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Journal of Mathematical Physics

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

Patrick Flandrin

Published Aug 1, 1998• 27 citations
Publisher

Abstract

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.

DOI
10.1063/1.532483

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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.
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.
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.
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.
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.
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.