Prerequisites & Notation

Before You Begin

Chapter 5 introduced the joint RIS optimization for single-user and multi-user scenarios at the framework level. Chapter 6 developed the passive-beamforming algorithms. This chapter specializes the machinery to multi-user problems — sum rate, max-min fairness, the active-passive interplay that distinguishes MU-RIS from SU-RIS.

MU-MIMO broadcast channel: capacity region, duality, precoder design(Review ch06)
Self-check: For a $K$ -user MISO downlink, can you state the sum-rate capacity in terms of the uplink dual and water-filling?
Linear precoders: MRT, ZF, regularized ZF, MMSE(Review ch17)
Self-check: Derive the ZF precoder $\mathbf{W}^{\text{ZF}} = \mathbf{H}^{\dagger}$ and its SINR floor.
SOCP / QCQP reformulations of communication problems(Review ch03)
Self-check: Can you rewrite a min power / QoS problem as an SOCP with second-order cone constraints?
RIS AO framework from Chapter 5 and passive algorithms from Chapter 6(Review ch05)
Self-check: State the bilinear structure of the joint problem and the three passive-update algorithms.
Bisection search for target-SINR problems
Self-check: Why does bisecting over a feasibility oracle solve a max-min problem in polynomial time?

Notation for This Chapter

Multi-user-specific notation builds on Chapter 5's conventions; per-user subscripts $k$ are added throughout.

Symbol	Meaning	Introduced
$K$	Number of users (scalar)	s01
$\mathbf{v}_{k}$	BS beamforming vector for user $k$ , $\mathbf{v}_{k} \in \mathbb{C}^{N_t}$	s01
$\mathbf{W}$	Stacked precoder $[\mathbf{v}_{1}, \ldots, \mathbf{v}_{\ntn{nusers}}] \in \mathbb{C}^{N_t \times K}$	s01
$\mathbf{h}_{k,\text{eff}}^H$	Effective channel for user $k$ : $\mathbf{h}_{k,d}^H + \mathbf{h}_{k,2}^H \boldsymbol{\Phi} \mathbf{H}_1$	s01
$\text{SINR}_k$	Signal-to-interference-plus-noise ratio for user $k$	s01
$R_k$	User $k$ 's achievable rate, $R_k = \log_2(1 + \text{SINR}_k)$	s01
$R_{\text{sum}}, R_{\text{mm}}$	Sum rate $\sum_k R_k$ and max-min rate $\min_k R_k$	s02
$\gamma$	Common SINR target in the max-min bisection	s03
$w_k$	WMMSE weight for user $k$ at a stationary point	s02
$\mathbf{u}_k$	MMSE combiner at user $k$ 's virtual receiver (scalar for MISO)	s02

← Ch 6 The Multi-User RIS Signal Model