Prerequisites & Notation

Before You Begin

This chapter studies the broadcast channel (BC) — the dual of the multiple access channel from Chapter 14. A single transmitter sends independent messages to multiple receivers, each observing a different noisy version of the transmitted signal. We focus on the degraded broadcast channel, where the receivers can be ordered by channel quality, and develop the capacity region through superposition coding and the corresponding converse.

Joint and conditional typicality, jointly typical sequences(Review ch03)
Self-check: Can you state the joint AEP and bound the probability that independently drawn sequences are jointly typical?
Channel coding theorem for DMCs: achievability and Fano's inequality(Review ch04)
Self-check: Can you sketch the random coding argument and state Fano's inequality?
Gaussian channel capacity and differential entropy(Review ch05)
Self-check: Can you write the AWGN capacity $C = \frac{1}{2}\log(1 + \text{SNR})$ and explain why Gaussian inputs are optimal?
Mutual information chain rule, data processing inequality(Review ch01)
Self-check: Can you state the data processing inequality: if $X \to Y \to Z$ then $I(X;Z) \leq I(X;Y)$ ?
Entropy power inequality(Review ch05)
Self-check: Can you state the entropy power inequality: $e^{2h(X+Y)} \geq e^{2h(X)} + e^{2h(Y)}$ for independent $X,Y$ ?
Multiple access channel capacity region(Review ch14)
Self-check: Can you describe the MAC capacity region as a pentagon with three mutual information constraints?

Notation for This Chapter

We use subscripts 1 and 2 for the strong and weak receivers, respectively. In the degraded BC, user 1 has the better channel: $X \to Y_1 \to Y_2$ forms a Markov chain. The auxiliary random variable $U$ plays a central role in the capacity region characterization.

Symbol	Meaning	Introduced
$X$	Channel input from the (single) transmitter	s01
$Y_1, Y_2$	Channel outputs at receivers 1 (strong) and 2 (weak)	s01
$R_{1}, R_{2}$	Rates (bits per channel use) for users 1 (strong) and 2 (weak)	s01
$W_1, W_2$	Independent messages for users 1 and 2	s01
$\mathcal{C}$	Capacity region of the broadcast channel	s01
$U$	Auxiliary random variable (cloud center) used in superposition coding	s02
$P$	Transmit power constraint: $\frac{1}{n}\sum_{i=1}^n x_i^2 \leq P$	s03
$\alpha$	Power-splitting parameter: fraction $\alpha P$ allocated to user 2 (weak user)	s03
$N_1, N_2$	Noise variances at receivers 1 and 2, with $N_1 < N_2$ (user 1 is stronger)	s03
$Z_{1}, Z_{2}$	Additive noise random variables at receivers 1 and 2	s03
$N(X)$	Entropy power of $X$ : $N(X) = \frac{1}{2\pi e} e^{2h(X)}$	s04

← Ch 14 The Broadcast Channel Model