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) 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
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