  We study a binary system composed of point particles of unequal masses in
eccentric orbits in the linear regime of the characteristic formulation of
general relativity, generalising a previous study found in the literature in
which a system of equal masses in circular orbits is considered. We also show
that the boundary conditions on the time-like world tubes generated by the
orbits of the particles can be extended beyond circular orbits. Concerning the
power lost by the emission of gravitational waves, it is directly obtained from
the Bondi's News function. It is worth stressing that our results are
completely consistent, because we obtain the same result for the power derived
by Peters and Mathews, in a different approach, in their seminal paper of 1963.
In addition, the present study constitutes a powerful tool to construct
extraction schemes in the characteristic formalism to obtain the gravitational
radiation produced by binary systems during the inspiralling phase.
