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Does the noisy-channel coding theorem apply to automatic repeat request systems? Some people seem to imply that, when a system has 2-way communication, such systems can have far lower error rates at the same data bit rate compared to one-way communication. The implication is that transmitting in both directions allows the system to do things that are simply impossible for one-way communication.

The presentations of the noisy-channel coding theorem (also called Shannon's theorem) that I've heard or seen all seem to assume one transmitter and one receiver (1-way communication), using some sort of forward error correction code, without any retransmissions.

What books or papers discuss the Shannon–Hartley theorem or Shannon's theorem with two-way communication with retransmission requests?

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  • \$\begingroup\$ Consider that the retransmissions lower the rate you can transfer useful data at. \$\endgroup\$
    – user253751
    Jan 31, 2015 at 10:03

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Yes. A retransmission is equivalent to sending parity bits in an FEC code. One advantage is that it can deal with varying channel conditions - more retransmits when the channel is worse, fewer when it is better. However, a frame must be thrown out if even a single bit is flipped (presuming no FEC) so retransmits end up being highly inefficient as you end up sending the same data multiple times, lowering the overall rate. If your frames are 1000 bits long and your bit error rate is 1 in 1000 bits, then on average every single frame has an error and nothing can be transmitted unless some sort of FEC is used. So yes, you can make a system have 'zero errors' by retransmitting anything that gets garbled, but this comes at the expense of bandwidth.

If you have two-way communications, you can use the back channel to adjust the FEC parameters (e.g. puncturing) or modulation style (BPSK, QPSK, QAM) to better take advantage of the current channel conditions. There are so called 'rateless' codes that take advantage of this back-and-forth to achieve the maximum rate possible under the channel conditions with a fixed bit error rate.

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