An average pixel

The scatter of the mixer performances, which translate into a scatter of receiver temperatures, is the first thing to deal with. Instead of computing the sensitivity associated with each mixer, we introduce an average pixel, which will represent all the other ones. In Eqs.  and , the caracteristics of the mixers are hidden into the system temparature, . We will thus define an average system temperature, , which will represent the receiver average pixel.

Among the different ways to define such an average system temperature, we priviledge the one which will give the right sensitivity in the case where the same point of the sky is seen by all the different pixels. This choice is made because 1) the same point of the sky is at least seen by two pixels (one per polarization) and 2) it is a good idea when mapping to try to cover the mapped area as many time as possible with sligthly different observing configuration of HERA (e.g. rotations by 90$\deg$) to homogenize the noise distribution and to ensure that bad pixels see different part of the mapped area.

It is well-known that the optimal way to combine (e.g. to average or to grid) spectra is to weight them by $\ensuremath{w}= 1/\ensuremath{\sigma_\ensuremath{\mathrm{}}}^2$ before combination, where is their rms noise. In this case, it can be shown that the weight of the combination is the linear sum of the weights. From this, it is easy to define as

$$\frac{\ensuremath{n_\ensuremath{\mathrm{pol}}}\,\ensuremath... ...thrm{pix}}}} \frac{1}{T^2_{\ensuremath{\mathrm{sys}}_{ij}}}.$$ (6)