  The epoch of reionization power spectrum is expected to evolve strongly with
redshift, and it is this variation with cosmic history that will allow us to
begin to place constraints on the physics of reionization. The primary obstacle
to the measurement of the EoR power spectrum is bright foreground emission. We
present an analysis of observations from the Donald C. Backer Precision Array
for Probing the Epoch of Reionization (PAPER) telescope which place new limits
on the HI power spectrum over the redshift range of $7.5<z<10.5$, extending
previously published single redshift results to cover the full range accessible
to the instrument. To suppress foregrounds, we use filtering techniques that
take advantage of the large instrumental bandwidth to isolate and suppress
foreground leakage into the interesting regions of $k$-space. Our 500 hour
integration is the longest such yet recorded and demonstrates this method to a
dynamic range of $10^4$. Power spectra at different points across the redshift
range reveal the variable efficacy of the foreground isolation. Noise limited
measurements of $\Delta^2$ at $k=$0.2hMpc$^{-1}$ and z$=7.55$ reach as low as
(48mK)$^2$ ($1\sigma$). We demonstrate that the size of the error bars in our
power spectrum measurement as generated by a bootstrap method is consistent
with the fluctuations due to thermal noise. Relative to this thermal noise,
most spectra exhibit an excess of power at a few sigma. The likely sources of
this excess include residual foreground leakage, particularly at the highest
redshift, and unflagged RFI. We conclude by discussing data reduction
improvements that promise to remove much of this excess.
