This answer is focussing on radio receivers such as AM and FM.
If you are only interested in receiving a signal from one station you may not need to have or use an intermediate frequency. You can build your receiver to tune in to just that frequency - the tuning needs to be sharp - you need to reject all possible other sources that may pollute the signal you want.
This is done by a bunch of band pass filters that together, have a passband that is wide enough to cope with the signal you wish to receive but not so wide that it lets others in.
Now say you wanted to tune in to 2 stations - you'd have to re-align all this filtering to coincide with a new station. Historically radios were simple and moving a bunch of tuned band pass filters to a new centre frequency would be hard.
It was a lot easier to have a bunch of fixed band-pass filters that did the majority of all the unwanted channel rection rather than trying to align them as you tuned the dial.
Thus super-heterodyne receivers were conceived. The incoming broad range of many radio stations were "mixed" with an oscillator that can be simply tuned with a dial - this produced sum and difference frequencies and usually the difference frequency became the new "wanted" frequency. So for FM (88MHz to 108MHz), the I.F. frequency became 10.7MHz and the oscillator would be (typically) at 98.7MHz for tuning 88MHz signals and at 118.7MHz for tuning 108MHz signals.
Don't hang me on this - it could equally be at 77.3MHz rising to 97.3MHz to produce the same set of difference frequencies. Maybe someone can modify my answer or advise me on this.
It's a small matter though because the point is that once you were able to manipulate the incoming signal's carrier frequency you can feed the result through a tightly tuned fixed set of band-pass filters before you demodulate.
A bit more info about the VHF FM band
It goes from 88MHz to 108MHz and has an IF that is just slightly bigger (10.7MHz) than half the frequency range it covers. There is a sensible reason - if the oscillator were exactly tuned to pick up 88MHz (i.e. osc = 98.7MHz) the difference frequency it would produce from the top of the band at 108MHz would be 9.3MHz and this would be just out of band of the tuning centred at 10.7 MHz and therefore "rejected".
Of course if someone started transmitting just outside the FM band you may pick this up but I believe that legislation prevents this.
Following recent activity in this question I remembered that there is another valid reason for using an intermediate frequency. Consider that the signal from an antenna might be in the order of 1 uV RMS and then consider that you'll probably want the radio circuit to amplify this to something like 1V RMS (forgive the hand waving) at the demodulator. Well, that's a gain of 1 million or 120 dB and, no matter how hard you might try, having a circuit board with a gain of 120 dB is a recipe for feedback disaster i.e. it will oscillate and turn into a "theramin".
What an IF gets you is a break in the signal chain which prevents oscillation. So, you might have 60 dB of RF gain then convert to your IF and have 60 dB of IF gain - the signal at the end of the chain is no longer frequency compatible with what happens at the antenna and therefore, there is no theramin effect!
Some radios might have two intermediate frequencies - for just this reason alone you can reduce the RF gain to 40 dB and each IF stage can have a gain of 40 dB and NO theramin.
The passive terminations \$Γ_g\$ and \$Γ_l\$ must be added so that the input and output resonate at the same frequency of oscillation. For this purpose, the 2 and 3 conditions are used. In other words, if the oscillator is oscillating at one port then it has to oscillate at the other port too.
there's a proof too for two port oscillator design. Suppose oscillation condition is met at port 1 then from second condition,
\$ \dfrac{1}{Γ_{in}}=Γ_g \$
Now we know,
\$Γ_{in} = s_{11} +
\dfrac{s_{12} \cdot s_{21} \cdot Γ_l}
{1-(s_{22} \cdot Γ_l)} \$
substituting Γin formula in the equation, we can easily get, \$ \dfrac{1}{Γ_{out}}=Γ_l\$.
And hence if one port is oscillating then the other port has to oscillate simultaneously. And thats what condition 2 and 3 convey.
Best Answer
The early satellite TV receivers used a 70 MHz IF, to filter and amplify and limit the down converted signal.
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On the other hand, in the 1970s, there were satellite to satellite laser links, and the data streams ended up on earth and were recorded onto photographic film. To handle the downlink (not the space-based laser links), some radio methods were needed to handle clouds and other non-laser friendly issues.
The datarate was 700,000,000 bits per second.
I don't think 70MHz IFs (with +-5MHz bandwidth?) would suffice.
I do recall various MODEMS (back then, we called them bit-syncs) crafted to handle 150,000,000 bits per second in the early 1970s using ECL; some in the late 1970s even were adaptively equalized, using coax-delays and Gilbert-cell multipliers; these also functioned at 150,000,000 bits per second.
In the late 1970s I recall boxes providing nearly 1/2 GigaBit spaceborne. These likely used 5x or 10x the bitrate as the carrier, just so the antenna bandwidth and the antenna beam-forming, were practical.