Prerequisites & Notation

Before You Begin

This chapter introduces multiuser information theory through the multiple access channel (MAC). We build on the single-user channel coding theory developed in earlier chapters and extend it to settings with multiple independent transmitters communicating to a common receiver. The key new ingredient is the notion of a rate region — a set of simultaneously achievable rate tuples — rather than a single capacity number.

Joint and conditional typicality, jointly typical sequences(Review ch03)
Self-check: Can you state the joint AEP and the probability that two independently drawn sequences are jointly typical?
Channel coding theorem for DMCs: achievability via random coding and converse via Fano's inequality(Review ch04)
Self-check: Can you sketch the random coding argument for a single-user DMC?
Gaussian channel capacity and water-filling(Review ch05)
Self-check: Can you write the capacity formula $C = \frac{1}{2}\log(1 + \text{SNR})$ for the scalar AWGN channel?
Mutual information, chain rule, and conditioning reduces entropy(Review ch01)
Self-check: Can you expand $I(X_1, X_2; Y)$ using the chain rule for mutual information?
Differential entropy and Gaussian maximizes entropy under a variance constraint(Review ch05)
Self-check: Why does $h(X) \leq \frac{1}{2}\log(2\pi e \sigma^2)$ with equality iff $X \sim \mathcal{N}(0, \sigma^2)$ ?
Basic linear algebra: eigenvalues, positive semidefinite matrices, trace
Self-check: Can you diagonalize a Hermitian matrix and interpret the eigenvalues of a covariance matrix?

Notation for This Chapter

We introduce notation for the multiuser setting. Subscripts $1, 2, \ldots, K$ index users. Rate tuples live in $\mathbb{R}_+^K$ . The capacity region $\mathcal{C}$ is a subset of the rate space.

Symbol	Meaning	Introduced
$X_1, X_2, \ldots, X_K$	Channel inputs from users $1, 2, \ldots, K$	s01
$Y$	Channel output at the receiver	s01
$R_{1}, R_{2}, \ldots, R_{K}$	Rates (bits per channel use) of users $1, 2, \ldots, K$	s01
$\mathcal{C}$	Capacity region (set of achievable rate tuples)	s01
$P_1, P_2$	Power constraints for users 1 and 2 in the Gaussian MAC	s03
$Z$	Additive noise random variable, typically $Z \sim \mathcal{N}(0, \sigma^2)$	s03
$\mathbf{H}_{k}$	Channel matrix for user $k$ in the MIMO MAC	s04
$\mathbf{K}_k$	Input covariance matrix $\mathbb{E}[\mathbf{x}_k \mathbf{x}_k^H]$ for user $k$	s04
$\mathcal{S}$	A subset of user indices, $\mathcal{S} \subseteq \{1, \ldots, K\}$	s05
$R(\mathcal{S}) = \sum_{k \in \mathcal{S}} R_{k}$	Sum rate of users in subset $\mathcal{S}$	s05
$h_k$	Fading coefficient for user $k$ (scalar fading MAC)	s06

← Ch 13 Channel Model and Capacity Region