Chapter Summary

Chapter 16 Summary

Key Points

• 1.

  For the general (non-degraded) broadcast channel, Marton's inner bound combines superposition coding with binning to achieve rate pairs beyond what superposition alone can reach. The binning cost $I(V_1; V_2 | U)$ is the price paid for correlating codewords destined for different receivers.

• 2.

  The Nair-El Gamal outer bound is the tightest known converse. It matches Marton's inner bound for degraded, deterministic, and binary-input BCs, but whether it is tight in general remains one of the major open problems in network information theory.

• 3.

  The capacity region of the Gaussian MIMO broadcast channel is achieved by dirty-paper coding (DPC), where the encoder pre-cancels interference sequentially using Costa's result. The converse uses the channel enhancement technique of Weingarten, Steinberg, and Shamai.

• 4.

  MAC-BC duality establishes that the MIMO BC and the dual MIMO MAC (with reversed channels and sum power constraint) have the same capacity region. This is both a conceptual insight (downlink and uplink are two sides of the same coin) and a computational tool (the MAC is a convex optimization).

• 5.

  The capacity region boundary can be computed via iterative water-filling on the dual MAC or the WMMSE algorithm on the BC directly. Both converge to the global optimum for sum-rate maximization.

• 6.

  In practice, DPC is too complex for real-time systems. Linear precoding (ZF, MMSE, RZF) approximates DPC with orders of magnitude less complexity, achieving within 1-3 dB of the DPC sum rate. For massive MIMO ($n_t \gg K$), the gap vanishes asymptotically.