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Turning Multipath into Positioning Information: Persistent Electromagnetic-Twin DoA Localization

Tuo Wu, Zhentian Zhang, Ye Tian, Hing Cheung So

arXiv:2610.39111v1eess.SP

TL;DR

When LoS links are blocked, reflected paths can support localization, but their usefulness depends on accurate predicted directions. The paper calibrates a persistent path-angle field from reference probes and jointly estimates positions and path assignments. Its analysis and simulations support positioning information from correctly associated reflected paths, conditional on physical calibration.

  • Problem

    Reflected paths may provide bearings when LoS links are blocked, but inaccurate stored direction predictions limit their usefulness for localization.

    Source: §I · p. 2

  • Method

    The method calibrates a persistent correction to nominal path-angle geometry and jointly estimates user position with one-to-one association of unordered DoAs.

    Source: §I · p. 2

  • Results

    Correctly associated reflected paths can add positioning information and preserve rank under LoS blockage; association exceeds 94% across three maps at 4° DoA error, with calibration gain mainly reflecting angle prediction rather than matching.

    Source: C. Effect of DoA Noise · p. 9, §VIII · p. 12

  • Takeaways & Limitations

    Reflected paths can add positioning information under LoS blockage, but their benefit depends on physical calibration.

    Source: §VIII · p. 12

    Full rank is a local condition, not global uniqueness under unknown associations, and ideal PEB does not certify the biased Huber estimator or robustness to unmodeled reflections.

    Source: B. Reflected-Path Information and… · p. 7

Abstract

from arXiv · show

Reflected paths can support localization when line-of-sight (LoS) links are blocked, but their usefulness depends on how accurately the stored model predicts their directions. This paper develops electromagnetic-twin (ET) direction-of-arrival (DoA) localization with a persistent, physically calibrated path-angle field. A nominal floor plan supplies virtual-path geometry, and sparse reference probes at known positions learn a compact directional correction. Field order and regularization are selected from calibration data alone. For each user, unordered DoAs are processed by joint position estimation and one-to-one path association, using assignment-aware multistart search and robust Gauss--Newton refinement. New trusted probes update the stored correction through recursive least squares (RLS), without reprocessing all previous measurements. The analysis establishes calibration rank conditions, objective descent, and conditional information gains from reflected paths, while also distinguishing the decay of initialization bias from the convergence of localization error. Synthetic experiments use 500 queries per operating point. At 50% LoS blockage and $1.5^\circ$ DoA error, the calibrated ET achieves 0.39 m root-mean-square error, versus 0.55 m for nominal geometry. After a wall change, its final error is 0.43 m, versus 0.63 m for a frozen calibrated map. Equal-evidence comparisons show competitive calibrated alternatives and faster abrupt-change recovery with sliding-window updates than with the configured RLS. The results support reusable directional calibration, not a universally superior map or update rule, and validation is limited to synthetic first-order reflections with trusted calibration labels.

I. INTRODUCTION

Reflected paths can preserve useful positioning information when LoS links are blocked, but a nominal floor plan may predict their directions inaccurately. The paper proposes a persistent, calibrated path-angle model and joint position-and-association estimation for unordered DoAs.

I. INTRODUCTION

  • Prior work supports using suitable reflected paths for information beyond direct bearings, while the paper targets calibration of useful paths in an existing floor plan rather than rediscovering the environment.The maintained state is path-angle information for localization, distinct from broader twins representing scene objects, devices, and network operation.Source: §I · pp. 1–2
  • A small wall-location error can bias predicted directions across the room, while sparse probes and unordered DoAs make correction complexity, path assignment, and trusted updates central challenges.The setting distinguishes known-position, path-labeled reference probes from queries with unknown positions and path labels.Source: §I · p. 2
  • The framework calibrates nominal reflection geometry with a compact correction, jointly estimates user position and one-to-one path assignments, and updates the persistent field using trusted probes.Field order and regularization are selected from reference measurements alone; updates use RLS.Source: §I · p. 2

II. SYSTEM MODEL

  • The study assumes known AP poses and resolvable LoS and first-order reflected paths, with known-position, labeled probes but unknown query positions and path labels.This separates directional calibration from autonomous reflector discovery.Source: §I · p. 2, A. Propagation Geometry and DoA… · p. 3

I. INTRODUCTION

  • The paper evaluates paired synthetic experiments with 500 localization queries per operating point, comparing nominal geometry, calibrated maps, robust losses, and online update policies.The comparisons distinguish the value of directional calibration from the choice of memory policy.Source: §I · p. 2

A. Propagation Geometry and DoA Measurements

  • The angle-domain model allows LoS to be absent and assumes no false detections, so it does not establish that a given array can resolve all modeled paths.The formulation allows heterogeneous angle variances, although the experiments use a common variance.Source: A. Propagation Geometry and DoA… · p. 3

B. Calibrated Directional Field

The directional field retains nominal virtual-path geometry while fitting a compact correction from reference probes. Its selected basis is used for persistent updates, with model selection based on calibration data rather than test-user localization errors.

B. Calibrated Directional Field

  • The affine correction models angular offset and spatial slope, while the quadratic field also captures curvature; both preserve nominal reflection geometry and are linear in their coefficients.The selected order has dimension 3 or 6, respectively.Source: B. Calibrated Directional Field · p. 3

C. Persistent State and Update Protocol

  • The stored directional state carries coefficients and inverse weighted normal matrices; after users are localized, trusted labeled probes update it for subsequent users.Estimated user positions are not reused as calibration labels, and freezing the coefficients yields the frozen ET.Source: C. Persistent State and Update Protocol · p. 3

A. Batch Estimation and Model Selection

  • Reference path labels are supplied by the simulator, while obtaining those labels from real measurements remains a separate calibration requirement.The innovations form a local angular model, so values crossing a wrapping discontinuity require unwrapping or circular regression.Source: A. Batch Estimation and Model Selection · pp. 3–4
  • Leave-one-out cross-validation selects field order and ridge strength from reference data, with the quadratic basis eligible only when K ≥8.This leaves at least seven probes for six coefficients in each held-out fit; LOOCV predicts held-out angles, not unknown-user positions, and does not guarantee lower localization RMSE.Source: A. Batch Estimation and Model Selection · p. 4

B. Recursive Update

Recursive least squares continues the selected calibration fit as new probes arrive, without repeating the batch fit. Its analysis concerns decay of initialization bias under stated conditions, not guaranteed convergence of localization error.

B. Recursive Update

  • RLS assimilates subsequent reference probes using the fixed selected basis, continuing the fit without repeating batch estimation.With a forgetting factor below one, earlier normal-equation terms, including the initial ridge contribution, are progressively discounted.Source: B. Recursive Update · p. 4
  • For 0 < λf < 1, initialization bias decays under exact post-change representability, bounded deterministic features, conditionally zero-mean calibration noise, and persistent excitation.The proposition concerns expected bias decay, not monotonic sample-path RMSE or zero steady-state error; measurement noise and finite-basis approximation error remain.Source: B. Recursive Update · p. 4
  • The implemented recursion accepts only trusted probes and has no change detector or innovation gate; residual patterns may not distinguish outliers from stale maps under systematic sensor error.An uncertainty-based gate would require separate validation and is not an established benefit of this method.Source: B. Recursive Update · p. 4

C. Calibration Accuracy and Identifiability

The section relates calibration quality to probe geometry and describes joint localization with explicit one-to-one path labels. It states conditions for identifying a correction field and cautions that conditional covariance does not capture all calibration error.

C. Calibration Accuracy and Identifiability

  • The stated sampling covariance assumes a fixed basis, exact reference positions, correct labels, and homoscedastic noise; it is not the full MSE.The ridge bias also contributes, and the covariance does not account for selecting the basis and ridge from the same noisy batch.Source: C. Calibration Accuracy and… · p. 4
  • A correction field is uniquely identifiable in the noise-free unregularized model only when the probe feature matrix has full column rank.At least D probes are necessary, but collinear probes leave directions unobserved; ridge yields a unique numerical fit without restoring that missing information.Source: C. Calibration Accuracy and… · p. 4
  • With a fixed full-rank basis and an exactly specified correction, well-conditioned feature directions let additional probes mainly average noise rather than remove finite-basis bias.This distinction motivates the probe-budget and wall-mismatch experiments.Source: C. Calibration Accuracy and… · pp. 4–5

IV. JOINT LOCALIZATION AND PATH ASSOCIATION; A. Association Constraints; B. Problem Formulation

V. PROPOSED LOCALIZATION ALGORITHM

The algorithm searches candidate positions, alternates path assignment with robust position refinement, and selects among refined candidates using a common objective. Backtracking tests the original robust cost before accepting position updates.

V. PROPOSED LOCALIZATION ALGORITHM; A. Assignment-Aware Initialization

  • The algorithm retains the nearest-path solution as a start, adds separated grid starts, gives each start the same refinement budget, and selects by the common objective.The grid scan retains at most three starts, separated by at least 2 m.Source: §V · p. 5, A. Assignment-Aware Initialization · p. 5

1) Path assignment:

  • For a fixed candidate position, path assignment minimizes the robust costs subject to one-to-one constraints; the four-path implementation enumerates at most 24 injections.The passage also identifies the Hungarian algorithm as a solution for the rectangular assignment problem.Source: 1) Path assignment · p. 5

2) Position refinement:

  • For fixed labels, position refinement uses a local residual linearization and tests the original robust cost through backtracking.A trial is accepted only if the fixed-assignment objective does not increase.Source: 2) Position refinement · pp. 5–6

V. PROPOSED LOCALIZATION ALGORITHM

  • The method requires DoAs, a fixed calibrated field, a grid, and query scales, and returns a position with path labels.Its refinement caps each direction at 1.5 m, tests up to eight halved steps, and stops after ten iterations or a displacement below 10^-4 m.Source: 2) Position refinement · p. 6

C. Convergence and Complexity

The section establishes nonincreasing objective values under exact assignment and accepted non-increasing position updates, and provides a retained-start safeguard. It distinguishes this optimization analysis from local information analysis under correct association and a specified error model.

C. Convergence and Complexity

  • Exact assignment and position updates accepted only when their fixed-assignment cost does not increase make the objective sequence nonincreasing and convergent.This result does not establish global optimality or convergence to a stationary point under all stopping rules.Source: C. Convergence and Complexity · p. 6
  • For a fixed map, loss, and per-start budget, selecting among refined candidates cannot give a higher fitting cost than refining the nearest-path start alone.A lower fitting cost need not mean lower position error under noisy or mismatched angles.Source: C. Convergence and Complexity · p. 6

VI. POSITIONING INFORMATION ANALYSIS; C. Convergence and Complexity

  • The analysis of positioning information assumes correct association and a specified error model, distinct from finite-sample performance of the robust estimator.The section gives complexity bounds for grid scanning, assignment, candidate screening, and per-start local solves, but notes that enumeration scalability is not established.Source: §VI · p. 6, C. Convergence and Complexity · p. 6

A. Angular Uncertainty and Fisher Information

The analysis connects angular uncertainty to Fisher information and local position bounds, then examines how an additional reflected path can affect information. Its conclusions are conditional and do not certify the robust estimator or unknown path matching.

A. Angular Uncertainty and Fisher Information

  • When front-end, map, and residual model errors are independent, their variances add, but the residual term represents random error rather than deterministic wall bias.This uncertainty model is not used to weight the simulated queries.Source: A. Angular Uncertainty and Fisher… · p. 6
  • The full Gaussian Fisher information includes a variance-gradient term when variances depend on position, and the stated CRB applies to locally unbiased estimation under its assumed likelihood.The PEB is specified for nonsingular information with constant variances.Source: A. Angular Uncertainty and Fisher… · p. 6

B. Reflected-Path Information and Local Identifiability

  • With existing paths and variances unchanged, adding an independent, correctly associated path adds positive-semidefinite information and cannot increase the CRB when the original FIM is positive definite.The result is conditional on the stated path and variance assumptions.Source: B. Reflected-Path Information and… · p. 7
  • Reflections can provide noncollinear gradients that enable local identifiability with one AP, while a path with larger modeled variance contributes less for a fixed gradient.The gain depends jointly on gradient orientation and calibration variance.Source: B. Reflected-Path Information and… · p. 7
  • Full rank is local, not global uniqueness under unknown associations, and the ideal CRB does not certify the biased Huber estimator or robustness to unmodeled reflections.The section reports ideal PEB and actual localization RMSE to distinguish these effects.Source: B. Reflected-Path Information and… · p. 7

VII. NUMERICAL RESULTS

  • The numerical-results section evaluates calibration efficiency, localization with blocked or corrupted paths, and adaptation to environmental change.Comparisons use the same observations and solver budgets where applicable.Source: §VII · p. 7

A. Scenario and Baselines

The evaluation compares calibrated electromagnetic-twin localization with geometric and learned alternatives in a fixed room scenario. Baselines share a solver and calibration-only selection procedures to support controlled comparisons.

A. Scenario and Baselines

  • The nominal 20 × 14 m room has two access points, one line-of-sight candidate per access point, and left-, right-, and top-wall reflections; calibration and query positions share a fixed rectangle.Physical wall changes do not alter the query population or place queries outside the optimizer’s feasible region.Source: A. Scenario and Baselines · p. 7
  • Unless stated otherwise, experiments use 1.5° DoA noise, 50% LoS blockage, and 20 calibration probes, averaging 500 independently drawn queries per operating point.Noise and probe-budget sweeps use paired queries and exact nested prefixes of a shared 80-probe pool.Source: A. Scenario and Baselines · p. 7, B. Compared Representations · p. 8
  • Comparisons include LoS bearing intersection with room-center fallback, uncalibrated nominal geometry, calibrated ET, and oracle physical geometry; map-based methods share solver settings.The fallback is not a successful angle-based position estimate, and gains are attributed to calibration and trusted-probe updates rather than untested solver extensions.Source: A. Scenario and Baselines · p. 7

B. Compared Representations

  • WKNN fingerprinting stores observed angle sets at reference locations and estimates position from nearby references after minimum-cost one-to-one query-to-reference matching.The neighbor count is chosen by leave-one-reference-out position error across four LoS-visibility patterns, with no test-user positions used.Source: B. Compared Representations · p. 8
  • Wall-NLS directly fits the left, right, and top reflection-plane coordinates, then uses the same Huber association and refinement as ET.Its three-variable search starts from the nominal floor plan and has no access to physical-wall truth.Source: B. Compared Representations · p. 8
  • GP mapping replaces finite-order correction with a squared-exponential spatial kernel, while ET-LS and ET-Cauchy change only the query loss and retain selected calibration.GP hyperparameters are selected using calibration data, and predictive variance does not alter the comparison.Source: B. Compared Representations · p. 8

C. Effect of DoA Noise

Calibration reduces nominal-map bias, particularly when LoS links are blocked or the floor plan is mismatched, while calibrated alternatives remain competitive. Robust-loss benefits depend on gross angle contamination, and calibration can add estimation cost when the map is already exact.

C. Effect of DoA Noise

  • At 1.5° DoA error, Wall-NLS, GP mapping, and ET-Huber give 0.373, 0.379, and 0.380 m RMSE, respectively.The association diagnostic reports correct-association probabilities above 94% even at 4° DoA error, with little separation among maps.Source: C. Effect of DoA Noise · p. 9

D. LoS Blockage

  • At 50% LoS blockage, static geometry gives 0.55 m RMSE and calibrated ET gives 0.39 m; with complete blockage, they give 0.65 m and 0.49 m.The all-user success rates at 50% blockage are 58.20% for the nominal map and 84.80% for calibrated ET; with all LoS paths blocked, they are 40.00% and 70.20%.Source: D. LoS Blockage · p. 9

E. Calibration Budget and Model Mismatch

  • At the largest wall mismatch, static geometry gives 1.01 m RMSE and calibrated ET gives 0.38 m; calibration can add estimation cost when mismatch is zero.At 80 probes, WKNN gives 0.768 m, compared with 0.372 m for GP and 0.384 m for selected ET.Source: E. Calibration Budget and Model Mismatch · pp. 9–10

F. Robustness to Gross Angle Errors

  • At 20% gross-angle contamination, quadratic, Huber, and Cauchy query losses give 1.932, 1.384, and 0.973 m RMSE, respectively.Cauchy has the lowest RMSE at the tested nonzero contamination levels, while the fixed default elsewhere remains Huber.Source: F. Robustness to Gross Angle Errors · p. 10

G. Number of APs and Paths

Across changing environments and AP/path configurations, calibrated reflected paths can improve positioning when their geometry is reliable. Persistent updates help after wall changes, but update-policy performance depends on how quickly stale evidence is discounted.

G. Number of APs and Paths

  • Mean ideal-geometry PEB decreases from 0.45 m with one path per AP to 0.27 m with all four.These values assume correct associations and physical geometry, without random LoS blockage; they are not matched lower bounds for blocked, biased experiments.Source: Figure 9 · p. 11, G. Number of APs and Paths · p. 11
  • With one AP, the calibrated multipath map gives 0.65 m RMSE, while three APs reduce its error to 0.27 m.The one-AP result depends on a known reflector model and resolvable reflected paths; it does not establish generic single-AP identifiability.Source: G. Number of APs and Paths · pp. 10–11

H. Persistent Recovery after Environmental Change

  • At epoch 16, online map error is 2.03° versus 1.91° for the frozen map, then falls to 0.90° by epoch 30.The change occurs before that epoch’s two trusted probes are assimilated; RLS uses a fixed basis, λf = 0.96, and no change-triggered reset.Source: H. Persistent Recovery after… · p. 11, Figure 10 · p. 11
  • After walls change at epoch 16, position RMSE is 0.43 m at epoch 30, versus 0.63 m for the frozen map and 0.99 m for nominal geometry, on the same calibration sequence.Each epoch uses 500 queries paired across methods and epochs through common random numbers.Source: H. Persistent Recovery after… · p. 11, Figure 10 · p. 11

I. Comparison of Online Update Policies

  • After the wall change, 12-probe-window LS reaches 0.45 m post-change RMSE versus 0.49 m for RLS.Across ten calibration/query histories, the window removes stale evidence sooner; all-history LS remains pulled toward the previous environment.Source: Figure 11 · p. 12, §I · p. 12
  • A short window discards stale evidence faster but can also lose useful probes, while RLS discounts older evidence gradually.The configured RLS was not the best tested policy for this abrupt change; the comparison does not determine the best policy for gradual motion, noisier probes, or other scenes.Source: §I · pp. 11–12

VIII. CONCLUSION

  • The framework combines nominal virtual-anchor geometry, calibration-only correction-field selection, trusted updates, and joint path association with robust position refinement.Its conclusion is conditional: correctly associated reflected paths can add positioning information and preserve rank under LoS blockage, but their benefit depends on physical calibration.Source: §VIII · p. 12

APPENDIX A VIRTUAL-PATH JACOBIAN

The appendices connect the virtual-path Jacobian and map uncertainty to localization information, while specifying when added paths can improve the position bound.

APPENDIX A VIRTUAL-PATH JACOBIAN

  • The ET correction contributes an additional position derivative to the virtual-path Jacobian, and geometric and correction derivatives must use consistent angular units.The implementation uses symmetric numerical differentiation for the quadratic correction and GP baseline.Source: Appendix A · p. 12

APPENDIX B MAP-UNCERTAINTY PROPAGATION

  • Under independent front-end angle, map, and residual-model errors, their variances add, while nonzero mean correction error remains a separate bias.The mean-square error also includes the squared correction-error mean term.Source: Appendix B · p. 12

APPENDIX C INFORMATION MONOTONICITY

  • Adding a finite-variance path to an initially positive-definite information matrix cannot increase squared PEB, with strict reduction for a nonzero gradient.If the initial matrix is singular, the rank condition must be checked separately.Source: Appendix C · pp. 12–13

APPENDIX D CALIBRATION RANK, ERROR, AND RECURSIVE FORM

The appendix states rank conditions for identifying calibration coefficients and describes recursive updating and its uncertainty caveat.

APPENDIX D CALIBRATION RANK, ERROR, AND RECURSIVE FORM

  • A unique unregularized batch calibration requires the probe design matrix to have full column rank.When that condition holds, ordinary least squares has the stated error bound.Source: Appendix D · p. 13
  • Persistent excitation prevents the recursive normal matrix from losing rank during RLS operation.The appendix defines an exponentially weighted normal matrix and gives the associated gain and inverse-normal recursion.Source: Appendix D · p. 13
  • The RLS inverse normal matrix alone is not a calibrated uncertainty estimate.The stated covariance assumes independence between the initial fit and subsequent probe noise and excludes deterministic model bias.Source: Appendix D · p. 13

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