Visibility-based Power Spectrum Estimation for Low-Frequency Radio Interferometric Observations

We present a visibility based estimator namely, the Tapered Gridded Estimator (TGE) to estimate the power spectrum of the diffuse sky signal. The TGE has three novel features. First, the estimator uses gridded visibilities to estimate the power spectrum which is computationally much faster than individually correlating the visibilities. Second, a positive noise bias is removed by subtracting the auto-correlation of the visibilities which is responsible for the noise bias. Third, the estimator allows us to taper the field of view so as to suppress the contribution from the sources in the outer regions and the sidelobes of the telescope's primary beam. We first consider the two dimensional (2D) TGE to estimate the angular power spectrum $C_{\ell}$. We have also extended the TGE to estimate the three dimensional (3D) power spectrum $P({\bf k})$ of the cosmological 21-cm signal. Analytic formulas are presented for predicting the variance of the binned power spectrum. Both the estimators and their variance predictions are validated using simulations of $150 \, {\rm MHz}$ GMRT observations. We have applied the 2D TGE to estimate $C_{\ell}$ using visibility data for two of the fields observed by TIFR GMRT Sky Survey (TGSS). We find that the sky signal, after subtracting the point sources, is likely dominated by the diffuse Galactic synchrotron radiation across the angular multipole range $240 \le \ell \lesssim 500$.