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Generalized spectrograms and τ-Wigner transforms

Boggiatto Paolo, Giuseppe De Donno, Alessandro Oliaro, Bui Kien Cuong

Published Oct 1, 2010• 0 citations
OpenAlex

Abstract

We consider in this paper Wigner type representations WigÏ„ depending on a parameter Ï„ ∈ [0,1] as defined in [2]. We prove that the Cohen class can be characterized in terms of the convolution of such WigÏ„ with a tempered distribution. We introduce furthermore a class of “quadratic representations” SpÏ„ based on the Ï„-Wigner, as an extension of the two window Spectrogram (see [2]). We give basic properties of SpÏ„ as subclasses of the general Cohen class.

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We consider in this paper Wigner type representations WigÏ„ depending on a parameter Ï„ ∈ [0,1] as defined in [2]. We prove that the Cohen class can be characterized in terms of the convolution of such WigÏ„ with a tempered distribution. We introduce furthermore a class of “quadratic representations” SpÏ„ based on the Ï„-Wigner, as an extension of the two window Spectrogram (see [2]). We give basic properties of SpÏ„ as subclasses of the general 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 propose classes of quadratic time-frequency distributions that retain the inner structure of Cohen's class. Each of these classes is based on a pair of conjugate unitary operators producing time-frequency displacements. The classes satisfy covariance and marginal properties corresponding to these operators. For each class, we define a central member generalizing the Wigner distribution and the Q-distribution, and we specify a transformation by which the class can be derived from Cohen's 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.
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.