Description of the received signal

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

$$\ensuremath{C}= \ensuremath{c_\ensuremath{\mathrm{dark}}}+ ... ...m{rec}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}),$$ (2)

where

• is the signal delivered when the (front-end + back-end) system is blind;
• is the (front-end + back-end) system gain;
• the noise added by the receiver and the optics, usually called receiver temperature;
• the signal seen by the receiver, usually called antenna temperature.

As heterodyne receivers mix input signal from two different frequencies, can be written as

$$\ensuremath{T_\ensuremath{\mathrm{ant}}\ifthenelse{\equal{t... ...h{\mathrm{ima}}}}}{1+\ensuremath{G_\ensuremath{\mathrm{im}}}},$$ (3)

where

• and are respectively the signal in the signal and image bands;
• and is the ratio of the receiver gain in the signal ( $\ensuremath{g}_\ensuremath{\mathrm{sig}}$) and image $\ensuremath{g}_\ensuremath{\mathrm{ima}}$ bands (at least 0.1 for (sub-)millimeter receivers at IRAM).

is a combination of losses and of the sum of the atmospheric emission ( ) and astronomical emission ( ) attenuated by the atmospheric absorption

$$\ensuremath{T_\ensuremath{\mathrm{ant}}\ifthenelse{\equal{}... ...thrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}$$ (4)

where

• is the forward efficiency, describing the forward coupling to of the receiver to the signal;
• $\tau$ is the atmopsheric zenith opacity at the observed frequency;
• is the airmass, which relate the zenith opacity to the opacity at the elevation of the observation:
  

  $$\ensuremath{a}= 1/\sin(\ensuremath{\mathrm{elevation}});$$ (5)

  
  

The losses come from the imperfect coupling of the receiver to the sky. It is here important to note that comes from two sources: 1) the (sub-)millimeter emission inside the cabin, picked up by the receiver optic and proportional to the cabin temperature and 2) the (sub-)millimeter emission of the ground, picked up by the antenna and proportional to the outside ambient temperature . Thus

$$\ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{... ...thrm{amb}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}},$$ (6)

with $\alpha$ being a largely unknown coupling factor.

It is usually supposed that and are identical for both bands of an heterodyne receiver. This maybe should be revisited with the large IF of today receivers.