We propose emulation of Hawking radiation (HR) by means of acoustic excitations propagating on top of persistent current in an atomic Bose-Einstein condensate (BEC) loaded in an annular confining potential. The setting is initially created as a spatially uniform one, and then switches into a nonuniform configuration, while maintaining uniform BEC density. The eventual setting admits the realization of sonic black and white event horizons with different slopes of the local sound speed. A smooth slope near the white-hole horizon suppresses instabilities in the supersonic region. It is found that tongue-shaped patterns of the density-density correlation function, which represent the acoustic analog of HR, are strongly affected by the radius of the ring-shaped configuration and number of discrete acoustic modes admitted by it. There is a minimum radius that enables the emulation of HR. We also briefly discuss a possible similarity of properties of the matter-wave sonic black holes to the known puzzle of the stability of Planck-scale primordial black holes in quantum gravity.