  (abridged) In this paper, we express the relativistic propagational delay of
light in the space-time of a binary system (commonly known as the "Shapiro
delay") as a sum of harmonics of the orbital period of the system. We do this
first for near-circular orbits as a natural expansion of an existing orbital
model for low-eccentricity binary systems. The amplitudes of the 3rd and higher
harmonics can be described by two new post-Keplerian (PK) parameters
proportional to the amplitudes of the third and fourth harmonics (h_3, h_4).
For high orbital inclinations we use a PK parameter proportional to the ratio
of amplitudes of successive harmonics (sigma) instead of h_4. The new PK
parameters are much less correlated with each other than r and s and provide a
superior description of the constraints introduced by the Shapiro delay on the
orbital inclination and the masses of the components of the binary (...). We
extend the h_3,sigma parameterisation to eccentric binaries with high orbital
inclinations. For some such binaries we can measure extra PK parameters and
test general relativity using the Shapiro delay parameters. In this case we can
use the measurement of h_3 as a test of general relativity. We show that this
new test is not only more stringent than the r test, but it is even more
stringent than the previous s test. Until now this new parametric test could
only be derived statistically from an analysis of a probabilistic chi2 map.
