Exercises

$B_n = 28 \text{ GHz}/20 = 1.4$ GHz (5% fractional).

Narrowband holds for signal $B < 10\% B_n \sim 140$ MHz. Beyond this, Theorem 18.1 $\mathrm{sinc}^2$ losses at band edges start degrading performance. 5G 400 MHz signals at 28 GHz are already at the edge — TTD or per-subcarrier design helps.

Edge offset = $B/2 = 400$ MHz. $\eta_{\text{edge}} = \mathrm{sinc}^2(\pi \cdot 0.4) = \mathrm{sinc}^2(1.256) \approx 0.66 \cdot 0.66 = 0.44$ ($-3.6$ dB).

TTD: flat response → $\eta = 1$ ($0$ dB).

TTD recovers $3.6$ dB at edge, about $2$ dB averaged across band.

$\nu_D = 1 \cdot 28 \times 10^9 / 3 \times 10^8 = 93.3$ Hz.

$\pi \nu_D \tau = \pi \cdot 93.3 \cdot 0.01 = 2.93$ rad — not small. Must use exact Bessel: $J_0(2.93) \approx -0.26$, $\eta = J_0^2 = 0.067$. Essentially decorrelated!

Even pedestrian mobility at mmWave is not negligible for 10 ms control-loop latency. Need $\tau < 1$ ms for adequate performance.

$R = 30 \cdot 512 \cdot 2 \cdot 100 = 3.07 \times 10^6$ = 3.07 Mbps.

Within typical 5G PDCCH capacity (50-100 Mbps). At higher refresh ($f_r = 1$ kHz = typical target), $R = 30.7$ Mbps — getting tight, would want compressed encoding.

$12 = 2\log_2 64 + C_0 = 12 + C_0$ → $C_0 = 0$.

$C(256) = 2\log_2 256 + 0 = 16$ bits/s/Hz. Adds 4 bits/s/Hz from $4 \times$ more elements — the $2\log_2 N$ scaling at work.

$(\pi \nu_D \tau)^2 \leq 0.1$ → $\tau \leq 0.316/(\pi \nu_D)$.

$\nu_D = 10$ Hz: $\tau \leq 10$ ms. $\nu_D = 100$ Hz: $\tau \leq 1$ ms. $\nu_D = 1$ kHz: $\tau \leq 100 \mu s$ — challenging.

Log-log plot: $\tau$ scales as $1/\nu_D$. The plot is a straight line with slope $-1$.

$\nu_D = 30 \cdot 28 \times 10^9 / 3 \times 10^8 = 2800$ Hz. $T_c \approx 0.4/2800 \approx 140 \mu s$.

Need $\tau_{\text{RIS}} \ll T_c$. Standard 1 kHz RIS has $\tau = 1$ ms $\gg 140 \mu s$ — infeasible without sub-carrier-latency control.

(i) Use RIS for slow-varying components only (angular spread). (ii) Use C-V2X sidelink (no RIS). (iii) Drop to sub-6 GHz where $T_c$ is much larger (hundreds of ms).

$16 \times \lambda/2 = 16 \cdot 5.36/2 = 43$ mm in one dimension, $64 \cdot 5.36/2 = 172$ mm in other. Max aperture = 172 mm.

$\tau_{\max} = 0.172 \cdot \sin(45°)/3\times 10^8 = 0.122/3\times 10^8 \cdot \sqrt 2 = 405$ ps.

Need $\sim 400$ ps tunable delay per element — achievable with switched transmission-line segments but costly.

Rel-18: 2024. Rel-19: 2025 (study item for RIS). Rel-20: late 2025/early 2026 (spec complete). Rel-21: 2027.

Rel-20 standardization late 2025/early 2026. Vendor implementation: 12 months. First commercial deployments: 2027. Scale: 2028-2029.

• Peak rate: RIS aperture → higher capacity (yes).
• Connection density: RIS coverage fill → more simultaneous connections (yes).
• Energy efficiency: passive RIS → lower power per dB gain (yes).
• AI-native: RIS control is ML-driven (yes).
• Integrated sensing: RIS-ISAC (yes, Chapter 13).

• Latency: RIS control adds latency (only partially helpful). RIS is central to 5 of the 6 pillars — a principal 6G technology.

$|A_k|^2 = N + \sum_{n \neq m} \cos[(\alpha_n - \alpha_m)(f_k - f_c)]$.

$\mathbb{E}[\cos((\alpha_n - \alpha_m) \Delta)]$ where $\Delta = f_k - f_c$. Since $\alpha_n - \alpha_m$ has a triangular distribution on $[-2A, 2A]$ (sum of two i.i.d. Uniform): $\mathbb{E}[\cos(X\Delta)] = \mathrm{sinc}^2(A\Delta/\pi)$.

$\mathbb{E}[|A_k|^2] = N + N(N-1) \mathrm{sinc}^2(A(f_k - f_c)/\pi)$. Agrees with Theorem 18.1.

Coherent combining at the RIS adds $N$ signals in amplitude → $N^2$ power gain. At high SNR: $\log_2(\text{SNR}) \propto \log_2(N^2) = 2\log_2 N$.

One $\log_2 N$ comes from the RIS-side coherent combining; the other comes from the BS-side transmit beamforming (which scales as $\log_2 N$ of effective antennas if the BS has $\sim N$ antennas matched to RIS rank).

In log-domain, multiplication becomes addition: SNR scales multiplicatively with the two factors, capacity scales additively. Same reason MIMO capacity scales with $\min(N_t, N_r)$ added to SNR.

Signal: $|\hat{\mathbf{h}}_{\text{eff}}^H \boldsymbol{\Phi} \mathbf{h}_{\text{eff}}|^2$. Estimation-error noise: $\sigma_e^2 \|\mathbf{h}_{\text{eff}}\|^2$. Effective SNR: $\text{SNR}_{\text{eff}} = \frac{P_t |\hat{\mathbf{h}}_{\text{eff}}^H \boldsymbol{\Phi} \mathbf{h}_{\text{eff}}|^2} {\sigma^2 + P_t \sigma_e^2 \|\mathbf{h}_{\text{eff}}\|^2}$.

$R \geq \log_2(1 + \text{SNR}_{\text{eff}})$. Note the noise term $P_t \sigma_e^2 \|\mathbf{h}_{\text{eff}}\|^2$ grows with power — causing a ceiling effect similar to aging.

The TIGHT capacity (upper and lower bounds matching) under arbitrary $\mathbf{e}$ distribution is still open for RIS channels.

Cascade: $\text{rank}(\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1) \leq \min(N_t, N, N_r)$. DoF = rank of effective channel = $\min(N_t, N, N_r)$.

If $N_t = N_r = 16$ and $N = 1024$: DoF = 16. The RIS provides aperture gain but does NOT add spatial multiplexing beyond the minimum of $N_t, N_r$.

Tighter analysis under near-field, correlation, and finite- blocklength regimes is ongoing research.

(1) Product path loss $d_1^2 d_2^2$ drives placement (Ch 1, 17): the $N^2$ gain is real, but only if the RIS is close to an endpoint. This is counterintuitive from point-to-point thinking and shapes deployment economics.

(2) Joint active-passive beamforming is non-convex but tractable (Ch 5, 6): the unit-modulus constraint on $\boldsymbol{\Phi}$ looks intimidating, but AO + SDR + manifold optimization reach provably near-optimal solutions. The algorithms lesson generalizes to many other wireless problems.

(3) The 4-6 dB theory-practice gap is persistent but manageable (Ch 16): hardware calibration, thermal drift, and 2-bit quantization collectively cost 4-6 dB relative to ideal. The algorithmic response is robustness: design algorithms that remain near-optimal under realistic imperfections. This matches the CommIT Group's research thread on robust optimization.