We characterize by electrical techniques uniformly Si-doped AlAs layers and short-period GaAs-AlAs superlattices grown in the same conditions by molecular-beam epitaxy. Deep level transient spectroscopy shows that both the layers and the superlattices contain the DX center. The AlAs layers contain, in addition, a distribution of electron traps emitting in the temperature range 50-200 K. Using electron irradiation to introduce defects as probes we verified that the band structures of the superlattices we deduce are consistent with theoretical calculations using a widely accepted value of the band offset (67%). Finally, we observe that the DX center remains present, with the same ionization energy, when the energy position of the first Γ miniband varies while the first L miniband remains practically at the energetical position of the L band in the AlAs barriers. From this result we conclude that the DX center is linked to the L band and thus must be ascribed to an L effective-mass state.