Ferkans — Interactive Telecom Tutor

Why Positioning Is a RIS Killer App

Conventional positioning uses trilateration from multiple base stations (GPS-style) or fingerprinting. Both struggle indoors and in dense urban: GPS signals don't penetrate, and fingerprint databases require expensive calibration. RIS-aided localization offers a third way: a known RIS panel acts as a reference point with precisely known geometry. UE signals bouncing off the RIS encode positional information in their phase patterns.

The golden thread: the RIS programs the propagation environment. For localization, it literally shapes the signal to reveal position information. Near-field RIS operation is especially powerful because wavefront curvature across the aperture directly encodes UE distance — a strong "range" measurement that conventional far-field positioning cannot exploit.

Definition:
RIS-Aided Positioning Signal Model

Consider a BS with $N_t$ antennas, a known RIS with $N$ elements at fixed position $\mathbf{p}_R$ , and a UE at unknown position $\mathbf{p} \in \mathbb{R}^3$ . The BS transmits a known pilot $\mathbf{x}$ , reaching the UE via the cascaded path through the RIS.

The received signal at the UE is

$y(\mathbf{p}) = \mathbf{h}_{\text{eff}}^H(\mathbf{p}) \mathbf{v} + w, \qquad w \sim \mathcal{CN}(0, \sigma^2)$

where the effective channel explicitly depends on UE position $\mathbf{p}$ :

$\mathbf{h}_{\text{eff}}^H(\mathbf{p}) = \mathbf{h}_d^H(\mathbf{p}) + \mathbf{h}_2^H(\mathbf{p}) \boldsymbol{\Phi} \mathbf{H}_1.$

The $\mathbf{p}$ -dependence lives in both the direct channel and the RIS-to-UE channel $\mathbf{h}_2(\mathbf{p})$ ; in the near-field regime, this dependence includes quadratic phase terms that encode distance, not just angle.

Conventional far-field positioning: $\mathbf{h}_2(\mathbf{p})$ depends on $\mathbf{p}$ only through AOA $\Omega(\mathbf{p})$ . Estimate AOA → get a bearing line; intersect with multiple bearings → triangulate.

Near-field RIS positioning: $\mathbf{h}_2(\mathbf{p})$ depends on $\mathbf{p}$ through AOA and distance. A single near-field measurement gives direct position information; no triangulation needed. This is the central algorithmic advantage.

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Theorem: Near-Field RIS: Direct Range + Angle Estimation

For a UE at position $\mathbf{p}$ in the near-field of the RIS ( $d(\mathbf{p}, \mathbf{p}_R) < d_F$ ), the per-element RIS-to-UE phase is

$\phi_n^{(\text{RIS-UE})}(\mathbf{p}) = \frac{2\pi}{\lambda} \|\mathbf{p} - \mathbf{r}_n\|,$

where $\mathbf{r}_n$ is the position of RIS element $n$ . Expanding in the near-field (but not far-field) regime:

$\phi_n^{(\text{RIS-UE})}(\mathbf{p}) \approx \frac{2\pi}{\lambda}\bigg[d(\mathbf{p}, \mathbf{p}_R) + \mathbf{n}^T (\mathbf{p} - \mathbf{p}_R)/d - \frac{|(\mathbf{p} - \mathbf{p}_R)|^2}{2d} + \frac{\mathbf{n}_2^T (\mathbf{p} - \mathbf{r}_n)^2}{2 d}\bigg],$

where $d = d(\mathbf{p}, \mathbf{p}_R)$ is the RIS-UE distance and $\mathbf{n}$ is a unit vector. The first two terms are the far-field linear phase (gives AOA). The third term is a quadratic in $(\mathbf{p} - \mathbf{p}_R)/d$ — this encodes distance $d$ .

The simultaneous presence of linear + quadratic terms means a single RIS panel in the near-field can estimate both position components. In the far-field, only AOA is observable — one bearing line only.

In the far-field, the RIS-to-UE channel is a plane wave from the RIS's perspective: per-element phases are linear in element index. Only angle is observable. In the near-field, per-element distances to the UE differ, producing a quadratic (not linear) phase pattern across the aperture. Both angle and distance are encoded.

Proof

Exact per-element phase

$\phi_n = (2\pi/\lambda)\|\mathbf{p} - \mathbf{r}_n\|$ (free-space propagation). Each element's distance to the UE depends on its position.

Taylor expansion

Expand $\|\mathbf{p} - \mathbf{r}_n\|$ around the RIS center $\mathbf{p}_R$ . First-order: linear in $\mathbf{r}_n$ (gives AOA). Second-order: quadratic in $\mathbf{r}_n$ (gives range).

Observability

Far-field: only first-order term dominates; single panel gives AOA only. Near-field: second-order term is significant; single panel gives range + AOA → full 3D position from one measurement. $\blacksquare$

Key Takeaway

Near-field RIS gives single-panel 3D positioning. The wavefront curvature across the RIS aperture encodes UE distance; phase variation across elements encodes angle. A single panel in the near-field observes both — no triangulation needed. This is a fundamentally different (and simpler) positioning architecture than GPS-style trilateration.

Example: Indoor Positioning with a RIS Panel

Indoor warehouse with a RIS panel on the ceiling. BS is external; UE is at unknown position $\mathbf{p}$ inside the warehouse (walls blocking direct path). RIS: $N = 256$ , $28$ GHz. Warehouse dimensions: $30 \times 30 \times 10$ m.

Solution

Near-field check

RIS aperture: $\sim 0.3$ m² at $N = 256$ , $\lambda/2$ spacing. $D \approx 0.5$ m. $d_F = 2 D^2 / \lambda = 2(0.25)/0.0107 \approx 47$ m. Warehouse size $\sim 30$ m $< d_F$ . UEs are near-field. ✓

Single-panel positioning

Near-field enables position estimation from one RIS alone. Per-element phase pattern encodes both angle (direction from panel) and distance (range from panel).

Expected accuracy

Angular CRB: $\sim \lambda / D = 2°$ (from aperture). Range CRB: depends on SNR and aperture; at $0$ dB SNR, $\sim$ few centimeters. Position CRB: combined angular + range gives cm-level accuracy in the warehouse. Compared with GPS ( $\sim 5$ m indoors, often unusable): orders-of-magnitude improvement.

Deployment

Single RIS panel on ceiling serves as reference point. UEs send pilots; BS processes them through the panel and estimates position via FIM methods (Section 14.2). $\blacksquare$

Near-Field RIS Positioning Geometry — A UE in the near-field of a RIS panel. Each RIS element $n$ sees the UE at a slightly different distance $d_n = \|\mathbf{p} - \mathbf{r}_n\|$ . The wavefront curvature across the aperture (dashed arcs) encodes UE range, not just angle. A single RIS panel in near-field thus gives a full 3D position fix.

Near-Field vs. Far-Field Positioning: Phase Pattern

Visualize the RIS-side phase pattern for a UE at different distances from the panel. Near-field: curved (quadratic) phase pattern encoding distance. Far-field: linear phase (AOA only). The transition happens around the Fraunhofer distance.

Parameters

RIS elements

N

128

UE distance from RIS (m)20

Carrier frequency (GHz)28

UE angle (deg)15

Handling NLoS via RIS

A remarkable feature of RIS localization: the UE and BS need not have a direct LoS. The RIS can act as a "virtual LoS" bouncing signals around blockages. For a UE at $\mathbf{p}$ :

Direct BS-UE channel is blocked (no direct LoS).
BS-RIS and RIS-UE channels are LoS (geometry-engineered).
BS-RIS-UE cascaded path reaches the UE with known RIS phase and position information.

This is the NLoS-robust feature of RIS positioning: as long as the RIS has LoS to both BS and UE, positioning works. Compare with GPS-denial in indoor/urban canyon scenarios, where every direct path is blocked.

Common Mistake: Don't Assume Far-Field in Localization

Mistake:

"RIS localization works like GPS — get bearings from each panel, triangulate."

Correction:

Near-field positioning is fundamentally different from far-field AOA-only positioning. A single near-field RIS panel gives a full 3D position fix via distance + angle; far-field panels require multiple measurements or triangulation.

Checking the regime:

At 28 GHz with a 30-cm RIS aperture: $d_F \sim 17$ m. Indoor UEs typically at $< 20$ m — near-field is the rule, not exception.
At 100 GHz (sub-THz): $d_F$ grows; near-field range shrinks. Design the RIS aperture carefully.

Assuming far-field when in near-field discards the distance information; accuracy falls from cm-level to meter-level.

RIS Localization Signal Model