Chapter Summary

Chapter Summary

Key Points

• 1.

  Multi-user RIS is about interference shaping, not just SNR boosting. For $K$ users, the RIS simultaneously amplifies desired signals and nulls cross-interference through $\boldsymbol{\Phi}$. The inter-user interference $\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2$ is the new ingredient; it determines the $K$-user rate region.

• 2.

  Sum-rate via AO + WMMSE. Each outer iteration holds $\boldsymbol{\Phi}$ fixed and runs WMMSE to maximize $\sum_k \log_2(1 + \text{SINR}_k)$ over $\mathbf{W}$; each passive update solves the rank-$K$ unit-modulus QCQP using Chapter 6 methods. Monotone convergence; typically 10-20 outer iterations.

• 3.

  Max-min fairness via AO + SOCP bisection. The min-SINR problem is polynomially tractable for fixed $\boldsymbol{\Phi}$: bisect on the common SINR target $\gamma$, solve a second-order cone program at each bisection step. The max-min problem is thus algorithmically easier than sum rate — each inner solve is global-optimal.

• 4.

  RIS is an SNR-boost, not a DoF-multiplier. The multiplexing gain of a $K$-user MU-RIS system is $\min(N_t, K)$, determined by the active antenna count. The RIS contributes $N^2$ per-user SNR gain but does not create new spatial DoF. To serve more users, add more active antennas; RIS improves each served user.

• 5.

  Biggest RIS wins: correlated users, low $N_t$, blocked UEs. The RIS recovers ZF-friendly channels when UEs are spatially correlated; it acts as a virtual aperture when the BS has few antennas; it provides the only link for blocked users. In massive-MIMO favorable-propagation regimes, the RIS marginal gain is smaller (already near-optimal without it).