  We model the gravitational-wave background created by double compact objects
from isolated binary evolution across cosmic time using the
\textbf{\textit{StarTrack}} binary population code. We include population I/II
stars as well as metal-free population III stars. Merging and non-merging
double compact object binaries are taken into account.
  In order to model the low frequency signal in the band of the space antenna
LISA, we account for the evolution of the redshift and the eccentricity. We
find an energy density of $\Omega_{GW} \sim 1.0 \times 10^{-9}$ at the
reference frequency of 25 Hz for population I/II only, making the background
detectable at 3 $\sigma$ after about 7 years of observation with the current
generation of ground based detectors, such as LIGO, Virgo and Kagra, operating
at design sensitivity. The contribution from population III is one order of
magnitude below the population I/II for the total background, but dominates the
residual background, after detected sources have been removed, in 3G detectors.
It modifies the shape of the spectrum which starts deviating from the usual
power law $\Omega_{GW}(f) \sim f^{2/3}$ after $\sim 10$ Hz. The contribution
from the population of non merging binaries, on the other hand, is negligible,
being orders of magnitude below. Finally, we observe that the eccentricity has
no impact in the frequency band of LISA or ground based detectors.
