Localization and the Heisenberg-Gabor principle A simple way to characterize a signal simultaneously in time and in frequency is to consider its mean localizations and dispersions in each of these representations. This can be obtained by considering $\vert x(t)\vert^2$ and $\vert X(\nu)\vert^2$ as probability distributions, and looking at their mean values and standard deviations : $$\begin{array}{rcll} t_m &=& \frac{1}{E_x}\ \int_{-\infty}^{+... ...(\nu)\vert^2\ d\nu & \mbox{\it frequency spreading} \end{array}$$ where $E_x$ is the energy of the signal, assumed to be finite (bounded) : $$E_x = \int_{-\infty}^{+\infty} \vert x(t)\vert^2\ dt < + \infty.$$ Then a signal can be characterized in the time-frequency plane by its mean position $(t_m, \nu_m)$ and a domain of main energy localization whose area is proportional to the time-bandwidth product $T\times B$.
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