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Joint Beam Training and Positioning For Intelligent Reflecting Surfaces Assisted Millimeter Wave Communications

Wei Wang, Wei Zhang

arXiv:2009.03536v2cs.ITeess.SP

TL;DR

MmWave blockage makes accurate beam, IRS reflection-pattern, and blockage estimation necessary, while IRS deployment complicates beam training. The paper combines random beamforming and ML AoA/AoD estimation with iterative positioning and location-aided parameter refinement. Numerical results report centimeter-level positioning accuracy, superior beam training, and gains from location information.

  • Problem

    MmWave blockage and IRS sensing constraints make jointly estimating optimal beams, reflection patterns, and blockage challenging for IRS-assisted communications.

  • Method

    The paper decomposes beam training into equivalent sub-problems, applies random beamforming and ML estimation to AoA/AoD, then uses iterative positioning for refinement and blockage prediction.

  • Results

    Centimeter-level positioning accuracy is achieved, while numerical results show superior beam training and performance gains from location information.

  • Takeaways & Limitations

    Location information supports cross-verification and enhancement of AoA/AoD estimates and facilitates blockage prediction within the proposed beam-training scheme.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) offers a cost effective solution to link blockage problem in mmWave communications, and the prerequisite of which is the accurate estimation of (1) the optimal beams for base station/access point (BS/AP) and mobile terminal (MT), (2) the optimal reflection patterns for IRSs, and (3) link blockage. In this paper, we carry out beam training design for IRSs assisted mmWave communications to estimate the aforementioned parameters. To acquire the optimal beams and reflection patterns, we firstly perform random beamforming and maximum likelihood estimation to estimate angle of arrival (AoA) and angle of departure (AoD) of the line of sight (LoS) path between BS/AP (or IRSs) and MT. Then, with the estimate of AoAs and AoDs, we propose an iterative positioning algorithm that achieves centimeter-level positioning accuracy. The obtained location information is not only a fringe benefit but also enables us to cross verify and enhance the estimation of AoA and AoD, and facilitates the prediction of blockage indicator. Numerical results show the superiority of our proposed beam training scheme and verify the performance gain brought by location information.

I. INTRODUCTION

IRSs are introduced to mitigate severe mmWave blockage, but their deployment creates heavier training overhead, blockage-estimation needs, and sensing constraints. The paper addresses these challenges with random-beamforming beam training, ML path-parameter estimation, positioning, and location-aided refinement.

  • Motivation: IRSs provide a cost-effective structure for mitigating mmWave blockage and extending coverage, but their passive hardware cannot sense signals directly.The lack of RF chains complicates training for IRS-assisted paths.
  • Motivation: Multiple IRSs make beam training more overhead-intensive than traditional mmWave training and require accurate blockage estimation for beam selection.These requirements arise alongside the inability of IRSs to sense signals.
  • Proposed beam training: The proposed method decomposes IRS-assisted mmWave MIMO beam training into mathematically equivalent sub-problems and uses random beamforming with ML estimation to jointly estimate dominant-path AoA and AoD.The scheme targets beam selection without requiring MT feedback during training, so overhead does not increase with MT number.
  • Proposed beam training: Uniqueness is proved for the AoA/AoD estimate, and larger training length almost surely reduces the pairwise error probability of the estimated pair.The analysis explicitly studies training-length effects.
  • Positioning and refinement: The algorithm sorts AoA/AoD-pair reliability to estimate the MT position, achieving centimeter-level positioning accuracy through numerical analysis.Three unblocked links are identified as the minimum sufficient number for 3-D position estimation, represented by intersecting right circular cones.
  • Positioning and refinement: Estimated MT position is used to cross-verify and enhance AoA/AoD through geometric relationships and to estimate blockage by comparing ML- and position-based pathloss estimates.The channel model includes LoS and IRS-reflected paths whose blockage indicators and reflection coefficients affect the received response.

III. FRAMEWORK OF JOINT BEAM TRAINING AND POSITIONING

The framework decomposes IRS-assisted mmWave beam training into mathematically equivalent sub-problems for direct and IRS-reflected links. Random transmit/receive beamforming collects measurements for ML estimation of path parameters, including blockage, gain, AoA, and AoD.

  • Measurement collection: Random transmit and receive beamforming produces the channel measurements used for parameter estimation.The beamforming vectors use phase-only complex variables with invariant amplitude, and measurements are concatenated across N time slots.
  • Measurement collection: Because NLoS paths are much weaker than the LoS path, the dominant LoS parameters are estimated while NLoS contributions are treated as interference.The analysis assumes the interference follows a complex Gaussian distribution for tractability.
  • Framework breakdown: Each sub-problem estimates blockage, equivalent path gain, cosine AoA, and equivalent cosine AoD from collected measurements.The reflected-link AoD is represented as φ_RiM ⊖ θ_BRi.
  • Framework breakdown: Beam training is split into a BS/AP–MT sub-problem and separate BS/AP–IRS–MT reflected-link sub-problems.The direct link deactivates all IRSs, while each reflected-link procedure activates one IRS and deactivates the others.
  • Framework breakdown: The framework uses flexible IRS control to convert complicated non-sparse channel estimation into a set of simplified sub-problems.The direct and reflected-link formulations are mathematically equivalent.

B. Protocol of Joint Beam Training and Positioning

The proposed protocol combines channel measurement, parameter estimation, and positioning with location-aided parameter refinement. Random beamforming supports broadcasting so multiple MTs can collect measurements without increasing training overhead with MT number.

  • Protocol stages: The protocol has three stages: channel measurement, parameter estimation, and positioning with location-aided parameter enhancement.Stages II and III estimate path parameters, position the MT, and refine path parameters using position information.
  • Protocol stages: In practice, random beamforming is triggered periodically before mmWave transmission to initialize and maintain beam alignment.Prior information such as beamforming sequences and BS/AP and IRS positions can be delivered over a lower-frequency link.
  • Broadcasting mechanism: Quasi-omnidirectional random beamforming allows pilots from the BS/AP to reach MTs from all directions simultaneously.Each MT can collect and process its measurements individually without causing interference.
  • Broadcasting mechanism: Training overhead does not increase with MT number, making the broadcasting scheme suitable for multi-user scenarios.The protocol uses the broadcasting property of random beamforming to serve multiple MTs concurrently.

IV. BEAM TRAINING WITH RANDOM BEAMFORMING – PARAMETER ESTIMATION AND FEASIBILITY STUDY

The parameter-estimation method applies ML estimation to random-beamforming measurements and solves the joint AoA/AoD search using coarse enumeration followed by gradient-based refinement. The resulting complexity is explicitly characterized.

  • Maximum likelihood estimation: ML estimation derives LoS/VLoS path parameters δ, θ, and φ from measurements collected with random receive and transmit beamforming.The unified model treats ζ as the blockage indicator and defines θ as cosine AoA and φ as equivalent cosine AoD.
  • Maximum likelihood estimation: Parameter estimation is performed only when the blockage indicator is ζ = 1, because blocked measurements contain no information about δ, θ, or φ.The parameters are estimated by maximizing the log-likelihood under ζ = 1.
  • Two-step search: The joint AoA/AoD solver first performs a coarse search over quantized candidates, then applies gradient descent for fine refinement.The best refined candidate is selected, after which the path gain estimate is obtained by substitution.
  • Complexity: The overall complexity is O(2^2ZφZθ + niterNpkNBNMN) or O(2^2ZφZθ + niterNpkNRiNMN), depending on the link.The gradient-search iteration count is generally less than 20.

B. Uniqueness of The Estimated AoA and AoD Pair

The feasibility analysis studies whether random-beamforming measurements uniquely represent channel parameters and therefore support unique ML-based AoA/AoD estimation. Random projections preserve manifold-point separation with high probability under sufficient dimension.

  • Uniqueness conditions: The analysis studies accurate AoA/AoD estimation from noiseless, interference-free measurements by examining uniqueness conditions.The channel response lies on a nonlinear array manifold parameterized by δ, θ, and φ.
  • Uniqueness conditions: Measurement-signal uniqueness means that a different AoA/AoD pair cannot generate the same measurement signal.This property is primarily determined by the sensing matrix, whereas estimated-pair uniqueness also depends on the estimation method.
  • ML estimation guarantee: As long as measurement-signal representation is unique, ML estimation can accurately estimate the AoA/AoD pair.The theorem equates the required uniqueness of the estimated pair with uniqueness of the measurement representation.
  • Random projection: With sufficient projected dimension, a random orthoprojector preserves pairwise distances on the channel manifold with high probability.The isometry constant ε measures the degree of distance preservation under projection.
  • Random projection: Because the sensing matrix can be reduced to an orthoprojector and nonsingular scaling, random sensing has a large probability of guaranteeing unique ML-based joint AoA/AoD estimation.The argument requires distinct channel-manifold points to remain distinguishable after sensing.

C. On The Impact of Training Length N

The section analyzes how training length affects random-beamforming ML estimation of the AoA/AoD pair. Longer training separates the authentic peak from erroneous peaks and lowers pairwise error probability, although noise and interference can still cause peak-selection errors.

  • C. On The Impact of Training Length N: Random beamforming yields a unique authentic ML peak with high probability in the noiseless setting.Other local-optimum peaks remain possible, but the global peak is separated from them with high probability.
  • C. On The Impact of Training Length N: The sufficient number of random measurements depends on ε and manifold-related factors such as condition number, volume, and geodesic covering regularity.In practice, noise and online adjustment of M limit the significance of an exact closed-form relationship.
  • C. On The Impact of Training Length N: Noise and interference can shift the highest peak or let another peak surpass it, with the latter causing significant AoA/AoD error and beam misalignment.The paper distinguishes mild Error Type 1 from severe Error Type 2.
  • C. On The Impact of Training Length N: Longer training increases the distance gap between the global peak and other local-optimum peaks.The contour-plot study reports an increasing first-to-second-peak gap as training length grows.
  • C. On The Impact of Training Length N: The first peak remains at the actual AoA/AoD pair as training length changes, while the second peak moves.This behavior supports uniqueness of the ML-based joint AoA/AoD estimate.
  • C. On The Impact of Training Length N: Pairwise error probability decreases almost surely as training length N increases.The paper links this trend to the monotonic increase of the distance measure governing pairwise error probability.

V. INTERPLAY BETWEEN POSITIONING AND BEAM TRAINING

The proposed positioning stage selects trustworthy links, uses their geometric AoD constraints, and iteratively estimates the MT position with a Taylor-series least-squares method. At least three unblocked links suffice ideally, while additional reliable anchors can improve practical accuracy.

  • V. INTERPLAY BETWEEN POSITIONING AND BEAM TRAINING: Reliability sorting selects links for positioning because blockage, insufficient training, and low SNR can corrupt estimated path parameters.The residual signal power ratio η is smaller for accurate reconstructions and larger under blockage or Error Type 2.
  • V. INTERPLAY BETWEEN POSITIONING AND BEAM TRAINING: Known anchor positions and array directions connect estimated AoDs to the MT location through the direction vector from each anchor to the MT.The geometric relation enables position estimation from selected links.
  • V. INTERPLAY BETWEEN POSITIONING AND BEAM TRAINING: Three unblocked anchors are minimally sufficient for ideal 3-D MT positioning, while more anchors can improve accuracy in practice.Each AoD constraint forms a right circular cone, and three cones can determine the position in the noiseless case.
  • V. INTERPLAY BETWEEN POSITIONING AND BEAM TRAINING: AoD-based positioning minimizes geometric angle errors over the indoor MT position range using a Taylor-series approximation and iterative local least-squares corrections.The objective is non-convex, so the method starts from a rough position estimate and repeatedly updates it.
  • V. INTERPLAY BETWEEN POSITIONING AND BEAM TRAINING: The algorithm iteratively adds anchors in ascending η order and retains the largest selected set whose least-squares cost is below a preset threshold.The cost is evaluated after each iteration to construct the reliable link set.

C. Parameter Estimation With The Aid of MT Position

The estimated MT position is used to refine AoD and AoA parameters through geometric consistency. AoA refinement additionally estimates the MT-side ULA direction from reliable links using constrained optimization.

  • C. Parameter Estimation With The Aid of MT Position: The estimated position ˆp refines channel parameters through their geometric relationship.Position information is fed back into path-parameter estimation rather than used only for localization.
  • C. Parameter Estimation With The Aid of MT Position: AoD estimates are updated using the estimated MT position and the known anchor geometry.The update enforces consistency between the estimated position and each anchor-to-MT path.
  • C. Parameter Estimation With The Aid of MT Position: AoA refinement requires estimating the MT-side ULA direction from the selected reliable links.The paper formulates this as an optimization problem and solves it with projected gradient descent.
  • C. Parameter Estimation With The Aid of MT Position: The numerical implementation uses √ξ_th = 0.005 as a threshold value that gives good performance.The threshold is selected through offline Monte Carlo experiments.
  • C. Parameter Estimation With The Aid of MT Position: After estimating the MT-side array direction, the method updates AoA estimates using the refined direction.The updated AoA is produced after projected-gradient optimization.

3) Estimation of Blockage:

The blockage estimator uses position-refined path parameters because the initial ML estimates assume an unblocked path. A heuristic alternative compares pathloss estimates, and simulations examine blockage across user densities.

  • 3) Estimation of Blockage:: Position-refined AoA and AoD estimates support blockage and path-gain estimation because multiple anchors cross-verify them.Directly using initial ML angles can be misleading when the actual blockage indicator is zero.
  • 3) Estimation of Blockage:: The blockage decision compares likelihoods associated with the path-gain estimates under blocked and unblocked hypotheses.The decision is based on comparing the conditional probabilities of the estimated gain under ζ_η = 0 and ζ_η = 1.
  • 3) Estimation of Blockage:: The heuristic blockage method compares pathloss estimated from geometry with a preset pathloss-distance threshold.The simulations set PL_th = 6 dB.
  • 3) Estimation of Blockage:: The numerical study models an indoor 20 m × 20 m × 5 m lecture hall with uniformly distributed users and 0.6 m × 0.4 m × 1.7 m user obstacles.The setup treats other MT holders as potential blockers.

B. Relationship Between User Density and Blockage Probability

The section evaluates random-beamforming beam training, positioning, blockage estimation, and location-aided path-parameter refinement in IRS-assisted mmWave communications under varying training lengths and transmit powers.

  • User density and blockage probability: With 12 IRSs, at least one unblocked link guarantees uninterrupted mmWave communication, while at least three enable positioning and parameter enhancement.For 20, 50, and 100 MTs, most channel realizations retain multiple unblocked links.
  • Beam training with random beamforming: Random beamforming estimates LoS AoA/AoD through channel measurements and maximum likelihood estimation, with higher-SNR performance approaching the Cramér–Rao bound.At N = 16, empirical MSE is significantly above the bound from 0 to 6 dBm, but the gap becomes marginal from 6 dBm upward.
  • Beam training with random beamforming: 0.559 misalignment rate is achieved by random beamforming at N = 256 and −20 dBm, outperforming exhaustive and hierarchical beam sweeping.The corresponding directional-beamforming rates are 0.606 and 0.966, respectively.
  • Joint beam training and positioning: 0.02 meter RMSE is reached for positioning with N = 16 from 15 dBm to 30 dBm, while N = 8 converges to 0.04 meter.At 0 dBm, RMSE is 0.13 meter for N = 16 and 0.45 meter for N = 8.
  • Blockage estimation: Position-aided blockage estimation becomes errorless as transmit power increases, whereas residual-ratio clustering stays below 0.1 error rate and received-power clustering is nearly random.K-means partitions 13 observations into blocked and unblocked links without requiring an absolute received-power range or an optimal residual-ratio threshold.
  • Location-aided refinement: Location information produces more accurate AoA/AoD estimates than random-beamforming beam training by refining parameters through geometric relationships.The refinement uses location information derived from multiple anchors.

APPENDIX A PARTIAL DERIVATIVES OF g(θ, φ)

The appendix develops derivative and identifiability steps for the joint AoA/AoD estimation objective, reducing uniqueness to the absence of proportional measurement vectors for distinct parameter pairs.

  • Partial derivatives: The receive and transmit steering-vector phase terms are differentiated with respect to the angular parameters θ and φ.The displayed expressions define array-dependent phase vectors used in derivatives of g(θ, φ).
  • Uniqueness argument: In the noiseless model y = Db(θ, φ), the proof uses the Cauchy–Schwarz inequality to analyze the estimation objective.The argument reduces uniqueness to showing that distinct angle pairs cannot produce proportional measurement vectors.
  • Pairwise error probability: The appendix formulates the pairwise error probability and simplifies its high-SNR expression using Gaussian noise and the Q function.The derivation neglects a noise-related component in the high-SNR regime before applying the Q-function definition.

APPENDIX D PROOF OF PROPOSITION 1

The appendix expands the squared-distance expression used in the proof of Proposition 1 by introducing shorthand variables and collecting its algebraic terms.

  • Proof setup: The proof begins by writing the expression for d2(Dn, θ, φ, eθ, eφ).This quantity is the distance term used in the pairwise-error analysis.
  • Notation: Auxiliary shorthand variables are introduced to make the subsequent algebraic expansion more concise.The passage explicitly states that the notation is adopted for conciseness.
  • Algebraic expansion: The resulting expression combines squared magnitudes, products of auxiliary terms, and a real-part cross term.The expansion includes Re{ˇb∗ˇd} alongside terms involving ˇa, ˇc, |ˇb|2, and |ˇd|2.
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