Presentation Among the desirable properties of an energy time-frequency distribution, two of them are of particular importance: time and frequency covariance. Indeed, these properties guaranty that, if the signal is delayed in time and modulated, its time-frequency distribution is translated of the same quantities in the time-frequency plane. It has been shown that the class of energy time-frequency distributions verifying these covariance properties possesses the following general expression: $$C_x(t,\nu;f)=\int\int\int_{-\infty}^{+\infty} e^{j2\pi \xi(s-... ...s+\tau/2)\ x^*(s-\tau/2)\ e^{-j2\pi \nu \tau}\ d\xi\ ds\ d\tau,$$ where $f(\xi,\tau)$ is a two-dimensional function called the parameterization function. This class of distributions is known as the Cohen's class, which can also be written: $$C_x(t,\nu;\Pi)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \Pi(s-t,\xi-\nu)\ W_x(s,\xi)\ ds\ d\xi,$$ (4.6) where $$\Pi(t,\nu)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(\xi,\tau)\ e^{-j2\pi(\nu \tau+\xi t)}\ dt\ d\nu$$ is the two-dimensional Fourier transform of the parameterization function $f$. This class is of significant importance since it includes a large number of the existing time-frequency energy distributions. Of course, the WVD is the element of the Cohen's class for which the function $\Pi$ is a double Dirac: $\Pi(t,\nu)=\delta(t)\ \delta(\nu)$, i.e. $f(\xi,\tau)=1$. In the case where $\Pi$ is a smoothing function, expression (4.6) allows one to interpret $C_x$ as a smoothed version of the WVD; consequently, such a distribution will attenuate in a particular way the interferences of the WVD. Before considering different kinds of smoothing functions $\Pi$, let us point out the different advantages of such a unified formulation: by specifying the parameterization function $f$ arbitrarily, it is possible to obtain most of the known energy distributions; it is easy to convert a constraint that we wish for the distribution in an admissibility condition for the parameterization function; it is possible, by using such admissibility arguments, to check a priori the properties of a particular definition, or to construct a class of solutions according to a specified schedule of conditions.
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