With respect to observations in a given sideband, one can usually improve sensitivity by rejecting the image sideband, as indicated in Equation 10 above. Nevertheless, spectral lines of interest are sometimes present in both sidebands. Given that one usually pays a noise penalty by observing in double sideband mode, at what point do we break even if lines of interest can be observed simultaneously in both sidebands? Put another way, when would we be better off from a noise standpoint to make two single sideband observations of the two sidebands rather than one simultaneous observation in double sideband mode? Thus, we want to find the condition in which
where t is integration time and the notation ``BE'' means ``break even''. From the radiometer equation,
where K is a constant determined by the switching mode
and spectrometer efficiency, 
is the rms noise of the
spectrum, and 
is the bandwidth. Hence, we want to find the
condition in which
From Equations 8 and 10, we find that
Solving for 
yields
Thus, if the DSB receiver temperature is below this
break-even point, 
is dominated by sky noise and it is always
more efficient to observe SSB.
This relation can also be expressed usefully in terms of the break-even sky noise, i.e., if sky noise exceeds this value, you could make two SSB observations in less time than one simultaneous DSB observation with lines in both sidebands:
In practical cases, even if the SSB noise exceeds the break-even point, DSB observing may not be the most efficient mode if a choice between SSB and DSB observing is available. It will usually be the case that the lines in the two sidebands are not of the same strength or may not be of the same importance to the observer. Hence, integration times and signal-to-noise ratios may get skewed in a less than optimum way - i.e., too much signal-to-noise in one sideband or not enough in the other. It will usually not be the case that DSB observing gives twice as much information as SSB. On the other hand, there are cases in which maximizing signal-to-noise is not as critical as minimizing observing overhead. DSB observing may be very useful in that case. There may be other cases in which having lines in both sidebands provides a useful system check or improves relative calibration.
In MMA Memo 70, Kerr outlined three options for sideband separation and/or rejection. Most current SSB systems use either quasi-optical filters or reactive termination in the mixer block using mechanical backshorts. Neither of these options would seem practical for the number of receivers in the MMA. If the RF performance can be achieved, image rejection mixers (MMA Memo 151; Kerr & Pan) would be more practical from an operations and reliability standpoint. In principle, such systems provide the ideal situation in which the information from both sidebands can be retained and analyzed separately, while the noise from the opposite sideband is rejected. Analyzing both sidebands independently would require doubling the size of the MMA correlator and may not be practical. The technique can probably be implemented at many single-dish facilities without requiring additional backend hardware.
