Positioning Error Bound (PEB) Overview

• Positioning Error Bound (PEB) is a scalar metric that quantifies the minimum achievable localization error, defined via the trace of the inverse Fisher information matrix or CRB.
• PEB underpins various estimation approaches, from CRB-based methods to deterministic worst-case bounds, and is key in systems like RIS-enabled and mmWave localization.
• PEB serves as both a diagnostic for geometric observability and an optimization constraint linking sensing fidelity to resource allocation in integrated sensing and communication designs.

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 (Sun et al., 2023). Across recent arXiv literature, the same label also appears in broader forms: as the positional sub-block of a joint position–orientation CRLB (Tercas et al., 25 Aug 2025), as a confidence-factor combination of horizontal and vertical error variances (Jiang et al., 2011), and as a deterministic geometric upper bound over feasible positions or clock-bias-driven inequalities (Gholami et al., 2012). 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 $\mathrm{CRB}_p=\mathbf J_p^{-1}$ (Sun et al., 2023). The same trace-of-inverse structure is used for two-dimensional target localization in cell-free mMIMO-OTFS ISAC, where $$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$ (Fan et al., 14 Apr 2025), for fluid-antenna-system localization, where $$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$ (Salem et al., 15 Dec 2025), for optical wireless 3D localization, where $$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$ (Acuna-Condori et al., 31 Mar 2026), and for distributed MIMO positioning, where $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$ (Xu et al., 9 Nov 2025).

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 (Tercas et al., 25 Aug 2025). 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

$\mathbf J_p$0

with $\mathbf J_p$1 giving $\mathbf J_p$2 confidence under Gaussian assumptions (Jiang et al., 2011). In range-based wireless sensor networks with positively biased range errors, two worst-case geometric bounds are defined: $\mathbf J_p$3 where $\mathbf J_p$4 is the feasible set formed by the intersection of measurement balls (Gholami et al., 2012).

Formulation	Expression	Representative sources
CRB/FIM-based RMS lower bound	$\mathbf J_p$5	(Sun et al., 2023, Fan et al., 14 Apr 2025)
Positional sub-block of joint CRLB	$\mathbf J_p$6	(Tercas et al., 25 Aug 2025)
Confidence-factor bound	$\mathbf J_p$7	(Jiang et al., 2011)
Feasible-set worst-case bound	$\mathbf J_p$8, $\mathbf J_p$9 over $\mathrm{CRB}_p=\mathbf J_p^{-1}$0	(Gholami et al., 2012)

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

$\mathrm{CRB}_p=\mathbf J_p^{-1}$1

and a scalar delay FIM

$\mathrm{CRB}_p=\mathbf J_p^{-1}$2

The position EFIM then becomes

$\mathrm{CRB}_p=\mathbf J_p^{-1}$3

from which $\mathrm{CRB}_p=\mathbf J_p^{-1}$4 and the PEB follow directly (Sun et al., 2023).

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 $\mathrm{CRB}_p=\mathbf J_p^{-1}$5, with $\mathrm{CRB}_p=\mathbf J_p^{-1}$6 and $\mathrm{CRB}_p=\mathbf J_p^{-1}$7. After forming the path FIM, the EFIM for $\mathrm{CRB}_p=\mathbf J_p^{-1}$8 is obtained by the Schur complement,

$\mathrm{CRB}_p=\mathbf J_p^{-1}$9

and the position FIM is

$$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$0

where $$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$1 (Fan et al., 14 Apr 2025).

Fluid antenna systems use the same elimination principle in an explicitly localization-oriented notation. If $$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$2 with $$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$3 the user position and $$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$4 the nuisance channel parameters, then

$$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$5

The resulting EFIM is a sum of per-base-station ToA and AoA contributions,

$$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$6

which makes the information decomposition geometrically transparent (Salem et al., 15 Dec 2025).

Distributed MIMO positioning likewise expresses the total FIM as a weighted sum of per-link contributions. After forming a per-AP EFIM for $$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$7, the local position FIM is mapped through a Jacobian and rotated into the global coordinate system, yielding

$$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$8

(Xu et al., 9 Nov 2025). 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

$$\mathrm{PEB}_{p_t}=\sqrt{\mathrm{Tr}\{F_{p_t}^{-1}\}}$$9

which places all position information along the direction from the RIS to the user (Sun et al., 2023). 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 $$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$0 is

$$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$1

and the compact bound

$$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$2

depends on $$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$3; the paper states that $$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$4 when the target lies near the line joining the two BSs, implying $$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$5 (Mao et al., 17 Feb 2025). In distributed MIMO, the two-dimensional geometry condition is summarized by the “AP geometry factor”

$$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$6

and the PEB is minimized when

$$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$7

(Xu et al., 9 Nov 2025). 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 $$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$8 and the RIS-assisted excess delay $$\mathrm{PEB}(\mathcal S)=\sqrt{\mathrm{trace}[J_x(\mathcal S)^{-1}]}$$9. With

$$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$0

the determinant obeys

$$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$1

so the PEB depends on the linear independence of the two delay gradients (Khalil et al., 21 Apr 2026). A similar geometric message appears in fluid antenna systems, where the AoA information term

$$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$2

grows with the projected spatial second moment of the selected ports, formalizing the synthetic-aperture effect (Salem et al., 15 Dec 2025).

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,

$$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$3

and the text states that in far-field the steering vector depends only on angles, so one cannot estimate $$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$4, whereas in near-field the path curvature makes the $$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$5 position submatrix full rank (Rahal et al., 2022). 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: $$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$6 subject to

$$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$7

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 (Sun et al., 2023).

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

$$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$8

and is solved through the quadratic/fractional transform followed by convex optimization (Fan et al., 14 Apr 2025). In bistatic hybrid-beamforming ISAC, the sensing bound is recast as a beamforming-gain threshold

$$\mathrm{PEB}(\mathbf r)=\sqrt{\operatorname{tr}\{\mathcal I(\mathbf r)^{-1}\}}$$9

which makes the spectral-efficiency/PEB trade-off explicit: tighter $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$0 implies larger $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$1, more restrictive beamforming, and lower spectral efficiency (Mao et al., 17 Feb 2025).

Optimization around PEB also appears outside conventional beamforming. In fluid antenna systems, the port-selection problem is cast as D-optimal design through $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$2, implemented either by greedy marginal log-det increments or by convex continuous relaxation with selection weights $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$3 (Salem et al., 15 Dec 2025). In beam-steered optical wireless positioning, the steering directions $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$4 are chosen by minimizing

$$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$5

using a genetic algorithm over a 3D testbed (Acuna-Condori et al., 31 Mar 2026). 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 (Cheng et al., 9 Aug 2025). Near-field RIS localization frames the phase-profile problem as an SDP in the RIS covariance $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$6, whose optimal solution is rank at most $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$7 and can be implemented by time-sharing the four beams induced by $$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$8 and its spatial derivatives (Rahal et al., 2022).

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,

$$\mathrm{PEB}(\theta)=\sqrt{\mathrm{trace}[J(\theta)^{-1}]}$$9

and then defines

$\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$0

for a confidence factor $\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$1 (Jiang et al., 2011). 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

$\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$2

contains the target, and the bounds $\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$3 and $\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$4 are obtained by maximizing distance over $\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$5 (Gholami et al., 2012). In safety-critical satellite navigation, horizontal position error is bounded by inequalities such as

$\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$6

where the magnification coefficients depend only on satellite geometry (Iwamoto, 2014). 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

$\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$7

and the PEB is defined from the position block of this sum: $\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$8 (Ma et al., 8 Nov 2025). 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)

$\zeta=[x_M,y_M,z_M,\alpha_M,\beta_M,\gamma_M]^T$9

and a multi-subcarrier collaborative PELB (MSCPEB)

$$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$0

with the proved inequality that MSCPEB never exceeds the arithmetic mean of the individual PELBs (Guo et al., 31 Aug 2025). 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 $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$1 dramatically reduces PEB, with a roughly $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$2 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 $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$3 bits achieves near-continuous-phase performance for $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$4; and joint active-plus-passive optimization outperforms benchmarks that fix $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$5 or use random beams (Sun et al., 2023). In multi-RIS mmWave sensing, PEB decreases roughly $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$6 at low transmit power, $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$7 with the number of measurements, and $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$8 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 (Cheng et al., 9 Aug 2025). 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 $$\mathrm{PEB} = \sqrt{ \mathrm{tr} \bigl\{ [\,\mathrm{CRLB}\,]_{1:3,1:3} \bigr\} }$$9 and $6\times 6$0; the shadowing-aware robust selection reduces PEB by up to $6\times 6$1 and by $6\times 6$2 on average relative to the non-robust joint design (Khalil et al., 21 Apr 2026).

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 (Fan et al., 14 Apr 2025). In distributed MIMO tracking, using all eight panels yields an average RMSE of $6\times 6$3, while activating only five panels selected by the PEB-aware strategy yields an average RMSE of approximately $6\times 6$4 (Xu et al., 9 Nov 2025). In RIS-aided Bayesian localization with channel-estimation errors, the paper reports $6\times 6$5 with RIS and $6\times 6$6 without RIS at $6\times 6$7, with simulated RMSE tracking the PEB closely for small channel-estimation errors in the RIS-aided case (Tercas et al., 25 Aug 2025).

Hardware reconfigurability and waveform diversity also tighten the bound measurably. In fluid antenna systems, user-side random port selection at SNR $6\times 6$8 dB yields $6\times 6$9, while greedy selection attains $\mathbf J_p$00; for BS-side FAS at SNR $\mathbf J_p$01 dB, random activation gives $\mathbf J_p$02, whereas optimal selection reaches $\mathbf J_p$03, and convex relaxation matches greedy performance within $\mathbf J_p$04–$\mathbf J_p$05 (Salem et al., 15 Dec 2025). In beam-steered optical wireless positioning, the genetic-algorithm steering patterns reduce median PEB by $\mathbf J_p$06–$\mathbf J_p$07 compared with random steering; the worst optimized median PEB is $\mathbf J_p$08 cm at $\mathbf J_p$09, reaching $\mathbf J_p$10 cm at $\mathbf J_p$11; and the $\mathbf J_p$12th-percentile PEB decreases from approximately $\mathbf J_p$13 cm at $\mathbf J_p$14 to approximately $\mathbf J_p$15 cm at $\mathbf J_p$16 (Acuna-Condori et al., 31 Mar 2026). In wideband mmWave ping-pong positioning, the AO-optimized MSCPEB beamformers improve positioning RMSE by at least $\mathbf J_p$17, with $\mathbf J_p$18 reduction for $\mathbf J_p$19 at SNR $\mathbf J_p$20 dB, while requiring only approximately one-quarter of the slot resources (Guo et al., 31 Aug 2025).

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.