Properties As for the Cohen's class, it can be useful to impose further constraints on the class defined by (4.10), to obtain a sub-class of distributions which validate particular properties (see page ). We detail here some of the most important ones. Energy conservation: by integrating $\Omega_x$ all over the time-scale plane, we obtain the energy of $x$: $$E_x = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Omega_x(t,a;\Pi)\ dt\ \frac{da}{a^2}$$ Marginal properties: the energy spectral density and the instantaneous power can be obtained as marginal distributions of $\Omega_x$: $$\begin{eqnarray*} \int_{-\infty}^{+\infty} \Omega_x(t,a;\Pi)\ dt = \vert X(\frac... ...{+\infty} \Omega_x(t,a;\Pi)\ \frac{da}{a^2} = \vert x(t)\vert^2 \end{eqnarray*}$$ Real-valued: $$\Omega_x(t,a;\Pi) \in {\cal R},\ \forall t,\ a$$ Time localization: $$X(\nu)=\frac{1}{\sqrt{\nu}}\ e^{-j2\pi\nu t_0}\ U(\nu)\ \Righ... ... \Omega_x(t,\frac{\nu_0}{\nu};\Pi)=\nu\ \delta(t-t_0)\ U(\nu)$$ where $U(\nu)$ is the Heaviside step function. Unitarity: conservation of the scalar product from the time domain to the time-scale domain (apart from the squared modulus): $$\left\vert\int_{-\infty}^{+\infty} x(t)\ y^*(t)\ dt\right\ver... ...fty} \Omega_x(t,a;\Pi)\ \Omega_y^*(t,a;\Pi)\ dt\ \frac{da}{a^2}$$ Group delay: we may want to obtain the group delay of $x$ as the first order moment in time of $\Omega_x$: $$t_x\left(\frac{\nu_0}{a}\right)={\int_{-\infty}^{+\infty} t\ ... ...,a;\Pi)\ dt\over\int_{-\infty}^{+\infty} \Omega_x(t,a;\Pi)\ dt}$$ Narrow-band limit: it can also be desirable that, for narrow-band signals, the affine distribution $\Omega_x$ coincides with the Wigner-Ville distribution: $$\Omega_x(t,a;\Pi)=W_x\left(t,\frac{\nu_0}{a}\right).$$
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