Summary

Chapter 26 Summary: Multiuser Information Theory

Key Points

• 1.

  The Gaussian MAC capacity region is a pentagon defined by individual rate constraints $R_k \leq \frac{1}{2}\log(1 + P_k/N)$ and a sum-rate constraint $\sum R_k \leq \frac{1}{2}\log(1 + \sum P_k/N)$. The sum capacity equals the single-user capacity with total power $\sum P_k$ — there is no loss from multiple access when SIC is used. The region has a polymatroidal structure with $K!$ corner points corresponding to SIC decoding orders.

• 2.

  The degraded BC capacity region is achieved by superposition coding: the transmitter encodes messages at different power levels, and stronger receivers decode and strip weaker users' messages first. For the MIMO BC, dirty-paper coding (DPC) achieves the capacity region, and the MAC-BC duality transforms the non-convex BC problem into the convex dual MAC problem. Costa's DPC theorem shows that known interference at the transmitter can be pre-cancelled with no rate loss.

• 3.

  The interference channel capacity is unknown in general. The channel has three distinct regimes: strong (decode interference, capacity known), weak (treat as noise, approximately optimal), and moderate (Han-Kobayashi scheme with message splitting). The Etkin-Tse-Wang result shows that a simple HK power split achieves within 1 bit/s/Hz of capacity for all parameter values, providing engineering-accuracy characterisation.

• 4.

  The relay channel is bounded above by the cut-set bound and below by decode-forward (optimal near source) and compress-forward (optimal near destination). Noisy network coding generalises CF to arbitrary multi-relay networks, achieving within a constant gap of the cut-set bound for Gaussian networks. The exact capacity remains open even for the simple single-relay Gaussian channel.

• 5.

  Degrees of freedom (DoF) capture the high-SNR capacity scaling. The Cadambe-Jafar result shows the $K$-user SISO IC has sum DoF $= K/2$, achieved by interference alignment — each user gets half a degree of freedom regardless of $K$. This fundamental result shows that interference does not limit the scaling of multiuser capacity, overturning the conventional wisdom that TDMA ($d_\Sigma = 1$) is the best one can do with single-antenna nodes.

• 6.

  Finite blocklength theory shows that the achievable rate at blocklength $n$ and error probability $\epsilon$ is $R^* \approx C - \sqrt{V/n}\,Q^{-1}(\epsilon)$, where $V$ is the channel dispersion. For URLLC ($n \sim 200$, $\epsilon = 10^{-5}$), the rate penalty can exceed 40% of capacity — a fundamental limit that system designers must account for through conservative MCS selection and HARQ retransmissions.