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Reconfigurable Intelligent Surfaces for Localization: Position and Orientation Error Bounds

Ahmed Elzanaty, Anna Guerra, Francesco Guidi, Mohamed-Slim Alouini

arXiv:2009.02818v1cs.ITeess.SP

TL;DR

The paper addresses RIS-assisted 3D localization and orientation estimation when near-field propagation and asynchronous signaling make simplified models inadequate. It derives CRLB and GDOP analyses for a general gNB–RIS–UE setting and proposes a closed-form spherical-wavefront-aware RIS phase profile. The proposed RIS-assisted scheme substantially improves localization performance, including under asynchronous signaling, while the phase design approaches the numerical CRLB-minimizing design.

  • Problem

    Existing RIS localization studies do not provide a general 3D near-field model for orientation estimation with the RIS controlling multipath, while far-field assumptions can be inaccurate for large surfaces.

  • Method

    The paper derives CRLB performance bounds and GDOP geometry analysis for synchronous and asynchronous signaling, then designs a closed-form RIS phase profile accounting for spherical wavefronts.

  • Results

    The proposed RIS phase design approaches the numerical optimal design that minimizes the CRLB and achieves remarkable localization performance, including in asynchronous signaling.

  • Takeaways & Limitations

    RISs can support joint communication and gNB-based UE localization and orientation estimation in near- and far-field propagation conditions.

Abstract

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Next-generation cellular networks will witness the creation of smart radio environments (SREs), where walls and objects can be coated with reconfigurable intelligent surfaces (RISs) to strengthen the communication and localization coverage by controlling the reflected multipath. In fact, RISs have been recently introduced not only to overcome communication blockages due to obstacles but also for high-precision localization of mobile users in GPS denied environments, e.g., indoors. Towards this vision, this paper presents the localization performance limits for communication scenarios where a single next-generation NodeB base station (gNB), equipped with multiple-antennas, infers the position and the orientation of the user equipment(UE) in a RIS-assisted SRE. We consider a signal model that is valid also for near-field propagation conditions, as the usually adopted far-field assumption does not always hold, especially for large RISs. For the considered scenario, we derive the Cramer-Rao lower bound (CRLB) for assessing the ultimate localization and orientation performance of synchronous and asynchronous signaling schemes. In addition, we propose a closed-form RIS phase profile that well suits joint communication and localization. We perform extensive numerical results to assess the performance of our scheme for various localization scenarios and RIS phase design. Numerical results show that the proposed scheme can achieve remarkable performance, even in asynchronous signaling and that the proposed phase design approaches the numerical optimal phase design that minimizes the CRLB.

I. INTRODUCTION

The paper develops a general 3D RIS-assisted framework in which a gNB estimates UE position and orientation under near- and far-field propagation. It derives performance bounds for synchronous and asynchronous signaling and proposes RIS phase designs for joint communication and localization.

  • I. INTRODUCTION: RIS-assisted localization lets the gNB infer both the UE’s 3D position and orientation, supporting communication and surveillance applications.The architecture uses the RIS as a passive reflector assisting the gNB rather than having the UE estimate its own position.
  • I. INTRODUCTION: The model supports arbitrary 3D multi-antenna array geometries at the gNB, RIS, and UE, including planar arrays.This generalizes analyses that commonly use linear arrays and simplified 2D beamforming.
  • I. INTRODUCTION: The signal model accounts for spherical wavefronts and is valid in both near-field and far-field localization conditions.The near-field model is motivated by large RISs for which the planar-wavefront approximation may fail and location and orientation information can be discarded.
  • I. INTRODUCTION: The paper analyzes both synchronous and asynchronous signaling and derives CRLB-based localization performance bounds alongside GDOP-based geometry analysis.The CRLB characterizes minimum estimation variance, while GDOP evaluates the effect of gNB, RIS, and UE geometry on localization.
  • I. INTRODUCTION: The proposed closed-form RIS phase design accounts for spherical wavefronts and is compared with alternative strategies, including quantization errors.The broader evaluation uses simulations and numerical results to examine RIS benefits and phase-design behavior.

B. Signal Model for Incident Spherical Wavefronts

The model represents RIS-assisted uplink signals with spherical wavefronts, incorporating direct and reflected paths, antenna geometry, propagation delays, and synchronization effects. It supports gNB-based estimation of UE position and orientation in synchronous and asynchronous systems.

  • The gNB estimates UE position and orientation from ranging and angular information in a spherical waveform model.
  • The received signal combines direct gNB–UE and RIS-relayed paths with transmitted data, RIS phase shifts, propagation attenuation, and thermal noise.
  • Asynchronous localization can still exploit relative phases and spherical-wave information even when time-of-arrival information is unavailable.
  • The model assigns antenna-pair delays from inter-element distances and angles, with distances converted to delays using the speed of light.
  • Spherical-wave modeling jointly captures ranging and bearing information and remains applicable when planar-wave assumptions fail in large-array near-field settings.

2) Asynchronous System:

For asynchronous signaling, unknown synchronization offsets prevent direct time-of-arrival ranging, so localization exploits spherical-wave curvature and relative phases. The paper derives CRLB-based position and orientation limits using direct and two-stage estimation formulations.

  • 2) Asynchronous System:: Unknown phase offsets represent clock asynchrony, removing direct distance information from time-of-arrival measurements.
  • A. The CRLB on UE position and orientation: The CRLB is obtained from the inverse Fisher information matrix, while GDOP analysis separates geometry effects from receiver-noise effects.
  • 2) Asynchronous System:: The asynchronous model estimates UE position and orientation through signal features such as relative phases, angles of arrival, and received signal strength indicators.
  • 2) Asynchronous System:: Direct localization is used for asynchronous signaling, whereas the two-stage approach extracts intermediate signal features before estimating the UE state.
  • A. The CRLB on UE position and orientation: The spherical-wavefront CRLB remains valid for near-field localization and accounts for both receiver noise and localization geometry.

B. Localization Algorithm

The paper considers maximum-likelihood estimation and alternative feature-based estimators for recovering UE location and orientation. The MLE approaches the CRLB at asymptotically high SNR, although iterative optimization is not guaranteed to reach the global maximum.

  • The MLE estimates UE location and orientation by maximizing the received-signal log-likelihood.
  • Grid search or Newton-Raphson and expectation-maximization methods can solve the optimization, but iterative methods lack guaranteed global convergence.
  • At asymptotically high SNR, the MLE approaches the CRLB derived for the localization problem.
  • With many measurements or high SNR, location-and-orientation estimation errors tend toward a zero-mean Gaussian distribution with covariance Λ(s).
  • A two-stage alternative estimates attenuation from RSSI and bearing angles through MUSIC variants or compressive-sensing methods.

C. Geometry Impact on Direct RIS-aided Localization

The geometry analysis uses GDOP to isolate how localization geometry affects position and orientation performance beyond receiver noise. It also compares numerically optimal RIS phase configurations with a closed-form design for spherical wavefronts.

  • C. Geometry Impact on Direct RIS-aided Localization: GDOP measures the geometry-only contribution to localization error by relating position RMSE to ranging-measurement RMSE.
  • C. Geometry Impact on Direct RIS-aided Localization: For direct localization, measurement-noise standard deviation is common across antennas and equals the thermal-noise standard deviation σ.
  • C. Geometry Impact on Direct RIS-aided Localization: Position and orientation errors are proportional to σ times their respective GDOP metrics.
  • IV. RIS PHASE DESIGN: RIS phase design is challenging because conventional approaches often assume planar incident wavefronts, whereas the paper considers spherical-wave alternatives.
  • 1) Optimal RIS Phase Design:: The numerical optimal phase design minimizes position or orientation error bounds but is complex because it requires minimizing an inverse FIM.

2) Proposed RIS Phase Design:

The proposed RIS phase design first optimizes a convex surrogate for phase alignment, then adds a constant shift to coherently combine direct and reflected paths at the gNB.

  • Phase optimization: The design maximizes the sum of SNRs across the gNB antennas using a convexified phase-alignment objective.The objective minimizes squared phase distances from their centroid because the RIS lacks enough controllable degrees of freedom for perfect alignment.
  • Direct and reflected paths: A constant phase shift is then selected so the direct and RIS-reflected paths are coherently summed at each gNB antenna.This follows from the Cauchy–Schwarz bound, whose equality requires coincident direct- and reflected-path phases.
  • Phase optimization: The convex objective yields a closed-form, optimal-but-nonunique RIS phase profile for the RIS elements.The phases are obtained by solving N_R linear equations in the N_R RIS phase shifts.
  • Design property: Adding the same constant phase to all RIS elements leaves the objective value unchanged, so the phase solution is not unique.The invariance follows from the absolute operator in the objective.
  • Direct and reflected paths: The resulting closed-form RIS phases account jointly for the direct and reflected paths.The design combines the per-element optimized phases with the additional constant phase shift.

V. NUMERICAL RESULTS

The numerical evaluation considers planar gNB, RIS, and UE arrays in a 3D localization setting, with position and orientation errors evaluated over a 20 × 20 m2 region.

  • Simulation setup: The simulations use planar arrays, supporting compact deployment and 3D beamfocusing at the gNB and UE, while also matching practical wall-mounted RIS geometry.The performance analysis focuses on planar antenna configurations because of these deployment considerations.
  • Simulation setup: The gNB and RIS are fixed on the XZ- and YZ-planes, whereas the UE can freely rotate about the x-, y-, and z-axes.The UE orientation is parameterized by roll, pitch, and yaw angles.
  • Position and orientation errors: Fig. 4 reports PEB in meters and OEB in degrees for mobile locations across a 20 × 20 m2 XY-plane area at φM = (π/6, π/6, π/6).The figure evaluates how position and orientation error vary with UE location.
  • Phase-design comparison: The phase-design comparison includes Mirror, Random, Proposed, Optimized CRLB, and Quantized RIS configurations.Quantized denotes four phase-quantization levels.

B. Numerical Results

The numerical results evaluate position and orientation error bounds across geometry, RIS configurations, UE orientation, signaling synchronization, and staged parameter estimation. They show that geometry and orientation affect performance, while accurate TOA estimation and suitable RIS phase design can substantially improve localization.

  • PEB and OEB for Different Mobile Positions: PEB and OEB are lower near both the gNB and RIS, with an error of about 6 × 10−6 in the reported setting.The experiment uses the proposed RIS phase design and evaluates varying UE locations.
  • Analysis of the UE Orientation: The orientation of the mobile impacts localization performance in the evaluated synchronous and asynchronous signaling scenarios.The study compares fixed and averaged UE orientations using PEB evaluations.
  • Synchronous vs. Asynchronous Signaling: Increasing UE distance from the gNB increases localization error faster than moving along the x-axis away from the RIS.This trend is observed after averaging over several mobile orientations.
  • Two-stage Localization: Discarding RSSI measurements has negligible impact on PEB, whereas relying on RSSI without TOA measurements increases localization error up to two orders of magnitude.The comparison uses RSSI and AOA, TOA and AOA, and all parameters as three estimation cases.
  • Parameter Estimation: The parameter-estimation errors for gNB-dependent quantities are smallest near the gNB location.The evaluated quantities include time and angular parameters associated with the gNB and RIS paths.
  • Geometric dilution of precision: GDOP measures geometry-driven amplification of estimation error, with smaller values indicating more favorable mobile geometry relative to the gNB and RIS.The study evaluates GDOP for both mobile position and orientation as a function of RIS element count.

VI. CONCLUSIONS

The paper develops a RIS-assisted architecture for joint UE localization and orientation estimation under near- and far-field propagation. Its proposed RIS phase design focuses the incident spherical wavefront toward the gNB, yielding substantial reductions in position and orientation error relative to a conventional system without RIS.

  • VI. CONCLUSIONS: The architecture jointly estimates UE position and orientation in a RIS-assisted environment while accounting for near- and far-field propagation conditions.The paper derives ultimate performance bounds in terms of PEB and OEB.
  • VI. CONCLUSIONS: The RIS phases are designed to maximize SNR toward the desired UE for communication and localization enhancement.The design focuses the incident spherical wavefront from the UE toward the gNB.
  • VI. CONCLUSIONS: Up to two orders and one order of magnitude reductions in PEB and OEB, respectively, are achieved compared with a conventional system without RIS.The reported gains depend on the considered geometry and UE orientation.
  • VI. CONCLUSIONS: Localization accuracy strongly depends on the considered geometry and the orientation of the UE.The conclusion identifies analysis with multiple RISs as a future step.

APPENDIX A THE JACOBIAN MATRIX

Appendix A presents the Jacobian matrix used in the CRLB derivation for the mobile location parameters. It defines the relationships needed to differentiate distances and related geometric quantities with respect to mobile coordinates.

  • APPENDIX A THE JACOBIAN MATRIX: The appendix reports the Jacobian matrix elements for the CRLB derivation of the mobile location.The location derivatives are given for each mobile coordinate and for the gNB and RIS.
  • APPENDIX A THE JACOBIAN MATRIX: The coordinate derivatives are simplified through algebraic manipulation before being used in the CRLB calculations.Equations (66)–(67) are simplified after the initial relationships are introduced.

APPENDIX B FIM ELEMENTS

Appendix B derives the derivatives and Fisher information matrix elements for direct and two-stage localization under synchronous and asynchronous signaling. The derivation accounts for position, orientation, path distances, angles, amplitudes, and RIS-relayed propagation terms.

  • APPENDIX B FIM ELEMENTS: The FIM elements are obtained from derivatives of the mean received signal with respect to the parameters in Γ.The appendix rewrites the signal model and defines the relevant synchronous and asynchronous quantities.
  • Two-stage localization: The appendix provides derivatives for the two-stage localization approach and for all UE rotational angles.It separately identifies derivatives for indirect localization and α_M, β_M, and γ_M.
  • TOA derivatives: TOA derivatives are written for each gNB or RIS path and its corresponding antenna index.The notation distinguishes the propagation source S and antenna index s.
  • Direct localization: The direct-localization derivation uses nonlinear functions depending on the parameters to be estimated, with separate synchronous and asynchronous distance definitions.The signal is rewritten before defining f_bm and g_brm and their associated distances.
  • Assumptions: The phase-design convenience assumption neglects dependence on UE location and orientation, while another derivation note drops synchronization mismatches and array errors.These assumptions apply to the indicated phase-design and derivative expressions.
  • RIS-relayed path: The RIS-relayed derivative terms include distance, angle, and path-loss-amplitude derivatives between array centers.The appendix develops separate gradient expressions for direct and RIS-relayed paths.
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