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Error Bounds for Uplink and Downlink 3D Localization in 5G mmWave Systems

Zohair Abu-Shaban, Xiangyun Zhou, Thushara Abhayapala, Gonzalo Seco-Granados, Henk Wymeersch

arXiv:1704.03234v2cs.IT

TL;DR

The paper asks how accurately 5G mmWave systems can localize a UE in 3D, including orientation, and whether uplink and downlink have different limits. It derives position and orientation error bounds for multipath channels and finds distinct uplink–downlink scaling and orientation behavior, with sub-meter and sub-degree accuracy feasible in the considered scenarios.

  • Problem

    The paper addresses the need to understand 3D position and orientation localization limits for uplink and downlink mmWave multipath channels, whose differences had not been investigated.

  • Method

    The paper derives and analyzes position and orientation error bounds using exact and approximate approaches, including arbitrary array geometries and multi-beam directional beamforming.

  • Results

    Uplink and downlink have different antenna-number scaling factors; uplink depends on UE orientation, whereas downlink does not, and NLOS paths generally improve localization when LOS exists.

  • Takeaways & Limitations

    For BS–UE separations up to 50 m, the considered mmWave systems can achieve PEB < 1 meters and OEB < 1° under mmWave assumptions.

Abstract

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Location-aware communication systems are expected to play a pivotal part in the next generation of mobile communication networks. Therefore, there is a need to understand the localization limits in these networks, particularly, using millimeter-wave technology (mmWave). Towards that, we address the uplink and downlink localization limits in terms of 3D position and orientation error bounds for mmWave multipath channels. We also carry out a detailed analysis of the dependence of the bounds of different systems parameters. Our key findings indicate that the uplink and downlink behave differently in two distinct ways. First of all, the error bounds have different scaling factors with respect to the number of antennas in the uplink and downlink. Secondly, uplink localization is sensitive to the orientation angle of the user equipment (UE), whereas downlink is not. Moreover, in the considered outdoor scenarios, the non-line-of-sight paths generally improve localization when a line-of-sight path exists. Finally, our numerical results show that mmWave systems are capable of localizing a UE with sub-meter position error, and sub-degree orientation error.

I. INTRODUCTION

The paper studies 3D uplink and downlink localization limits in multipath 5G mmWave systems, focusing on position and orientation error bounds. It addresses the previously uninvestigated uplink–downlink difference and derives theoretical bounds for arbitrary arrays using directional beamforming.

  • Approach: The paper derives and analyzes position error bounds and orientation error bounds for uplink and downlink 3D mmWave multipath localization.The analysis uses multi-beam directional beamforming and arbitrary array geometry.
  • Key asymmetry: The channel-parameter Fisher information matrix is transformed differently for uplink and downlink, producing different position and orientation error bounds.Although the channel-parameter matrices are structured similarly, the location-parameter matrices are not.
  • Analytical contributions: Under relevant mmWave conditions, multipath parameter estimation can be approximately reduced to multiple single-path estimation problems.The conditions involve channel sparsity, many antennas, and very large bandwidth.
  • Analytical contributions: The paper derives closed-form single-path channel-parameter CRLBs and closed-form 3D and 2D LOS position and orientation error bounds.These results apply to arbitrary array geometries and relate channel-parameter bounds to position and orientation bounds.
  • Evaluation: Analytical scaling results and URA simulations demonstrate asymmetry between uplink and downlink localization.The study evaluates general uplink and downlink bounds using exact and approximate approaches.

B. Channel Model

The channel model represents sparse multipath propagation between arbitrary transmit and receive arrays with directional beamforming and pilot signals. The localization analysis derives channel-parameter bounds and transforms them into position-domain bounds.

  • B. Channel Model: Each path is described by directions of departure, directions of arrival, time of arrival, and a complex path gain.The array responses use arbitrary known element geometries and spherical-coordinate angles.
  • C. Transmission Model: The transmitter uses a directional beamforming matrix with NB beams and pilot symbols transmitted over those beams.The observation duration is approximately To ≈ NsTs, and transmitted power is kept fixed with NT and NB.
  • C. Transmission Model: The received signal contains spatially white Gaussian noise with power spectral density N0 at the receive-beamformer input.Low-noise amplifiers and passband filters are assumed at each receive antenna.
  • D. 3D Localization Problem: The analysis first derives bounds for channel parameters, then transforms them into position-domain bounds for both uplink and downlink.The target channel parameters are DOA, DOD, TOA, and path gains.
  • D. 3D Localization Problem: Exact Fisher information matrix expressions are followed by path-orthogonality conditions and closed-form single-path CRLBs for 3D and 2D localization.The derivations cover arbitrary arrays.

A. Exact Expression

The paper constructs the channel-parameter FIM from receiver, transmitter, and signal factors, then simplifies its structure under typical mmWave conditions. The resulting approximation exploits large arrays, bandwidth, and directional beamforming behavior.

  • Factorized FIM: The FIM is assembled from receiver, transmitter, and signal factors under an i.i.d.-symbol assumption.The receiver and transmitter factors capture array and beamforming effects, while the signal factor captures pilot-signal path correlations.
  • Factorized FIM: The signal factor represents frequency-domain correlation between different multipath components.Path-delay differences and signal bandwidth determine this correlation through Parseval’s theorem.
  • FIM scaling: The FIMs scale linearly with SNR, so channel-parameter CRLBs decrease as SNR increases.This relation follows from the FIM structure under additive white Gaussian noise.
  • Approximation: The approximation reorganizes FIM submatrices into diagonal or nearly diagonal forms using the almost-diagonal matrix structure.Almost-diagonal matrices are defined as a nonzero diagonal component plus a nonzero hollow component.
  • Transmit-side behavior: As the number of transmit antennas increases, directional beams become narrower and can reduce cross-correlation between distinct departure directions.For extremely narrow beams, coverage of another departure direction can approach zero, potentially making the corresponding FIM contribution negligible.

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Under typical mmWave conditions, large arrays and bandwidth make multipath components approximately orthogonal, yielding a block-diagonal FIM after path-wise reordering. The resulting channel bounds separate several parameter-estimation effects.

  • Path separation: Multipath components become approximately orthogonal when receive-array size or bandwidth is sufficiently large.Large mmWave arrays make this assumption reasonable, allowing paths to be treated separately after reordering.
  • Path separation: The reordered approximate FIM becomes block diagonal, with each block associated with an individual path.Figure 2 illustrates the nonzero entries associated with separate paths.
  • Single-path structure: The time-of-arrival parameter is estimated independently of the other parameters for each path.This follows from the corresponding zero delay-coupling terms.
  • Single-path structure: Receiver-side angles are independent of other parameters, whereas transmitter-side angles depend on the channel gain under transmit-only beamforming.The dependence arises because power gain contains both channel gain and antenna directional gain.
  • CRLB interpretation: The CRLB combines an SNR component with spatial information after accounting for nuisance-parameter uncertainty.The equivalent FIM removes dependence on other unknown parameters, including the channel gain.
  • CRLB interpretation: The paper provides closed-form CRLBs for DOA, DOD, and TOA for arbitrary array geometries.These bounds are obtained from the channel-parameter FIM.

IV. FISHER INFORMATION OF THE LOCATION PARAMETERS

The localization analysis transforms multipath channel-parameter information into position and orientation information for uplink and downlink systems. UE angles are obtained through coordinate translation and rotation, with the transformation differing by transmission direction.

  • FIM transformation: The PEB and OEB are derived by transforming exact and approximate channel-parameter FIMs into location-parameter FIMs.The transformation maps channel parameters to position and orientation, followed by a Schur-complement reduction.
  • FIM transformation: The position and orientation EFIM is obtained from the transformed FIM after eliminating nuisance location parameters.The paper then defines squared position and orientation error bounds from this EFIM.
  • LOS geometry: For the LOS path, BS elevation and azimuth angles are obtained from the UE position using spherical-coordinate relations, while delay equals distance divided by c.The relations use the position components and its Euclidean norm.
  • Angle transformation: UE angles are derived by shifting the coordinate origin to the UE and rotating the system opposite to the UE orientation.The BS location in the transformed coordinates provides the UE angles through spherical coordinates.
  • Angle transformation: In 3D, the rotation consists of a rotation by φ0 around the z-axis followed by a rotation by θ0 around the negative x′-axis.The resulting rotation matrix is orthogonal.

B. PEB and OEB: Approximate Approach

The approximate approach aggregates path-wise localization information while accounting for uncertainty in channel gains and NLOS cluster locations. It yields different uplink/downlink position behavior and distinct antenna-scaling results.

  • Approximate EFIM: The approximate position-and-orientation EFIM sums transformed information from the individual multipath components.Each path contributes through its DOA, DOD, and TOA CRLBs.
  • Approximate EFIM: The useful TOA, DOA, and DOD information from multiple paths accumulates positively in the approximate EFIM.Unknown channel gains and NLOS cluster locations contribute negative information terms because they are nuisance quantities.
  • Approximate EFIM: Unknown cluster locations introduce an information loss term for NLOS paths beginning with m = 2 when m = 1 is LOS.The corresponding negative term represents uncertainty in the NLOS cluster locations.
  • LOS closed form: For the LOS-only 3D case, position error depends on BS angles, whereas orientation error depends on both UE and BS angles with different uplink/downlink weights.Consequently, the squared orientation error is generally asymmetric between uplink and downlink.
  • LOS closed form: Position error is asymmetric between uplink and downlink because uplink uses DOA information and downlink uses DOD information.The two channel-parameter CRLBs have different expressions.
  • Scaling laws: For URAs, CRLB(φR) scales with 1/N_R^2, whereas for ULAs it scales with 1/N_R^3.The scaling factors combine SNR improvement with spatial resolution and differ between array geometries.
  • Scaling laws: DOA and DOD have different antenna-number scaling laws, so SPEB scales differently in uplink and downlink; SOEB’s scaling factor is unchanged.The coefficients b1 through b6 do not depend on the number of antennas.

V. NUMERICAL RESULTS AND DISCUSSION

The numerical study evaluates 3D mmWave localization in uplink and downlink across modeled outdoor sectors, channel scenarios, antenna configurations, and path separations. It derives error bounds using exact and approximate Fisher information analyses and examines how system parameters affect localization.

  • Propagation scenarios: The modeled configuration produces a maximum of M = 6 contributing paths at any location in the studied sector.Clusters contribute only when their power exceeds 10% of the LOS power.
  • Propagation scenarios: Five propagation scenarios are evaluated: LOS, LOS+R, LOS+S, LOS+C, and NLOS.The scenarios distinguish free-space propagation, reflected paths, scattered paths, combined NLOS paths, and blocked LOS conditions.
  • Uplink–downlink comparison: The study compares uplink and downlink localization through UE orientation, transmit-antenna count, and receive-antenna count under a symmetric array setup.Equal UE and BS array sizes are used for a fair comparison, although larger BS arrays could improve the bounds subject to beam count.
  • Approximate FIM behavior: With a 10° path separation, the receiver factor falls below −20 dB at NR = 100 and the transmitter factor at NT = 16.At higher separation angles, both factors fall below −20 dB with fewer antennas, making the approximate FIM nearly diagonal when NR ≥100 or NT ≥16.

C. Downlink PEB and OEB

Downlink position and orientation error bounds are evaluated across UE locations and propagation scenarios using exact and approximate Fisher information approaches. The results show benefits from suitable NLOS paths and a beam-count trade-off driven by coverage, complexity, and antenna-dependent beamwidth.

  • NLOS contributions: Scatterers and reflectors generally improve localization relative to LOS-only propagation, but their effects differ across locations.Scatterers provide smaller PEB improvements over many locations, reflectors provide modest improvements over fewer locations, and combined paths show both effects.
  • FIM approximation and singularities: The approximate FIM closely follows the exact PEB and OEB, while producing slightly lower bounds because it assumes independent paths.Spatially localized singularities can occur when a scatterer blocks LOS and violates the unique-parameters assumption.
  • Error bounds across scenarios: At 90% CDF, downlink PEB is 0.23 m for LOS, 0.21 m for LOS+R, 0.19 m for LOS+S, and 0.18 m for LOS+C.These values show lower position error bounds when reflected and/or scattered paths accompany LOS.
  • Error bounds across scenarios: NLOS-only localization is unreliable, reaching 0.5 m PEB at 13% CDF and an irrelevant value at 90% CDF for mmWave localization.The reported degradation applies when the LOS path is blocked and only scattered and reflected paths remain.
  • Beam-count selection: Increasing the beam count first lowers bounds through better coverage, then reaches a floor where additional beams add complexity with negligible improvement.With fixed total transmitted power, increasing beam count reduces power per beam; overlapping beams eventually leave impinging power approximately constant.

1) UE Orientation impact on PEB and OEB:

Uplink and downlink localization respond differently to antenna counts and UE orientation. Multipath generally improves localization when LOS exists, while mmWave can achieve sub-meter position and sub-degree orientation errors.

  • UE orientation and PEB: Downlink PEB is independent of UE orientation, whereas uplink PEB depends strongly on orientation because fixed UE beams may miss the base station.A 10° orientation causes more frequent beam misalignment and degrades uplink PEB.
  • Effect of NR and NT: Very large receive-antenna counts favor uplink PEB, which scales as 1/NR, compared with downlink PEB scaling as 1/√NR.The different exponents cause the uplink and downlink curves to cross, so the better scheme depends on NR.
  • Effect of NR and NT: Uplink position estimation is limited more by angle estimation than by range estimation because CRLB(DOA) dominates CRLB(TOA).This explains the faster uplink PEB decay with NR.
  • Effect of NR and NT: For relatively large NR, OEB scales as 1/√NR, while for small NR it scales as 1/NR in both uplink and downlink.OEB therefore has a different receive-antenna scaling behavior from PEB.
  • Effect of NR and NT: Both PEB and OEB scale non-linearly with NT: increasing NT first helps, but overly narrow beams worsen bounds, especially in uplink.Small NT gives poor spatial resolution and SNR; larger NT increases SNR but eventually narrows beams excessively.
  • Multipath and achievable accuracy: NLOS clusters improve localization when a LOS path exists, with scatterers helping more locations and reflectors providing modest PEB improvement in some locations.Under the approximate mmWave model, resolvable multipath components are treated as orthogonal and their FIMs accumulate.
  • Multipath and achievable accuracy: PEB < 1 meters and OEB < 1° are feasible for BS-UE separations up to 50 m in the considered outdoor mmWave scenarios.The study analyzes 3D position and orientation error bounds for uplink and downlink multipath localization.

APPENDIX A

Appendix A develops the channel-parameter Fisher information structure and its component submatrices for the localization analysis.

  • APPENDIX A: The appendix derives Fisher information submatrices using transmit-side and receive-side channel quantities and Hadamard-product notation.It defines the signal-to-noise-related quantity γ ≜ NRNTNsEs/N0 and path-delay differences Δτuv = τv − τu.

APPENDIX B

Appendix B derives effective Fisher information for location and angular parameters by separating individual paths and eliminating nuisance parameters with Schur complements.

  • APPENDIX B: For angular parameters, the Fisher information matrix includes DOA, DOD, and complex path-gain components, whose CRLBs are obtained through their effective Fisher information.The derivation also exploits independence between DOA and DOD to simplify inversion.
  • APPENDIX B: The effective Fisher information for the mth path is derived first, then the overall location information is expressed as a sum over paths.The derivation explicitly focuses on combining all M paths.
  • APPENDIX B: The appendix uses Schur’s complement to eliminate nuisance parameters when forming the effective Fisher information matrix.The resulting expressions rely on the block structure of the multipath information matrix.

CLOSED-FORM PEB AND OEB FOR LOS-ONLY

The LOS-only analysis derives closed-form position and orientation error bounds for uplink and downlink. The position expression is shared across the two links, while orientation error depends on angular bounds.

  • CLOSED-FORM PEB AND OEB FOR LOS-ONLY: Closed-form LOS bounds are obtained by transforming channel-parameter information into position and orientation error bounds.The uplink channel-parameter vector includes BS and UE angles together with propagation delay.
  • CLOSED-FORM PEB AND OEB FOR LOS-ONLY: The downlink SPEB follows the uplink expression after exchanging angle columns and swapping the roles of BS and UE angles.The resulting SPEB expression is identical for uplink and downlink.
  • CLOSED-FORM PEB AND OEB FOR LOS-ONLY: The LOS orientation-error derivation uses spherical-coordinate relationships between the path direction and BS and UE angular parameters.The analysis differentiates these relationships to obtain the angular contributions to SOEB.
  • CLOSED-FORM PEB AND OEB FOR LOS-ONLY: SOEB is a weighted sum of angular bounds because the derivatives involving propagation delay are zero.The same SOEB expression applies to both uplink and downlink after the corresponding column-swapping procedure.
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