Exercises

$|r_n|^2 + |t_n|^2 \leq 1$.

Real hardware has insertion loss (dielectric absorption, ohmic loss). Some incident energy is lost rather than reflected or transmitted. Typical practical loss: 5-10%. The strict equality is the lossless-idealized upper bound.

One real variable per element ($\alpha_n \in [0, \pi/2]$); total $N = 32$ amplitude variables. Plus $2N = 64$ phase variables (one for each of $\boldsymbol{\Phi}^r, \boldsymbol{\Phi}^t$). Total: 96 real variables.

$N = 32$ phase variables. 32 real variables.

3× more optimization variables for ES STAR-RIS. Corresponds to the 3×-ish compute cost per iteration. $\blacksquare$

Sub-slot 1 ($\tau = 0.5$): serve user 1 with full passive RIS on the reflect side. Rate contribution: $0.5 R_1^{\text{passive}}$. Sub-slot 2 ($1-\tau = 0.5$): similar for user 2: $0.5 R_2^{\text{passive}}$. Total: $0.5 (R_1 + R_2)$.

Half-energy to each side: per-side SNR is $\eta_B N^2/4$ instead of $N^2$ (energy split factor $1/\sqrt{2}$ in amplitude, $1/2$ in power). Rate: $R_1^{\text{ES}} = \log_2(1 + \text{SNR}/2)$. Both users served simultaneously: $R_1 + R_2$ full coherence.

ES: $2 \log_2(1 + \text{SNR}/2)$. TS: $\log_2(1 + \text{SNR})$. For high SNR: ES $\approx 2 \log_2(\text{SNR}/2) = 2\log_2(\text{SNR}) - 2$. TS $\approx \log_2(\text{SNR})$. ES wins by $\log_2(\text{SNR}) - 2$ bits at high SNR. $\blacksquare$

MS constraint: $(a_n^r, a_n^t) \in \{(1,0), (0,1)\}$. ES constraint: $|a_n^r|^2 + |a_n^t|^2 = 1$ with $a_n^r, a_n^t \in [0,1]$. $(1,0)$: $1^2 + 0^2 = 1$ ✓. $(0,1)$: same. Hence every MS solution satisfies ES's constraints.

$R_{\text{ES}}^\star \geq R_{\text{MS}}^\star$ because ES optimizes over a superset. MS is ES restricted to binary amplitudes. $\blacksquare$

The total rate $R(\tau) = \tau R_r + (1-\tau) R_t$ is linear in $\tau$. The maximum occurs at an endpoint of $[0, 1]$ (unless $R_r = R_t$, giving the same rate for all $\tau$).

If $R_r > R_t$: $\tau^\star = 1$ (all-reflect, ignore transmit side). If $R_r < R_t$: $\tau^\star = 0$ (all-transmit). If $R_r = R_t$: any $\tau$ is optimal.

For fair TS (both sides served), add a fairness constraint (e.g., max-min or proportional-fair objective). With max-min, $\tau^\star = R_t / (R_r + R_t)$ balances the two sides. $\blacksquare$

Element sits in front of a ground plane. The ground plane blocks the wave entirely; only the reflected wave reaches the forward half-space.

Transparent (or semi-transparent) substrate replaces the ground plane. Some of the incident wave is reflected back (by the metasurface elements); some passes through and is re-radiated on the other side (controlled by transmission coefficients). Both half-spaces receive energy.

Lossless STAR-RIS element cannot give full reflection amplitude AND full transmission amplitude — by energy conservation, the total is $|r|^2 + |t|^2 \leq 1$. $\blacksquare$

The desired $(|r_n^{\text{raw}}|, |t_n^{\text{raw}}|)$ lies in $\mathbb{R}_+^2$; we need to scale it to the unit circle. Normalization: $a_n^r = |r_n^{\text{raw}}| / \sqrt{|r_n^{\text{raw}}|^2 + |t_n^{\text{raw}}|^2}$, $a_n^t = |t_n^{\text{raw}}| / \sqrt{|r_n^{\text{raw}}|^2 + |t_n^{\text{raw}}|^2}$.

The phases of $r_n, t_n$ are kept from the raw signal — normalization only adjusts the magnitudes.

Elements for which $|r_n^{\text{raw}}| \gg |t_n^{\text{raw}}|$ (stronger reflection demand) get $a_n^r$ close to 1; the opposite for strong transmission demand. This is the element-level load-balancing automatic in the optimization. $\blacksquare$

$t_n = a_n^t e^{j\theta_n^t} = a_n^t e^{j(\theta_n^r + \pi/2)} = j a_n^t e^{j\theta_n^r}$. So $t_n = j (a_n^t / a_n^r) r_n$ when $a_n^r \neq 0$.

The transmitted wave is always $90^\circ$ phase-shifted relative to the reflected wave from the same element. Beams formed by $\boldsymbol{\Phi}^r$ and $\boldsymbol{\Phi}^t$ have related phase profiles.

The degrees of freedom in phase are reduced: $\theta_n^r$ is free, but $\theta_n^t$ is determined. Total phase DoF: $N$ (instead of $2N$ for independent phases). Objective value typically 0.5-1 dB worse than independent-phase case. $\blacksquare$

UEs in front of the RIS (half-plane): covered. UEs behind: not covered. Fraction: $180/360 = 0.5$.

Both sides covered. Fraction: up to $1.0$ (full circle). With QoS threshold, slightly less due to per-side energy split.

$\sim 1.5$-$2\times$, depending on QoS threshold. At tight thresholds (low SINR target), near $2\times$; at loose thresholds, closer to $1.5\times$ due to the per-side 3-dB penalty. $\blacksquare$

Under balanced user sets and identical channel statistics, the objective is symmetric in the roles of reflection and transmission. If $(a_n^r, a_n^t) = (x, y)$ is optimal, so is $(y, x)$ by swapping.

If $x \neq y$, the average of the two symmetric solutions — $(x+y)/2, (y+x)/2)$ — is also on the unit circle after renormalization, and by convexity of the log-SINR on the appropriate variables, achieves at least the same value.

The symmetric choice $a_n^r = a_n^t = 1/\sqrt{2}$ is optimal by symmetry + averaging. Phases still vary per user demand; amplitudes are uniform across the panel. $\blacksquare$

Each element chooses reflect or transmit: $2^N$ combinations. For $N = 100$: $2^{100} \approx 10^{30}$ — infeasible.

Each element has $2^B$ phase states; for chosen mode, total space is $2^N \cdot 2^{BN}$.

Use relaxation-then-round or greedy assignment: relax mode choice to continuous in $[0, 1]$, solve continuous problem, threshold. Loses $< 1$ dB vs. exhaustive MS at large $N$. $\blacksquare$

Both serve users on both sides. Each uses passive RIS technology. Single-panel TS uses existing passive RIS hardware with just a time-switching controller.

• Deployment: TS uses 1 panel at one location; two-panel uses 2 panels at two locations (possibly on opposite walls).
• Spectral efficiency: TS loses a factor of 2 in time; two-panel serves both continuously.
• Hardware cost: 1 panel for TS; 2 panels for the two-panel scheme.
• Channel estimation: TS estimates both sides of the same panel; two-panel estimates two separate RISs (more pilots).

TS is cheaper but less efficient. Two-panel is more expensive but serves both sides at full speed. For coverage-critical deployments: two-panel or STAR-RIS ES. For cost-sensitive: TS. $\blacksquare$

At $N = 256, K = 8, N_t = 16$: $\sim 15$-$30$ ms per AO convergence with manifold passive update.

Three-block AO: active + reflect + transmit + amplitude. Each iteration $\sim 1.5$-$2\times$ passive. Total: $\sim 25$-$60$ ms.

For 20-ms coherence time: feasible with warm-starting and reduced iteration count (5-8 outer instead of 15-30). Borderline for 10-ms coherence. $\blacksquare$

In TS, during sub-slot 1 (fraction $\tau$), all elements reflect ($a_n^r = 1, a_n^t = 0$). During sub-slot 2 ($1 - \tau$), all elements transmit. Over the coherence block, the average per-element power in reflection is $\tau \cdot 1 + (1-\tau) \cdot 0 = \tau$. The average in transmission is $1 - \tau$.

An ES configuration with $a_n^r = \sqrt{\tau}, a_n^t = \sqrt{1-\tau}$ has the same per-element time-averaged energy split. Since rate is a function of averaged quantities (at long coherence time), the ES rate is at least as large as the TS rate.

ES can additionally diversify across elements: some elements at high $a_n^r$, others at high $a_n^t$. TS cannot. Hence ES $\geq$ TS always, usually strict. $\blacksquare$

Each element: active amplifier + bidirectional radiating structure. The amplified signal is split between reflection and transmission paths (via ES or MS). Each path has its own phase shifter.

Per element: amplifier gain $g_n$, reflection amplitude $a_n^r$, transmission amplitude $a_n^t$ (with $|a_n^r|^2 + |a_n^t|^2 \leq 1$ referring to the split post-amplification), phases $\theta_n^r, \theta_n^t$.

1. Full-space coverage (like STAR-RIS).
2. Breaks the product-path-loss ceiling (like active RIS).
3. Can serve users on both sides simultaneously at long range.

1. Much higher cost (active hardware × 2).
2. Amplifier noise on both sides.
3. Power consumption: $\sim 2\times$ active RIS.
4. Research-only, no commercial demos yet.

An interesting research direction for 6G and beyond. A few preliminary papers from 2024-2025 explore this combination. Worth watching as hardware matures. $\blacksquare$