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

AIP conference proceedings

Cohen operators associated with signal representations

Paolo Boggiatto, Evanthia Carypis, Alessandro Oliaro

Published Jan 1, 2017• 3 citations
Publisher

Abstract

In this paper we study the relationships between general quantization rules which yield Cohen operators and the signal time-frequency representations in the Cohen class. After defining a suitable functional setting, we focus on the particular case of the concentration operators which, as the multipliers and Fourier multipliers in the Donoho-Stark uncertainty principle, furnish estimates of the energy concentration on subsets of the time-frequency space. Finally we study the case of positive representations in the Cohen class.

DOI
10.1063/1.5012677

Related papers

A quantum harmonic analysis approach to nonlinear time-frequency concentration

Erling A. T. Svela, S. Ivan Trapasso

We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the existence of optimizers in terms of the ``window operator'' and explicit examples where the supremum is never attained. We also study the structural properties of window operators, in particular operators that yield weakly continuous concentration functionals and operators for which the nonlinear concentration problem admits an optimizer, also beyond the Heisenberg representation. We then consider generalizations to the study of concentration problems for phase space representations of operators. We consider generalized Husimi distributions via quantum convolution, and their optimization problem when optimizing over Hilbert--Schmidt and density operators. Lastly, we consider representations of operators on double phase space, in the spirit of quantum time-frequency analysis, and give a full solution in terms of the Weyl symbols.

arXiv:2605.28786•May 27, 2026•0 citations

Windowed-Wigner Representations, Interferences and Operators

Paolo Boggiatto, Evanthia Carypis, Alessandro Oliaro

Windowed-Wigner" representations, denoted by Wig ψ and Wig * ψ , were introduced in [2] in connection with uncertainty principles and interferences problems.In this paper we present a more precise analysis of their behavior obtaining an estimate of the L 2 -norm of interferences of couples of "model" signals.We further define a suitable functional framework for the associated operators and show that they form a class of pseudodifferential operators which define a natural "path" between the multiplication, Weyl and Fourier multipliers operators.

•Jan 1, 2012•0 citations

Separability, positivity, and minimum uncertainty in time–frequency energy distributions

Patrick Flandrin

Gaussian signals play a very special role in classical time–frequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty, and positivity and separability of Wigner–Ville distributions. We investigate here some of the logical connections which exist between these different features, and we discuss some examples and counterexamples of their extension to more general joint distributions within Cohen’s class and the affine class.

Journal of Mathematical Physics•Aug 1, 1998•27 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

Integral transforms covariant to unitary operators and their implications for joint signal representations

A.M. Sayeed, Douglas L. Jones

Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the "covariant" transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual.

IEEE Transactions on Signal Processing•Jun 1, 1996•33 citations

A general approach for obtaining joint representations in signal analysis. I. Characteristic function operator method

Leon W. Cohen

A general approach for obtaining joint representations for arbitrary physical quantities is presented. The characteristic function operator method of Moyal (1949) and Ville (1960) and generalized by Cohen (1966, 1976) and Scully and Cohen (1987) for the time-frequency case is developed for arbitrary variables.

IEEE Transactions on Signal Processing•May 1, 1996•33 citations
In this paper we study the relationships between general quantization rules which yield Cohen operators and the signal time-frequency representations in the Cohen class. After defining a suitable functional setting, we focus on the particular case of the concentration operators which, as the multipliers and Fourier multipliers in the Donoho-Stark uncertainty principle, furnish estimates of the energy concentration on subsets of the time-frequency space. Finally we study the case of positive representations in the Cohen class.
We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the existence of optimizers in terms of the ``window operator'' and explicit examples where the supremum is never attained. We also study the structural properties of window operators, in particular operators that yield weakly continuous concentration functionals and operators for which the nonlinear concentration problem admits an optimizer, also beyond the Heisenberg representation. We then consider generalizations to the study of concentration problems for phase space representations of operators. We consider generalized Husimi distributions via quantum convolution, and their optimization problem when optimizing over Hilbert--Schmidt and density operators. Lastly, we consider representations of operators on double phase space, in the spirit of quantum time-frequency analysis, and give a full solution in terms of the Weyl symbols.
Windowed-Wigner" representations, denoted by Wig ψ and Wig * ψ , were introduced in [2] in connection with uncertainty principles and interferences problems.In this paper we present a more precise analysis of their behavior obtaining an estimate of the L 2 -norm of interferences of couples of "model" signals.We further define a suitable functional framework for the associated operators and show that they form a class of pseudodifferential operators which define a natural "path" between the multiplication, Weyl and Fourier multipliers operators.
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.
Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the "covariant" transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual.