## USB and LSB frequency axes in the observatory frame

Heterodyne receivers mix a local oscillator frequency, , to the astronomical signal to transfer the frequency axis from high radio frequencies, , to low intermediate frequencies, , which are then suitable for easy electronic processing. A consequence is that two bands of radio frequencies, namely the lower side band (LSB) and the upper side band (USB) are mixed in the observed spectrum, i.e., the set of brightnesses, , are associated to one IF frequency axis and two RF frequency axes linked through

(11)
with for the RF frequency axis belonging to the lower side band (LSB) tuning and for the RF frequency axis belonging to the upper side band (USB). The measured spectra is thus in the IF axis and Eq. must be modified as
(12)
with
(13)
The constancy of is a property of the heterodyne receivers. Combining the last 3 equations, we yield
(14)