  The study of the charge excitations in cuprates is presently an interesting
topic because of the development of new and precise x-ray experiments. Based on
a large-$N$ formulation of the two-dimensional $t$-$J$ model, which allows us
to consider all possible charge excitations on an equal footing, we investigate
the charge spectrum for both electron- and hole-doped cases. In both cases, the
instability toward phase separation, which has momentum modulation ${\bf
q}=(0,0)$, is found to be robust in a large region of the doping-temperature
phase diagram. If a short-range Coulomb repulsion is included the phase
separation region shrinks, but the instability remains at ${\bf q}=(0,0)$. If
on the other hand a two-dimensional long-range Coulomb interaction is included
the instability sets in at ${\bf q}$ momenta forming a ring around ${\bf
q}=(0,0)$. The computed charge spectrum in the translation-invariant phase
shows well-formed rings. We discuss our results in the light of recent x-ray
experiments in electron- and hole-doped cuprates, where ring-like shaped charge
modulations have been reported.
