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Position and Orientation Estimation through Millimeter Wave MIMO in 5G Systems

Arash Shahmansoori, Gabriel E. Garcia, Giuseppe Destino, Gonzalo Seco-Granados, Henk Wymeersch

arXiv:1702.01605v2cs.IT

TL;DR

Accurate positioning from a single millimeter-wave transmitter remains underexplored, so the paper derives performance bounds and proposes an estimation algorithm that supports localization even when line of sight is blocked.

  • Problem

    The potential for accurate receiver localization from a single millimeter-wave transmitter, including blocked-line-of-sight conditions, remains insufficiently established.

  • Method

    The paper derives fundamental performance bounds and develops a two-stage estimation approach for receiver position and rotation angle.

  • Results

    Accurate localization is possible even when the line of sight is blocked.

  • Takeaways & Limitations

    A single transmitter can support receiver localization in obstructed propagation conditions.

Abstract

from arXiv · show

Millimeter wave signals and large antenna arrays are considered enabling technologies for future 5G networks. While their benefits for achieving high-data rate communications are well-known, their potential advantages for accurate positioning are largely undiscovered. We derive the Cramér-Rao bound (CRB) on position and rotation angle estimation uncertainty from millimeter wave signals from a single transmitter, in the presence of scatterers. We also present a novel two-stage algorithm for position and rotation angle estimation that attains the CRB for average to high signal-to-noise ratio. The algorithm is based on multiple measurement vectors matching pursuit for coarse estimation, followed by a refinement stage based on the space-alternating generalized expectation maximization algorithm. We find that accurate position and rotation angle estimation is possible using signals from a single transmitter, in either line-of- sight, non-line-of-sight, or obstructed-line-of-sight conditions.

I. INTRODUCTION

The paper establishes mm-wave and large-MIMO positioning and orientation estimation with one base station, including when the line-of-sight path is blocked. It derives accuracy bounds and proposes an estimator that attains them at average-to-high SNR.

  • Motivation: Mm-wave’s severe path loss necessitates directional beamforming, creating a close relationship between channel estimation, localization, and communication.Mm-wave channels are sparse in the angular domain because LOS and single-bounce reflections dominate received power.
  • Research gap: Prior compressed-sensing studies primarily used narrow-band channel models, motivating wide-band modeling that accounts for path delays.The limitation arises because larger bandwidths require delays of different propagation paths to be included.
  • Contributions: The paper demonstrates accurate position and device-orientation estimation using mm-wave and large MIMO with only one BS, even when LOS is blocked.The analysis covers LOS, NLOS, and obstructed-line-of-sight conditions.
  • Contributions: The derived fundamental bounds show that NLOS links provide information for estimating the mobile station’s location and orientation.The bounds address position and orientation estimation accuracy under LOS, NLOS, and OLOS propagation conditions.
  • Contributions: A novel three-stage position-and-orientation estimator attains the derived bounds at average-to-high SNR by combining sparse angular estimation, per-path delay estimation, and refinement.DCS-SOMP exploits frequency-stable angular sparsity, while refinement addresses its predefined AOA/AOD grid.

II. SYSTEM MODEL · A. Transmitter Model

The system models a hybrid-precoded OFDM MIMO link between a multi-antenna base station and a single mobile station, with the base-station location known while the mobile station’s position and rotation are estimated. The transmitter supports general beamformers and signals under a normalized power constraint, using fewer beams than antenna elements in the mm-wave setting.

  • II. SYSTEM MODEL: The MIMO system comprises a base station with Nt antennas and a mobile station with Nr antennas, operating at carrier frequency fc and bandwidth B.The carrier wavelength is denoted λc.
  • II. SYSTEM MODEL: The base-station location q is known, whereas the mobile-station position p and array rotation angle α are unknown parameters.The locations are represented in R2, with α ∈ [0, 2π).
  • A. Transmitter Model: The transmitter uses OFDM, with a hybrid analog/digital precoder at the base station communicating with a single mobile station.The transmission model follows the OFDM formulation in [37].
  • A. Transmitter Model: G sequential transmissions are sent, and each g-th transmission carries Mt simultaneously transmitted symbols on every subcarrier n = 0, . . . , N − 1.The symbol vector is x(g)[n] = [x1[n], . . . , xMt[n]]T ∈ CMt.
  • A. Transmitter Model: Symbols are precoded, converted to the time domain with an N-point IFFT, given a cyclic prefix, and then RF-precoded.The cyclic-prefix duration is TCP = DTs, Ts = 1/B, and TCP exceeds the channel delay spread.
  • A. Transmitter Model: The transmitted signal is F(g)[n]x(g)[n], where the beamforming matrix factors as F[n] = FRFFBB[n].FRF uses analog phase shifters with entries ejφm,n, while FBB[n] is the digital beamformer.
  • A. Transmitter Model: The analog and digital beamformers satisfy the total-power constraint ∥FRFFBB[n]∥F = 1, while mm-wave channel sparsity typically permits Mt ≪ Nt.The model also extends to multi-user mm-wave downlink systems with limited feedback from mobile stations.
  • A. Transmitter Model: The analysis provides general expressions for studying different choices of beamformers and signals, without assuming a specific beamformer.The approach is compatible with beam reference signal procedures and can be complemented by Bayesian recursive tracking with user-specific precoding; optimization is outside the paper’s scope.

B. Channel Model · C. Received Signal Model · III. POSITION AND ORIENTATION ESTIMATION: FUNDAMENTAL BOUNDS

The paper models wideband millimeter-wave MIMO channels and received signals across LOS and scattered paths, then derives fundamental bounds for estimating the mobile station’s position and orientation. The estimation uses frequency-domain observations under Gaussian noise, with the analysis considering LOS, NLOS, and OLOS conditions.

  • B. Channel Model: The channel describes each path using its angle of arrival, angle of departure, and path length derived from time of arrival.The path index is k = 0 for LOS and k > 0 for NLOS paths.
  • B. Channel Model: Each NLOS path includes a scatterer with unknown location, whose two segment distances determine the path geometry.The segment distances are defined from the transmitter to the scatterer and from the scatterer to the receiver.
  • B. Channel Model: The wideband channel model supports fractional bandwidth B/fc up to 50% and remains constant over G transmitted symbols.It uses frequency-dependent array responses and K+1 propagation paths.
  • B. Channel Model: For a ULA, the array response depends on the subcarrier wavelength λn, with antenna spacing set to d = λc/2.When B ≪fc, the model reduces to the standard narrow-band formulation.
  • C. Received Signal Model: After cyclic-prefix removal and FFT, the received signal is represented for each subcarrier n and transmission g with additive Gaussian noise.The noise has zero mean and variance N0/2 per real dimension.
  • C. Received Signal Model: The estimation objective is to recover the mobile station’s position p and orientation α from the received observations {y(g)[n]}∀n,g.The paper first derives a fundamental lower bound and then proposes a practical estimation method.
  • III. POSITION AND ORIENTATION ESTIMATION: FUNDAMENTAL BOUNDS: The fundamental-bounds analysis derives the Fisher information matrix and Cramér-Rao bound for position and orientation estimation under LOS, NLOS, and OLOS conditions.For notational simplicity, the derivation assumes G = 1, meaning one OFDM symbol is transmitted.

A. FIM Derivation for Channel Parameters · B. FIM for Position and Orientation

The paper derives a Fisher information matrix (FIM) for unknown channel parameters, then transforms it into a FIM for position and orientation using geometry-based parameter relationships.

  • A. FIM Derivation for Channel Parameters: The unknown channel parameter vector η_k comprises delay, angle of departure, angle of arrival, and channel coefficients.The channel coefficients are represented through their real and imaginary parts.
  • A. FIM Derivation for Channel Parameters: The mean squared error of an unbiased estimator is bounded using the expectation parameterized by η and the channel-parameter FIM J_η.The likelihood function f(y|η) describes the random vector y conditioned on the unknown parameters.
  • A. FIM Derivation for Channel Parameters: The likelihood-based FIM uses the mean received signal μ[n] ≜ H[n]F[n]x[n], with proportionality suppressing irrelevant constants.The FIM is organized into blocks Ψ(η_r, η_s), whose entries are derived in Appendix A.
  • B. FIM for Position and Orientation: The position-space FIM is obtained by transforming variables from η to a parameter vector containing position, orientation, and channel-related quantities.The blocked-LOS case is treated through a separate parameterization.
  • B. FIM for Position and Orientation: The transformed FIM uses a (4K + 5) × 5(K + 1) transformation matrix T.The matrix entries follow from geometric relationships between the original and transformed parameters.
  • B. FIM for Position and Orientation: T is structured in blocks T_k,k′, with separate expressions for k′ ≠ 0 and k′ = 0 that capture parameter dependencies on position and orientation.The listed derivatives include delay and angle derivatives with respect to position, orientation, and channel parameters.
  • B. FIM for Position and Orientation: The remaining transformation-matrix entries are zero, while the channel-coefficient derivative satisfies ∂˜h_k/∂˜h_k = I_2 for k ≥ 0.This completes the specified structure of T.

C. Bounds on Position and Orientation Estimation Error … A. Beamspace Channel Representation

The paper derives position and rotation error bounds, analyzes how multipath affects localization information, and proposes a sparse beamspace estimator that combines coarse, refined, and geometric estimation. Beamspace sparsity enables compressed-sensing-based channel parameter recovery while reducing complexity.

  • C. Bounds on Position and Orientation Estimation Error: The position error bound is obtained from the inverse Fisher information matrix by taking the square root of the sum of its first 2 × 2 diagonal entries.The rotation error bound is similarly extracted from the third diagonal entry.
  • D. The Effect of Multi-Path Components on Position and Orientation Estimation Error: Large Nt, Nr, and bandwidth make multipath components more separable and reduce their cross-correlation terms in the Fisher information matrix.Increasing antennas narrows beams, while increasing bandwidth helps resolve paths from different scatterers.
  • D. The Effect of Multi-Path Components on Position and Orientation Estimation Error: Multipath generally improves mobile-station localization because its additive information contributions increase the equivalent Fisher information and reduce the Cramér–Rao bound.The exact and approximate Fisher information matrices produce nearly identical position error bounds under the stated conditions.
  • D. The Effect of Multi-Path Components on Position and Orientation Estimation Error: Multipath degrades localization only when components heavily overlap, especially with the line of sight, in directional and time domains.In those cases, negative terms dominate the information contribution.
  • IV. POSITION AND ORIENTATION ESTIMATION: ESTIMATOR IN BEAMSPACE: The proposed beamspace channel transformation reduces estimation complexity by exploiting the sparsity of millimeter-wave MIMO channels.The transformation is introduced to estimate the channel parameters in (6).
  • A. Beamspace Channel Representation: Beamspace channel coefficients are approximately sparse, with strong components only at the paths’ transmit and receive directions, enabling compressed-sensing recovery.Nonzero beamspace entries provide coarse AOA/AOD estimates, while their values support delay estimation.
  • A. Beamspace Channel Representation: Because beamspace supports are approximately jointly (K +1)-sparse across subcarriers, DCS-SOMP can estimate channel vectors jointly and efficiently.The approach then applies SAGE for fine estimation and estimates position and orientation from the refined parameters.
  • A. Beamspace Channel Representation: A direct sparse representation over AOA/AOD/TOA triplets would have length Nt × Nr × N and significantly higher complexity because N is generally large.This motivates the selected beamspace representation.

B. Step 1: Coarse Estimation of Channel Parameters using DCS-SOMP

The coarse-estimation stage uses modified DCS-SOMP to estimate channel paths, angular parameters, and channel coefficients without assuming the number of paths. It then simplifies the frequency-dependent model and estimates each path’s delay and channel gain through least squares.

  • Coarse channel estimation: Modified DCS-SOMP provides coarse estimates of the number of paths, AOA/AOD parameters, and channel coefficients across subcarriers.Its stated outputs include estimates of K, θTx,k, θRx,k, and ˇh[n] for n = 0, . . . , N −1.
  • Coarse channel estimation: The algorithm is rank-blind because it does not require prior knowledge of the number of paths, K + 1.Since K + 1 is unknown, the procedure uses a change-based stopping criterion with threshold δ.
  • Simplified model: For coarse estimation, the method ignores the dependence on subcarrier index n, yielding a simplified model.This approximation supports subsequent estimation of path-specific delay and channel gain.
  • Path refinement: The simplified model estimates each path’s delay τk and channel gain ˜hk by solving a least-squares problem.The path model uses the delay steering vector a(τk) and a path-specific signal term z(k).

C. Step 2: Fine Estimation of Channel Parameters using SAGE

The fine-estimation stage iteratively refines channel parameters with SAGE, initialized by the estimates from Step 1. SAGE replaces computationally complex multidimensional likelihood maximization with expectation and maximization updates over a complete-data representation.

  • Step 2: Fine Estimation of Channel Parameters using SAGE: SAGE iteratively refines channel parameter estimates starting from the estimates produced in Step 1.The iterative procedure uses the Step 1 estimates for initialization.
  • Step 2: Fine Estimation of Channel Parameters using SAGE: SAGE provides a practical alternative to direct log-likelihood ascent, which requires computationally complex multidimensional minimization.The method is introduced to avoid the complexity of directly optimizing the model’s log-likelihood function.
  • Step 2: Fine Estimation of Channel Parameters using SAGE: At iteration m + 1, SAGE performs expectation and maximization steps using an incomplete-data model represented as the superposition of K + 1 complete-data spaces.The complete-data spaces correspond to the channel paths and are used to define the iterative updates.

Expectation step: · Maximization step: · D. Step 3: Conversion to Position and Rotation Angle Estimates

The maximization step updates the model parameters through sequential Gauss-Seidel-type iterations, while Step 3 converts refined AOA, AOD, and TOA estimates into position and rotation estimates across LOS, NLOS, OLOS, and unknown conditions.

  • Expectation step:: The expectation step constructs a function from the previous estimate ˆη(m) and the incomplete data space ˇy.This function provides the basis for the subsequent maximization step.
  • Maximization step:: The maximization step seeks ηk that maximizes (58), but updates ˆθ(m+1) sequentially because direct optimization is analytically complex.The difficulty arises from computing the gradient and Hessian with respect to ηk; Gauss-Seidel-type iterations are used.
  • D. Step 3: Conversion to Position and Rotation Angle Estimates: Step 3 recovers MS position and orientation from refined AOA/AOD/TOA estimates under LOS, NLOS, OLOS, and unknown channel conditions.The conversion uses different mappings and optimization procedures depending on whether a LOS path is available.
  • D. Step 3: Conversion to Position and Rotation Angle Estimates: In LOS, the expressions (17), (19), and (22) define η = f los(˜η), enabling direct estimates of ˆp and ˆα through ML invariance.The classical invariance principle establishes equivalence between minimizing the ML criterion in η0 and ˜η0.
  • D. Step 3: Conversion to Position and Rotation Angle Estimates: In NLOS, EXIP provides an asymptotically ML-equivalent estimator of ˜η, with LMA initialized from the LOS path and its smallest-delay path.Replacing Jη with the identity matrix remains meaningful but may produce slightly larger RMSE.
  • D. Step 3: Conversion to Position and Rotation Angle Estimates: In OLOS, trial rotation values ˆαtrial are evaluated using linear equations and LMA, retaining the solution with the smallest volos(˜ηolos).The trial range is [−αm, +αm] with resolution ∆α, creating a performance/complexity trade-off; at least three scatterers are needed to estimate all parameters.
  • D. Step 3: Conversion to Position and Rotation Angle Estimates: When LOS status is unknown, the receiver runs the NLOS and OLOS techniques separately and retains the solution with the lowest cost.The two candidate solutions are compared using the costs in (63) and (64).

V. SIMULATION RESULTS · A. Simulation Setup · B. Results and Discussion

The simulations evaluate the proposed estimators and bounds in an indoor 60 GHz localization scenario, showing how sequential-beam count and antenna count affect coverage, accuracy, and complexity. In LOS conditions, at least 20 randomly selected beams approximately achieves the same localization accuracy at CDF = 0.9, with similar behavior in NLOS conditions.

  • A. Simulation Setup: The simulation models an indoor conference-room scenario with a maximum MS–BS distance of 4 meters, fc = 60 GHz, B = 100 MHz, and N = 20.NLOS paths contain one reflector, and geometry-based statistical path loss is used.
  • B. Results and Discussion: Increasing G improves the probability of covering the target location with a specified accuracy, while localization accuracy converges once spatial coverage is sufficient.Further beams then increase complexity without improving the coverage effect.
  • B. Results and Discussion: Larger Nt narrows the ULA 3 dB beam width, increasing the number of beams required to cover the same target area with the same probability.Reducing Nt decreases the required G for area coverage.
  • B. Results and Discussion: Using more transmit antennas improves localization accuracy but requires transmitting more beams G to maintain the same coverage.Thus, antenna count trades off accuracy against beam-transmission cost.
  • B. Results and Discussion: G ≥20 randomly selected beams approximately provides the same localization accuracy with CDF = 0.9 for the stated system parameters.A well-chosen deterministic strategy would require fewer beams, and the same behavior is observed under NLOS conditions.

Performance in LOS: · Performance in NLOS: · Performance in OLOS:

The proposed algorithm approaches the corresponding Cramér–Rao bounds for estimation of timing, angular, rotation, and position parameters across LOS, NLOS, and OLOS conditions. Performance remains strong at low received SNR, although OLOS estimation is more sensitive to rotation-angle grid resolution and yields higher errors than NLOS.

  • Performance in LOS:: After a few iterations, TOA and AOA/AOD RMSE converge to their corresponding bounds in LOS, even at SNR = −20 dB, −10 dB, 0 dB.The bounds are represented by red lines with corresponding markers.
  • Performance in LOS:: At SNR ≈−20 dB, TOA, AOA/AOD, rotation angle, and position RMSE approach their corresponding bounds in LOS.The algorithm performs well at very low received SNR, typical before beamforming in mm-wave systems.
  • Performance in NLOS:: In NLOS with a scatterer at sk [m] = [1.5, 0.4]T, TOA and AOA/AOD RMSE for LOS and reflected signals converge to the theoretical bounds.This convergence occurs even at very low received SNR.
  • Performance in NLOS:: At SNR ≈−5 dB in NLOS, TOA, AOA/AOD, rotation angle, and position approach their corresponding bounds.The result is reported for 1000 Monte Carlo realizations.
  • Performance in OLOS:: In OLOS with three scatterers, position and rotation-angle estimation approaches the bound even with rotation-angle initialization resolution ∆α [rad] = 0.05.The investigated scatterers satisfy sk [m] = [1.5, 0.4 + 0.5(k −1)]T for k = 1, 2, 3, with αm [rad] = 0.5.
  • Performance in OLOS:: A finer rotation-angle grid produces better initial estimates and lower final RMSE in OLOS.The algorithm’s performance depends on the resolution of the grid of points ∆α.
  • Performance in OLOS:: At SNR ≈−10 dB, OLOS position and rotation-angle RMSE approach the corresponding bounds, but fixed-SNR values are significantly higher than in NLOS.The comparison concerns OLOS and NLOS performance values at a fixed SNR.

UNKNOWN CONDITIONS · VI. CONCLUSION

The study shows that single-transmitter millimeter-wave MIMO can estimate receiver position and orientation across LOS, NLOS, and OLOS conditions. It derives uncertainty bounds and proposes a staged algorithm that uses multipath information, although OLOS incurs a significant performance penalty.

  • UNKNOWN CONDITIONS: UNKNOWN CONDITIONS: Assuming the shortest-delay path is LOS when it is actually a reflection produces a mean cost-function ratio Δv on the order of 5.The mismatch arises because the erroneous AOA/AOD-based rotation estimate causes a large initial-position error that propagates to the final solution.
  • UNKNOWN CONDITIONS: UNKNOWN CONDITIONS: Observing the ratio of cost functions identifies that the shortest-delay path is associated with a scatterer and that the LOS path does not exist.This enables correct identification of the OLOS condition.
  • UNKNOWN CONDITIONS: UNKNOWN CONDITIONS: Adding scatterers improves localization accuracy and robustness compared with using only the LOS path.The comparison considers LOS and NLOS at SNR = −5 dB and G = 20, with one and two scatterers.
  • UNKNOWN CONDITIONS: UNKNOWN CONDITIONS: OLOS performance is much worse than LOS or NLOS because of severe path loss.The paper demonstrates this comparison through the RMSE results in Figs. 5, 8, and 9.
  • VI. CONCLUSION: VI. CONCLUSION: The study determines receiver position and orientation using a single transmitter in a MIMO system under LOS, NLOS, and OLOS conditions.It thereby examines localization even when the LOS path is blocked.
  • VI. CONCLUSION: VI. CONCLUSION: The paper derives fundamental uncertainty bounds for each path’s delay, arrival angle, departure angle, and channel gain, as well as user position and orientation angle.These bounds characterize estimation uncertainty for the channel and receiver geometry.
  • VI. CONCLUSION: VI. CONCLUSION: The proposed three-stage algorithm obtains coarse sparse beamspace channel estimates, iteratively refines them, and converts them into position and orientation estimates.The algorithm exploits millimeter-wave sparsity in beamspace before refinement and geometric conversion.
  • VI. CONCLUSION: VI. CONCLUSION: Simulations show that position and orientation remain estimable in OLOS by exploiting multipath information, though at a significant performance penalty.This conclusion follows the demonstrated efficiency of the proposed algorithm across the studied propagation conditions.

APPENDIX A · APPENDIX B · COMPLEXITY ANALYSIS

Appendix A details the Fisher information matrix derivation through subcarrier-, array-, derivative-, and channel-coefficient terms. The complexity analysis decomposes the algorithm into coarse estimation, fine estimation, and position/orientation conversion, with coarse estimation identified as the dominant stage.

  • APPENDIX B: Appendix B has no substantive passage in the supplied excerpt, so its content cannot be summarized without adding unsupported claims.The input lists APPENDIX B as a merged section but provides no passage labeled APPENDIX B.
  • APPENDIX A: Appendix A derives Fisher information matrix entries for each subcarrier using Ψ_n(x_r, x_s) and delay- and angle-related parameter pairs.The derivation introduces Ψ_n(x_r, x_s) for {τ_r, τ_s} and {θ_r, θ_s}.
  • APPENDIX A: Appendix A defines auxiliary notation, A_k,n(τ_r, τ_s), and transmit-array vectors and derivatives through the frequency-domain matrix F_H[n].The vectors are specified as a_Tx,F,n(θ_Tx,r) = F_H[n]a_Tx,n(θ_Tx,r) and a_DTx,F,n(θ_Tx,r) = F_H[n]D_Tx,r[n]a_Tx,n(θ_Tx,r).
  • APPENDIX A: Appendix A also defines receive-array scalar terms and summarizes the channel-coefficient contributions used in the Fisher information matrix.The receive derivative matrix D_Rx,r[n] follows the transmit-side expression with Tx replaced by Rx and N_t by N_r.
  • COMPLEXITY ANALYSIS: The maximum coarse-estimation complexity is dominated by the term identified in the analysis.The supplied passage states that the maximum complexity from coarse estimation of channel parameters is dominated by a specific term, but that term is truncated.
  • COMPLEXITY ANALYSIS: Coarse estimation combines mutilated-basis QR factorization, channel-coefficient matrix inversion, and delay-grid computations across subcarriers.The supplied complexity terms include O(GN_r K̂^2) per subcarrier for QR factorization, O(NK̂^3) for matrix inversion, O(ND_oK̂), and O(NK̂).
  • COMPLEXITY ANALYSIS: Fine estimation is mainly affected by Gauss-Seidel-type iterations using first- and second-order derivatives for delay, AOA, and AOD variables.The derivative operations apply to a vector a(x) of length L_x and are followed by path refinement.
  • COMPLEXITY ANALYSIS: Position-and-orientation conversion is basic in LOS, while NLOS and OLOS use LMA; this algorithm is not considered the complexity driver.The analysis attributes LMA’s effectiveness to combining gradient-descent and Gauss-Newton methods, with delayed gratification yielding higher success rate and fewer Jacobian operations.
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