Exercises

Only linear phase term (AOA) is observable. A single panel gives a bearing line, not a point. Multiple bearings → triangulate.

Per-element phase has both linear (AOA) and quadratic (range) terms. A single panel observes both; gives a point estimate.

Near-field RIS is uniquely suited for single-panel deployment. Simpler infrastructure, faster convergence. $\blacksquare$

CRB $\propto \sigma^2 / (P_t\,N^2\,\alpha^2\beta^2)$ $\propto 1/(N^2 \cdot \text{SNR})$.

• Double $N$: CRB → CRB/4. Position accuracy improves by 2×.
• 10 dB SNR boost: CRB → CRB/10. Accuracy improves by √10 ≈ 3.16×.

Comm rate grows logarithmically in $N^2$ SNR. Position accuracy grows linearly in $N$. RIS localization benefits more from large $N$ than RIS comm does. $\blacksquare$

$\log p(\mathbf{y}|\mathbf{p}) = -(1/(2\sigma^2)) \|\mathbf{y} - \mathbf{h}(\mathbf{p})\|^2 + \text{const}$.

$\nabla_\mathbf{p} \log p = (1/\sigma^2) \Re\{(\partial \mathbf{h}/\partial \mathbf{p})^H (\mathbf{y} - \mathbf{h}(\mathbf{p}))\}$.

FIM = $-\mathbb{E}[\partial^2 \log p / \partial \mathbf{p} \partial \mathbf{p}^T]$ = $(1/\sigma^2) \Re\{\mathbf{J}_\mathbf{p}^H \mathbf{J}_\mathbf{p}\}$, where $\mathbf{J}_\mathbf{p} = \partial \mathbf{h}(\mathbf{p})/\partial \mathbf{p}^T$ is the Jacobian. $\blacksquare$

FIM $\sim N^2 \cdot \text{SNR} / \lambda^{2} = 512^2 \cdot 10 / (0.0107)^2 \approx 2.6\times 10^5 \cdot 10 / 10^{-4} = 2.6 \times 10^{10}$.

CRB $\sim 1/\sqrt{\text{FIM}} \approx 6\times 10^{-6}$ m $\approx 6\,\mu\text{m}$ per coordinate.

Sub-10-μm theoretical precision. In practice: $\sim 100$μm-$1$ mm after accounting for imperfections. Still cm-level or better. $\blacksquare$

$\mathbf{y}_m$ are conditionally independent given $\mathbf{p}$. $\log p(\{\mathbf{y}_m\} | \mathbf{p}) = \sum_m \log p(\mathbf{y}_m | \mathbf{p})$.

$\mathbf{J}_{\text{total}} = -\mathbb{E}[\nabla^2 \log p] = -\mathbb{E}[\sum_m \nabla^2 \log p_m] = \sum_m (-\mathbb{E}[\nabla^2 \log p_m]) = \sum_m \mathbf{J}_{m}$.

CRB$_\text{total} = (\sum_m \mathbf{J}_{m})^{-1}$. For equivalent panels, CRB scales as $1/M$. $\blacksquare$

Single: coherent FIM $\propto 1024^2 = 10^6$. Multi: $4 \cdot 256^2 = 2.6 \cdot 10^5$. Single is 4× stronger in FIM.

Single-panel positioning sees all coords well; multi gives less per-signal info. Single wins (4× better).

Single-panel: some coords poorly observed (small sensitivity). Multi-panel: different geometries cover all directions. CRB conditioning is poor for single; multi distributes well. Multi wins (possibly 10× better conditioning).

For well-conditioned scenarios: single panel. For challenging geometries: multi-panel. Deploy based on expected UE distribution. $\blacksquare$

Uses the Hessian (second derivative = FIM). Converges quadratically near the optimum: each iteration halves the error in $\log_{10}$ sense.

Uses only first derivative. Converges linearly: each iteration reduces error by a constant factor.

For 6-digit accuracy: Newton ~5-10 iterations; gradient descent ~50-100. Newton is 5-10× faster per convergence.

Newton requires Hessian to be PSD (convex near optimum). At low SNR, Hessian may be ill-conditioned — fallback to regularized Newton (Levenberg-Marquardt) or gradient descent. $\blacksquare$

$\lambda = 5$ mm. $D = 0.25$ m. $d_F = 2 \cdot 0.0625 / 0.005 = 25$ m. UE at 2 m $\ll d_F$. Near-field. ✓

At 20 dB SNR per element: $\text{SNR}_{\text{lin}} = 100$. FIM $\propto N^2 \cdot \text{SNR}/\lambda^{2} = 256^2 \cdot 100 / (0.005)^2 = 2.6 \cdot 10^{11}$.

$\sqrt{1/\text{FIM}} \approx 2\mu\text{m}$ per coord. Practical accuracy: $\sim 20\mu\text{m}$. Excellent for AR/VR. $\blacksquare$

If RIS element $n$ is actually at $\mathbf{r}_n + \delta \mathbf{r}_n$ (mismeasured by $\delta \mathbf{r}_n$), the observed phase is offset by $(2\pi/\lambda) (\mathbf{n}^T \delta \mathbf{r}_n + O(\ldots))$.

The position estimate absorbs this phase offset as a UE position error. A 1-mm element position error → 1-mm UE position error (rough 1:1).

For $\sim 1$-mm UE positioning: need sub-mm element position calibration. Achievable via laser total-station surveying or known-reference-transmitter calibration at deployment. Skipping calibration: 10-100x worse accuracy. $\blacksquare$

Range CRB $\propto d^2 / (N D_R)^2$. At longer distance, more $d^2$, worse CRB.

Range CRB at 20 m is $(20/1)^2 = 400\times$ worse than at 1 m.

Angular CRB relatively insensitive to distance (depends on aperture and wavelength). Becomes relatively dominant at larger distances.

Near the panel: range-dominated. Far from panel: angle- dominated. Best positioning near the panel (few meters). For longer ranges, multi-panel fusion or moving the UE closer helps. $\blacksquare$

• Pros: ubiquitous outdoor coverage; no infrastructure needed.
• Cons: blocked indoors by buildings ($-30$ dB penetration); $\sim 5$ m accuracy outdoors, unusable indoors.

• Pros: works indoors; cm-level accuracy; no UE modification.
• Cons: requires installed RIS panel(s); single room/area coverage per panel; needs calibration.

Use GPS outdoors, RIS indoors. Hybrid deployments in urban canyons can have both (GPS supplement where RIS isn't available). $\blacksquare$

$\mathbf{p}(t+1) = \mathbf{p}(t) + \mathbf{v}\,\Delta t + \mathbf{w}_{\text{motion}}$ (constant-velocity + process noise).

RIS-based MLE at each time step: $\hat{\mathbf{p}}_{\text{obs}}(t)$ with uncertainty Cov$_{\text{obs}}$ (from FIM).

Predict $\mathbf{p}(t|t-1)$ from previous estimate + motion model.

Combine prediction with observation via Kalman gain weighted by covariances: $\hat{\mathbf{p}}(t|t) = \hat{\mathbf{p}}(t|t-1) + K (\hat{\mathbf{p}}_{\text{obs}}(t) - \hat{\mathbf{p}}(t|t-1))$.

Smoother trajectories, predicts during sensor outages, fuses multi-panel observations across time. Standard tool for practical positioning. $\blacksquare$

1000 Hz → 1 ms per update.

Newton MLE: ~1-10 ms on CPU. Might not fit. Kalman-style incremental update: much faster, sub-ms.

Feasible with warm-started incremental Kalman filter. Each update re-estimates one step using previous state + new RIS observation. Per-step compute: $\sim 100\mu\text{s}$.

High-performance robotics with RIS tracking: feasible. Typical robotic precision targets ($\sim 0.1$ mm) are well within RIS capability. $\blacksquare$

$D = (N-1) \lambda/2 = 255 \cdot 0.0107/2 = 1.37$ m (for a 1D array). For a $16 \times 16$ UPA: $D_{\text{max}} \approx 8 \lambda/\sqrt{2} = 0.086$ m (smaller, but 2D). Take 1D for simplicity: $D = 1.37$ m.

$\Delta\theta \sim \lambda/D = 0.0107/1.37 \approx 0.0078$ rad $= 0.45°$.

At 20 m distance: $\Delta x \sim 0.0078 \cdot 20 = 0.16$ m = 16 cm lateral resolution. Combined with range resolution (< mm in near-field), gives 3D positioning accuracy bounded by the angular term in far-field, range term in near-field. $\blacksquare$

Need mm-scale accuracy → sub-THz (140 GHz) for short wavelength. $\lambda = 2$ mm. Near-field at typical distances: ✓.

4 panels at corners, each $N = 1024$ elements, $D \sim 1$ m. Near-field range: $d_F = 2 \cdot 1 / 0.002 = 1000$ m $\gg$ factory size. All robots in near-field.

Each panel individually gives robot position. Multi-panel fusion yields sub-mm accuracy via diversity + FIM summing.

Robotic manipulation: 1 kHz. RIS with Kalman tracking: feasible. Each update: sub-millisecond compute.

4 panels at corners cover full 50×50 m². Backup: individual panels handle obstruction.

4 × $\$5K = \$20K$ per panel (sub-THz hardware). Comparable to laser-tracker or optical motion-capture alternatives, with RIS advantage of no line-of-sight camera requirement. $\blacksquare$