The problem that Cherenkov radiation may exist in the ion-channel when a relativistic electron beam (REB) is injected into plasma has been demonstrated and studied. The effect of the beam-wave interaction in ion background is analyzed by use of the linear perturbation theory. The dispersion equation and the simultaneous radiation condition of the system are derived. It is clearly shown that the electromagnetic instability of the system results from the coupling of the TM mode in ion-channel to the beam mode via the electron beam. Ion beam focusing is the microscopic mechanism. The ion-beam system in the moving plasma state is analyzed rigorously in theory. Then, the frequency shift of the radiation wave and the wave growth rate in ion-channel are presented. Finally, the effects of the parameters of the system on them are discussed by numerical calculation.