Chapter Summary

Chapter Summary

Key Points

• 1.

  Feedback enlarges the MAC capacity region. Unlike the point-to-point case (where feedback does not help), feedback in the MAC allows encoders to correlate their inputs through shared observations, enlarging the capacity region. The Cover-Leung inner bound uses an auxiliary variable to capture this common information.

• 2.

  Gaussian MAC with feedback: coherent combining. For the Gaussian MAC, Ozarow showed that feedback achieves a sum-rate of $\frac{1}{2}\log(1 + (\sqrt{P_1}+\sqrt{P_2})^2/N)$ — the full beamforming gain. Feedback effectively turns independent transmitters into a coherent array.

• 3.

  Feedback helps non-degraded BCs via retransmission. For non-degraded broadcast channels, the transmitter can XOR (or linearly combine) information that different receivers have decoded, serving multiple receivers with each retransmission. For degraded BCs, feedback does not help.

• 4.

  The erasure BC XOR trick. In the erasure BC, a single XOR retransmission of bits received by different users serves both users simultaneously, strictly improving the sum-rate. This idea underlies network coding in practical HARQ systems.

• 5.

  Shannon's two-way channel: interaction can help. When two users communicate simultaneously over a coupled channel, interactive encoding (adapting to past observations) can improve rates beyond independent coding. The general capacity region remains open after over sixty years.

• 6.

  The Gaussian two-way channel decomposes. For the additive Gaussian two-way channel, the two directions decouple because each user can cancel its own known interference. Each direction achieves full point-to-point capacity, and interaction does not help.