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Journal of Mathematical Physics

Positivity properties of phase-plane distribution functions

Augustus J. E. M. Janssen

Published Jul 1, 1984• 29 citations
Publisher

Abstract

The aim of this paper is to compare the members of Cohen’s class of phase-plane distributions with respect to positivity properties. It is known that certain averages (which are in a sense compatible with Heisenberg’s uncertainty principle) of the Wigner distribution over the phase-plane yield non-negative values for all states. It is shown in this paper that the Wigner distribution is unique in this respect among the members of Cohen’s class that have correct marginals or that satisfy Moyal’s formula for all states. The subset of members of Cohen’s class (not necessarily satisfying one of these two conditions) with positivity properties comparable with those for the Wigner distribution is shown to be rather small.

DOI
10.1063/1.526417

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The aim of this paper is to compare the members of Cohen’s class of phase-plane distributions with respect to positivity properties. It is known that certain averages (which are in a sense compatible with Heisenberg’s uncertainty principle) of the Wigner distribution over the phase-plane yield non-negative values for all states. It is shown in this paper that the Wigner distribution is unique in this respect among the members of Cohen’s class that have correct marginals or that satisfy Moyal’s formula for all states. The subset of members of Cohen’s class (not necessarily satisfying one of these two conditions) with positivity properties comparable with those for the Wigner distribution is shown to be rather small.
There is a theorem of Wigner that states that phase-plane distribution functions involving the state bilinearly and having correct marginals must take negative values for certain states. The purpose of this paper is to support the statement that these phase-plane distribution functions are for hardly any state everywhere non-negative. In particular, it is shown that for certain generalized Wigner distribution functions there are no smooth states (except the Gaussians for the Wigner distribution function itself) whose distribution function is everywhere non-negative. This class of Wigner-type distribution functions contains the Margenau–Hill distribution. Furthermore, the argument used in the proof of Wigner’s theorem is augmented to show that under mild conditions one can find for any two states f, g with non-negative distribution functions a linear combination h of f and g whose distribution function takes negative values, unless f and g are proportional.
The Wigner distribution is a milestone of Time–frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen’s class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781–786; Time–frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.
Here, we have considered the Husimi and Wigner distribution functions. We have, inparticular, shown how the uncertainty principle works for the two cases of simple and damped harmonicoscillators when either of these two distributions are used. The conclusion shows that the Husimidistribution function remains non-negative through the phase space but it does not always satisfies theuncertainty principle. On the other hand the Wigner distribution function while becoming negative incertain regions of the phase space do not violate the uncertainty principle.
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.
Marginal properties of arbitrary s-ordered quasiprobability distribution fucntions are investigated. The positivity of marginal distributions for all quasiprobabilities, interpolating between the Wigner function and the Q representation of the density operator (s\ensuremath{\le}0), is proven. We also show that, on the contrary, marginal distributions of the remaining quasiprobabilities, interpolating between the Wigner function and the P representation (s>0), can take on negative values. General formulas for the marginal distributions are given, and their relations to the actual quantum-mechanical probability distributions for position and momentum are established. Our results provide an interesting generalization of some recent results concerning marginal properties of the P representation [G. S. Agarwal, Opt. Commun. 95, 109 (1993)].
Various time-frequency pseudo-density functions used in signal analysis are compared with respect to spread. Among the members of Cohen's class of pseudo-density functions satisfying the finite support property as well as Moyal's formula, the Wigner distribution is the most well-behaved one in the sense that it has the least amount of global spread around its center of gravity. The Wigner distribution does not perform significantly better globally than the real part of Rihaczek's function; it does, though, if the global criterion is replaced by a local one, especially for signals f of the form f(t) equals a(t)exp(2 pi i phi (t)) where phi is a smooth real-valued function and a is a slowly varying positive function. A general principle is formulated according to which the various pseudo-density functions of f should be concentrated around the curve (t, phi prime (t)). A more detailed qualitative analysis of the behavior of the Wigner distribution of f around this curve is included.