Considering an electric and magnetic plane wave, $E(t)=E_p(\omega)e^{i\omega t}$ and $H(t)=H_p(\omega)e^{i\omega t}$, where $E_p(\omega)$ and $H_p(\omega)$ represent the frequency dependent peak amplitudes of the wave. The time-averaged Poynting vector is defined as $<S_x> = \frac{1}{2}\text{Re}\{E_p(\omega)H_p^*(\omega)\}$ with unit $\left[\frac{V}{m}\frac{A}{m} = \frac{W}{m^2}\right]$ (see: https://en.wikipedia.org/wiki/Poynting_vector#Time-averaged_Poynting_vector).
How to determine the total power if there is a whole radiation spectrum? For example, if you want to know the power carried by the solar spectrum? I would expect something like
$$ S_{tot} = \frac{1}{2} \text{Re}\{ \int\limits_0^\infty E_p(\omega) H_p^*(\omega) d\omega \} \,.$$
However, this expression has a wrong unit $\left[\frac{V}{m}\frac{A}{m}\frac{1}{s} = \frac{W}{m^2}\frac{1}{s}\right]$...
My question: what is the correct expression of the total power in function of the reciprocal electric and magnetic field?