We report some specific features of transient radiation from a periodic spatially modulated one-dimensional medium with a resonant response upon excitation by an ultrashort pulse. The case of ring geometry (with particle density distributed along the ring according to the harmonic law) is considered. It is shown that the spectrum of scattered radiation contains (under both linear and nonlinear interaction), along with the frequency of intrinsic resonance of the medium, a new frequency, which depends on the pulse velocity and the spatial modulation period. The case of superluminal motion of excitation, when the Cherenkov effect manifests itself, is also analysed.