We have performed the first self-consistent, large-scale simulation of cosmic reionization by stellar sources, including the Population III stars that emerged and were hosted by minihalos at very high redshifts (z~40). Based on this result, we calculate the redshifted radiation background from these stars and the relic H II regions which can be observed at near-infrared and infrared regime. Formation of the first stars inside minihalos are quenched by radiative feedback at z~15, while the relic H II regions have much longer lifetime due to the slow recombination rate. Therefore, the radiation output from the relic H II regions, dominated by Lyman alpha photons, will be observed both in the near-infrared and infrared regime. The estimated background from the first stars inside minihalos are still sub-dominant compared to that from stars inside larger halos, however, and thus complementary observations are necessary, such as redshifted 21-cm line observation.