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Philips Journal of Research

On the locus and spread of pseudo-density functions in the time-frequency plane

Ajem Guido Janssen

Published Jan 1, 1982• 111 citations
OpenAlex

Abstract

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.

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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.
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.
Cohen's (see Prentice Hall, Englewood Cliffs, NJ, 1995) class of time-frequency distributions for continuous signals has been to extended to discrete signals using both an axiomatic approach and an operator theory approach. We investigate the formulation of several classical time-frequency distributions (Wigner, Rihaczek, Margenau-Hill, Page, Levin, Born-Jordan, spectrogram) in the discrete Cohen's classes. The main result of this paper concludes that there does not exist a formulation of the Wigner distribution in all of the discrete Cohen's classes.
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.
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.
This contribution addresses the implications of Heisenberg's uncertainty relation on time-frequency energy distribution functions depending bilinearly on the signal. It is indicated that all these distribution functions must have a certain minimum spread in the time-frequency plane. Further-more such distribution functions can either have correct marginal distributions or be positive, but not both. Therefore a local interpretation of these energy distributions must be taken with care. Some facts determining the choice of an appropriate distribution function will be discussed. A comparison will be made between the spectrogram and the Wigner distribution.
We show that there is a way to unify distribution functions that describe simultaneously a classical signal in space and (spatial) frequency and position and momentum for a quantum system. Probably the most well known of them is the Wigner distribution function. We show how to unify functions of the Cohen class, Rihaczek's complex energy function, and Husimi and Glauber-Sudarshan distribution functions. We do this by showing how they may be obtained from ordered forms of creation and annihilation operators and by obtaining them in terms of expectation values in different eigenbases.