Time-frequency energy distribution

From equation (6) we can derive a time-frequency distribution of signal's energy by adding Wigner distributions of selected atoms [Mallat and Zhang, 1993]. Calculating the Wigner distribution directly from equations (2) and (6) would yield

$$\begin{array}{ll} W f = & \sum_{n=0}^\infty \vert<R^n f, \;g... ... f, \; g_{\gamma_m}>} W[g_{\gamma_n}, g_{\gamma_m}] \end{array}$$ (11)

The double sum, containing cross Wigner distributions of different atoms from the expansion given in eq. (6), corresponds to the cross terms generally present in Wigner distribution. These terms one usually tries to remove in order to obtain a clear picture of the energy distribution in the time-frequency plane. Removing these terms from eq. (11) is straightforward--we keep only the first sum; we can define a magnitude $E f (t,\omega)$:

$$E f (t, \omega) = \sum_{n=0}^\infty \vert<R^n f, \;g_{\gamma_n}>\vert^2 \; W g_{\gamma_n} (t, \omega)$$ (12)

Wigner distribution of a single time-frequency atom $g_\gamma$ satisfies

$$\int _{-\infty}^{+\infty} \int _{-\infty}^{+\infty} W g_{\ga... ...ega) \: d t \:d \omega = \vert\vert g_{\gamma}\vert\vert^2 = 1$$ (13)

Combining this with energy conservation of the MP expansion (eq. 6) yields

$$\int _{-\infty}^{+\infty} \int _{-\infty}^{+\infty} E f (t, \omega)\: d t \: d \omega = \vert\vert f\vert\vert^2$$ (14)

This justifies interpretation of $E f (t,\omega)$ as the energy density of signal $f(t)$ in the time-frequency plane.