  A compact binary in an eccentric orbit radiates gravitational waves (GWs) at
all integer harmonics of its orbital frequency. In this study, we investigate
the effect of orbital eccentricity on the expected gravitational background
radiation (GWBR) from supermassive black hole (SMBH) binaries in the nuclei of
galaxies. For this purpose, we formulate a power spectrum of the GWBR from
cosmological evolving eccentric binaries. Then, we apply this formulation to
the case of the GWBR from SMBH binaries. The key to doing this is to correctly
estimate the number density of coalescing SMBH binaries. In this study, we use
a semi-analytic model of galaxy and SMBH formation. We find that the power
spectrum of the GWBR from SMBH binaries on eccentric orbits is suppressed for
frequencies $f \lesssim 1~{\rm nHz}$ if the initial eccentricity, $e_0$,
satisfies $e_0 > 0.2$ and the initial semi-major axis is 300 times Scwarzschild
radius. Our model predicts that while the overall shape and amplitude of the
power spectrum depend strongly on the processes of galaxy formation, the
eccentricity of binaries can affect the shape of the power spectrum for lower
frequencies, i.e., $f \lesssim 1~{\rm nHz}$. Pulsar timing measurements, which
can detect GW in this frequency range, could constrain the effect of
eccentricity on the power spectrum of the GWBR from SMBH binaries.
