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SISO RIS-Enabled Joint 3D Downlink Localization and Synchronization

Kamran Keykhosravi, Musa Furkan Keskin, Gonzalo Seco-Granados, Henk Wymeersch

arXiv:2011.02391v2eess.SPcs.ITmath.IT

TL;DR

The paper studies 3D localization and synchronization with single-antenna BS and UE, a configuration in which RIS-aided localization had not been studied. It derives bounds and a low-complexity estimator, and reports submeter positioning for r < 30 with estimates close to theoretical bounds.

  • Problem

    RIS-aided localization had not been studied in the elemental configuration of single-antenna BS and UE, motivating 3D position and clock-bias estimation with a single RIS.

    Source: §I · p. 1

  • Method

    The paper analyzes Fisher information and CRBs and proposes a low-complexity estimator that replaces a four-dimensional problem with two 1D and one 2D problems.

    Source: §I · pp. 1–2

  • Results

    For r < 30, positioning with submeter accuracy is possible, and the proposed estimator operates close to the theoretical bounds.

    Source: §V · p. 5

  • Takeaways & Limitations

    The results indicate that a single RIS can enable 3D localization and synchronization in a system with single-antenna BS and UE, given multiple distinct RIS phase profiles.

    Source: §I · p. 1, §VI · p. 5

    The study leaves multi-user scenarios and the effects of a multi-antenna BS for future research.

    Source: §VI · p. 5

Abstract

from arXiv · show

We consider the problem of joint three-dimensional localization and synchronization for a single-input single-output (SISO) system in the presence of a reconfigurable intelligent surface (RIS), equipped with a uniform planar array. First, we derive the Cram\'er-Rao bounds (CRBs) on the estimation error of the channel parameters, namely, the angle-of-departure (AOD), composed of azimuth and elevation, from RIS to the user equipment (UE) and times-of-arrival (TOAs) for the path from the base station (BS) to UE and BS-RIS-UE reflection. In order to avoid high-dimensional search over the parameter space, we devise a low-complexity estimation algorithm that performs two 1D searches over the TOAs and one 2D search over the AODs. Simulation results demonstrate that the considered RIS-aided wireless system can provide submeter-level positioning and synchronization accuracy, materializing the positioning capability of Beyond 5G networks even with single-antenna BS and UE. Furthermore, the proposed estimator is shown to attain the CRB at a wide interval of distances between UE and RIS. Finally, we also investigate the scaling of the position error bound with the number of RIS elements.

I. INTRODUCTION

The introduction motivates RIS-aided localization as a route to 3D positioning and synchronization with a single-antenna BS and UE, then presents the paper’s system and contributions. The setup assumes known BS/RIS geometry and orientation, but an unknown UE position and clock bias.

I. INTRODUCTION

  • Prior RIS-localization work includes 2D and 3D bounds and positioning methods, but [15] presented no low-complexity estimation algorithm.The introduction notes that [15] considered 3D uplink localization in synchronous and asynchronous cases.Source: §I · p. 1
  • The paper studies 3D downlink localization and synchronization with single-antenna BS and UE, using the direct LOS signal and a RIS-reflected signal.The RIS has a uniform planar array.Source: §I · p. 1
  • A single RIS can enable estimation of UE position and clock bias in this configuration when multiple distinct RIS phase profiles are used.The authors state that localization and synchronization would not be possible in this scenario without the RIS.Source: §I · p. 1
  • The paper calculates Fisher information and CRBs for position and channel parameters, identifying RIS-to-UE AOD estimation as enabling localization.Source: §I · p. 1
  • The proposed low-complexity method transforms a four-dimensional problem into two 1D searches and one 2D search, and is reported to attain the CRB.Source: §I · pp. 1–2

Notation

  • The notation defines vectors and matrices in bold, uses ⊗ for the Kronecker product, and distinguishes complex conjugate, Hermitian, and transpose operators.Source: Notation · p. 2

A. System Setup

  • The system setup treats BS and RIS positions and RIS orientation as known, while UE position and asynchronous clock bias are unknown.Source: A. System Setup · p. 2

B. Signal Transmission

The signal model represents the received transmission as direct and RIS-reflected paths with delays, gains, and RIS response determined by the path angles. It assumes OFDM pilots, a nearly constant wavelength across the bandwidth, and an RIS uniform planar array.

B. Signal Transmission

  • The transmission uses T OFDM symbols with N subcarriers and equal pilot symbols of amplitude √Es; the received observations are arranged by OFDM symbol.Source: B. Signal Transmission · p. 2
  • The two paths have delays τb and τr that include propagation distance and UE clock offset; their complex gains are gb and gr.Source: B. Signal Transmission · p. 2
  • The model assumes subcarrier spacing Δf and zero-mean circularly symmetric i.i.d. Gaussian noise with variance σ2.Source: B. Signal Transmission · p. 2
  • The RIS-to-UE departure angle φ is represented by azimuth and elevation, while θ denotes the known BS-to-RIS arrival angles.Source: B. Signal Transmission · p. 2
  • The RIS response for an angle is formed as the Kronecker product of row and column response vectors.The response vectors use the RIS element spacing d and components of the angle-dependent wavenumber vector.Source: B. Signal Transmission · p. 2
  • The wavenumber vector represents azimuth and elevation as a wavelength-scaled direction vector, with wavelength assumed relatively constant across the transmission bandwidth.Source: B. Signal Transmission · p. 2

C. Goal

The goal is to estimate UE position and clock bias from the received observations, given the transmitted signals, RIS phase profiles, and known BS/RIS geometry. The paper derives position bounds from channel-parameter Fisher information and a Jacobian mapping between channel and positional parameters.

C. Goal

  • The estimator targets UE position and clock bias from received observations, given RIS phase profiles, transmitted signals, and known BS/RIS locations and orientation.Source: C. Goal · p. 2

A. FIM Derivation

  • The position error bound (PEB) is a lower bound on the position RMSE of any unbiased estimator based on the observations.Source: A. FIM Derivation · p. 3
  • The positional Fisher information matrix is obtained from the channel-parameter FIM through the Jacobian transformation Fpo = J⊤FchJ.The channel parameter vector contains two delays, two RIS-to-UE AODs, and real and imaginary parts of both path gains.Source: A. FIM Derivation · p. 3
  • The channel FIM is derived from derivatives of the noiseless received signal, and the paper derives the channel FIM and Jacobian in appendices.Source: A. FIM Derivation · p. 3
  • The paper also calculates CRB bounds for clock bias and channel parameters.Source: A. FIM Derivation · p. 3

B. FIM Interpretation

The FIM analysis establishes identifiability conditions and a geometric mapping from channel parameters to position and clock bias. The estimator separates channel-parameter estimation into stages, including an IFFT-based estimate of the LOS delay.

B. FIM Interpretation

  • Identifiability requires TN ≥ 4, N > 1 for delay estimation, T > 1 for AOD estimation, and nonparallel RIS phase profiles across transmissions.The FIM is a sum of rank-2 matrices over transmissions and subcarriers.Source: B. FIM Interpretation · p. 3
  • The two delays and two AODs determine the four positional parameters, with an invertible Jacobian linking the channel-geometric and positional parameters.The position is described geometrically as the intersection of the AOD line and one half of a two-sheet hyperboloid.Source: B. FIM Interpretation · p. 3

IV. LOW-COMPLEXITY ESTIMATION

  • The estimator first estimates τb and τr, then estimates φ, refining each step with a quasi-Newton algorithm.This staged process estimates channel parameters from the received signal before estimating UE position and clock bias.Source: §IV · p. 3

A. Estimation of τb

  • Summing the received-signal columns forms yc, coherently combining the T LOS transmissions but not the RIS-reflected signal.The method assumes T times the BS received power is much greater than the RIS received power, particularly when both BS and UE are in the RIS far field.Source: A. Estimation of τb · p. 3
  • The LOS delay estimate uses an IFFT peak, then adds an artificial delay δ and refines the estimate when the peak condition does not hold.A quasi-Newton search starts at δ = 0 and constrains δ to [0, 1/(NF ∆f)].Source: A. Estimation of τb · p. 3

B. Estimation of τr

The reflected-path delay procedure first removes the BS contribution from the received signal, then applies an IFFT-based estimate and refinement.

B. Estimation of τr

  • The method removes the BS contribution from Y before estimating the reflected-path delay.The resulting signal is denoted Yr.Source: B. Estimation of τr · p. 4
  • The reflected-path delay is estimated with an optimization analogous to the LOS-delay procedure, using an IFFT of Yr.The IFFT operation extracts a row from the transformed Yr.Source: B. Estimation of τr · p. 4
  • The refined reflected-path delay is calculated as τ̃r = k̃/(NF ∆f) − δ̃.Source: B. Estimation of τr · p. 4

C. Estimation of φ

The AOD estimator removes reflected-path delay effects, uses a two-dimensional IFFT to obtain an initial direction estimate, and refines the result.

C. Estimation of φ

  • The estimator first removes the reflected-path delay effects from Yr before estimating φ.Source: C. Estimation of φ · p. 4
  • After summing the rows of Yφ, the method uses the Kronecker product and vec operator to form a 2D-IFFT search over φ.The matrices used in the transform can be calculated offline because they do not depend on the received signal.Source: C. Estimation of φ · p. 4
  • The method selects an initial grid estimate from wℓ,m and uses quadratic interpolation to obtain continuous values for ℓ̃ and m̃.It then estimates φ from those values and refines the estimate with a quasi-Newton algorithm.Source: C. Estimation of φ · p. 4
  • For a given ηℓ,m, h(ηℓ,m) is the scalar that minimizes the right-hand side of the stated expression.Source: C. Estimation of φ · p. 4

D. Estimation of UE position and clock bias

The method converts the estimated AOD into a UE position estimate and then estimates clock bias. It also contrasts the staged channel-parameter search with a four-dimensional search and presents errors alongside CRB bounds.

D. Estimation of UE position and clock bias

  • The UE position estimate is calculated from the RIS position, the estimated AOD, and the numerically found κ̃; the method then estimates clock bias.The position expression is p̃ = pr + κ̃R⊤k(φ̃), and κ̃ is found by gradient descent.Source: D. Estimation of UE position and clock… · p. 4
  • Fig. 2 presents estimation errors and CRB bounds for position, clock bias, both AOD angles, and the LOS and reflected-path delays.Source: Figure 2 · p. 5
  • The channel parameters could be estimated with a four-dimensional search, but the proposed method uses two one-dimensional searches and one two-dimensional search.Source: D. Estimation of UE position and clock… · p. 5

V. SIMULATION RESULTS

The simulations show submeter positioning for r < 30 m, with the estimator operating near the theoretical bounds. Across the considered RIS sizes, the PEB decreases approximately with the square root of the number of RIS elements.

V. SIMULATION RESULTS

  • Submeter positioning is possible for r < 30 m, and the proposed estimator operates close to the theoretical bounds.The simulations average over 1,000 noise realizations; the UE is placed 10 meters below the BS and RIS.Source: §V · p. 5
  • For r > 10 m, estimation performance deteriorates as SNR decreases; at low r, the UE is nearly beneath the RIS and angle estimation is relatively poor.The clock-bias estimation error is close to the PEB bound because the error in τb is much less than the position error.Source: §V · p. 5
  • The PEB decreases in proportion to the square root of the number of RIS elements across the considered locations.The reflected-path SNR increases with the number of elements, while random phase shifts provide no significant beamforming gain.Source: §V · p. 5, Figure 3 · p. 5

VI. CONCLUSION

The paper derives CRBs and a low-complexity estimator for localization and synchronization with single-antenna BS and UE and a RIS. The estimator is efficient, and 3D localization and synchronization are possible in this considered system.

VI. CONCLUSION

  • CRBs and a low-complexity estimator are developed to estimate RIS AOD and direct and reflected signal delays, then infer UE position and clock bias.The estimator's errors are compared with the CRBs.Source: §VI · p. 5
  • The estimator is efficient, and 3D localization and synchronization are possible in the considered single-antenna BS, single-antenna UE, and RIS system.The results point to RIS potential for radio localization in simple wireless networks.Source: §VI · p. 5
  • Future work includes multi-user scenarios and studying multi-antenna BSs, different RIS phase profiles, and different pilot energy levels.Source: §VI · p. 5

APPENDIX A CALCULATING F ch

The appendices describe derivative calculations for the channel Fisher information and the Jacobian that maps positional parameters to channel parameters.

APPENDIX A CALCULATING F ch

  • The appendix derives ∂[E]n,t/∂[ζch]r and gives the final derivatives for gains.Source: Appendix A · p. 6

APPENDIX B CALCULATING J

  • To calculate J, the appendix expresses each channel parameter as a function of the positional parameters and then differentiates.Source: Appendix B · p. 6
  • The appendix defines ∂f/∂p as the gradient with respect to p; the remaining derivatives are zero.Source: Appendix B · p. 6
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