Exercises

$y = (\mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{H}_1) \mathbf{v}\,s + \mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{w}^{\text{RIS}} + w$.

Amplified RIS noise at UE: $\mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{w}^{\text{RIS}}$, with variance $\sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2$.

Local UE noise $w \sim \mathcal{CN}(0, \sigma^2)$, independent of the RIS noise. Total noise at UE: sum of the two. $\blacksquare$

Coherent: $P_t |N \cdot 2 \cdot 0.1 \cdot 0.1|^2 = P_t \cdot (32 \cdot 0.01)^2 = P_t \cdot 0.102$.

$\sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2 = \sigma^2_{\text{RIS}} N \cdot 0.1^2 \cdot 2^2 = 0.64 \sigma^2_{\text{RIS}} = 1.92 \sigma^2$.

$\sigma^2 + 1.92 \sigma^2 = 2.92 \sigma^2$.

$\text{SNR} = 0.102 P_t / 2.92 \sigma^2 = 0.102/2.92 \cdot 10 = 0.349 = -4.6\text{ dB}$. (Note: coherent gain $(32)^2 = 1024 \times (0.01) = 10.24$ on signal — but noise is also boosted.) $\blacksquare$

$|\psi_n| \leq g_{\max}$ is a closed complex disk of radius $g_{\max}$. Take two feasible $\psi_n^{(1)}, \psi_n^{(2)}$. Midpoint: $(\psi_n^{(1)} + \psi_n^{(2)})/2$. By triangle inequality: $|(\psi_n^{(1)} + \psi_n^{(2)})/2| \leq (|\psi_n^{(1)}| + |\psi_n^{(2)}|)/2 \leq g_{\max}$. ✓

$|\psi_n| = 1$: midpoint of $\psi_n = 1$ and $\psi_n = -1$ is $0$, with $|0| = 0 \neq 1$. Not convex.

Active RIS's convex constraint allows interior-point SOCP solutions; passive RIS requires non-convex methods (SDR, manifold). $\blacksquare$

$|\mathbf{h}_{\text{eff}}|^2 \cdot P_t \sim N^2 g_{\max}^2 \alpha^2 \beta^2 \cdot P_t$.

$\sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2 \sim \sigma^2_{\text{RIS}} N g_{\max}^2 \beta^2$.

UE noise becomes negligible: $\text{SNR} \to (P_t N^2 g_{\max}^2 \alpha^2 \beta^2) / (\sigma^2_{\text{RIS}} N g_{\max}^2 \beta^2) = P_t N \alpha^2 / \sigma^2_{\text{RIS}}$. $\beta^2$ cancels; the RIS-UE path loss is compensated by the amplifier. $\blacksquare$

$\lambda = 3 \cdot 10^8 / 28 \cdot 10^9 = 1.07 \cdot 10^{-2}$ m.

NF = 3 dB → $\sigma^2_{\text{RIS}}/\sigma^2 = 10^{0.3} \approx 2$.

$d^\star = 0.0107 \cdot \sqrt{512/2} = 0.0107 \cdot 16 = 0.17$ m. Still very short. $\blacksquare$

$g_n^2 = g^2 / N = 100 / 256 \approx 0.39$, so $g_n \approx 0.625$, or $-4$ dB.

With 256 elements, per-element gain is actually below unity. The active RIS uses many weak amplifiers rather than one strong one. This is what enables coherent combining without hitting per-amplifier saturation limits. $\blacksquare$

$N$ elements each contribute $g_n \alpha \beta e^{j\theta_n}$. Coherent sum: $|N g_n \alpha \beta|^2 = N^2 g_n^2 \alpha^2 \beta^2 = N^2 (P_{\text{RIS}}/N) \alpha^2 \beta^2 = N P_{\text{RIS}} \alpha^2 \beta^2$. AF relay: $g^2 \alpha^2 \beta^2 = P_{\text{relay}} \alpha^2 \beta^2$. Ratio: $N$ (assuming $P_{\text{RIS}} = P_{\text{relay}}$).

Active RIS: noise from $N$ amplifiers adds independently; total at UE: $\sigma^2_{\text{RIS}} g_n^2 N \beta^2 = \sigma^2_{\text{RIS}} (P_{\text{RIS}}/N) N \beta^2 = \sigma^2_{\text{RIS}} P_{\text{RIS}} \beta^2$. AF relay: $\sigma^2_{\text{relay}} g^2 \beta^2 = \sigma^2_{\text{relay}} P_{\text{relay}} \beta^2$. Same (under equal NF).

Relay needs half the time; active RIS is full-duplex. Factor of 2 in effective rate.

SNR ratio (active RIS / AF relay) = $N$ (coherent) $\times$ 2 (full-duplex) $= 2N$. $\blacksquare$

$10$ mW per element; $N = 128$ elements: total RF = $1.28$ W.

$P_{\text{DC,ampl}} = 1.28 / 0.30 = 4.27$ W.

$\sim 5$ mW × 128 = $0.64$ W bias current.

$\sim 4.9$ W DC for the active RIS panel. Modest but requires wired DC supply at the deployment site. $\blacksquare$

Passive $|\psi_n| = 1$ defines the unit circle — non-convex (the interior is excluded). Product of circles is the torus, non-convex.

Active $|\psi_n| \leq g_{\max}$ defines a closed disk — convex (all interior points allowed). Product of disks is a polydisc, convex.

Convex feasibility + convex constraints + block-convex objective (WMMSE) → globally optimal inner solution. Active RIS optimization is much easier theoretically than passive. In practice, still requires AO outer loop due to bilinear structure. $\blacksquare$

Objective: maximize $|\sum_n g_n \alpha_n \beta_n|^2 /\sigma^2_{\text{RIS}} \sum_n |g_n|^2 |\beta_n|^2$ subject to $\sum_n |g_n|^2 (|\alpha_n P_t|^2 + \sigma^2_{\text{RIS}}) \leq P_{\text{RIS}}$.

Forming the Lagrangian and setting $\partial/\partial g_n = 0$: $g_n \propto \alpha_n \beta_n / (\mu \beta_n^2 + \ldots)$, where $\mu$ is the Lagrange multiplier for the power constraint.

$|g_n|^2 = \max(0, (\alpha_n \beta_n / \nu - \beta_n^2 \sigma^2_{\text{RIS}})/(\alpha_n^2 P_t + \sigma^2_{\text{RIS}}))$ for some water level $\nu$. Strong-channel elements receive more gain. $\blacksquare$

$\text{SNR}_{P} \propto N^2 \alpha^2 \beta^2$ with $\alpha = \beta \sim 10^{-3}$ at 28 GHz, 15 m each. $\sim 128^2 \cdot 10^{-12} = 1.64 \cdot 10^{-8}$ (relative to unit Tx power). Poor.

High-gain regime: $\text{SNR}_{A} \to P_t N \alpha^2 / \sigma^2_{\text{RIS}}$. Roughly $N \cdot 10^{-6} / (10^{0.5} \sigma^2) \sim 10^{-5}$ — much better.

AF relay SNR with single amplifier: $\text{SNR}_{R} \propto g^2 \alpha^2 \beta^2 / \sigma^2_{\text{relay}} \beta^2 \sim g^2 \alpha^2 / \sigma^2$. $\sim 10^{1.5} \cdot 10^{-6} \approx 3 \cdot 10^{-5}$. Comparable to active RIS at $N = 128$ (active wins by $\sim 128/2 = 64 \times \sim 18$ dB).

Active RIS >> AF relay >> Passive RIS. At 28 GHz with 128 elements, active RIS is the clear winner. $\blacksquare$

Indoor WiFi at 5 GHz, UE $5\text{ m}$ from BS and RIS each. Per-hop path loss $\alpha^2 = \beta^2 = \lambda^{2} / (4\pi \cdot 5)^2 = (0.06)^2 / 3943 \approx 1 \cdot 10^{-6}$.

$N^2 \alpha^2 \beta^2 \sim 10^4 \cdot 10^{-12} = 10^{-8}$. With coherent combining and small UE noise, SNR $\sim 20\text{ dB}$ over noise — tolerable.

High-gain active: $N \alpha^2 / \sigma^2_{\text{RIS}} \sim 100 \cdot 10^{-6} / 3 \cdot 10^{-6} \approx 30$ — worse than passive! Because $N^2 \alpha^2 \beta^2 / \sigma^2 = 10^4 \cdot 10^{-12} / 10^{-6} = 10^{-2}$ vs. active's $N \alpha^2 / \sigma^2_{\text{RIS}} \approx 10^{-4}$ if we normalize properly. Passive wins by ~$20\text{ dB}$ in this regime.

Short-distance, high-coherence, low-path-loss scenarios are where passive shines. Active RIS is overkill and adds noise. $\blacksquare$

$\lambda = 0.086$ m. NF $= 5$ dB, so ratio $= 3.16$.

$d^\star = \lambda \sqrt{N/\text{NF}} = 0.086 \sqrt{256/3.16} = 0.086 \cdot 9.0 \approx 0.77$ m.

$d_0 = 70\,\text{m} \gg d^\star$. Active RIS dominates by a very large margin at this geometry. In fact, passive RIS is essentially useless at this distance in mmWave/sub-6 GHz mix; the active version is necessary.

For medium-to-long-range RIS deployments (> few meters), always go active at moderate or high frequencies. Passive is only practical for short-range indoor or very short mmWave. $\blacksquare$

Single amplifier, receives on one antenna and retransmits on another. Self-interference (transmit leak into receiver) forces half-duplex operation: receive in one slot, transmit in the next.

$N$ distributed amplifiers, each per-element. Incoming wave at element $n$ and outgoing amplified wave are separated spatially (via the metasurface structure) and temporally (inherent amplifier delay). No self-interference to kill.

Active RIS saves the 2× time factor of the AF relay, doubling rate at the same physical resources. Combined with coherent combining, this explains the $2N$ SNR advantage. $\blacksquare$

E.g., $N = 256$ panel: 64 active elements (25%) + 192 passive elements (75%). Each has own phase control; active ones additionally have per-element amplifier. The RIS controller selectively uses active elements for long-range UEs, passive for short-range.

• Lower total DC power (active consumes DC; passive free).
• Graceful fallback: if active amplifiers fail, passive elements still work.
• Per-element control allows adaptive operation.

• More complex hardware (mixed design).
• Non-uniform phase response across active/passive elements (harder to calibrate).
• Control software must differentiate which elements to active vs. passive per user.

Hybrid architectures are a natural research direction. Caire et al. have explored variants. Most production panels remain uniform (all-active or all-passive), but hybrid is a reasonable engineering compromise for heterogeneous deployments. $\blacksquare$