Demonstrating the Tapered Gridded Estimator (TGE) for the Cosmological HI 21-cm Power Spectrum using $150 \, {\rm MHz}$ GMRT observations

We apply the Tapered Gridded Estimator (TGE) for estimating the cosmological 21-cm power spectrum from $150 \, {\rm MHz}$ GMRT observations which corresponds to the neutral hydrogen (HI) at redshift $z = 8.28$. Here TGE is used to measure the Multi-frequency Angular Power Spectrum (MAPS) $C_{\ell}(\Delta\nu)$ first, from which we estimate the 21-cm power spectrum $P(k_{\perp},k_{\parallel})$. The data here are much too small for a detection, and the aim is to demonstrate the capabilities of the estimator. We find that the estimated power spectrum is consistent with the expected foreground and noise behaviour. This demonstrates that this estimator correctly estimates the noise bias and subtracts this out to yield an unbiased estimate of the power spectrum. More than $47\%$ of the frequency channels had to be discarded from the data owing to radio-frequency interference, however the estimated power spectrum does not show any artifacts due to missing channels. Finally, we show that it is possible to suppress the foreground contribution by tapering the sky response at large angular separations from the phase center. We combine the k modes within a rectangular region in the `EoR window' to obtain the spherically binned averaged dimensionless power spectra $\Delta^{2}(k)$ along with the statistical error $\sigma$ associated with the measured $\Delta^{2}(k)$. The lowest $k$-bin yields $\Delta^{2}(k)=(61.47)^{2}\,{\rm K}^{2}$ at $k=1.59\,\textrm{Mpc}^{-1}$, with $\sigma=(27.40)^{2} \, {\rm K}^{2}$. We obtain a $2 \, \sigma$ upper limit of $(72.66)^{2}\,\textrm{K}^{2}$ on the mean squared HI 21-cm brightness temperature fluctuations at $k=1.59\,\textrm{Mpc}^{-1}$.