  We present a fast and versatile method to calculate the characteristic
spectrum $h_c$ of the gravitational wave background (GWB) emitted by a
population of eccentric massive black hole binaries (MBHBs). We fit the
spectrum of a reference MBHB with a simple analytic function and show that the
spectrum of any other MBHB can be derived from this reference spectrum via
simple scalings of mass, redshift and frequency. We then apply our calculation
to a realistic population of MBHBs evolving via 3-body scattering of stars in
galactic nuclei. We demonstrate that our analytic prescription satisfactorily
describes the signal in the frequency band relevant to pulsar timing array
(PTA) observations. Finally we model the high frequency steepening of the GWB
to provide a complete description of the features characterizing the spectrum.
For typical stellar distributions observed in massive galaxies, our calculation
shows that 3-body scattering alone is unlikely to affect the GWB in the PTA
band and a low frequency turnover in the spectrum is caused primarily by high
eccentricities.
