Overcoming real-world obstacles in 21 cm power spectrum estimation: A method demonstration and results from early Murchison Widefield Array data

We present techniques for bridging the gap between idealized inverse covariance weighted quadratic estimation of 21 cm power spectra and the real-world challenges presented universally by interferometric observation. By carefully evaluating various estimators and adapting our techniques for large but incomplete data sets, we develop a robust power spectrum estimation framework that preserves the so-called "Epoch of Reionization (EoR) window" and keeps track of estimator errors and covariances. We apply our method to observations from the 32-tile prototype of the Murchinson Widefield Array to demonstrate the importance of a judicious analysis technique. Lastly, we apply our method to investigate the dependence of the clean EoR window on frequency---especially the frequency dependence of the so-called ``wedge" feature---and establish upper limits on the power spectrum from $z=6.2$ to $z=11.7$. Our lowest limit is Δ(k)<0.3 Kelvin at 95% confidence at a comoving scale k=0.046 Mpc^-1 and $z=9.5$.