Description of the received signal

A perfectly linear (front-end + back-end) system delivers a signal ( \ensuremath{C}) in counts which is defined as

  $\displaystyle \ensuremath{C}= \ensuremath{c_\ensuremath{\mathrm{dark}}}+ \ensur...
...\ensuremath{\mathrm{rec}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}),
$ (2)
where As heterodyne receivers mix input signal from two different frequencies, \ensuremath{T_\ensuremath{\mathrm{ant}}\ifthenelse{\equal{tot}{}}{}{^\ensuremath{\mathrm{tot}}}} can be written as
  $\displaystyle \ensuremath{T_\ensuremath{\mathrm{ant}}\ifthenelse{\equal{tot}{}}...
...}}{}{^\ensuremath{\mathrm{ima}}}}}{1+\ensuremath{G_\ensuremath{\mathrm{im}}}},
$ (3)
where \ensuremath{T_\ensuremath{\mathrm{ant}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} is a combination of losses \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} and of the sum of the atmospheric emission ( \ensuremath{T_\ensuremath{\mathrm{atm}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}) and astronomical emission ( \ensuremath{T_\ensuremath{\mathrm{astro}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}) attenuated by the atmospheric absorption
  $\displaystyle \ensuremath{T_\ensuremath{\mathrm{ant}}\ifthenelse{\equal{}{}}{}{...
...T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}$ (4)
where The losses come from the imperfect coupling of the receiver to the sky. It is here important to note that \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} comes from two sources: 1) the (sub-)millimeter emission inside the cabin, picked up by the receiver optic and proportional to the cabin temperature \ensuremath{T_\ensuremath{\mathrm{cab}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} and 2) the (sub-)millimeter emission of the ground, picked up by the antenna and proportional to the outside ambient temperature \ensuremath{T_\ensuremath{\mathrm{amb}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}. Thus
  $\displaystyle \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}...
..._\ensuremath{\mathrm{amb}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}},
$ (6)
with $\alpha$ being a largely unknown coupling factor.

It is usually supposed that \ensuremath{F_\ensuremath{\mathrm{eff}}} and \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} are identical for both bands of an heterodyne receiver. This maybe should be revisited with the large IF of today receivers.