the visibilities at $\ell = 500$.

The APS of visibility data is computed as [@2012ApJ...757..101T], $$\begin{aligned}
C_{l} = \frac{\displaystyle{\sum_{2\pi|u|=l}}N_{uv}|V_{uv}|^2}{\displaystyle{\sum_{2\pi|u|=l}}N_{uv}},
\label{Clvis}\end{aligned}$$ where $N_{uv}$ is the number of visibility samples at a uv-grid and $V_{uv}$ is the averaged visibility. The units of visibility is Jy, and we convert Jy into K as, $$\begin{aligned}
{\rm K} = \frac{\lambda^2}{2\times10^{26}k_{\rm B}\Omega}[\rm Jy]\label{Jy2K}\end{aligned}$$ where $\lambda$ is observed wavelength, $k_{\rm B}$ is the Boltzmann constant and $\Omega$ is the solid angle of the MWA’s field-of-view.

The APS using the visibility data can be used to estimate the error power of the 21cm observation. For the error estimation, we split the data into “even” and “odd” samplings at 8s time intervals, and calculate the APS of the difference between the samplings. Ideally, the 21cm signal and the foreground radiation are common in those two groups while the thermal noise is time variant. Therefore, the difference should be dominated by the thermal noise.

Cross Power Spectrum
--------------------

Using the Polspice, we calculate the CPS from images of the MWA radio map and the CMB temperature fluctuation map. The definition is written as $$\begin{aligned}
\langle a^{21}_{lm}a^{\rm CMB,*}_{l'm'} \rangle = \delta_{ll'}\delta_{mm'} \langle C^{21,\rm CMB}_l \rangle,\end{aligned}$$ where $a^{21}_{lm}$ and $a^{\rm CMB}_{lm}$ are 