Exercises

Desired signal at user $k$ from precoder $\mathbf{v}_{k}$: $|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k}|^2$.

Interference: $\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2$. Noise: $\sigma^2$.

$\text{SINR}_k = |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k}|^2 / (\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2)$. $\blacksquare$

Sum-rate inner subproblem: WMMSE, which is non-convex (reformulated as a block-convex problem with local optima). Max-min inner subproblem: SOCP bisection — each SOCP is convex and globally solvable in polynomial time.

Max-min AO gives globally optimal active updates at each outer iteration; sum-rate AO is locally optimal. The outer AO loop has the same local-minimum issue in both, but the quality of the inner solve is different. $\blacksquare$

$\text{SINR}_k \geq \gamma$ becomes $|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k}|^2 \geq \gamma (\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2)$.

Pick a phase so $\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k} \geq 0$ (real positive). Then the LHS is $(\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k})^2$ (without absolute value), the sqrt of which is linear in $\mathbf{v}$. SOC: $\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k} \geq \sqrt{\gamma} \cdot \| [\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}]_{j \neq k}; \sqrt{\sigma^2}\|$.

The SOCP is: minimize $\|\mathbf{W}\|_F^2$ subject to the $K$ SOC constraints and an imaginary-part equality (the phase rotation). Standard SOCP solvers handle this. $\blacksquare$

Correlation $\rho = 0.95$. Inflation factor $\sim 1/(1-\rho^2) = 1/(1 - 0.9025) = 10.26$, or $10\text{ dB}$ per-user SNR loss to ZF.

The RIS can steer reflected beams in different directions, reducing effective correlation to $< 0.3$. Inflation $\sim 1.1$, $< 1\text{ dB}$ loss.

Combined with $N^2 = 64^2 = 4096$ SNR boost (~$36\text{ dB}$) minus the $1\text{ dB}$ residual inflation, the RIS gain is $\sim 10 + 36 - 1 = 45\text{ dB}$ in per-user SNR, roughly $3 \log_2 N = 18$ bits/s/Hz per user at high SNR, or $\sim 6$ bits/s/Hz in the sum rate. Substantial. $\blacksquare$

Start with interval $[\gamma_{\text{lo}}, \gamma_{\text{hi}}]$ of width $W_0$. After $t$ iterations, width $= W_0 / 2^t$.

Stop when $W < \epsilon$: $t > \log_2(W_0/\epsilon)$.

$O(\log(1/\epsilon))$ bisection calls, each a single SOCP. Total inner cost per AO outer iteration: $O(\log(1/\epsilon) \cdot C_{\text{SOCP}})$. For $\epsilon = 10^{-2}$: 20 iterations. $\blacksquare$

$d = \min(N_t, K)$. Sum rate at high SNR: $R_{\text{sum}} \sim d \log_2 \text{SNR}$.

RIS adds SNR (via $N^2$) but does not change the pre-log factor. At high SNR: $\Delta R = d \cdot 2 \log_2 N$ (additional bits/s/Hz from RIS SNR gain).

RIS is a power/SNR amplifier, not a DoF generator. To serve more users, need more active antennas. $\blacksquare$

The WMMSE objective as a function of $w_k$ alone: $f(w_k) = w_k e_k - \log_2 w_k$.

$df/dw_k = e_k - 1/(w_k \ln 2)$. Setting to zero: $w_k = 1/(e_k \ln 2)$. (Equivalently $w_k = 1/e_k$ if using natural log; the extra $\ln 2$ factor is conventional.)

The weight $w_k$ is the inverse MSE of user $k$'s estimate — higher weight for better-served users. The sum-rate objective rewards reducing the "worst MSE" because its log-derivative is steepest there. $\blacksquare$

For any set $\{R_k\}$, $\min_k R_k \leq (1/K) \sum_k R_k$.

$R_{\text{mm}}(\mathbf{W}, \boldsymbol{\Phi}) \leq R_{\text{sum}}(\mathbf{W}, \boldsymbol{\Phi})/K$. Maximizing LHS over $(\mathbf{W}, \boldsymbol{\Phi})$: $R_{\text{mm}}^\star \leq R_{\text{sum}}^\star / K$.

A sharp upper bound when users are symmetric (e.g., i.i.d. Rayleigh channels); slack otherwise. In practice, max-min is roughly $0.7$-$0.9 \times R_{\text{sum}}/K$ in MU-RIS scenarios. $\blacksquare$

Per-user: $\mathbb{E}\|\mathbf{h}_{k,\text{eff}}\|^2 \approx \mathbb{E}\|\mathbf{h}_{k,d}\|^2 + N^2 \alpha^2 \beta^2$ under coherent RIS alignment. For $\alpha = \beta = 1$ (normalized) and $\mathbb{E}\|\mathbf{h}_{k,d}\|^2 = N_t = 2$: $\mathbb{E}\|\mathbf{h}_{k,\text{eff}}\|^2 \approx 2 + 1024 = 1026$.

No-RIS sum rate (MRT): $2 \log_2(1 + 10 \cdot 2 / 2) = 2 \log_2 11 \approx 6.9$ bits/s/Hz. RIS sum rate: $2 \log_2(1 + 10 \cdot 1026 / 2) = 2 \log_2 5131 \approx 24.6$ bits/s/Hz.

$\Delta R \approx 17.7$ bits/s/Hz — dramatic, but note the assumption of coherent alignment. Under imperfect CSI or finite iterations, the realized gain is $\sim 70$-$80\%$ of this. $\blacksquare$

Sum rate is concave in per-user SNR (at each fixed $K$), but the marginal return is non-decreasing in channel strength: $d R_k / d P_k = (\log_2 e) / (\sigma^2/h_k^2 + P_k)$. Stronger $h_k$ → larger marginal return → allocate more power.

The RIS can further amplify strong users via matched-filter phases. Sum-rate optimization naturally chooses $\boldsymbol{\Phi}$ to over-amplify the strong user, widening the gap.

Weak users get less power and less RIS steering. Their rates may approach zero in extreme cases. Max-min fairness (Section 7.3) reverses this tendency. $\blacksquare$

For $\epsilon = 10^{-3}$ relative convergence: 10-20 outer AO iterations are typical.

More users means more local minima in the passive subproblem (higher-rank QCQP). Iteration count grows roughly linearly with $K$: $\sim 10 + 2K$ outer iterations.

Across coherence blocks, warm-started AO converges in 3-5 iterations regardless of $K$. The one-time cold-start cost is amortized. $\blacksquare$

Gives $+N_t$ new DoF (until $N_t > K$), plus SNR boost of $2\times$ (3 dB per user).

Gives $N^2$ per-user SNR gain (no new DoF).

If $N_t \leq K$: doubling antennas buys new DoF — probably wins. If $N_t \gg K$: additional antennas are marginal (favorable propagation); RIS's $N^2$ gain is more valuable. If link is blocked: RIS is the only option (direct path gives no SNR).

RIS wins when: blocked direct path, $N_t \geq K$ already, or power-cost constraints make active antennas expensive. Doubling antennas wins when: $N_t < K$, no blockage, minor added hardware cost. $\blacksquare$

If one SINR is strictly above $\gamma^\star$, we can reduce power for that user and redistribute, improving the min. Contradiction; hence all constraints are equality at optimum.

All SOC constraints are active (on the boundary). The feasibility oracle operates at the boundary of the SOC cone, which is numerically sensitive but well-handled by modern SOCP solvers.

The bisection converges to this tight boundary; tolerance $\epsilon$ determines the precision of the "all equal SINR" condition at the solution. $\blacksquare$

Suppose at $\gamma = \gamma^\star$, user 1 has $\text{SINR}_1 > \gamma^\star$ strict. Reduce $\|\mathbf{v}_{1}\|^2$ slightly → SINR$_1$ drops a bit but remains $\geq \gamma^\star$. Feasibility still holds. Now redistribute the freed power to the worst-served user (say user $k^\star$): $\|\mathbf{v}_{k^\star}\|^2$ grows, $\text{SINR}_{k^\star}$ grows above $\gamma^\star$.

We just constructed a feasible solution at a higher $\gamma$, contradicting optimality. Hence at $\gamma^\star$, all users achieve exactly $\gamma^\star$. $\blacksquare$

Compute pairwise channel correlations; group users with high correlation into clusters. Within a cluster, a single $\boldsymbol{\Phi}$ serves all users jointly.

In each coherence block, serve one cluster with its dedicated $\boldsymbol{\Phi}^{(c)}$ and precoder. Round-robin across clusters.

• More clusters: finer granularity, but pilot overhead per cluster × number of clusters can overflow coherence.
• Fewer clusters: less RIS specialization, but lower overhead.
• Cluster size affects fairness — large clusters dilute per-user RIS gain.

This is the hierarchical scheduler of the Caire et al. 2022 CommIT contribution. In practice, 2-4 clusters with 4-8 users each works well for $K \leq 32, N = 256$. $\blacksquare$