  We have theoretically shown that the yield of diffuse radiation generated by
relativistic electrons passing random multilayered systems can be increased
when a resonant condition is met. Resonant condition can be satisfied for the
wavelength region representing visible light as well as soft X-rays. The
intensity of diffusive soft X-rays for specific multilayered systems consisting
of two components is compared with the intensity of Cherenkov radiation. For
radiation at photon energy of $99.4eV$, the intensity of Resonant Diffusive
Radiation (RDR) generated by $5MeV$ electrons passing a $Be/Si$ multilayer
exceeds the intensity of Cherenkov radiation by a factor of $\approx 60$ for
electrons with the same energy passing a $Si$ foil. For a photon energy of
$453eV$ and $13MeV$ electrons passing $Be/Ti$ multilayer generate RDR exceeding
Cherenkov radiation generated by electrons passing a $Ti$ foils by a factor
$\approx 130$.
