  A study of binary systems composed of two point particles with different
masses in the linear regime of the characteristic formulation of general
relativity in a Minkowski background is provided. The present paper generalizes
a previous study find in the literature. The boundary conditions at the world
tubes generated by the particle's orbits are explored, when the metric
variables are decomposed in spin-weighted spherical harmonics. The power lost
by the emission of gravitational waves is computed using the News Bondi's
functions. The power found is nothing but the well-known result for the power
obtained by Peters and Mathews in a different approach. This agreement shows
the consistency of the approach considered here. The contribution to the
gravitational radiation of several multipole terms is also shown.
