Second class of solutions : the energy distributions In contrast with the linear time-frequency representations which decompose the signal on elementary components (the atoms), the purpose of the energy distributions is to distribute the energy of the signal over the two description variables: time and frequency. The starting point is that since the energy of a signal $x$ can be deduced from the squared modulus of either the signal or its Fourier transform, $$E_x = \int_{-\infty}^{+\infty} \vert x(t)\vert^2\ dt\ =\ \int_{-\infty}^{+\infty} \vert X(\nu)\vert^2\ d\nu,$$ (4.1) we can interpret $\vert x(t)\vert^2$ and $\vert X(\nu)\vert^2$ as energy densities, respectively in time and in frequency. It is then natural to look for a joint time and frequency energy density $\rho_x(t,\nu)$, such that $$E_x = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \rho_x(t,\nu)\ dt\ d\nu,$$ (4.2) which is an intermediary situation between those described by (4.1). As the energy is a quadratic function of the signal, the time-frequency energy distributions will be in general quadratic representations. Two other properties that an energy density should satisfy are the following marginal properties: $$\int_{-\infty}^{+\infty} \rho_x(t,\nu)\ dt = \vert X(\nu)\vert^2$$ (4.3) $$\int_{-\infty}^{+\infty} \rho_x(t,\nu)\ d\nu = \vert x(t) \vert^2,$$ (4.4) which mean that if we integrate the time-frequency energy density along one variable, we obtain the energy density corresponding to the other variable. The main references for this chapter are [Fla93], [Coh89], [Aug91], [Hla91] and [HBB92].
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