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Robustness Guarantees for Optimal RIS Placement in Self-Localization Under Placement Errors
Mahdi Koloushani, Omid Rezaei, Seyed Mohammad Karbasi, Mohammad Mahdi Naghsh, Mohammad Mahdi Nayebi
TL;DR
The paper asks whether uncertain RIS installation changes optimal RIS placement for self-localization, a design question motivated by deployment errors. Using a CRB analysis, it shows that under moderate i.i.d. Gaussian placement errors and i.i.d. Gaussian ToF noise, uncertainty scales the bound independently of geometry, leaving A-, D-, and E-optimal placements unchanged.
Problem
Existing RIS placement studies generally assume ideal deployment, although installation inaccuracies can displace RISs from intended locations and degrade localization performance.
Source: §I · p. 2Method
The paper analyzes the Hybrid CRB while treating uncertain RIS positions as nuisance random parameters in a CRB framework.
Source: §I · p. 2, §III · p. 3Results
Under moderate RIS position errors, i.i.d. Gaussian placement errors, and i.i.d. Gaussian ToF noise, the source-position CRB scales by a geometry-independent factor, leaving A-, D-, and E-optimal placements unchanged.
Source: §V · p. 5Takeaways & Limitations
Under the considered conditions, placements optimized without position errors remain optimal when those errors are considered, simplifying deployment design while preserving the configuration.
Source: §I · p. 2The analysis leaves non-i.i.d. or non-Gaussian RIS errors, correlated ToF noise, model mismatch, and augmented AoA or RSS measurements for future work.
Source: §V · p. 5
Abstract
from arXiv · showhide
This paper studies the effect of Reconfigurable Intelligent Surface (RIS) placement errors on optimal RIS deployment for Time-of-Flight (ToF)-based self-localization. We consider a setup in which a source transmits a signal, receives the RIS-reflected echo, and estimates its own position from the corresponding ToF measurements. Under the conditions of moderate, i.i.d. Gaussian RIS position errors and i.i.d. Gaussian ToF noise, we demonstrate that the placement uncertainty manifests as a geometry-independent scaling factor in the Cram\'er-Rao Bound (CRB). Consequently, we prove that the optimal RIS placement remains unchanged by the presence of RIS position errors. This result is established for A-, D-, and E-optimality criteria. Simulation results verify the analysis and show that optimized RIS deployments retain their advantage even in the presence of placement uncertainty.
I. INTRODUCTION
The introduction motivates robust RIS placement for self-localization, where deployment errors may undermine geometry-sensitive designs. It argues that under the paper’s considered conditions, placement errors scale the localization bound without changing the optimal configuration.
I. INTRODUCTION
- RISs can shape wireless propagation by steering, reflecting, or scattering signals, and passive architectures are commonly adopted for practical deployment reasons.The passage cites lower hardware complexity, reduced power consumption, and easier deployment as practical considerations.Source: §I · p. 1
- RIS-enabled self-localization estimates a user’s own position from RIS reflections without access points or base stations, complementing infrastructure-assisted localization.Prior work considered OFDM pilots and CRB-based estimators, and FMCW radar self-localization was experimentally demonstrated.Source: §I · pp. 1–2
- Existing RIS placement studies generally assume ideal deployment, although installation inaccuracies can move RISs from intended positions and degrade localization.Earlier work found localization accuracy depends strongly on RIS deployment position,.Source: §I · p. 2
- Under mild conditions, RIS position errors scale the localization error bound independently of geometry, leaving the error-free optimal placement unchanged.The paper states this result for A-, D-, and E-optimality and says it simplifies deployment design while preserving the optimal configuration.Source: §I · p. 2
II. SIGNAL MODEL
The signal model considers a source that estimates its own position from round-trip ToF echoes reflected by multiple passive RIS panels. It models the actual RIS positions as uncertain around their intended locations and accounts for measurement noise.
II. SIGNAL MODEL
- A source estimates its position from round-trip ToF measurements of echoes reflected by multiple passive RIS panels.The model considers a D-dimensional environment with D = 2 or 3.Source: §II · p. 2
- The actual RIS position vector is modeled as Gaussian around the intended positions, with covariance capturing placement errors.The unknown source location is denoted by u, while the RIS positions are treated as uncertain.Source: §II · p. 2
- ToF measurement noise and RIS position errors are assumed to be independent, i.i.d. Gaussian components.The measurement-noise covariance is σ_n^2 I_N and the RIS-error covariance is σ_p^2 I_DN.Source: §II · p. 2
III. CRB DERIVATION AND UNCERTAINTY ANALYSIS
Treating uncertain RIS locations as nuisance random parameters, the derivation shows that moderate placement errors scale the CRB without changing the optimal placement under A-, D-, and E-optimality.
III. CRB DERIVATION AND UNCERTAINTY ANALYSIS
- The uncertain-position CRB is a geometry-independent scalar multiple of the error-free CRB, so the optimal RIS placement is unchanged under A-, D-, and E-optimality.The derivation gives ΔCRB = γ CRB0 and α = 1 + γ.Source: §III · pp. 3–4
- RIS positions are treated as nuisance random parameters, making the Hybrid CRB the relevant bound for estimating the source location.The full unknown vector contains both source location and RIS positions.Source: §III · p. 3
- The CRB for the source location is obtained from the HCRB using the moderate-position-error approximation and the matrix inversion lemma.The error-free bound CRB0 is used as the reference in the resulting scaling relation.Source: §III · p. 3
- The derivation structures the source-location and RIS-position information matrices, then decomposes S before substituting into the CRB expression.The Jacobians are with respect to source location u and RIS positions p.Source: §III · p. 3
IV. NUMERICAL ANALYSIS
Numerical analysis verifies that placement uncertainty scales the CRB independently of geometry, leaving optimal RIS placement invariant across tested scenarios.
IV. NUMERICAL ANALYSIS
- The Monte Carlo study used 2 × 10^4 realizations in three-dimensional scenarios, sampling source and RIS positions within a 100 m-radius sphere, ToF noise from 0.1–10 ns, and placement errors from 5 mm–1 m.The two-dimensional case is described as analogous; the sampled noise and error standard deviations were drawn logarithmically.Source: §IV · p. 4
- Table I reports mean and maximum relative scaling errors for the A-, D-, and E-optimality criteria.The numerical evaluation compares the scaling relations for the three criteria; the supplied table passage does not give the cell values.Source: Table I · p. 4, §IV · p. 4
- Relative scaling errors were negligibly small across simulated conditions, while the optimal nominal positions remained unchanged with placement errors.The numerical results support the geometry-independent scaling relation; Fig. 2 contrasts the optimal nominal positions with one realization of deployed positions.Source: Figure 2 · p. 4, §IV · p. 4
- The robustness property held across varied RIS configurations, measurement-noise levels, and position-error levels, supporting design from the error-free CRB under moderate uncertainty.The simulations report that placement uncertainty degrades performance without changing the optimal deployment under A-, D-, and E-optimality.Source: §IV · pp. 4–5
V. CONCLUSION
The paper analyzes ToF-based self-localization with RIS deployment errors and proves a geometry-independent CRB scaling under moderate, i.i.d. Gaussian errors and noise. Simulations validate that the optimal placement remains unchanged, while broader error and model settings remain future work.
V. CONCLUSION
- Under moderate i.i.d. Gaussian RIS position errors and i.i.d. Gaussian ToF noise, placement errors multiply the source-position CRB by a factor tied to the ratio of error power to ToF-noise power, leaving A-, D-, and E-optimal placements unchanged.The factor is independent of RIS geometry.Source: §V · p. 5
- Numerical simulations validated the result: placement errors increase the absolute error bound but do not shift optimal RIS deployment coordinates.The paper concludes that ideal-condition placement strategies remain robust to practical deployment uncertainty within the studied setting.Source: §V · p. 5
- Future work includes non-i.i.d. or non-Gaussian RIS errors, correlated ToF noise, model mismatches such as multipath and NLOS, and adding AoA or RSS observables.Source: §V · p. 5