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

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Space-time-frequency moment densities

David H. Hughes, Bruce W. Suter

Published Dec 1, 2002• 0 citations
Publisher

Abstract

The frequency operator, Ω equivalent to i (partial derivative) δ/δt, is not necessarily Hermitian when acting on nonstationary signals. Central moment densities of Ω and its conjugate, temporal operator Τ, are proportional to the real parts of local central moments derived from the Wigner distribution. A nonrelativistic space-time-frequency Wigner distribution forms the backdrop and motivation for the present investigation.

DOI
10.1117/12.455783

Related papers

variant Time-Frequency Distributions ased on Conjugate Operators

Franz Hlawatsch, Helmut Bölcskei

We propose classes of quadratic time-frequency distributions that retain the inner structure of Cohen's class. Each of these classes is based on a pair of conjugate unitary operators producing time-frequency displacements. The classes satisfy covariance and marginal properties corresponding to these operators. For each class, we define a central member generalizing the Wigner distribution and the Q-distribution, and we specify a transformation by which the class can be derived from Cohen's class.

•Jan 1, 1996•0 citations

Wigner-type a-b and time-frequency analysis based on conjugate operators

Franz Hlawatsch, T. Twaroch, Helmut Bölcskei

We extend the Wigner distribution (WD) to conjugate unitary operators A/sub /spl alpha// and B/sub /spl beta//. The resulting "AB-WD" is defined both as an a-b representation and as a time-frequency representation. Important properties and relations of the WD are generalized to the AB-WD.

IEEE International Conference on Acoustics Speech and Signal Processing•Dec 23, 2002•4 citations

Mutual space-frequency distribution of Gaussian signal

Yura Kozlovskii

Mutual space-frequency distribution is proposed and it is shown that Wigner and Weyl distribution functions are only particular cases of these distribution. Mutual distribution for Gaussian signal is analytically obtained. The simple connection between Wigner and Weyl distributions is established. It is shown that Wigner distribution forms as the rotational displacement of Weyl distribution on informational diagram of conjugate coordinates $(x;p)$ on an angle proportional to the mutual parameter $t$. The results of direct calculations of mutual distribution for Gaussian signal in the mutual domain are presented.

arXiv:0810.2559•Oct 14, 2008•0 citations

I. On the formation of quantum mechanical operators. II. The Wigner distribution function

John Robert Shewell

Although it was in fact the study of the Wigner distribution function that suggested to the author the general problem discussed in Part I, a consideration of the question of the formation of quantum mechanical operators corresponding to dynamical variables, being of fundamental importance to quantum theory, should properly precede a discussion of the Wigner distribution function. In both Parts I and II spin and relativistic effects will be ignored.\nA word about notation is in order. A symbol such as A will denote the quantum mechanical operator corresponding to the dynamical quantity A. For example, H is the Hamiltonian operator; q is the coordinate operator and p the conjugate momentum operator. Whenever there is a possibility of confusion, O(A) is used to represent the operator corresponding to the dynamical quantity A. For example, O(pq) is the operator corresponding to the product pq, and O(H2) is the operator corresponding to the square of the classical Hamiltonian.\nA subscript appearing on a differential operator, such as in 66PH , indicates the function on which the operator acts. Unless otherwise indicated all integrations are to be taken from -infinity to infinity. As usual, h denotes Planck's constant divided by 2pi.\nSince any physical investigation is directly dependent on the quality of the tools used in the investigation, mathematical rigor is much to be desired in physical theory. That rigor has not been achieved in this work is all too obvious, but it was nonetheless a constant goal.

•Jan 1, 1958•0 citations

Moments of the Wigner Distribution and a Generalized Uncertainty Principle

Richard M Simon, Narasimhaiengar Mukunda

The nonnegativity of the density operator of a state is faithfully coded in its Wigner distribution, and this places constraints on the moments of the Wigner distribution. These constraints are presented in a canonically invariant form which is both concise and explicit. Since the conventional uncertainty principle is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. Possible application in quantum state reconstruction using optical homodyne tomography is noted.

arXiv:quant-ph/9708037•Aug 22, 1997•4 citations

The fractional fourier transform and the wigner distribution

David Mustard

Abstract The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms.

ANZIAM Journal•Oct 1, 1996•82 citations
The frequency operator, Ω equivalent to i (partial derivative) δ/δt, is not necessarily Hermitian when acting on nonstationary signals. Central moment densities of Ω and its conjugate, temporal operator Τ, are proportional to the real parts of local central moments derived from the Wigner distribution. A nonrelativistic space-time-frequency Wigner distribution forms the backdrop and motivation for the present investigation.
We propose classes of quadratic time-frequency distributions that retain the inner structure of Cohen's class. Each of these classes is based on a pair of conjugate unitary operators producing time-frequency displacements. The classes satisfy covariance and marginal properties corresponding to these operators. For each class, we define a central member generalizing the Wigner distribution and the Q-distribution, and we specify a transformation by which the class can be derived from Cohen's class.
Mutual space-frequency distribution is proposed and it is shown that Wigner and Weyl distribution functions are only particular cases of these distribution. Mutual distribution for Gaussian signal is analytically obtained. The simple connection between Wigner and Weyl distributions is established. It is shown that Wigner distribution forms as the rotational displacement of Weyl distribution on informational diagram of conjugate coordinates (x;p)(x;p)(x;p) on an angle proportional to the mutual parameter ttt. The results of direct calculations of mutual distribution for Gaussian signal in the mutual domain are presented.
Although it was in fact the study of the Wigner distribution function that suggested to the author the general problem discussed in Part I, a consideration of the question of the formation of quantum mechanical operators corresponding to dynamical variables, being of fundamental importance to quantum theory, should properly precede a discussion of the Wigner distribution function. In both Parts I and II spin and relativistic effects will be ignored.\nA word about notation is in order. A symbol such as A will denote the quantum mechanical operator corresponding to the dynamical quantity A. For example, H is the Hamiltonian operator; q is the coordinate operator and p the conjugate momentum operator. Whenever there is a possibility of confusion, O(A) is used to represent the operator corresponding to the dynamical quantity A. For example, O(pq) is the operator corresponding to the product pq, and O(H2) is the operator corresponding to the square of the classical Hamiltonian.\nA subscript appearing on a differential operator, such as in 66PH , indicates the function on which the operator acts. Unless otherwise indicated all integrations are to be taken from -infinity to infinity. As usual, h denotes Planck's constant divided by 2pi.\nSince any physical investigation is directly dependent on the quality of the tools used in the investigation, mathematical rigor is much to be desired in physical theory. That rigor has not been achieved in this work is all too obvious, but it was nonetheless a constant goal.
The nonnegativity of the density operator of a state is faithfully coded in its Wigner distribution, and this places constraints on the moments of the Wigner distribution. These constraints are presented in a canonically invariant form which is both concise and explicit. Since the conventional uncertainty principle is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. Possible application in quantum state reconstruction using optical homodyne tomography is noted.
Abstract The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms.