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)$^2$ ($1sigma$). 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.
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