Exercises

$\mathbf{h}_{k,\text{eff}}^H = \mathbf{h}_{d,k}^H + \sum_{m=1}^M \mathbf{h}_{2,km}^H \boldsymbol{\Phi}_m \mathbf{H}_{1,m}$

• $\mathbf{h}_{d,k}$: direct BS-to-UE-$k$ channel.
• $\mathbf{H}_{1,m}$: BS-to-panel-$m$ channel.
• $\boldsymbol{\Phi}_m$: phase matrix of panel $m$.
• $\mathbf{h}_{2,km}$: panel-$m$-to-UE-$k$ channel. Sum the cascaded paths across all $M$ panels. $\blacksquare$

$M^2 N^2 = 9 \cdot 10^4 = 90000$.

$N_{\text{total}}^2 = 300^2 = 90000$.

At equal geometry (same distances to all panels), multi-RIS with $M$ panels of $N$ elements each delivers the SAME coherent gain as a single panel of $MN$ elements. The advantage of multi-RIS is not higher gain under uniform geometry — it's geometric diversity and graceful degradation. $\blacksquare$

Signal bounces: BS → panel 1 → panel 2 → UE. Each transition applies coherent combining.

At panel 1: $N_1$ amplitude, $N_1^2$ power. At panel 2 (receiving amplified signal): $N_2$ amplitude, $N_2^2$ power. Cascade multiplies: $N_1^2 N_2^2$.

Two parallel panels ($M = 2$): $M^2 N^2 = 4 N^2$ for same $N$. Double-RIS ($N_1 = N_2 = N$): $N^4$ — much larger for large $N$. But four-hop path loss vs. two-hop: path loss is much more severe for double-RIS. $\blacksquare$

Distributed access points, coordinated by CPU via fronthaul, provide active aperture.

Distributed reflecting surfaces, coordinated by CPU via control links, provide passive aperture.

Both are: distributed, centrally coordinated, contributing aperture to user signals. The distinction between "transmitter" and "channel shaper" blurs. The CPU optimizes all of them jointly as a network-wide resource. $\blacksquare$

Active update: same as single-RIS. Passive updates: $M = 4$ panel subproblems, each similar complexity to single-RIS's single passive update. Total per-iter cost: $(1 + M) = 5\times$ single-RIS.

AO iteration count similar. Total: $\sim 5 \times$ single-RIS compute. For $N = 256, K = 4$: single-RIS $\sim 30$ ms $\to$ multi-RIS $\sim 150$ ms.

At $M = 20$: $\sim 20 \times$ cost. Hierarchical scheduling (CommIT contribution) keeps this manageable by parallelizing across cluster boundaries. $\blacksquare$

Amplitude: $\alpha_1 = \lambda / (4\pi d_1)$. Aperture factor from panel 1 (combined into amplitude): $\alpha_1 \cdot \sqrt{N_1}$.

Amplitude: $\alpha_{12} = \lambda / (4\pi d_{12})$. Similar aperture factor from panel 2.

Amplitude: $\alpha_2 = \lambda / (4\pi d_2)$.

End-to-end amplitude: $\alpha_1 \alpha_{12} \alpha_2 = \lambda^{3} / [(4\pi)^3 d_1 d_{12} d_2]$. Power: $(amp)^2 \propto 1/(d_1 d_{12} d_2)^2$ — four-hop path loss in power. $\blacksquare$

$O(L^3 N_t^{3} K) + O(M N^3)$. With $L = 16, N_t = 4, K = 8, M = 8, N = 256$: $16^3 \cdot 64 \cdot 8 + 8 \cdot 256^3 \approx 2.1 \cdot 10^6 + 1.3 \cdot 10^8 \approx 10^8$ flops.

$10^8$ flops on a GHz CPU: $\sim 100$ ms. Borderline for real-time (10 ms coherence) but feasible at 100 ms coherence.

For $L = 64, M = 32, K = 100$: $> 10^{10}$ flops per block. Infeasible centralized. Must use hierarchical scheduling (CommIT approach) to split into cluster-level sub-problems. $\blacksquare$

For the sum of panel contributions to have amplitude $M \cdot N \alpha \beta$ (coherent) instead of $\sqrt{M} \cdot N \alpha \beta$ (incoherent), all panel outputs must be phase-aligned at the UE.

Panels need a common phase reference — distribute a high- quality clock signal + calibrate per-panel offsets. Without this, each panel's phase is in an arbitrary offset, and the summed signals add in power (incoherent) rather than amplitude (coherent).

Incoherent: SNR $\propto M N^2$ instead of $M^2 N^2$. Factor of $M$ in SNR. For $M = 4$: $6$ dB loss. Unacceptable for high-SNR 6G targets. Sync is mandatory. $\blacksquare$

$L \cdot N_t = 32 \cdot 4 = 128$ effective active antennas (for ZF-like precoding).

Per panel: $N^2 = 256^2 = 65536$. $M$ panels: $M \cdot N^2 = 10 \cdot 65536 = 655360$.

RIS aperture gain: ~5000× the AP aperture gain. But AP gain is active (actually transmitted power); RIS gain is passive (reshaped, not amplified). In the SNR formula, RIS appears inside the log as an additive term. The total SNR: $P_t (L N_t + M N^2) / \sigma^2$. RIS contribution dominates the additive SNR improvement. $\blacksquare$

A UE is "served by" panels within its service radius (limited by QoS threshold). Typical service radius at 28 GHz, $N = 256$: $R \sim 100$ m.

By PPP: expected panels in $\pi R^2$ = $\lambda_{\text{RIS}} \pi R^2$. $= 3 \cdot \pi \cdot 0.1^2 = 3 \cdot 0.0314 = 0.09$ panels expected in service range.

At this density, most UEs see fewer than one RIS panel. Coverage is spotty: some UEs see 1-2 panels, others none. Achieving $> 95\%$ coverage requires $\lambda_{\text{RIS}} \geq 10$ panels/km² at this $R$. This informs the deployment density design. $\blacksquare$

Single large BS with $N_t$ antennas at one site. Coverage radius limited by path loss; $N_t$ grows quadratically with coverage size.

$L$ APs + $M$ RIS panels, distributed across coverage area. Total aperture spread. Each user close to some AP/RIS: short per-link distances, modest per-site hardware.

Centralized: $N_t \propto \text{area}^2$ (too expensive at scale). Cell-free: $L \cdot N_t \propto \text{area}$ (linear). RIS-aided adds $M N^2$ gain at $\sim \lambda_{\text{RIS}} \text{area}$ panel cost. Linear scaling dominates at city scale. $\blacksquare$

$\alpha_1 = \lambda/(4\pi d_1) = 0.0107 / 125 = 8.6 \cdot 10^{-5}$. $\alpha_{12} = 0.0107 / 62.8 = 1.7 \cdot 10^{-4}$. $\alpha_2 = 0.0107 / 188 = 5.7 \cdot 10^{-5}$.

$(\alpha_1 \alpha_{12} \alpha_2)^2 (N_1 N_2)^2 = (8.6 \cdot 10^{-5})^2 (1.7 \cdot 10^{-4})^2 (5.7 \cdot 10^{-5})^2 \cdot 128^4$ $\approx 7.5 \cdot 10^{-24} \cdot 2.7 \cdot 10^8 = 2 \cdot 10^{-15}$.

$\alpha_d = 0.0107 / 377 = 2.8 \cdot 10^{-5}$. SNR contribution: $\alpha_d^2 = 8 \cdot 10^{-10}$.

Direct link far stronger than double-RIS at this geometry. Double-RIS wins only when direct is blocked (absorbed) with $> 20$ dB loss. Otherwise: direct is better. $\blacksquare$

$N$ per panel × $M$ panels × $K$ users + AP pilots. For $M = 10, N = 256, K = 100$: $256000$ pilots per coherence block. Infeasible.

Cluster users into $M_c = 2$-$3$ panels per cluster, $K_c = 10$ users per cluster. Cluster pilots: $256 \cdot 3 \cdot 10 = 7680$ pilots. $\sim 30\times$ reduction.

Pilot reuse across clusters, compressed sensing within clusters, two-timescale CSI (slow BS-panel, fast panel-UE) further reduces overhead by $\sim 5$-$10\times$. Total: $100$-$300\times$ reduction. Feasible at 6G coherence times. $\blacksquare$

Panel $m$ contributes amplitude $N \alpha_m \beta_m$ at the UE (under coherent phase alignment), where $\alpha_m, \beta_m$ are per-hop path losses for panel $m$.

Under synchronized phase alignment (all panels in phase at UE): Total amplitude = $\sum_m N \alpha_m \beta_m = N \sum_m \alpha_m \beta_m$. Under symmetric geometry ($\alpha_m \beta_m = \alpha \beta$ for all $m$): Total = $M N \alpha \beta$.

SNR $\propto (M N \alpha \beta)^2 = M^2 N^2 \alpha^2 \beta^2$. Matches Theorem 12.1. $\blacksquare$

$\sim 20$ APs at $\sim 150$ m spacing, each with $N_t = 4$-$8$ antennas. Lamp-post mounting, building corner integration.

$\sim 10$-$20$ panels at major building facades, shadow boundaries, street junctions. Each panel $N = 128$-$256$.

APs uniform on grid for active coverage. RIS panels strategic: fill shadow zones, corners, high-density UE zones. Panels complement APs in geometric coverage.

Central compute site handles all AP and RIS coordination. Fronthaul: $\sim 10$ Gbps per AP, $\sim 10$ kbps per RIS.

Per-user: $\sim 5$-$10$ Gbps under coherent operation. Coverage: $> 95\%$ with this density.

APs: $\sim 20 \times \$10K = \$200K$. RIS: $\sim 20 \times \$1K = \$20K$. Fronthaul: $\sim \$50K$. Total: $\sim \$270K$ per km² — competitive with 4G/5G cell costs while delivering 6G speeds. $\blacksquare$