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RIS-aided Joint Localization and Synchronization with a Single-Antenna Receiver: Beamforming Design and Low-Complexity Estimation

Alessio Fascista, Musa Furkan Keskin, Angelo Coluccia, Henk Wymeersch, Gonzalo Seco-Granados

arXiv:2204.13484v1eess.SP

TL;DR

RIS-aided localization can reduce infrastructure needs, but high propagation losses can require long integration times that limit mobility; tailored beamforming for localization and synchronization is less explored. This paper jointly designs BS and RIS beams and develops a low-complexity estimator, with reported gains over benchmark approaches and bound-attaining estimation under challenging conditions.

  • Problem

    RIS-aided localization can reduce infrastructure needs, but high propagation losses can require long coherent processing intervals that limit mobility, while beamforming strategies tailored to localization and synchronization are less investigated.

    Source: §I · p. 1

  • Method

    The paper derives localization and synchronization information bounds, jointly designs BS precoding and RIS phases, then uses a codebook-based strategy for location uncertainty and a reduced-complexity ML estimator.

    Source: §I · p. 2, §VIII · p. 14

  • Results

    The proposed designs outperform literature benchmarks in PEB, CEB, and localization and synchronization RMSEs, while the reduced-complexity estimator attains CRLBs at low SNR and is robust to uncontrollable multipath.

    Source: §I · p. 2

  • Takeaways & Limitations

    Directional and derivative beams at both the BS and RIS are required to make the joint beamforming optimization feasible, unlike the corresponding communication problem.

    Source: §I · p. 2

    The considered setup assumes a LoS path and a controllable NLoS path through the RIS.

    Source: §VIII · p. 14

Abstract

from arXiv · show

Reconfigurable intelligent surfaces (RISs) have attracted enormous interest thanks to their ability to overcome line-of-sight blockages in mmWave systems, enabling in turn accurate localization with minimal infrastructure. Less investigated are however the benefits of exploiting RIS with suitably designed beamforming strategies for optimized localization and synchronization performance. In this paper, a novel low-complexity method for joint localization and synchronization based on an optimized design of the base station (BS) active precoding and RIS passive phase profiles is proposed, for the challenging case of a single-antenna receiver. The theoretical position error bound is first derived and used as metric to jointly optimize the BS-RIS beamforming, assuming a priori knowledge of the user position. By exploiting the low-dimensional structure of the solution, a novel codebook-based robust design strategy with optimized beam power allocation is then proposed, which provides low-complexity while taking into account the uncertainty on the user position. Finally, a reduced-complexity maximum-likelihood based estimation procedure is devised to jointly recover the user position and the synchronization offset. Extensive numerical analysis shows that the proposed joint BS-RIS beamforming scheme provides enhanced localization and synchronization performance compared to existing solutions, with the proposed estimator attaining the theoretical bounds even at low signal-to-noise-ratio and in the presence of additional uncontrollable multipath propagation.

I. INTRODUCTION

The paper addresses joint localization and synchronization with a single-antenna UE, where prior RIS beamforming work had not jointly optimized the BS precoder and RIS phase profiles for localization. It models a single-BS, RIS-assisted downlink and proposes beamforming designs and an estimator for that setting.

I. INTRODUCTION

  • Prior RIS beamforming studies used SNR-maximizing heuristics that may not be localization-optimal, while joint BS-RIS design for localization and synchronization had not been studied.RISs can add a location reference and measurements independent of uncontrolled multipath.Source: §I · pp. 1–2
  • The proposed approach jointly designs BS precoding and RIS phase profiles using a position error bound, and targets a single-antenna UE served by one BS.The design uses a priori UE location information.Source: §I · p. 2

A. RIS-aided Localization Scenario

  • The scenario has a known-location BS and RIS, an unknown-location single-antenna UE, and an unknown UE clock offset.The UE receives signals over a direct LoS path and a controllable RIS-reflected NLoS path.Source: A. RIS-aided Localization Scenario · p. 2

B. Signal Model

C. Joint Localization and Synchronization Problem

The paper frames joint localization and synchronization as designing BS and RIS transmissions and estimating UE position and clock offset from received downlink signals. It derives Fisher information measures and uses them to guide joint beamforming design.

C. Joint Localization and Synchronization Problem

  • The task is to design BS precoding and RIS phase profiles for accurate UE position and clock-offset estimation, then estimate both unknowns from the received signals.The signal model omits uncontrolled multipath for deriving bounds and algorithms; robustness to such paths is evaluated in simulations.Source: C. Joint Localization and… · p. 3

III. FISHER INFORMATION ANALYSIS

  • The Fisher information analysis derives a performance measure for localization and synchronization to support the subsequent BS and RIS design.The channel-domain Fisher information matrix is formed for unknown channel parameters.Source: §III · p. 3, A. FIM in the Channel Domain · p. 3

A. FIM in the Channel Domain

  • The channel-domain Fisher information depends on the BS precoder and RIS phase profiles, with submatrices corresponding to the LoS and RIS-reflected paths.The cross-correlation submatrix represents interaction between the LoS and NLoS paths.Source: A. FIM in the Channel Domain · pp. 3–4

B. FIM in the Location Domain

  • A transformation from channel parameters to location parameters yields the location-domain Fisher information matrix while preserving its linearity and bilinearity properties.The location-domain matrix is denoted Jη and is obtained using a transformation matrix.Source: B. FIM in the Location Domain · p. 4

IV. JOINT TRANSMIT PRECODING AND RIS PHASE PROFILE DESIGN

  • For known UE parameters, the design jointly optimizes BS precoding and RIS phases, using convex relaxation and alternating optimization to obtain subproblems and reveal low-dimensional structure.That structure supports codebook design when UE location is imperfectly known.Source: §IV · p. 4

A. Problem Formulation for Joint Optimization

The joint design uses PEB minimization to choose BS precoders and RIS phase profiles. The authors reformulate the problem using covariance variables and relax it to expose tractable subproblems and low-dimensional transmission structure.

A. Problem Formulation for Joint Optimization

  • The PEB-based objective is motivated by the coupling of positioning and synchronization, with the text pointing to simulations comparing position and clock-offset RMSE.The cited discussion refers to Fig. 6 for improvements over benchmark schemes.Source: A. Problem Formulation for Joint… · p. 4
  • The PEB is the objective for jointly optimizing BS precoders and RIS phase profiles under perfect knowledge of the UE parameters.The PEB depends on the unknown parameters and the beam and phase variables.Source: A. Problem Formulation for Joint… · p. 4
  • The formulation uses BS and RIS covariance variables because the Fisher information is linear in those variables, unlike its quadratic dependence on the original beam variables.The RIS cross-term is linear in its phase vector, while the RIS-path term is linear in its phase covariance.Source: A. Problem Formulation for Joint… · pp. 4–5

B. Relaxed Problem for PEB Minimization

  • The relaxed PEB problem approximates the channel-domain Fisher information as block diagonal by assuming negligible LoS–NLoS cross-correlation, justified for large bandwidth and arrays with non-interfering paths.The relaxation also drops non-convex rank constraints while retaining positive-semidefinite and unit-diagonal constraints for the RIS covariance.Source: B. Relaxed Problem for PEB Minimization · p. 5
  • The resulting relaxed problem is convex in the BS covariance variables for fixed RIS variables, and its optimization can use off-the-shelf solvers.The authors say the relaxations reveal transmission structure; simulations and power optimization instead use the true Fisher information matrix.Source: B. Relaxed Problem for PEB Minimization · p. 5, 1) Optimize BS Precoders for Fixed RIS… · p. 5
  • Alternating optimization updates BS precoders with RIS phases fixed, then updates RIS phases with the BS precoders fixed.This is the proposed approach to solving the relaxed joint problem.Source: B. Relaxed Problem for PEB Minimization · p. 5

1) Optimize BS Precoders for Fixed RIS Phase Profiles:

  • For fixed RIS phase profiles, the BS-precoder update is a convex problem, and the optimal precoder covariance structure is used to motivate low-complexity optimization.The supplied proposition passage identifies a positive-semidefinite matrix in the structure result but does not provide the full characterization.Source: 1) Optimize BS Precoders for Fixed RIS… · p. 5

2) Optimize RIS Phase Profiles for Fixed BS Precoders:

The optimal beam structures motivate a practical robust design for uncertain UE location: a codebook-based alternative to an intractable unconstrained worst-case PEB optimization. The section also explains the roles of directional and derivative beams and the assumptions that bound the design.

2) Optimize RIS Phase Profiles for Fixed BS Precoders:

  • The RIS optimal phase-profile covariance has a low-dimensional form without the unit-modulus constraints, using the RIS response at the UE direction and a positive semidefinite 2 × 2 matrix.The structure is used to expose analytical insights for later codebook design, not to solve the perfect-location PEB minimization itself.Source: 2) Optimize RIS Phase Profiles for… · p. 5, . bRIS · p. 5

D. Interpretation of Proposition 1 and Proposition 2

  • At the BS, the UE direction uses directional and derivative beams for angle diversity, while the known BS-to-RIS direction uses one directional beam to maximize received SNR.The BS serves both the RIS and UE; the BS-to-RIS angle need not be estimated because both positions are known.Source: D. Interpretation of Proposition 1 and… · pp. 5–6
  • For UE angle estimation, the RIS uses both a directional beam and its derivative, whose sharp beampattern curvature makes small angular deviations produce large amplitude changes.The derivative-beam principle is compared with sum-and-difference beams in monopulse radar.Source: D. Interpretation of Proposition 1 and… · p. 6, Figure 2 · p. 6

V. ROBUST JOINT DESIGN OF BS PRECODER AND RIS PHASE PROFILES UNDER LOCATION UNCERTAINTY

  • The unconstrained robust design minimizes worst-case PEB over the UE location uncertainty region, but its optimization is intractable, motivating a low-complexity codebook alternative.The robust formulation accounts for imperfect location knowledge arising from measurement noise and tracking errors.Source: A. Optimal Unconstrained Design · p. 6, §V · p. 6

A. Optimal Unconstrained Design

  • The design assumes perfectly known channel gains; uncertain gains are excluded because LoS–NLoS correlations make the PEB expressions intractable.The clock offset need not be known to solve the PEB minimization, although the Fisher information matrix depends functionally on it.Source: A. Optimal Unconstrained Design · p. 6

B. Low-Complexity Codebook-Based Design

The low-complexity design uses beam codebooks derived from the optimal low-dimensional beam structure and spans the UE location uncertainty region. It optimizes BS-beam power allocation through a convex problem and is noniterative.

B. Low-Complexity Codebook-Based Design

R,U }LRIS

  • The design optimizes BS-beam power allocation across the BS–RIS signal-pair transmissions under a total-power constraint, yielding a convex problem.The number of transmissions is G = (2LBS + 1)2LRIS.Source: R,U }LRIS · p. 7
  • The proposed noniterative design is more efficient than even individual iterations of alternating optimization in the stated simulation setup.The comparison is reported for the setup in Sec. VII-A.Source: R,U }LRIS · p. 7
  • The robust beam design can in principle extend to 3D arrays, requiring directional, azimuth-derivative, and elevation-derivative beams.The 2D scenario uses only directional and derivative beams.Source: R,U }LRIS · p. 7

VI. MAXIMUM LIKELIHOOD JOINT LOCALIZATION AND SYNCHRONIZATION

The paper formulates joint maximum-likelihood estimation of UE position and clock offset, then reduces the initialization cost by decoupling delay and angle dependencies. The resulting initial estimates seed an iterative refinement of the joint ML objective.

A. Joint Position and Clock Offset Maximum Likelihood Estimation

  • The joint ML estimator seeks UE position and clock offset, but its nonlinear cost may have multiple local minima, making a good initialization necessary for practical iterative optimization.The stated refinement uses a numerical method such as Nelder–Mead rather than exhaustive search over the full 3D parameter space.Source: A. Joint Position and Clock Offset… · p. 8
  • The signal model is parameterized by UE position and clock offset, with channel gains retained as nuisance parameters and noise variance assumed known.The noise variance can be estimated after the remaining parameters have been estimated.Source: A. Joint Position and Clock Offset… · p. 8

1) Relaxed Maximum Likelihood Position Estimation:

  • The relaxed ML position method decouples delay and angle dependencies by treating the delay-dependent vector as unstructured, enabling a 2D position search for initialization.The received observations are stacked across transmissions, and the relaxed inner problem is separable across subcarriers.Source: 1) Relaxed Maximum Likelihood Position… · pp. 8–9

2) FFT-based Clock Offset Estimation:

The estimator derives delay and clock-offset initial values from the relaxed position estimate, then refines position and offset jointly with the ML cost. Its complexity replaces a cubic-in-grid-size search with a quadratic search plus FFT and refinement steps.

2) FFT-based Clock Offset Estimation:

  • The relaxed position estimate reconstructs complex-exponential vectors whose dominant FFT peaks estimate the two delays, from which the clock offset is obtained in closed form.The resulting position and clock-offset estimates initialize iterative optimization of the joint ML objective.Source: 2) FFT-based Clock Offset Estimation · p. 9

C. Complexity Analysis

  • The FFT step costs O(NF log NF), while Nelder–Mead refinement typically takes fewer than 30 iterations in the reported trials.The overall complexity is practically dominated by the two-dimensional relaxed position search.Source: C. Complexity Analysis · p. 10
  • The proposed estimator reduces the plain JML search’s cubic cost in grid size Q to a quadratic position search plus FFT and iterative refinement.The refinement contribution is practically negligible compared with the initial position search.Source: C. Complexity Analysis · pp. 9–10

VII. SIMULATION ANALYSIS AND RESULTS

The simulations compare the proposed beamforming and estimation approach with theoretical bounds and benchmark precoding schemes. They show that optimized beam power allocation improves localization and synchronization accuracy over uniform allocation.

A. Simulation Setup

  • The evaluation uses 1,000 Monte Carlo trials and a 3 m-by-3 m UE-position uncertainty region, with 180 transmitted OFDM symbols in the described setup.The setup uses 28 GHz carrier frequency and 100 MHz bandwidth.Source: §VII · p. 10, A. Simulation Setup · p. 10

B. Benchmark Precoding Schemes

1) Comparison Between Uniform and Proposed Beam Power Allocation:

  • The proposed power allocation yields more accurate position and clock-offset estimates than uniform allocation and brings the corresponding lower bounds closer.Uniform allocation can waste power on directions that contribute little to estimation, a concern intensified by the RIS path's tandem-channel losses.Source: 1) Comparison Between Uniform and… · p. 11

2) Comparison Between State-of-the-art and Proposed Joint BS-RIS Signal Design:

The results show improved joint localization and synchronization against the considered benchmarks, with the estimator approaching theoretical limits at relatively low SNR. Robustness and reduced-beam evaluations also reveal how position uncertainty and propagation conditions affect performance.

2) Comparison Between State-of-the-art and Proposed Joint BS-RIS Signal Design:

  • The JML estimator reaches theoretical bounds once its RML initialization is sufficiently accurate, whereas random initialization can trap it in erroneous local minima.The RML estimate provides satisfactory initial position and clock-offset estimates, with a 2D-search estimation process.Source: 2) Comparison Between State-of-the-art… · pp. 11–12
  • The robust joint precoding scheme outperforms directional and DFT codebooks, achieving position error below 10 cm and synchronization precision below 1 ns for SNR ≥ −5 dB.The comparison attributes this performance to codebooks that cover the uncertainty region and power allocation that minimizes worst-case PEB.Source: 2) Comparison Between State-of-the-art… · p. 12

3) Robustness Analysis:

  • At SNR = 0 dB, robust-design PEB stays similar across the uncertainty region, while the non-robust design has errors almost three times as large away from the nominal position.The non-robust comparison shrinks the design region to 0.1 m around the nominal UE position.Source: 3) Robustness Analysis · p. 12

4) Performance Assessment for Reduced Number of Transmitted Beams:

  • The reduced-beam heuristic drops RIS-profile variations for UE-directed BS beams while retaining them for RIS-directed beams, reducing transmissions to G = 2LBS + 2LRIS + 1.This uses the premise that UE-directed beams deliver negligible power along the BS-RIS path.Source: 4) Performance Assessment for Reduced… · pp. 12–13
  • The heuristic reduces transmissions by 85%, with slightly increased RMSEs and bounds; the JML estimator reaches bounds at 0 dB rather than at lower SNR.The reported trade-off is for the evaluated setup.Source: 4) Performance Assessment for Reduced… · p. 13

B,U }2LBS

  • The full- versus reduced-beam performance gap grows with greater position uncertainty, but shrinks when UE and RIS AoDs are widely separated and the uncertainty region is sufficiently small.The heuristic becomes less suitable when a wider set of UE-directed beams illuminates the RIS path with non-negligible power.Source: B,U }2LBS · p. 13

5) Performance Assessment in Presence of Uncontrollable Multipath:

  • With two additional uncontrollable NLoS paths, localization degrades only slightly at low SNR, and similar behavior is reported for clock-offset estimation.The comparison tests a propagation environment mismatched to the design model.Source: 5) Performance Assessment in Presence… · p. 14

VIII. CONCLUSION

  • At SNR = −10 dB, the proposed joint BS-RIS beamforming improves localization and synchronization over state-of-the-art benchmarks; the estimator attains theoretical limits around −5 dB and is resilient to uncontrolled multipath.The conclusion reports improvements on the order of meters for localization and nanoseconds for synchronization at −10 dB.Source: §VIII · p. 14

APPENDIX A PROOF OF PROPOSITION 1

The proof identifies a covariance component that does not affect the FIM and shows why an optimal solution excludes it. It also relates equivalent covariance solutions of different dimensions.

APPENDIX A PROOF OF PROPOSITION 1

  • The optimal covariance excludes f_Xg: it leaves the FIM unchanged while consuming nonnegative power.The proof uses this irrelevance and power cost to rule out that component in an optimal solution.Source: Appendix A · pp. 14–15
  • The proof rules out an alternative solution that scales the FIM upward and thereby achieves a smaller PEB.That contradicts the assumed optimality of X⋆_g.Source: Appendix A · p. 15
  • A full-rank representation Υ_g yields the same covariance as X_g through the relation in (65).The two forms are equivalent solutions of (21), despite having different dimensions.Source: Appendix A · p. 15

APPENDIX B PROOF OF PROPOSITION 2

The appendices derive the FIM entries used in the channel domain and describe their dependence on the transmit covariance and RIS phase profile.

APPENDIX B PROOF OF PROPOSITION 2

  • The Proposition 2 proof reduces dependence on Ψ_g to the 2×2 matrix B_RIS^H Ψ_g B_RIS.The proposition's claim is proved using arguments similar to Appendix A.Source: Appendix B · p. 15

APPENDIX C FIM IN THE CHANNEL DOMAIN

B. FIM Entries as a Function of Xg, Ψg and ωg

1) FIM Submatrix for the LoS Path:

3) FIM Submatrix for the LoS-NLoS Cross-Correlation:

APPENDIX D ENTRIES OF TRANSFORMATION MATRIX IN (15)

The transformation matrix organizes derivatives with respect to position and clock offset. The appendix also compares clock-bound and position-bound beam power allocation.

APPENDIX D ENTRIES OF TRANSFORMATION MATRIX IN (15)

  • The rows of T collect derivatives with respect to p_x, p_y, and Δ, while the remaining entries are zero.The columns correspond to delays, angles of departure, and RIS phase, as represented in the displayed matrix.Source: Appendix D · p. 17

CEB AS PERFORMANCE METRIC

  • PEB- and CEB-based power allocation produce practically the same PEB and CEB results.Fig. 11 compares the two allocations; the authors interpret their near-equality as evidence that positioning and synchronization are strongly coupled.Source: Figure 11 · p. 18, CEB AS PERFORMANCE METRIC · p. 18

APPENDIX F OBTAINING CODEBOOKS IN (28) FROM PROPOSITION 1

The appendices explain how the proposed codebooks implement rank-one BS/RIS beam choices and cover location uncertainty. Additional simulations examine beam-count and grid-resolution trade-offs and estimation performance under increased uncertainty.

APPENDIX F OBTAINING CODEBOOKS IN (28) FROM PROPOSITION 1

  • Rank-one codebook construction makes each transmission select one beam from the BS basis and assign it power; sweeping beam indices spans that basis.The same construction applies to rank-one RIS phase-profile matrices.Source: Appendix F · p. 18
  • For uncertain UE locations, the codebook adds BS and RIS vectors whose UE-directed AoDs cover the uncertainty interval.The BS-to-RIS AoD remains a known constant and therefore stays a single beam.Source: Appendix F · pp. 18–19

R,U }LRIS

  • The proposed codebook beams need not be orthogonal: each non-orthogonal solution has an equivalent orthogonal solution with the same precoder covariance.Source: R,U }LRIS · p. 19

APPENDIX H SETTING OF NUMBER OF GRID POINTS M IN (31)

  • For the power-allocation grid, M = 9 is the best accuracy–complexity trade-off: lower values lose accuracy, while 16 or 25 points take two or three times longer without appreciable accuracy gains.The analysis used 1,000 independent Monte Carlo trials.Source: Appendix H · p. 19

APPENDIX I ADDITIONAL SIMULATION RESULTS FOR INCREASED UE

  • With 5 m uncertainty along each direction, the proposed estimator reaches theoretical bounds at −5 dB with proposed precoding, while other schemes do so at about 0 dB; robust precoding also outperforms directional and DFT codebooks.The increased-uncertainty setup uses 378 transmitted beams.Source: Appendix I · pp. 19–20, Figure 13 · p. 20, Figure 14 · p. 20
  • Reducing transmissions from G = (2L_BS + 1)L_RIS to G = 2L_BS + 2L_RIS + 1 worsens estimation, especially at low SNR, under increased uncertainty.The comparison also shows a more evident PEB/CEB gap when RIS phase-profile configurations are not considered.Source: Appendix I · p. 20
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