  Identifying the source parameters from a gravitational-wave measurement alone
is limited by our ability to discriminate signals from different sources and
the accuracy of the waveform family employed in the search. Here we address
both issues in the framework of an adapted coordinate system that allows for
linear Fisher-matrix type calculations of waveform differences that are both
accurate and computationally very efficient. We investigate statistical errors
by using principal component analysis of the post-Newtonian (PN) expansion
coefficients, which is well conditioned despite the Fisher matrix becoming ill
conditioned for larger numbers of parameters. We identify which combinations of
physical parameters are most effectively measured by gravitational-wave
detectors for systems of neutron stars and black holes with aligned spin. We
confirm the expectation that the dominant parameter of the inspiral waveform is
the chirp mass. The next dominant parameter depends on a combination of the
spin and the symmetric mass ratio. In addition, we can study the systematic
effect of various spin contributions to the PN phasing within the same
parametrization, showing that the inclusion of spin-orbit corrections up to
next-to-leading order, but not necessarily of spin-spin contributions, is
crucial for an accurate inspiral waveform model. This understanding of the
waveform structure throughout the parameter space is important to set up an
efficient search strategy and correctly interpret future gravitational-wave
observations.
