Power and sensitivity measured at the correlator output for one baseline

After the atmopheric calibration that converts the measurement scale from the correlator output (in counts) to the $T_\ensuremath{\mathrm{A}}^\star$ scale, the output of the correlator for one correlation is a power equivalent temperature (in the Rayleigh-Jeans domain), which is sampled at a rate of $2\ensuremath{d\nu}$, where $d\nu$ is the frequency bandwidth over which the power is measured. As explained in the Sect. [*], the standard deviation of each power measurement is given by the system temperature power ( $T_\ensuremath{\mathrm{sys}}$). During the integration time ( $\Delta t_\ensuremath{\mathrm{}}$), $2\ensuremath{d\nu}\,\ensuremath{\Delta t_\ensuremath{\mathrm{}}}$ independent samples of the signal power are measured to ensure the Nyquist sampling of the signal in the bandwidth $d\nu$. The signal power is averaged over these independent samples. The uncertainty on the averaged signal power, named sensitivity ( $\sigma_\ensuremath{\mathrm{K}}$), is thus standard deviation of the average or

$\displaystyle \ensuremath{\sigma_\ensuremath{\mathrm{K}}} = \frac{\ensuremath{T...
...}}}{\sqrt{2\,\ensuremath{d\nu}\,\ensuremath{\Delta t_\ensuremath{\mathrm{}}}}}.$ (1)

One subtlety is that the system temperature characterize one baseline between, eg., antennas $i$ and $j$. Instead of just $T_\ensuremath{\mathrm{sys}}$, we should write in all generality $\ensuremath{T_\ensuremath{\mathrm{sys}}}^{ij}$ with

$\displaystyle \ensuremath{T_\ensuremath{\mathrm{sys}}}^{ij} = \sqrt{\ensuremath{T_\ensuremath{\mathrm{sys}}}^{i}\,\ensuremath{T_\ensuremath{\mathrm{sys}}}^{j}},$ (2)

where $\ensuremath{T_\ensuremath{\mathrm{sys}}}^{i}$ and $\ensuremath{T_\ensuremath{\mathrm{sys}}}^{j}$ are the single dish system temperature of antenna $i$ and $j$. For simplicity, we will keep the notation $T_\ensuremath{\mathrm{sys}}$ hereafter.