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I'm curious how AM and FM repeaters work.

Is the received signal demodulated and then remodulated at the new frequency or do they use some other technique?

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It is possible to repeat a signal without demodulating it. This is done on some amateur satellites where it is called a linear transponder. The advantage of this method is that it can carry multiple signals, and any mode, including SSB and CW. The disadvantage is that it retransmits all the noise it hears and cannot clean up the modulation along the way — very much like how putting an amplifier before your receiver can't improve the signal-to-noise ratio.

A typical FM repeater consists of a fairly ordinary receiver and transmitter, with the audio output of the receiver connected to the (microphone) audio input of the transmitter.

The second most critical element is a duplexer, which is a pair of very narrow filters which ensure that the receiver is not overloaded by the transmitted signal on the same antenna. (The less-usual case of a cross-band repeater, which has far more frequency separation between transmit and receive frequencies, does not need a duplexer.)

Finally, you usually have a repeater controller, which can be connected to the audio, PTT, etc. signals from/to the radios in order to implement additional features (repeater ID, time-out timer, repeater linking/phone patch, etc.).

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    $\begingroup$ Just wanted to add that for digital transmissions (which OP didn't ask about), this is even clearer in favor of decode&forward (instead of just amplify&forward): Since these typically have forward error correction, you'd always want to demodulate and error correct on the repeater. $\endgroup$ – Marcus Müller Dec 11 '16 at 17:28

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