---
title: Positioning Error Bound (PEB) Overview
url: https://www.emergentmind.com/topics/positioning-error-bound-peb
type: topic
---

# Positioning Error Bound (PEB) Overview

Searching arXiv for recent and foundational papers on Positioning Error Bound (PEB) across localization and ISAC contexts.
{"query":"Positioning Error Bound localization Fisher information PEB arXiv", "max_results": 10}
Positioning Error Bound (PEB) is a scalar metric for localization accuracy. In many estimation-theoretic formulations, it is defined as the square-root of the trace of the inverse position Fisher information matrix (FIM), or equivalently the square-root of the trace of the position-domain Cramér–Rao bound (CRB), and therefore represents the minimum achievable root-mean-square position error under unbiased estimation [2305.12094]. Across recent arXiv literature, the same label also appears in broader forms: as the positional sub-block of a joint position–orientation CRLB [2508.18009], as a confidence-factor combination of horizontal and vertical error variances [1111.6226], and as a deterministic geometric upper bound over feasible positions or clock-bias-driven inequalities [1201.2513]. This diversity makes PEB both a unifying concept and a source of terminology drift.

## 1. Canonical definitions and notational variants

In the classical localization literature represented here, the most common form of the bound is
\[
\mathrm{PEB}
=\sqrt{\mathrm{tr}\bigl\{\mathrm{CRB}_{p}\bigr\}}
=\sqrt{\mathrm{tr}\bigl\{\mathbf J_{p}^{-1}\bigr\}},
\]
where $\mathbf J_p$ is a position-domain FIM or equivalent FIM (EFIM) and $\mathrm{CRB}_p=\mathbf J_p^{-1}$ [2305.12094]. The same trace-of-inverse structure is used for two-dimensional target localization in cell-free mMIMO-OTFS ISAC, where $\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$ [2504.10137], for fluid-antenna-system localization, where $\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$ [2512.13941], for optical wireless 3D localization, where $\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$ [2603.29400], and for distributed MIMO positioning, where $\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$ [2511.06440].

When position is estimated jointly with other quantities, the bound is often extracted from a larger CRLB matrix. In RIS-aided 6D Bayesian localization with channel-estimation errors, the full parameter vector is $\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$, and the three-dimensional PEB is
\[
\mathrm{PEB}
=
\sqrt{
\mathrm{tr}
\bigl\{
[\,\mathrm{CRLB}\,]_{1:3,1:3}
\bigr\}
}
\]
after inversion of the $6\times 6$ FIM [2508.18009]. This same block-extraction logic also underlies orientation-aware localization works in which positional and angular variables are jointly estimated.

Other papers use the term in non-CRB senses. In airborne three-dimensional laser imaging, the combined bound is defined by
\[
\mathrm{PEB}=k\sqrt{\sigma_h^2+\sigma_v^2},
\]
with $k=2$ giving $\approx95\%$ confidence under Gaussian assumptions [1111.6226]. In range-based wireless sensor networks with positively biased range errors, two worst-case geometric bounds are defined:
\[
B_1=\max_{x\in F}\|x-\hat x\|_2,\qquad
B_2=\max_{x,y\in F}\|x-y\|_2,
\]
where $F$ is the feasible set formed by the intersection of measurement balls [1201.2513].

| Formulation | Expression | Representative sources |
|---|---|---|
| CRB/FIM-based RMS lower bound | $\sqrt{\mathrm{tr}(J_p^{-1})}$ | [2305.12094], [2504.10137] |
| Positional sub-block of joint CRLB | $\sqrt{\mathrm{tr}([CRLB]_{1:3,1:3})}$ | [2508.18009] |
| Confidence-factor bound | $k\sqrt{\sigma_h^2+\sigma_v^2}$ | [1111.6226] |
| Feasible-set worst-case bound | $B_1$, $B_2$ over $F$ | [1201.2513] |

This suggests that “PEB” is best understood as a family of scalar position-error summaries, with the CRB-derived version being dominant but not exclusive.

## 2. Fisher-information construction and equivalent information

A recurring pattern in the literature is the construction of a FIM in a measurement domain, elimination of nuisance parameters through a Schur complement, and transformation into Cartesian position coordinates through a Jacobian. In the RIS-enabled integrated positioning and communication framework, the starting point is ToA estimation from the RIS-aided path, with propagation delay
\[
\tau(p_U)=\frac{d_{BR}+d_{RU}}{c},
\]
and a scalar delay FIM
\[
J_{\tau\tau}=\frac{2}{\sigma^2}\sum_{n=1}^N |\alpha_n|^2 (2\pi n\Delta_f)^2.
\]
The position EFIM then becomes
\[
\mathbf{J}_{\rm pos}
=\frac{2}{\sigma^2 c^2}\sum_{n=1}^N |\alpha_n|^2 (2\pi n\Delta_f)^2
\frac{(p_U-p_R)(p_U-p_R)^T}{\|p_U-p_R\|^2},
\]
from which $\mathrm{CRB}_{p_U}=\mathbf J_{\rm pos}^{-1}$ and the PEB follow directly [2305.12094].

The same architecture appears in multi-target CF mMIMO-OTFS ISAC, but with a richer path-parameter vector. There, the unknowns for each bi-static path are partitioned as $\theta_{p,r,t}=[\theta^{(1)};\theta^{(2)}]$, with $\theta^{(1)}=[\omega^r,\omega^t,\tau,\nu]^T$ and $\theta^{(2)}=[\mathrm{Re}\{\beta\},\mathrm{Im}\{\beta\}]^T$. After forming the path FIM, the EFIM for $\theta^{(1)}$ is obtained by the Schur complement,
\[
F^{e}_{\theta^{(1)}}=F_{11}-F_{12}F_{22}^{-1}F_{12}^T,
\]
and the position FIM is
\[
F_{p_t}
=
\sum_{p=1}^{N_{\rm tx}}\sum_{r=1}^{N_{\rm rx}}
J^T F^{e}_{\theta^{(1)}} J,
\]
where $J=\nabla_{p_t}\theta^{(1)}$ [2504.10137].

Fluid antenna systems use the same elimination principle in an explicitly localization-oriented notation. If $\eta=[x^T,\nu^T]^T$ with $x\in\mathbb R^2$ the user position and $\nu$ the nuisance channel parameters, then
\[
J_x\triangleq J_{xx}-J_{x\nu}J_{\nu\nu}^{-1}J_{\nu x}.
\]
The resulting EFIM is a sum of per-base-station ToA and AoA contributions,
\[
J_x(\mathcal S)
=
\sum_{b=1}^B
\left[
\lambda_b^{(\tau)}u_bu_b^T+
\lambda_b^{(\theta)}(\mathcal S)\frac{u_b^\perp(u_b^\perp)^T}{r_b^2}
\right],
\]
which makes the information decomposition geometrically transparent [2512.13941].

Distributed MIMO positioning likewise expresses the total FIM as a weighted sum of per-link contributions. After forming a per-AP EFIM for $\xi_k=[\theta_k,\phi_k,\tau_k]$, the local position FIM is mapped through a Jacobian and rotated into the global coordinate system, yielding
\[
F_p=\sum_{k=1}^K r_k^{AP}\,F_{p,k}^{Local}\,(r_k^{AP})^T,
\qquad
PEB=\sqrt{\mathrm{trace}[F_p^{-1}]}
\]
[2511.06440]. Across these works, the PEB is therefore not a primitive quantity; it is the scalar reduction of an information matrix whose structure depends on the sensing modality and the nuisance-parameter treatment.

## 3. Geometry, observability, and what the bound measures

The meaning of a given PEB is inseparable from geometry. In ToA-only RIS-aided IPAC, the EFIM is proportional to
\[
\frac{(p_U-p_R)(p_U-p_R)^T}{\|p_U-p_R\|^2},
\]
which places all position information along the direction from the RIS to the user [2305.12094]. This suggests an observability limitation for single-path ToA-only configurations: without additional geometric diversity, the information matrix is structurally anisotropic.

Several recent works make this dependence explicit. In bistatic MIMO-OFDM ISAC with AoA/ToA positioning, the position CRB of target $l$ is
\[
C_{p_l}=\Upsilon_l\,(J^e_{\eta_l})^{-1}\,\Upsilon_l^T,
\]
and the compact bound
\[
SPEB_l
=
\frac{o\cdot CRB_{\theta^r}+p\cdot CRB_{\phi^r}+q\cdot CRB_\tau}{4\omega^4}
\]
depends on $\omega=c\tau_l-D\sin\theta_l^r\sin\phi_l^r$; the paper states that $\omega\to0$ when the target lies near the line joining the two BSs, implying $PEB\to\infty$ [2502.11446]. In distributed MIMO, the two-dimensional geometry condition is summarized by the “AP geometry factor”
\[
D\triangleq \sum_{k=1}^K u_k e^{j2(\phi_k+\omega_k)},
\]
and the PEB is minimized when
\[
\sum_{k=1}^K u_k e^{j2(\phi_k+\omega_k)}=0
\]
[2511.06440]. The role of balanced angular spread is therefore explicit rather than heuristic.

RIS-assisted non-terrestrial positioning uses a reduced two-dimensional model based on the direct delay $\tau_d$ and the RIS-assisted excess delay $\Delta\tau_r$. With
\[
J(x)=\frac{1}{\sigma^2_{\tau_d}}g_dg_d^T+\frac{1}{\sigma^2_{\Delta\tau_r}}g_rg_r^T,
\]
the determinant obeys
\[
\det(J)=\frac{1}{\sigma^2_{\tau_d}\sigma^2_{\Delta\tau_r}}\|g_d\times g_r\|^2,
\]
so the PEB depends on the linear independence of the two delay gradients [2604.19388]. A similar geometric message appears in fluid antenna systems, where the AoA information term
\[
\lambda_b^{(\theta)}(\mathcal S)
=
\frac{(2\pi/\lambda)^2}{\sigma_\phi^2}
\sum_{m\in\mathcal S}\bigl((u_b^\perp)^T r_m\bigr)^2
\]
grows with the projected spatial second moment of the selected ports, formalizing the synthetic-aperture effect [2512.13941].

Near-field RIS localization sharpens the distinction between far-field and near-field observability. In the near-field formulation, the optimal information-bearing subspace is spanned by four beams,
\[
\mathbf a(\mathbf p),\quad
\dot{\mathbf a}_\rho(\mathbf p),\quad
\dot{\mathbf a}_\theta(\mathbf p),\quad
\dot{\mathbf a}_\phi(\mathbf p),
\]
and the text states that in far-field the steering vector depends only on angles, so one cannot estimate $\rho$, whereas in near-field the path curvature makes the $3\times 3$ position submatrix full rank [2203.07269]. PEB is thus a condensed statistic of observability, but the observability itself is determined by geometry, modality, and wavefront model.

## 4. PEB as an optimization objective and constraint

A major contemporary use of PEB is as an explicit design variable in joint sensing, communication, and localization problems. In RIS-enabled IPAC, the base-station beamformers and RIS phases are chosen to minimize transmit power subject to both rate and positioning constraints:
\[
\min_{W,\Phi}\sum_{n,k}\|w_{n,k}\|^2
\]
subject to
\[
R_k(W,\Phi)\ge r_k,\qquad
\sqrt{\mathrm{tr}\{J_{{\rm pos},k}(W,\Phi)^{-1}\}}\le \delta_k,\qquad \forall k.
\]
The coupling of active and passive beamforming makes the rate and PEB constraints non-convex; the reported solution is a two-stage procedure using exhaustive search for discrete RIS phases and semidefinite relaxation (SDR) for continuous phases [2305.12094].

This use of PEB as a resource-allocation constraint extends across ISAC architectures. In CF mMIMO-OTFS ISAC, the power-allocation problem maximizes the minimum user communication SINR while imposing
\[
CRLB_{p_t}\le \gamma_s,\qquad t=1,\ldots,T_g,
\]
and is solved through the quadratic/fractional transform followed by convex optimization [2504.10137]. In bistatic hybrid-beamforming ISAC, the sensing bound is recast as a beamforming-gain threshold
\[
G\ge \kappa_l,
\]
which makes the spectral-efficiency/PEB trade-off explicit: tighter $\Gamma$ implies larger $\kappa_l$, more restrictive beamforming, and lower spectral efficiency [2502.11446].

Optimization around PEB also appears outside conventional beamforming. In fluid antenna systems, the port-selection problem is cast as D-optimal design through $\max \log\det J_x$, implemented either by greedy marginal log-det increments or by convex continuous relaxation with selection weights $x_m\in[0,1]$ [2512.13941]. In beam-steered optical wireless positioning, the steering directions $\{\mathbf n_{t,i}\}$ are chosen by minimizing
\[
f(\{\mathbf n_{t,i}\})
=
\sqrt{\frac{1}{|\mathcal R|}\sum_{\mathbf r\in\mathcal R}\mathrm{PEB}(\mathbf r)^2},
\]
using a genetic algorithm over a 3D testbed [2603.29400]. In multi-RIS mmWave sensing, continuous phase-shift design is handled on the complex circle manifold, while discrete phase-shift design uses an improved grey wolf optimizer to minimize the PEB directly [2508.06958]. Near-field RIS localization frames the phase-profile problem as an SDP in the RIS covariance $\mathbf X$, whose optimal solution is rank at most $4$ and can be implemented by time-sharing the four beams induced by $\mathbf a(\mathbf p)$ and its spatial derivatives [2203.07269].

The recent literature therefore treats PEB not merely as an evaluation metric, but as a control variable linking sensing fidelity to power, spectral efficiency, codebook choice, and hardware activation.

## 5. Beyond the classical unbiased matched-model bound

Although CRB-based PEB is dominant, several works make clear that it is not universal. The airborne laser imaging analysis derives horizontal and vertical error variances through first-order differential propagation,
\[
\sigma_h^2
=
\sum_i\Bigl(\frac{\partial h}{\partial \varepsilon_i}\Bigr)^2\sigma_{\varepsilon_i}^2,
\qquad
\sigma_v^2
=
\sum_i\Bigl(\frac{\partial v}{\partial \varepsilon_i}\Bigr)^2\sigma_{\varepsilon_i}^2,
\]
and then defines
\[
\mathrm{PEB}=k\sqrt{\sigma_h^2+\sigma_v^2}
\]
for a confidence factor $k$ [1111.6226]. This is a propagated uncertainty bound rather than a CRB from a stochastic likelihood.

The geometric-upper-bound literature departs even further from estimation-theoretic lower bounds. For range-based sensor networks with positively biased ranges, the feasible set
\[
F=\bigcap_{i=1}^N\{x\in\mathbb R^n:\|x-a_i\|_2\le \hat d_i\}
\]
contains the target, and the bounds $B_1$ and $B_2$ are obtained by maximizing distance over $F$ [1201.2513]. In safety-critical satellite navigation, horizontal position error is bounded by inequalities such as
\[
|\delta u|\le M_u|\delta b|,\qquad
|\delta v|\le M_v|\delta b|,
\]
where the magnification coefficients depend only on satellite geometry [1406.4952]. These are deterministic upper bounds, not lower bounds on unbiased estimation error.

A different departure occurs under model mismatch. In LEO positioning under orbital errors, the misspecified CRB (MCRB) yields
\[
\mathrm{MSE}(\bar{\boldsymbol\xi})
\succeq
\mathbf J_M^{-1}\mathbf J_B\mathbf J_M^{-1}
+
(\bar{\boldsymbol\xi}-\boldsymbol\xi_0)(\bar{\boldsymbol\xi}-\boldsymbol\xi_0)^T,
\]
and the PEB is defined from the position block of this sum:
\[
\mathrm{PEB}(\bar{\boldsymbol\xi})
=
\sqrt{
\mathrm{tr}
\Bigl(
[
\mathbf J_M^{-1}\mathbf J_B\mathbf J_M^{-1}
+\Delta\Delta^T
]_{1:3,1:3}
\Bigr)
}
\]
[2511.06060]. Here the model-bias term does not vanish with increasing SNR, so the classical unbiased matched-model interpretation no longer applies.

Terminology also shifts in wideband mmWave positioning. The ping-pong positioning framework defines a per-subcarrier positioning-error-lower-bound (PELB)
\[
e_n(\mathbf p)=\mathrm{tr}\bigl(\mathbf J_n^{-1}\bigr)
\]
and a multi-subcarrier collaborative PELB (MSCPEB)
\[
e(\mathbf p)=\mathrm{tr}\bigl(\mathbf J^{-1}\bigr),\qquad
\mathbf J=\frac1{N_c}\sum_{n=1}^{N_c}\mathbf J_n,
\]
with the proved inequality that MSCPEB never exceeds the arithmetic mean of the individual PELBs [2509.00727]. This suggests that scalar trace-of-inverse measures remain structurally stable even when nomenclature changes from PEB to PELB.

## 6. Reported behavior across systems and architectures

The reported numerical behavior of PEB is highly system-specific, but several recurrent trends appear. In RIS-enabled IPAC, increasing the number of RIS elements $N$ dramatically reduces PEB, with a roughly $\sim 1/N$ gain; higher SNR obtained by raising transmit power leads to a steeper drop in PEB until the bound saturates at a nuisance-parameter floor; finite quantization with $q=2,3,4$ bits achieves near-continuous-phase performance for $q\ge3$; and joint active-plus-passive optimization outperforms benchmarks that fix $\theta=0$ or use random beams [2305.12094]. In multi-RIS mmWave sensing, PEB decreases roughly $\propto 1/\sqrt{P_0}$ at low transmit power, $\propto 1/\sqrt{N}$ with the number of measurements, and $\propto 1/\sqrt{\ell}$ with the number of RIS elements; the continuous-phase Riemannian method outperforms benchmark beam-sweeping by up to an order of magnitude and achieves sub-cm PEB with two RISs, while two-bit quantization is only slightly above the continuous bound [2508.06958]. In urban NTN joint communication and positioning, joint success probability increases with RIS size and phase resolution, but the gains exhibit diminishing returns beyond moderate sizes such as $N\ge256$ and $b\ge4$; the shadowing-aware robust selection reduces PEB by up to $\approx5.4\%$ and by $\approx4\%$ on average relative to the non-robust joint design [2604.19388].

Distributed and multi-static architectures repeatedly show geometric-diversity gains. In CF mMIMO-OTFS ISAC, the exact PEB from the full FIM and the low-complexity approximation closely match Monte-Carlo RMSE, the approximation acts as an upper bound, and adding APs reduces PEB much faster than simply enlarging a single-site array [2504.10137]. In distributed MIMO tracking, using all eight panels yields an average RMSE of $0.09\,\mathrm m$, while activating only five panels selected by the PEB-aware strategy yields an average RMSE of approximately $0.10\,\mathrm m$ [2511.06440]. In RIS-aided Bayesian localization with channel-estimation errors, the paper reports $PEB\approx0.53\,\mathrm m$ with RIS and $PEB\approx4.54\,\mathrm m$ without RIS at $\sigma^2=10^{-3}$, with simulated RMSE tracking the PEB closely for small channel-estimation errors in the RIS-aided case [2508.18009].

Hardware reconfigurability and waveform diversity also tighten the bound measurably. In fluid antenna systems, user-side random port selection at SNR $=0$ dB yields $PEB\approx3.0\,\mathrm m$, while greedy selection attains $PEB\approx2.0\,\mathrm m$; for BS-side FAS at SNR $=20$ dB, random activation gives $PEB\approx1.5\,\mathrm m$, whereas optimal selection reaches $\approx0.9\,\mathrm m$, and convex relaxation matches greedy performance within $1$–$2\%$ [2512.13941]. In beam-steered optical wireless positioning, the genetic-algorithm steering patterns reduce median PEB by $49$–$72\%$ compared with random steering; the worst optimized median PEB is $1.43$ cm at $K=3$, reaching $0.74$ cm at $K=9$; and the $90$th-percentile PEB decreases from approximately $5$ cm at $K=3$ to approximately $1.5$ cm at $K=9$ [2603.29400]. In wideband mmWave ping-pong positioning, the AO-optimized MSCPEB beamformers improve positioning RMSE by at least $16\%$, with $36\%$ reduction for $N_t=128$ at SNR $=25$ dB, while requiring only approximately one-quarter of the slot resources [2509.00727].

Taken together, these results indicate that PEB is simultaneously a fundamental-limit metric, a geometry diagnostic, and a systems-design objective. Its numerical value can shrink through bandwidth, aperture, multi-static diversity, RIS configurability, waveform structure, and optimized scheduling, but its interpretation remains contingent on the underlying statistical model, nuisance-parameter treatment, and whether the bound is lower, upper, matched, or misspecified.

Source: https://www.emergentmind.com/topics/positioning-error-bound-peb