Chapter Summary

Chapter Summary

Key Points

• 1.
  The broadcast channel models one-to-many communication

  A single transmitter sends independent messages to multiple receivers, each observing a different noisy version of the transmitted signal. The capacity region — the set of simultaneously achievable rate tuples — is known in closed form only for degraded broadcast channels, where $X \to Y_1 \to Y_2$ forms a Markov chain.

• 2.
  Superposition coding is the key achievability technique

  The weak user's message is encoded as a coarse "cloud center" and the strong user's message as a fine "satellite." The weak user decodes only the cloud; the strong user decodes both layers via successive decoding. The capacity region is $R_{2} \leq I(U; Y_2)$ and $R_{1} \leq I(X; Y_1 | U)$ over all $p(u)p(x|u)$.

• 3.
  The Gaussian BC has a clean closed-form capacity region

  For the scalar Gaussian BC with noise variances $N_1 < N_2$, the capacity region is parameterized by a power-splitting ratio $\alpha \in [0,1]$: $R_{2} \leq \frac{1}{2}\log(1 + \alpha P / ((1-\alpha)P + N_2))$ and $R_{1} \leq \frac{1}{2}\log(1 + (1-\alpha)P/N_1)$.

• 4.
  Superposition coding strictly outperforms TDMA

  The superposition coding boundary is a convex curve that lies strictly above the TDMA straight line connecting the corner points. The gain is largest when users have very different channel qualities. This is the information-theoretic foundation for NOMA.

• 5.
  The converse uses the entropy power inequality

  Bergmans' converse shows that Gaussian inputs are optimal for the Gaussian BC. The entropy power inequality — $N(X+Y) \geq N(X) + N(Y)$ with equality iff both are Gaussian — is the essential tool that forces Gaussian optimality in the converse argument.

• 6.
  MAC-BC duality connects uplink and downlink

  The capacity region of the Gaussian BC with total power $P$ equals the capacity region of the dual MAC with individual powers summing to $P$. This duality extends to MIMO and is the computational basis for practical beamforming design. The MIMO BC capacity is achieved by dirty paper coding.