  We present a new iterative method to reduce eccentricity in black-hole-binary
simulations. Given a good first estimate of low-eccentricity starting momenta,
we evolve puncture initial data for ~4 orbits and construct improved initial
parameters by comparing the inspiral with post-Newtonian calculations. Our
method is the first to be applied directly to the gravitational-wave (GW)
signal, rather than the orbital motion. The GW signal is in general less
contaminated by gauge effects, which, in moving-puncture simulations, limit
orbital-motion-based measurements of the eccentricity to an uncertainty of
$\Delta e \sim 0.002$, making it difficult to reduce the eccentricity below
this value. Our new method can reach eccentricities below $10^{-3}$ in one or
two iteration steps; we find that this is well below the requirements for GW
astronomy in the advanced detector era. Our method can be readily adapted to
any compact-binary simulation with GW emission, including black-hole-binary
simulations that use alternative approaches, and neutron-star-binary
simulations. We also comment on the differences in eccentricity estimates based
on the strain $h$, and the Newman-Penrose scalar $\Psi_4$.
