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 99.4 eV, the intensity of resonant diffusive radiation (RDR) generated by 5 MeV electrons passing a Be/Si multilayer exceeds the intensity of Cherenkov radiation by a factor of ˜60 for electrons with the same energy passing a Si foil. For a photon energy of 453 eV and 13 MeV electrons passing Be/Ti multilayer generate RDR exceeding Cherenkov radiation generated by electrons passing a Ti foils by a factor ˜130.