future applications are \ndiscussed.
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.
It is well known that the fractional Fourier transform (FrFT) is a non-trivial generalization of the classical Fourier transform. This transform has been extensively utilized in signal analysis and image processing. The uncertainty principle is one of the fundamental results related to the FrFT. In the present paper, we first propose the different method in proving a variant of Heisenberg uncertainty principles associated with the FrFT. This method combines the Heisenberg-type uncertainty principle pertaining to the FrFT and the additive property of the FrFT. We then provide an alternative proof of the logarithmic inequality for the FRFT and find that this result differs slightly that in the literature. Finally, we explicitly expand the variant of Heisenberg uncertainty principle to a more general case.