Ferkans — Interactive Telecom Tutor

The Rigorous Accuracy Limit

The Fisher Information Matrix (FIM) quantifies how much information a given measurement carries about an unknown parameter. For position estimation, the inverse FIM is the Cramér-Rao bound — a lower bound on the covariance of any unbiased estimator. Section 14.2 derives the FIM for RIS-aided positioning and shows how the RIS phase-shift matrix $\boldsymbol{\Phi}$ enters the FIM (and thus affects the achievable accuracy).

Definition:
Fisher Information Matrix for Position Estimation

For observation $\mathbf{y} = \mathbf{h}_{\text{eff}}(\mathbf{p}) \mathbf{v} + \mathbf{w}$ with AWGN, the FIM for position $\mathbf{p} \in \mathbb{R}^3$ is

$\mathbf{J}(\mathbf{p}) = \frac{2}{\sigma^2} \Re\left\{\mathbf{J}_\mathbf{p}^H(\boldsymbol{\Phi}) \mathbf{J}_\mathbf{p}(\boldsymbol{\Phi})\right\},$

where $\mathbf{J}_\mathbf{p}(\boldsymbol{\Phi}) = \partial \mathbf{h}_{\text{eff}}(\mathbf{p}) / \partial \mathbf{p}^T \cdot \mathbf{v}$ is the Jacobian of the signal with respect to position. The Cramér-Rao bound on position estimation is

$\mathbb{E}\big[(\hat{\mathbf{p}} - \mathbf{p})(\hat{\mathbf{p}} - \mathbf{p})^T\big] \succeq \mathbf{J}^{-1}(\mathbf{p}).$

The RIS phase matrix $\boldsymbol{\Phi}$ appears in $\mathbf{h}_{\text{eff}}$ , hence in $\mathbf{J}_\mathbf{p}$ , hence in $\mathbf{J}$ . Choosing $\boldsymbol{\Phi}$ well increases the FIM (and lowers the CRB), i.e., improves accuracy.

The $\Re\{\cdot\}$ is because position $\mathbf{p}$ is real. The FIM has units of inverse squared distance — larger FIM means smaller position variance.

,

Theorem: CRB Scaling with $N$

Under coherent RIS alignment and near-field regime, the CRB on each position coordinate scales as

$\text{CRB}(\mathbf{p}) \propto \frac{\sigma^2}{P_t\,N^2\,\alpha^2 \beta^2}.$

Thus CRB decreases as $1/N^2$ — position accuracy improves linearly in $N$ (not logarithmically as communication rate does).

For $N = 256$ vs. $N = 16$ : $(256/16)^2 = 256$ times better accuracy in position estimation. A stark illustration of why RIS localization benefits more from large $N$ than communication does.

The angular and range CRB scale differently:

Angular CRB: $\propto 1/(N \cdot D_{\text{RIS}})^2$ . Larger aperture → finer angular resolution.
Range CRB: $\propto d^2/(N \cdot D_{\text{RIS}})^2$ . Range accuracy degrades quadratically with distance.

Each RIS element contributes one phase measurement. More elements = more measurements = more information = lower CRB. The coherent combining gives the information $N$ -fold enhancement; additional aperture-size effects (larger baseline for angular resolution) give further scaling.

Proof

FIM per element

Each RIS element contributes one observation of the UE's signal with known phase. Per-element information: proportional to $\text{SNR}_n$ .

Coherent combining

Under optimal $\boldsymbol{\Phi}$ , all elements' information combines coherently: total FIM $\propto N^2 \cdot \text{SNR}_1$ .

Invert for CRB

CRB $= \mathbf{J}^{-1} \propto \sigma^2/(N^2 \cdot P_t\,\alpha^2\beta^2)$ . Scales as $1/N^2$ , matching the coherent-combining prediction. $\blacksquare$

Comm vs. Positioning Use Different Metrics

For communication, we want high SNR (or SINR). Metric: rate. For positioning, we want high FIM (low CRB). Metric: accuracy.

These are related but not identical. SNR depends on signal magnitude; FIM depends on signal sensitivity to position — how much the signal changes when the UE moves.

Consequence: the optimal $\boldsymbol{\Phi}$ for comm differs from the optimal for positioning. Comm wants coherent focusing on the UE; positioning wants a "sharp" gradient of signal with respect to $\mathbf{p}$ (which can correspond to high contrast at the focal spot, not just peak).

In practice, the two objectives roughly align in near-field — coherent focusing creates both high signal and high spatial sensitivity. But the CRB-minimizing $\boldsymbol{\Phi}$ is not exactly the same as the SINR-maximizing one.

CRB on Position vs. $N$

Plot the position CRB as a function of $N$ for single-RIS positioning. Compare near-field vs. far-field regimes. In the near-field, the $1/N^2$ scaling is fully realized.

Parameters

Max

N

512

Carrier frequency (GHz)28

UE distance (m)20

Per-element SNR (dB)0

Example: CRB Calculation for a 6G Positioning Scenario

Indoor 6G deployment at 140 GHz, RIS with $N = 512$ elements, aperture $D = 0.3$ m, UE at 10 m distance, pilot SNR = 0 dB per element. Compute the position CRB.

Solution

Fraunhofer check

$\lambda = c/f = 3 \times 10^8 / 1.4 \times 10^{11} = 2.14$ mm. $d_F = 2 D^2 / \lambda = 2 \cdot 0.09 / 0.00214 = 84$ m. UE at 10 m $\ll d_F$ . Near-field. ✓

Coherent FIM

At per-element SNR = 0 dB, coherent FIM per coordinate: FIM $\propto N^2 \cdot \text{SNR}_1 / \lambda^{2}$ . $= 512^2 \cdot 1 / (2.14 \times 10^{-3})^2 = 2.6 \times 10^5 / 4.58 \times 10^{-6} \approx 5.7 \times 10^{10}$ .

CRB

$\sigma_p \geq 1/\sqrt{\text{FIM}} \approx 4 \times 10^{-6}$ m per coordinate — sub-millimeter precision! Near-field RIS at sub-THz is capable of mm-level positioning, enabling applications like precise asset tracking, robotic control, AR/VR alignment.

Practical accuracy

The CRB assumes perfect CSI. In practice, $\sim 10$ - $100\times$ worse due to imperfect CSI and non-ideal conditions; but still $\sim 0.1$ - $1$ mm accuracy — far beyond GPS. $\blacksquare$

Joint Position + Orientation + Velocity

In full generality, the UE has position, orientation, velocity as unknowns. The FIM becomes a larger matrix covering all of them. Relevant for:

AR/VR: orientation tracking for head-mounted displays.
Autonomous navigation: velocity + position for dead-reckoning.
Robotic manipulation: full 6-DOF pose (position + orientation).

The CRB structure is the same (inverse FIM); the FIM grows with the number of observations and with the sensitivity of the signal to each parameter. Multi-RIS fusion (Section 14.3) naturally extends to multi-parameter estimation.

Common Mistake: The CRB Is a Lower Bound, Not Reality

Mistake:

"A 4-μm CRB means the system achieves 4-μm positioning."

Correction:

The CRB is a theoretical lower bound for unbiased estimators under perfect conditions. In practice, positioning systems achieve $10$ - $100\times$ the CRB due to: CSI imperfections, hardware non-idealities (phase noise, amplitude errors), SNR limits, and algorithmic suboptimality. Treat the CRB as a theoretical benchmark, not a deployment prediction. A 4-μm CRB might translate to 0.1-1 mm field accuracy — still excellent, but not the CRB figure itself.

Fisher Information Matrix and the CRB