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Up-converting mixers take two signals and (ideally) create a pair of signals at their sum and difference frequencies. Often, you only want one of those products, and you need to get rid of the other. Using single sideband (SSB) mixers will allow you to eliminate the need for a narrow-band filter to take out the unwanted sideband, or at least allow you to reduce the filter to a more-manageable specification. SSB transmissions are more efficient, providing longer-range transmission for the same power output and occupies less than half the spectrum of a full carrier AM signal.
The video below was made by Bell Laboratories in 1977. It gives a basic overview of single sideband transmission and explains why it was implemented in the "long lines" phone system starting in 1980. The spectrum analyzer sweeps make it very clear what SSB was trying to do: put two signals onto one carrier. The downside is that you lose sensitivity in the link, as you are only sending out half as much power for each signal. Those horn antennas used to be ubiquitous are now rare, having been killed off by fiber optics and satellites.
It boggles the mind that today anyone outside of electrical engineering would be interested in this topic, but the video seems to be produced so even the unwashed masses can understand it.. You can find more gems like this at ATT's Tech Channel Archive.
Even in 1977, SSB wasn't new - the first patent for the technology was filed by John Renshaw Carson back in 1915 (though it wasn't actually approved until 1923).