Normal galaxies and radio galaxies are associated with star-formation and active galactic nuclei (AGN), respectively. These galaxies are considered to be the primary source of extra-galactic radio emissions permeating throughout the universe. The dominant mechanism responsible for this radio emission is synchrotron radiation due to relativistic electrons interacting with the galactic magnetic field. In this study we calculate the radio spectra from these galaxies. We use observations from the source count population data, which influence the selection of the appropriate radio luminosity function. Finally, we calculate intensity of diffuse extra-galactic radio background using the lambda-CDM cosmology.