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

IEEE Transactions on Signal Processing

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

Leon W. Cohen

Published May 1, 1996• 33 citations
Publisher

Abstract

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.
DOI
10.1109/78.502322

Related papers

Joint distributions for arbitrary variables

Leon W. Cohen, Richard G. Baraniuk

We compare the methods of Cohen (1971) and of Baraniuk and Jones (see Proc. IEEE ICASSP-93, p.320-323, 1993) for obtaining joint distributions for arbitrary variables. We show that the two methods produce identical results for variables whose associated operators are obtained via a unitary transformation of the time and frequency operators. In addition, we generalize this result and show that all pairs of variables connected by a unitary transformation have joint distributions that are identical in functional form. This result shows how pairs of variables can be grouped into classes whose joint distributions are functionally equivalent and, therefore, share equivalent properties.>

IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis•Dec 17, 2002•2 citations

Topics in time-frequency representations

Leon W. Cohen

We discuss some issues relevant to time-frequency analysis. Among them are, time-frequency spatial/spatial-frequency representations, generalized marginals and generalized characteristic functions. We derive representations that jointly involve the four variables, time, frequency, space, and spatial frequency. We apply these representations to wave packet motion. For the study on generalized marginals we show that there is an inherent difference between the standard methods of transforming variables and in time- frequency analysis. We use the method of generalized characteristic function operator method to generalize Taylor's theorem.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Oct 22, 1996•5 citations

Transformation of operators and scale

Leon W. Cohen

Explicit expressions are given for the transformation of an operator from one representation to another. Both continuous and discrete representations are considered. Examples are given for the time-frequency and scale operators in various representations including the Hermite, Fourier, discrete Fourier, scale, time, and sample theorem representations. Also, we generalize the characteristics operator method where we use an arbitrary complete set rather than complex exponentials.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Oct 28, 1994•0 citations

Marginals vs. covariance in joint distribution theory

Richard G. Baraniuk

Cohen (1966, 1995) has proposed a method for constructing joint distributions of arbitrary physical quantities, in direct generalization of joint time-frequency representations. We investigate the covariance properties of this procedure and caution that in its present form it cannot generate all possible distributions. Using group theory, we extend Cohen's construction to a more general form that can be customized to satisfy specific marginal and covariance requirements.

IEEE International Conference on Acoustics Speech and Signal Processing•Nov 19, 2002•13 citations

Time-frequency representations and operators

Leon W. Cohen

We discuss the issues that arise in developing quasi-distributions for arbitrary operators in contrast to the usual case of time and frequency. The arbitrary operator case has many mathematical and physical challenges that have not been solved. We also discuss the connection with differential equations and pseudo-differential operators. In regard to differential equations we argue that the proper generalization of the constant coefficient case is not the variable coefficients case but an equation where the coefficients are kept constant and the differential operator is replaced by a Hermitian operator.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 28, 2009•0 citations

Beyond time-frequency analysis: energy densities in one and many dimensions

Richard G. Baraniuk

Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the "A" content of signals. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen's (see Englewood Cliffs, NJ: Prentice-Hall, 1995) general construction for joint distributions of arbitrary variables.

IEEE Transactions on Signal Processing•Jan 1, 1998•23 citations
We compare the methods of Cohen (1971) and of Baraniuk and Jones (see Proc. IEEE ICASSP-93, p.320-323, 1993) for obtaining joint distributions for arbitrary variables. We show that the two methods produce identical results for variables whose associated operators are obtained via a unitary transformation of the time and frequency operators. In addition, we generalize this result and show that all pairs of variables connected by a unitary transformation have joint distributions that are identical in functional form. This result shows how pairs of variables can be grouped into classes whose joint distributions are functionally equivalent and, therefore, share equivalent properties.>
We discuss some issues relevant to time-frequency analysis. Among them are, time-frequency spatial/spatial-frequency representations, generalized marginals and generalized characteristic functions. We derive representations that jointly involve the four variables, time, frequency, space, and spatial frequency. We apply these representations to wave packet motion. For the study on generalized marginals we show that there is an inherent difference between the standard methods of transforming variables and in time- frequency analysis. We use the method of generalized characteristic function operator method to generalize Taylor's theorem.
We discuss the issues that arise in developing quasi-distributions for arbitrary operators in contrast to the usual case of time and frequency. The arbitrary operator case has many mathematical and physical challenges that have not been solved. We also discuss the connection with differential equations and pseudo-differential operators. In regard to differential equations we argue that the proper generalization of the constant coefficient case is not the variable coefficients case but an equation where the coefficients are kept constant and the differential operator is replaced by a Hermitian operator.
Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the "A" content of signals. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen's (see Englewood Cliffs, NJ: Prentice-Hall, 1995) general construction for joint distributions of arbitrary variables.