  Besides the well-known negative refraction, a negative refractive-index
material can exhibit another two hallmark features, which are the inverse
Doppler effect and backward Cherenkov radiation. The former is known as the
motion-induced frequency shift that is contrary to the normal Doppler effect,
and the latter refers to the Cherenkov radiation whose cone direction is
opposite to the source's motion. Here we combine these two features and discuss
the Doppler effect inside the backward Cherenkov cone. We reveal that the
Doppler effect is not always inversed but can be normal in negative
refractive-index systems. A previously un-reported phenomenon of normal Doppler
frequency shift is proposed in a regime inside the backward Cherenkov cone,
when the source's velocity is two times faster than the phase velocity of
light. A realistic metal-insulator-metal structure, which supports metal
plasmons with an effective negative refractive index, is adopted to demonstrate
the potential realization of this phenomenon.
