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

Introduction

Accurate wireless localization supports location-aware communications, indoor navigation, and integrated sensing and communication services [1], [2]. Antenna arrays provide direction-of-arrival (DoA) measurements that relate a user’s position to the known locations of access points (APs). Combined angle and delay measurements can further support position and orientation estimation in millimeter-wave systems and three-dimensional indoor positioning [3], [4]. The accuracy of these estimates depends not only on measurement noise but also on the available propagation geometry, as established by information analysis of localization accuracy [5]. In indoor environments, however, walls and obstacles may block direct links while producing several reflected arrivals. How to use these arrivals reliably is therefore an important question for maintaining positioning service.

Multipath can be treated as either an impairment or a source of geometric information. Rejecting non-line-of-sight (NLoS) measurements protects an estimator designed for direct propagation, but may leave too few independent bearings when line-of-sight (LoS) links are blocked. Alternatively, a specular reflection can be represented through a mirrored anchor or user, allowing the reflected direction to constrain position. Leitinger et al. [6] quantified the position-related information in multipath components, while Meissner et al. [7] evaluated multipath-assisted indoor navigation using measured ultra-wideband signals. These results support using suitable reflected paths rather than discarding all NLoS observations [8]. For angle-based localization, multipath-aware reference models likewise provide information beyond direct bearings [9].

When reflector locations are unknown, simultaneous localization and mapping (SLAM) can estimate the user and environmental features together. Channel-SLAM [10] and belief-propagation-based multipath SLAM [11] exploit the consistency of propagating paths across observations. Related work extends this principle to situational awareness in millimeter-wave massive multiple-input multiple-output (MIMO) systems [12]. These studies establish that propagation structure can be learned and reused. A complementary situation arises when an initial floor plan is already available but does not accurately predict the measured directions. In that case, discovering the entire environment again may be unnecessary, while the immediate task is to calibrate the useful paths in the existing model.

Environment-dependent radio knowledge can also be retained without explicitly estimating reflector geometry. Channel charting constructs a radio representation from accumulated channel state information (CSI) [13], and channel knowledge maps (CKMs) organize location-indexed channel descriptors for subsequent communication tasks [14]. Digital-twin-assisted fingerprinting further uses a digital replica to generate location-dependent MIMO signatures [15]. These approaches offer different interfaces for reusing environmental information. Their accuracy in a physical site nevertheless depends on the agreement between the retained representation and the observations available during operation.

Digital-twin channel research addresses this agreement through synchronization between physical observations and a digital propagation model [16, 17]. Recent electromagnetic-twin (ET) studies connect sparse evidence to an updated state and subsequent queries. Wu et al. [18] instantiate this interface through channel-gain completion, while recurrent wireless-state updates support AP association and beam selection [19]. Persistent CSI estimation instead reuses angle–delay information across pilot blocks [20], and ET beamforming converts sparse probes into covariance-based sum-rate optimization [21]. These studies motivate retained, task-specific states but do not address the path-angle calibration and unknown DoA association considered here. We adopt this ET viewpoint for a directional field calibrated by reference measurements and queried for user localization. A CKM can provide storage and retrieval, while a broader twin may also represent scene objects, devices, and network operation. Here, the maintained state contains only the path-angle information needed for localization. The contribution is its calibration and use with unordered DoAs, rather than map updating itself.

Consider an indoor deployment in which the AP locations are known and a survey provides approximate wall positions. The nominal model predicts the directions of direct and reflected arrivals throughout the room. A small wall-location error can bias these predictions at many user locations, even when the current DoAs are measured accurately. Known-position reference probes reveal this discrepancy and can improve the model for users that were not part of the calibration survey. If a reflecting surface subsequently moves, the previously calibrated model can become stale. Reusing it unchanged preserves that error, whereas rebuilding from only a few new probes discards potentially useful earlier observations. This motivates a directional state that is retained across queries and corrected as new reference evidence becomes available.

Realizing this idea involves three coupled issues. First, a sparse probe set cannot support an arbitrarily detailed correction. A low-order field may miss spatial variation, whereas a richer field may fit calibration noise rather than improve predictions. Second, the current user supplies an unordered set of DoAs: the receiver must determine which modeled path generated each observation while estimating the unknown position. Allowing several observations to select the same path can give a misleading initial estimate. Third, persistent operation must distinguish the information used to locate a user from the information trusted to change the model. Otherwise, an erroneous position or path label can be propagated to subsequent queries.

To address these issues, we develop an ET-assisted multipath localization framework with two separate operations: directional calibration and user localization. The nominal floor plan supplies first-order reflection geometry, and labelled probes at known positions fit a compact correction to each predicted path angle. Calibration-only model selection chooses between affine and quadratic fields. For a new user, a joint optimization estimates position and one-to-one path assignments using the calibrated field. Assignment-aware initialization and robust Gauss–Newton refinement provide a practical solver for this mixed discrete–continuous problem. After the queries at an update epoch, new trusted probes modify the stored coefficients through recursive least squares (RLS). The

corrected model then serves the next users without requiring their positions to be known or their trajectories to be tracked.

The study considers a two-dimensional region with known AP poses and resolvable LoS and first-order reflected paths. Reference probes have known positions and path labels, whereas both quantities are unknown for user queries. This setting separates directional calibration from autonomous reflector discovery and tests whether a compact maintained model can improve localization under nominal-map mismatch, blockage, and environmental change. The contributions are summarized as follows.

• We develop a persistent path-angle representation that combines nominal reflection geometry, a calibrated spatial correction, and recursive update statistics. Field order and regularization are selected from reference measurements alone. Batch estimation and recursive updating provide a common directional model for successive localization queries.

• We formulate joint position estimation and one-to-one path association as a constrained robust optimization problem. The proposed solver combines assignment-aware multistart initialization, exact small-set association, and backtracked Gauss–Newton refinement. We establish objective descent and a retained-start safeguard under a common per-start budget.

• We analyze correction-field identifiability, calibration error, and conditional positioning information. The analysis explains how probe geometry affects calibration and when independent reflected paths add useful spatial constraints. It also distinguishes the decay of the initial calibration bias from convergence of localization error.

• We evaluate the framework using paired synthetic experiments with 500 localization queries per operating point. Comparisons with nominal geometry, calibrated geometric and learned maps, and alternative robust losses identify the benefit of directional calibration. Equal-evidence online comparisons further distinguish maintaining an updated state from the particular memory policy used to update it.

The remainder of this paper is organized as follows. Section II presents the system model. Section III develops directional calibration. Sections IV and V formulate and solve the localization problem. Section VI analyzes positioning information. Section VII presents numerical results, and Section VIII concludes the paper.

System Model

In this section, we describe the directional measurements and the environment-dependent state used for localization. We then specify how reference probes update this state without using unknown user positions.

Propagation Geometry and DoA Measurements

As illustrated in Figure 1, we consider a two-dimensional indoor region $\Omega\subset\mathbb{R}^{2}$ served by $A$ APs at known positions $\mathbf{q}_{a}$, $a=1,\ldots,A$. The AP array orientations and nominal floor plan $\mathcal{M}_{0}$ are known in a common coordinate frame. A user at unknown position $\mathbf{p}_t = [p_{x,t},p_{y,t}]^{T}$ is observed through LoS propagation and a finite set of first-order wall reflections. Let $\ell= 0$ denote the LoS path and $\ell= 1,\ldots,L_a-1$ the reflected-path candidates at AP $a$.

The array front end returns the unordered DoA set

where $z_{am,t}$ and $\sigma_{am,t}^{2}$ denote the measured angle and its noise variance, respectively. For a measurement generated by path $\ell$,

where $\mathcal{M}^{\star}$ is the physical geometry, which need not coincide with $\mathcal{M}_0$. Angles are wrapped consistently to $(-\pi,\pi]$. The numerical implementation uses the equivalent degree convention. The path labels are unknown at query time, and LoS may be absent. We assume resolvable modeled paths and no false detections, so that $M_{a,t} \leq L_a$. This angle-domain model isolates map calibration and positioning from array processing, and it does not establish that all paths are resolvable by a given array. The experiments use a common angular variance, while heterogeneous variances are allowed in the formulation. The epoch index is omitted below when a single query is considered.

Calibrated Directional Field

For a planar reflector, the virtual-user position is

The LoS path corresponds to $\mathbf{H}_{a0}=\mathbf{I}$ and $\mathbf{h}_{a0}=0$. Reflection in a vertical wall $x=x_w$ gives $\mathbf{v}_{a\ell} = [2x_w-p_x,p_y]^{T}$, and a horizontal wall is treated analogously. The corresponding arrival direction is

Instead of estimating every physical wall parameter, we calibrate the direction predicted from the nominal map:

Here, $\boldsymbol{\beta}_{a\ell}$ contains the correction coefficients. For nominal room width $W$ and height $H$, define $u=(p_x-W/2)/W$ and $v=(p_y-H/2)/H$. We consider

The affine field represents angular offset and spatial slope, while the quadratic field also captures curvature. Both preserve the nominal reflection geometry and remain linear in their coefficients. The order $r\in\{1,2\}$, and hence dimension $D\in\{3,6\}$, is selected from calibration data in Section III. We subsequently write $\boldsymbol{\phi}=\boldsymbol{\phi}_r$.

Persistent State and Update Protocol

At the beginning of epoch $t$, the implemented ET stores

where $\mathbf{B}_t=\{\boldsymbol{\beta}_{a\ell,t}\}$, $\lambda_{\beta}$ is the initial ridge strength, and $\mathbf{A}_{a\ell,t}$ is the inverse weighted normal matrix for recursive calibration, not a coefficient-error covariance. Users are first localized with the current field. New reference probes with known positions and path labels then update the coefficients and normal matrices for the next epoch. The same state can therefore serve different users without tracking a particular trajectory.

Only trusted reference probes update the reported ET. Estimated user positions are not reused as calibration labels. Self-calibration would require independently validated position confidence, association consistency, and innovation screening. It is not part of the tested method. Motion tracking, delay measurements [4], and controlled reconfigurable intelligent surface (RIS) paths are likewise extensions, not additional sources of information in the reported results. Setting $\mathbf{B}_t=0$ recovers the nominal map, while freezing $\mathbf{B}_t$ after initial calibration gives the frozen ET. The physical-geometry reference employs $\mathcal{M}^{\star}$ and serves only as a benchmark, not a guaranteed finite-sample error bound.

Sparse-Probe Directional Calibration

We first estimate the correction field from a reference batch, and then derive its recursive update. All model-selection decisions use reference measurements rather than localization errors at test users.

Batch Estimation and Model Selection

Suppose $K$ reference probes provide known positions $\mathbf{p}_k$ and path-labeled angles $\theta^{\mathrm{geo}}_{a\ell,k}$. Their angular innovations relative to the nominal geometry are

Stacking $\boldsymbol{\phi}^{T}(\mathbf{p}_k)$ as the rows of $\boldsymbol{\Phi}_K$ gives the regression model

This is a local angular model. Innovations crossing a wrapping discontinuity require unwrapping or circular regression. Reference path labels are supplied by the simulator, whereas query labels are unknown. Obtaining these reference labels from real measurements remains a separate calibration requirement. For fixed basis and regularization $\lambda_\beta> 0$, we solve

The objective is strictly convex. Setting its gradient to zero yields

Regularization stabilizes a sparse fit, while its strength and the basis order determine the bias–variance tradeoff. We select them jointly by leave-one-out cross-validation (LOOCV) [22]. For candidate $(r,\lambda)$, define the hat matrix

and residual $e_{a\ell} = (I-H_{r,\lambda})d_{a\ell}$. Removing probe $k$ subtracts $\phi_k\phi_k^{T}$ from the normal matrix. The Sherman–Morrison identity therefore gives the held-out residual

Hence one fit per candidate suffices to evaluate

We minimize (14) over $\lambda\in\{10^{-5},\ldots,10^{-1},1\}$ and eligible orders, using one selected pair $(\widehat r,\widehat\lambda)$ for all APs and paths. The affine basis is always eligible, while the quadratic basis requires $K \ge8$, leaving at least seven probes for six coefficients in each held-out fit. This is an eligibility rule, not a coverage guarantee. The selected model is refitted to the full initial batch and then kept at the same order during online operation. With no probes, its correction is zero. LOOCV predicts held-out angles, not unknown user positions. It therefore provides no guarantee of improved localization root-mean-square error (RMSE).

Recursive Update

For a fixed selected basis, subsequent reference probes can be assimilated without repeating the batch fit. Suppressing path indices and indexing individual probes by $k$, RLS [23] with forgetting factor $0 < \lambda_f \le1$ gives

The initial inverse normal matrix is $A_0=(\Phi_K^{T}\Phi_K+\widehat{\lambda}I)^{-1}$, consistent with (11). Thus RLS continues the same fit. For $\lambda_f < 1$, previous normal-equation terms, including the initial ridge contribution, are progressively discounted.

Proof. The weighted normal equation retains the initial state with weight $\lambda_f^k$, and taking expectation yields (19). The most recent $N_{\mathrm{pe}}$ probes imply $G_k\succeq\lambda_f^{N_{\mathrm{pe}}-1}\mu I$. Applying this bound to $G_k^{-1}$ proves the claim. Appendix Appendix § 7.4 gives the full recursion. ■

Proposition 1 concerns decay of initialization bias, not monotonic sample-path RMSE or zero steady-state error. Measurement noise and finite-basis approximation error remain.

Remark 1 (Trusted updates): The implemented recursion accepts only trusted probes and uses no change detector or innovation gate. With uncertain labels, isolated and persistent residuals may indicate outliers and a stale map, respectively, but systematic sensor error can produce either pattern. An uncertainty-based gate would require separate validation and must not be treated as an established benefit of the present ET.

Calibration Accuracy and Identifiability

For fixed basis and ridge, exact reference positions, correct labels, and homoscedastic noise variance $\sigma_c^2$, the sampling covariance is

This follows by substituting (9) into (11). The corresponding ridge bias is $-\lambda_\beta(\Phi_K^{T}\Phi_K+\lambda_\beta I)^{-1}\beta_{a\ell}$. Thus covariance alone is not mean-square error (MSE). Furthermore, (20) holds for a prespecified model. It does not account for choosing the basis and ridge from the same noisy batch. It is used for conditional analysis, not for weighting simulated queries.

Lemma 1 (Identifiable correction field): In the noise-free unregularized model (9), $\beta_a\in\mathbb{R}^D$ is uniquely identifiable if and only if

Proof. Two coefficient vectors produce identical innovations precisely when their difference belongs to the null space of $\Phi_K$. Uniqueness is therefore equivalent to a trivial null space. ■

At least $D$ probes are necessary, but spatial coverage is also needed. For example, collinear probes leave directions unobserved in either basis. Ridge ensures a unique numerical fit without restoring this missing information.

Let $\mu_K=\lambda_{\min}(\Phi_K^{T}\Phi_K)>0$. For a fixed full-rank basis, and an exactly specified error model, and no ridge,

The derivation is provided in Appendix D. This identity explains the role of probe geometry, rather than the MSE of the selected ridge estimator. Once the feature directions are well conditioned, additional probes mainly average noise. They do not remove finite-basis bias. This distinction motivates the probe-budget and wall-mismatch experiments.

Joint Localization and Path Association

Given a calibrated field, our objective is to estimate a user position and identify the paths that generated its unordered DoAs. The field is held fixed during each query.

Association Constraints

Let $x_{am\ell}=1$ if measurement $m$ at AP $a$ is assigned to path $\ell$, and zero otherwise. Each measurement has one label, whereas a path explains at most one measurement:

The inequality permits missing paths, including blocked LoS propagation. Denote the resulting assignment set by $\mathcal{X}$. Unlike probabilistic data association, which retains weighted origin hypotheses [24], we optimize explicit one-to-one labels. Unmodeled detections would require dummy assignments with an outlier penalty. Such detections are excluded from the present experiment.

Problem Formulation

The angular residual for candidate association $(a,m,\ell)$ is

We use the Huber loss

which is quadratic for small errors and grows linearly for large ones. For positive scales $\nu_{am\ell}^2$ fixed throughout a query, the joint problem is

Denote its objective by $f(\mathbf{p},\mathbf{x})$. The experiments use one common scale, which cancels from the minimizer. A weighted extension could evaluate the variance in (39) at a coarse initializer and freeze the weights, but is not tested here.

Problem (26) is nonconvex due to both nonlinear angle geometry and discrete association. Nevertheless, fixing position gives a linear assignment problem, and fixing association leaves a two-dimensional continuous problem. We exploit this structure below. The convexity of the Huber loss in its scalar argument does not make the joint problem convex.

Proposed Localization Algorithm

We initialize several candidate positions and alternate exact path assignment with robust position refinement. All candidates minimize the same objective in (26). Algorithm 1 summarizes the procedure.

Assignment-Aware Initialization

On a grid $\mathcal{G}\subset\Omega$, precompute $\theta_{a\ell}^{\mathrm{ET}}(\mathbf{g})$ for all AP–path pairs. A low-cost nearest-path initializer is

This score permits several measurements to select the same path and differs from the robust query objective. We retain its solution as one start, but rank additional starts using

Starting with $\mathbf{p}_{\mathrm{near}}^{(0)}$, we scan grid points in ascending order of $F$ and retain a point only if it is at least 2 m from every existing start, until $S\le3$ starts are obtained. This separation diversifies the local search rather than detecting environmental change. Each start receives the same refinement budget. For their final positions $\hat{\mathbf{p}}_s$, select

The precomputed angles remain valid until the field is updated, whereas assignment costs are query dependent. Nominal, calibrated-wall, Gaussian-process (GP), and physical-geometry maps use the same candidate rule and budget.

Alternating Assignment and Position Updates

Path assignment

1) Path assignment: For fixed $\mathbf{p}^{(i)}$, each AP independently minimizes its sum of costs $\rho_\delta(r_{am\ell}(\mathbf{p}^{(i)}))/\nu_{am\ell}^2$ subject to (23). This rectangular assignment can be solved by the Hungarian algorithm [25]. With four candidate paths, our implementation instead enumerates at most $4!=24$ injections. Let $\hat{\ell}_{am}^{(i)}$ denote the resulting labels.

Position refinement

2) Position refinement: For fixed labels, stack the residuals into $\mathbf{r}(\mathbf{p})$ and define

For a displacement vector $\mathbf{s}\in\mathbb{R}^2$, $\mathbf{r}(\mathbf{p}+\mathbf{s})\simeq\mathbf{r}(\mathbf{p})-\mathbf{J}\mathbf{s}$ locally. Iteratively reweighted least squares [26, 27] adopts

with its continuous limit at $r_j^{(i)}=0$. Writing $\mathbf{W}^{(i)}=\operatorname{diag}(w_j^{(i)})$, the damped linearized subproblem is

The minimizing displacement, denoted by $\Delta\mathbf{p}^{(i)}$, satisfies

Algorithm 1 ET-Assisted Multipath Localization
Require: DoAs, fixed calibrated field, grid, and query scales.
Ensure: Position $\hat{\mathbf{p}}$ and path labels.
1: Retain {xref}`(27)` and add separated starts using {xref}`(28)`.
2: **for** each retained start **do**
3:     **repeat**
4:         Minimize assignment cost for fixed position.
5:         Form weights {xref}`(31)` and direction {xref}`(33)`.
6:         Project and backtrack using {xref}`(34)`.
7:     **until** the stopping tolerance or iteration limit is met
8:     Store the refined position and its cost $F$.
9: **end for**
10: Select {xref}`(29)` and return its minimizing labels.

The geometric Jacobian is derived in Appendix A. Because (33) is a local approximation, we test the original robust cost by backtracking:

A trial is accepted only if the fixed-assignment objective does not increase. The implementation caps the direction norm at 1.5 m and tests up to eight successively halved steps. If none is accepted, position is unchanged. Each start stops after ten iterations or displacement below $10^{-4}$ m. Assignment, refinement, and acceptance use the same loss. The linearized solve alone is not a global descent guarantee.

Convergence and Complexity

Proposition 2 (Objective descent): If assignment is solved exactly and each position update is accepted only when its fixed-assignment cost does not increase, the sequence of objective values is nonincreasing and convergent.

Proof. Exact reassignment and accepted backtracking respectively imply

The objective is nonnegative, so the scalar sequence converges.

This result does not establish convergence to a globally optimal position or to a stationary point under all stopping rules.

Corollary 1 (Retained-start safeguard): For a fixed map, loss, and per-start budget, let $\mathbf{p}_{\mathrm{near}}$ be the refined result from (28) alone. The proposed selection satisfies

Proof. That same refinement is included among the candidates in (30).

A smaller fitting cost need not imply smaller position error under noisy or mismatched angles. The rule is fixed before evaluating queries.

For $G$ grid points, $A$ APs, and at most $L$ paths and $M\leq L$ observations, nearest-path scanning costs $O(GALM)$. Exact assignment adds $O(GAL^3)$ with Hungarian matching, or $O\left(GAML!/(L-M)!\right)$ with the enumeration used here. Sorting costs $O(G\log G)$ and candidate screening costs $O(GS)$. Each of the $S\leq3$ local solves uses a $2\times2$ linear system plus assignment and line-search evaluations per iteration. The improved initialization therefore has an explicit computational cost. The field stores $D\in\{3,6\}$ coefficients per path, and each RLS path update costs $O(D^2)$. These dimensions support the small-path implementation but do not establish scalability of enumeration.

Positioning Information Analysis

We next relate calibration quality and reflected-path geometry to local positioning information. The analysis assumes correct association and a specified error model. It is distinct from the robust estimator’s finite-sample performance.

Angular Uncertainty and Fisher Information

For the fixed-model coefficient covariance in (20), the induced angle variance is

If front-end, map, and residual model errors are independent, their variances add:

Here, the last term represents random residual error, not deterministic wall bias. Appendix B separates variance and squared bias. This uncertainty model is not used to weight the simulated queries.

For an available correctly associated AP–path pair $(a,\ell)\in\mathcal{A}$, define

Assume independent locally Gaussian errors away from wrapping discontinuities, with mean $\theta_{a\ell}^{\mathrm{cal}}(\mathbf{p})$ and variance $\nu_{a\ell}(\mathbf{p})=\sigma_{\mathrm{eff},a\ell}^{2}(\mathbf{p})$. Freezing the variances locally gives the mean-gradient information

This is the full Gaussian Fisher information matrix (FIM) for position-independent variances, as in the ideal-geometry experiment. Otherwise, the full Gaussian information is (29)

Here, $\boldsymbol{\eta}_{a\ell}(\mathbf{p})=\nabla_{\mathbf{p}}\nu_{a\ell}(\mathbf{p})$ is the variance-gradient vector. For nonsingular $\mathbf{F}$ and constant variances, the ideal-model position error bound (PEB) is

As the Cramér–Rao bound (CRB), it applies to locally unbiased estimation under the assumed likelihood.

Reflected-Path Information and Local Identifiability

Proof. The added outer product is positive semidefinite, and inversion reverses positive-definite order. Appendix C gives the corresponding trace reduction. The variance-gradient contribution of an additional independent path in (42) is also positive semidefinite.

For positive finite path variances, (41) further implies

A single LoS bearing supplies rank-one information and cannot determine range. Reflections can supply noncollinear gradients because their mirror transforms differ, enabling local identifiability even with one AP. The gain depends jointly on gradient orientation and calibration variance: a path with a large modeled variance contributes less for a fixed gradient.

Remark 2 (Scope of the information result). Full rank is a local condition, not global uniqueness under unknown associations. Correlated errors require a joint covariance, and deterministic bias cannot be absorbed into a variance term. Consequently, the ideal CRB does not certify the biased Huber estimator, path matching, or robustness to unmodeled reflections. We report both ideal PEB and actual localization RMSE to distinguish these effects.

Numerical Results

In this section, we evaluate calibration efficiency, localization under blocked or corrupted paths, and adaptation to environmental change. All comparisons use the same observations and solver budgets where applicable.

Scenario and Baselines

We consider the $20 \times14$ m nominal room in Figure 1. Two APs are located at $[3, 2]^{T}$ and $[17, 2]^{T}$ m. Each AP has one LoS candidate and reflections from the left, right, and top walls. The physical room is bounded by the left, right, and top reflection planes at $x = 0.35$ m, $x = 19.25$ m, and $y = 14.55$ m, respectively. Calibration probes have known positions and path labels, with independent $0.25^\circ$ angular noise. All calibration and localization positions are drawn from the same fixed rectangle $[1.5, 17.5] \times[3, 12.5]$ m, contained in every tested room. Changing physical walls therefore does not change the query population or place queries outside the optimizer’s feasible rectangle $\Omega= [0.5, 19.5] \times[0.5, 13.5]$ m. The ET selects the affine or quadratic correction in (6) and its ridge strength using (14). Selection uses only the labelled reference batch. For an online history, the selected basis is fixed before the first user query and shared by every update policy.

Unless stated otherwise, the DoA standard deviation is $1.5^\circ$, LoS blockage probability is $50\%$, and 20 calibration probes update the ET. Every point averages 500 independently drawn user positions, blockage events, angle errors, and path permutations, conditional on its calibration batch. Query realizations are paired across algorithms and reused across sweep points where applicable. The probe-count sweep uses exact prefixes of the same position and noise pool, rather than merely resetting a seed. No curve is smoothed or forced to be monotone. Current DoAs are generated from physical geometry plus Gaussian front-end error. Array-level validation using spatial channel models such as QuaDRiGa and 3GPP TR 38.901 [29], [30] remains outside this study. Table I collects the default parameters.

Parameter	Value	Parameter	Value
Room size	20 × 14 m	APs	2
Paths/AP	4	DoA std.	1.5°
Blockage	50%	Probes	20
Grid step	1 m	Realizations	500
Huber threshold	3°	GN iterations	10
Candidate starts	3	Separation	2 m
Field dimension	3 or 6	Online forgetting	0.96

The compared methods are:

• LoS + fallback: bearing intersection with known surviving LoS labels when at least two independent bearings are present, and a room-center decision otherwise. The latter is a fallback policy, not a successful angle-based position estimate.

• **static geometry: all available paths and nominal walls, without physical calibration.

• updated ET: nominal walls plus sparse-probe correction.

• **oracle geometry: all paths and physical wall parameters.

All map-based methods use the same Huber loss for assignment and position refinement, the same grid, three candidate starts with 2-m separation, ten iterations per start, and a common angular weight. The simulations do not enable covariance weighting, dummy outlier assignments, or innovation screening. Thus gains should be attributed to calibrated angle predictions and continued trusted-probe updates, not to these untested extensions. The RMSE sweeps use linear axes starting at zero. LoS availability and all-user positioning success are reported separately, rather than allowing the room-center fallback error to set the scale of every map-accuracy plot. Complete four-method RMSE data and plots remain available in the reproducibility files.

The experiments address three distinct questions: whether physical calibration removes nominal-map bias, whether reflected paths preserve positioning service under LoS blockage, and whether new probes correct a stale map after an environmental change. None of these questions requires the compact ET to outperform every calibrated representation. We therefore include strong geometric and nonlinear-map baselines in the main noise and probe-budget sweeps, rather than use separate, repeated sweeps to make the same comparison. The robust-loss experiment isolates a solver property, not an advantage unique to the ET representation.

Compared Representations

The comparison distinguishes measured fingerprints, model-generated multipath references [9], and learned spatial fields. The implementations, summarized in Table II, share this paper’s unordered-angle interface. They are not claimed to reproduce the array-spectrum processing, delay observations, or tracking systems of other papers.

Method	Calibration and prior	Main comparison
Measured WKNN	Probe angles/positions without floor plan	Direct fingerprint lookup
Wall-NLS	Same probes and nominal wall model	Correct geometry parameters
GP map	Same probes and nominal wall model	Nonlinear residual field
ET-LS	Same selected ET field	Quadratic query loss
ET-Cauchy	Same selected ET field	Alternative robust loss
ET-Huber	Same selected ET field	Reported query solver

Measured weighted K-nearest-neighbor (WKNN) fingerprinting stores the observed angle set at each reference location. Its distance $D_j$ is the average squared wrapped-angle difference after minimum-cost one-to-one matching of each query set to reference $j$. An absent LoS measurement need not be imputed. The estimate is

where the floor prevents division by zero. The neighbor count is selected from $\{1,3,5\}$ by leave-one-reference-out position error, averaging the four LoS-visibility patterns of the two APs. No test-user position is used. Unlike a dense fingerprint database, this baseline is given the same sparse physical probe budget.

Wall nonlinear least squares (Wall-NLS) calibrates the left, right, and top reflection-plane coordinates $\mathbf{w}$ directly:

The room must contain the known reference positions. A three-variable Nelder–Mead search starts from the nominal floor plan, without access to physical-wall truth. The fitted geometry then applies the same Huber association and refinement as the ET. This is a particularly strong baseline here: its parameterization matches the simulator’s planar reflection model.

Gaussian-process (GP) residual mapping replaces the finite-order correction with a squared-exponential spatial kernel. For each labelled path, its posterior-mean correction is [31]

The empirical mean and hyperparameters use calibration data only. Kernel lengths are selected from $\{2,4,8,16,32\}$ m and signal variances from $\{0.25,1,4,16\}\max\{\operatorname{var}(\mathbf{d}),\sigma_c^2\}$ by training marginal likelihood. Query positions are estimated by the same Huber solver. Predictive variance is not used to introduce a second change to the comparison.

ET-LS and ET-Cauchy keep the selected calibration unchanged and replace only the angular query loss. The alternatives are

respectively, with $c=3^\circ$. Each of the assignment, iteratively reweighted refinement, and backtracking stages uses the selected loss consistently. These are ablations of the query solver, not independent new map architectures. The nominal Huber map and physical-geometry Huber reference remain in the comparison. The “Selected ET” legend denotes ET-Huber with the field chosen by (14), not a per-query selection among competing methods.

The noise and probe-budget sweeps below use the same fixed 80-probe pool and 500 held-out queries per point. Probe budgets are exact nested prefixes, including measurement noise. Every method sees identical query positions, angles, and blockage events, while the map-based solvers use the same initialization grid, multistart rule, and iteration budget. WKNN selection, GP fitting, and ET model selection use calibration data only. The other controlled sweeps retain their original paired samples. Numerical differences across figures should not be interpreted as comparisons on one identical calibration batch.

Effect of DoA Noise

Figure 2 distinguishes two regimes. At low DoA noise, the nominal map is limited by geometry mismatch, whereas physical calibration substantially reduces this bias. At higher noise, the calibrated-map curves approach one another because angular measurement error increasingly controls the position estimate. Improving the map and the DoA front end therefore address different error sources. The ET lies close to the calibrated geometric and GP alternatives, but does not dominate them: at $1.5^\circ$, Wall-NLS, GP mapping, and ET-Huber give

0.373, 0.379, and 0.380 m, respectively. In particular, Wall-NLS is well matched to the three-plane simulator. This figure supports the use of a calibrated directional state, not a unique accuracy advantage of its polynomial parameterization. At high noise, choosing a richer field or a lower query objective need not improve position RMSE. The selection criterion does not optimize unknown test locations.

The common-batch association diagnostic gives correct-association probabilities above 94% for the nominal, calibrated, and physical-geometry maps even at 4° DoA error, with little separation between them. Thus, the calibration gain mainly reflects more accurate predicted angles, not a substantially better combinatorial matcher.

LoS Blockage

In Figure 3, static geometry and the calibrated ET obtain 0.55 and 0.39 m RMSE at 50% LoS blockage. With complete LoS blockage, their errors are 0.65 and 0.49 m. The relevant gain is the ability to use reflected directions after LoS constraints disappear, together with the correction of nominal-wall bias.

LoS performance must be interpreted separately. With two independently blocked LoS links and per-link blockage probability $b$, bearing availability is $(1-b)^2$, not one. At $b=0.5$, empirical availability is 27.60%, and conditional RMSE on those available cases is 0.42 m. The all-query value 4.55 m includes room-center fallback on unavailable cases. Its large value cannot be presented as the error of a functioning LoS estimator. Availability and conditional RMSE are exported separately with the numerical results.

Figure 4 asks a different, service-oriented question: what fraction of users can be positioned within a specified tolerance? For the same 500 queries used in Figure 3, we report

The 0.5-m tolerance is an illustrative indoor requirement, not a claimed standard or an optimized threshold. All users remain in the denominator, including unavailable LoS cases.

Even a coincidentally accurate fallback counts as a success. At 50% blockage, LoS with fallback, the nominal map, and the calibrated ET give 21.20%, 58.20%, and 84.80%, respectively. With all LoS paths blocked, the latter two give 40.00% and 70.20%. This comparison separates the value of retaining reflected-path constraints from the additional value of calibrating their directions. The ideal-geometry reference remains visible, so that the remaining calibration gap is not hidden.

Calibration Budget and Model Mismatch

Figure 5 asks how effectively each representation uses the same physical probes. The calibrated model-based methods work with relatively few probes, whereas sparse fingerprint lookup improves more gradually as reference locations cover the room. At 80 probes, WKNN gives 0.768 m, compared with 0.372 m for the GP and 0.384 m for the selected ET. This difference reflects both spatial interpolation and prior information: WKNN is not supplied with the floor plan. It is not evidence of superiority over dense fingerprinting.

The more important comparison is within the calibrated maps. Wall-NLS employs a correct physical parameterization, GP mapping learns a flexible residual field, and the ET retains only three or six coefficients per path. In this nested calibration pool, LOOCV selects the affine field with $\lambda_\beta= 0.1$ at six probes, and the quadratic field with $\lambda_\beta= 0.01$ at the other budgets. This change is decided from held-out angular innovations, not localization errors. Their close errors show that a compact correction captures much of the useful map information in this scene. A correctly specified wall model can still do better. Additional probes reduce estimation uncertainty but do not remove query noise or all approximation error. Small nonmonotonic changes are retained. Noisy finite calibration does not guarantee monotonically decreasing test RMSE.

Figure 6 scales physical wall displacement from zero to twice its default value. At the largest mismatch, static geometry gives 1.01 m and the calibrated ET gives 0.38 m. At zero mismatch, calibration is unnecessary and can introduce an estimation cost by fitting probe noise. Thus the useful regime is an informative but imperfect floor plan, not an already exact map. Increasing angular noise can likewise conceal the value of improved geometry, as Figure 2 demonstrates. Each scene receives its own trusted calibration probes, so this is a recalibration test rather than zero-shot generalization to unseen walls. The remaining gap to the perfect-map reference reflects both finite probe noise and the limited spatial flexibility of the correction field.

Robustness to Gross Angle Errors

Figure 7 isolates robustness from representation. Each observed query angle independently receives an additional zero-mean $20^\circ$-standard-deviation Gaussian perturbation with probability $p_o \in\{0, 0.05, 0.10, 0.20\}$, on top of the common $1.5^\circ$ noise. The physical probes remain clean, the path count is unchanged, and all methods receive exactly the same corrupted observations. This controlled gross-angle-error test is not a simulation of false detections or unresolved coherent sources. At $p_o = 0.20$, quadratic, Huber, and Cauchy query losses give 1.932, 1.384, and 0.973 m RMSE. The three losses are

close without gross errors, so Gaussian-noise curves alone provide little evidence for a robust-loss advantage. In this contamination test, Huber improves on quadratic loss, while the fixed-scale Cauchy alternative yields the lowest RMSE at the tested nonzero contamination levels. Its redescending influence suppresses very large angular residuals more strongly. Huber is retained as the fixed default throughout the other experiments, not switched to Cauchy after inspecting contaminated test cases. A convex residual penalty is convenient, but it does not make the joint position-and-association problem convex or make Huber uniformly optimal. This is a conditional robustness comparison, not an optimality claim across noise models or tuning choices. The residual error remains substantial because corrupted angles can still mislead the initialization or path assignment. Robust refinement does not solve arbitrary outlier association.

Number of APs and Paths

Figure 8 tests whether reflected paths compensate for fewer APs. With one AP, a single LoS bearing does not identify range. Its all-query baseline comes entirely from the fallback policy, and is omitted from this map-only RMSE plot. The calibrated multipath map instead gives 0.65 m because reflected-path gradients can supply independent spatial constraints. Three APs reduce its error to 0.27 m. This result depends on the known reflector model and resolvable reflected paths. It does not imply generic single-AP identifiability in an arbitrary environment.

The information calculation in Figure 9 uses correct associations and physical geometry. Mean PEB decreases from 0.45 m with one path per AP to 0.27 m with all four. The monotonic trend matches Proposition 3. These values use all retained paths without random LoS blockage and therefore must not be read as matched lower bounds on the blocked, biased, unknown-association experiments. They isolate the effect of additional ideal geometric constraints.

Persistent Recovery after Environmental Change

The preceding experiments recalibrate the ET for each fixed environment. Figure 10 instead retains one state across 30 epochs. Online and frozen ETs start from the same 12-probe batch calibration. At epoch 16, the left, right, and top reflection planes change to $x = 0.72$ m, $x = 18.55$ m, and $y = 15.15$ m. Localization and map evaluation occur before that epoch’s two new trusted probes are assimilated, so neither method can anticipate the change. RLS uses $\lambda_f = 0.96$ and the initial batch normal matrix, with no innovation gate or change-triggered reset. Its initially selected basis remains fixed throughout the 30 epochs. The online map error changes from 2.03° at the change to 0.90° at the final epoch, compared with 1.91° for the frozen map. Their different errors at the change reflect their different pre-change coefficients, not Monte Carlo variation in the deterministic map metric.

For the same calibration sequence, position RMSE decreases from 0.65 m at epoch 16 to 0.43 m at epoch 30, versus 0.63 m for the frozen map and 0.99 m for nominal geometry. Each epoch uses 500 queries paired across methods and epochs through common random numbers. This is one evolving calibration history, not 500 independent map histories or a tracked user trajectory. The next experiment tests localization over independent update histories.

Remark 3 (Querying without self-calibration): Current user DoAs are always used for positioning, but only trusted probes are used to modify the map in this experiment. A high-confidence user estimate would require the additional tests discussed in Section II-C. Keeping the query path and update path separate is essential: a doubtful position may benefit from the ET without being allowed to corrupt future positions.

Comparison of Online Update Policies

An online-versus-frozen comparison alone cannot establish an advantage over other ways of using new probes. We therefore compare three updates of the same correction field: all-history least squares (LS), LS on the most recent 12 probes, and RLS with $\lambda_f = 0.96$. All start from the same 12-probe calibration and receive the same two new labelled probes after each epoch. The frozen map and physical-geometry reference are retained as controls. The query grid, association, robust refinement, and DoA observations are identical for every method.

For all-history and sliding-window LS, let $\mathcal{I}_t$ contain all available probes or the last 12, respectively. Each solves

Here, $\hat{\lambda}$ and the basis are selected from each history’s initial batch and then shared and frozen across all update policies. The RLS recursion instead progressively discounts older information. Its initial ridge contribution decays with the initial normal matrix. There is no oracle change detector, reset at epoch 16, or access to future probes. The window length and forgetting factor are fixed rather than selected to favor one method on the test queries. All three adaptive methods implement the persistent-state framework. The comparison tests memory policies, not an ET label against otherwise identical LS.

Figure 11 averages squared error over 10 independent calibration/query histories with 50 queries each at every epoch, then takes the square root. All histories use the same prescribed wall-change schedule. They are not 10 different environments. The first 15 epochs provide an unchanged-scene control, while epochs 16–30 test the use of stale measurements. The close pre-change curves are a useful control: updating an already accurate state should not by itself create a large gain. Once the walls change, the frozen state retains an outdated angular field. All-history LS incorporates new evidence, but old observations continue to pull its fit toward the previous environment. RLS reduces this influence gradually, whereas the 12-probe window removes old evidence sooner. The post-change RMSE is 0.45 m for the window and 0.49 m for RLS.

Remark 4 (Representation versus memory policy): The frozen-versus-adaptive comparison supports maintaining the stored path-angle state. Differences among adaptive curves concern memory policy, not the ET representation itself. The configured RLS is not the optimum among the tested policies for this abrupt change. A short window discards stale evidence faster but also loses potentially useful probes, whereas RLS discounts them gradually. The present result does not determine the best policy for gradual motion, noisier probes, or other scenes.

Conclusion

This paper formulated multipath DoA localization as a query to a persistent electromagnetic state. Nominal virtual-anchor geometry provided physical structure, calibration-only model selection set the correction-field order and ridge strength, and continued trusted updates maintained that correction. Assignment-aware multistart search retained the original initialization while examining other association basins. Joint association and robust position refinement remained computationally simple because the path set was small and the continuous variable was two dimensional. The analysis and simulations support a precise conclusion: correctly associated reflected paths can add positioning information and preserve rank under LoS blockage, but their benefit depends on physical calibration.

Virtual-Path Jacobian

Let

For angle $\theta=\operatorname{atan2}(d_y,d_x)$,

Therefore,

and the ET correction adds $\nabla_{\mathbf{p}}(\boldsymbol{\phi}^{T}\boldsymbol{\beta}_{a_\ell})$. For the quadratic basis, suppressing path indices and numbering coefficients from one, the additional derivatives are

The affine case follows by setting $\beta_4=\beta_5=\beta_6=0$. The geometric derivatives above are in radians, while correction derivatives inherit the units of their coefficients. Both terms must use the same unit before addition. The implementation uses degrees for angular derivatives and noise standard deviations, preserving the same information after consistent unit conversion. The implementation employs symmetric numerical differentiation so that the same solver also supports the quadratic correction and the GP baseline.

Map-Uncertainty Propagation

Let $\widehat{\boldsymbol{\beta}}_{a_\ell}=\boldsymbol{\beta}_{a_\ell}+\widetilde{\boldsymbol{\beta}}_{a_\ell}$, with error mean $\mathbf{b}_{a_\ell}$ and covariance $\mathbf{P}_{\beta,a_\ell}$. Linear correction error at position $\mathbf{p}$ is

Hence

The mean-square error additionally contains $(\boldsymbol{\phi}^{T}\mathbf{b}_{a_\ell})^2$. If front-end angle error, map error, and residual model error are independent, their variances add, giving (39). A nonzero mean remains a separate bias and is not removed by variance weighting.

Information Monotonicity

Let $\mathbf{F}_0\succ\mathbf{0}$ be information from an existing correctly modeled path set and $\Delta\mathbf{F}\succeq\mathbf{0}$ be (44). Then

Matrix inversion reverses the positive-definite order:

Taking trace proves that the squared PEB cannot increase. For one finite-variance nonzero gradient, the reduction is strictly positive. The matrix inversion lemma gives

Equality requires $\mathbf{g}=\mathbf{0}$ for finite $v$, or is approached as $v\to\infty$. The rank condition for an initially singular matrix must be checked separately.

Appendix D: Calibration Rank, Error, and Recursive Form

For the unregularized batch model, suppose two coefficients $\beta_1$ and $\beta_2$ explain the same noise-free probe innovations. Then

Uniqueness requires a trivial null space of $\boldsymbol{\Phi}_K$, equivalently (21). Under full column rank, ordinary LS has error

and covariance $\sigma_c^2(\boldsymbol{\Phi}_K^{T}\boldsymbol{\Phi}_K)^{-1}$. Taking the trace and using $\operatorname{tr}[(\boldsymbol{\Phi}_K^{T}\boldsymbol{\Phi}_K)^{-1}]\le D/\mu_K$ proves (22).

For recursive operation, define the exponentially weighted normal matrix

Applying the matrix inversion lemma to $\mathbf{A}_k=\mathbf{G}_k^{-1}$ yields the gain and inverse-normal recursion in (17). Persistent excitation prevents $\mathbf{G}_k$ from losing rank. The right-hand side of the normal equation is $\mathbf{h}_k=\lambda_f^k\mathbf{G}_0\widetilde{\beta}_0+\sum_{i=1}^{k}\lambda_f^{k-i}\boldsymbol{\phi}_i d_i$. Substituting $d_i=\boldsymbol{\phi}_i^{T}\beta^\star+\epsilon_i$ and taking conditional expectation results in (19). For independent noise, the coefficient sampling covariance instead requires

This covariance assumes independence between the initial fit and subsequent probe noise and excludes deterministic model bias. Thus the RLS inverse normal matrix alone is not a calibrated uncertainty estimate.