A widely used statistical property of the sky brightness distribution is its power spectrum (Lazarian 1995; Bharadwaj & Sethi 2001, and others). As the redshifted 21-cm signal is expected to be faint and hard to detect with imaging, estimating its power spectrum or equivalently intensity mapping gives a possible probe of the evolution of the baryonic matter distribution over cosmic time. Bharadwaj & Sethi (2001) show that visibility correlation directly measures the power spectrum. This method and its variants (Datta, Choudhury & Bharadwaj 2007; Choudhuri et al. 2014; Choudhuri et al. 2016; Bharadwaj et al. 2019; Choudhuri et al.2019, and others) have been used to estimate the angular power spectrum of the diffused galactic foreground (Ghosh et al. 2012; Choudhuri et al. 2017b; Chakraborty et al. 2019a; Choudhuri et al. 2020) as well as the power spectrum of H i distribution in nearby galaxies (Dutta et al. 2009; Dutta & Bharadwaj 2013; Nandakumar & Dutta 2020). These works propagate the uncertainties in each visibility estimate and combine that with the sample variance error in measuring the power spectrum to quote uncertainties in the power spectrum estimates. In this work, we use the estimator discussed in Choudhuri et al. (2014), where visibilities are gridded before estimating the power spectrum. Given an angular field of view of θ0 to which the telescope is sensitive, it has been shown (Bharadwaj & Sethi 2001; Bharadwaj & Ali 2005; Choudhuri et al. 2014) that the visibilities in the nearby baselines remain correlated to a baseline separation of $\Delta U \lt \frac{1}{\pi \theta _0}$. The size of the uv-grids is chosen such that they are large enough to include a sufficient number of baselines in a given uv-grid and small enough to have all visibilities in the uv-grid correlated. In each uv-grid, they estimate the power spectrum by correlating visibilities only in nearby baselines, omitting the visibility autocorrelations. This drastically reduces the noise bias in estimates of the power spectrum in uv-grids. The contribution from each uv-grid within a given annulus in $U = \mid \vec{U} \mid$ is then combined, and the real part of it is used to quote the value of the isotropic power spectrum for the baseline separation U. We may schematically write it as
(4)$$\begin{eqnarray}
\mathcal {E} \lbrace P(U)\rbrace = \mathcal {R} [\langle \tilde{V}(\vec{U})^{*} \tilde{V}(\vec{U}+\Delta \vec{U}) \rangle].
\end{eqnarray}$$Here, the average is taken over the uv-grid first and then within the annulus, as explained above. Note that the power spectrum estimator here assumes that a perfect calibration is done and the gains are all unity. In such a case, the power spectrum estimate has no bias arising from instrumental noise, and its uncertainties can be written as (Ali et al. 2008; Dutta 2011)
(5)$$\begin{eqnarray}
\sigma _P^2 = \frac{P^2(U)}{N_\mathrm{ G}} + 2\frac{P(U)\sigma _N^2}{N_\mathrm{ B}} + 2\frac{\sigma _N^4}{N_\mathrm{ B}},
\end{eqnarray}$$where NG is the number of independent estimates of the power spectrum in a given annulus bin at U, NB is the total number of visibility pairs in the bin.
Citation: Chakraborty et al. 2019a
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Citation: Dutta & Bharadwaj 2013
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