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POPT: Physics-Informed Deep Learning for Phase-Only Positioning in Distributed MIMO

Fatih Ayten, Ossi Kaltiokallio, Jukka Talvitie, Akshay Jain, Mehmet C. Ilter, Musa Furkan Keskin, Elena Simona Lohan, Henk Wymeersch, Mikko Valkama

arXiv:2609.31218v1eess.SP

TL;DR

Narrowband phase-only D-MIMO positioning must handle unresolved ground reflections, which challenge methods designed for LoS-only channels. The paper combines GP-learned phase perturbations and synthetic training data with a transformer positioning model. With 50 GP training locations, it reports accuracy comparable to MLE at substantially lower computation and runtime.

  • Problem

    Prior phase-only D-MIMO approaches largely assume LoS-only propagation, although narrowband observations coherently superimpose unresolved reflected and LoS components.

    Source: A. Related Works and CPP Prior Art · p. 2

  • Method

    A GP models quasi-periodic reflection-induced phase perturbations and generates synthetic training data for POPT, an encoder-only transformer for phase-only positioning.

    Source: §VII · pp. 11–12

  • Results

    With L=50, POPT attains comparable accuracy to MLE using 81.10 × 10^6 FLOPs and 0.95 ms runtime versus MLE's 3771.04 × 10^6 FLOPs and 3716.05 ms.

    Source: E. Computational Complexity Results · p. 11

  • Takeaways & Limitations

    POPT's 0.95 ms runtime is below the reported 1–10 ms end-to-end service latency target for real-time digital twins.

    Source: E. Computational Complexity Results · p. 11

Abstract

from arXiv · show

This article develops a physics-informed deep learning framework for single-snapshot 3D phase-only positioning in narrowband phase-coherent distributed multiple-input multiple-output (D-MIMO) networks under two-ray propagation. Unlike prior phase-coherent D-MIMO localization methods that largely assume line-of-sight (LoS)-only channels, we consider a LoS path and a specular ground reflection. Since the narrowband single-snapshot observation cannot resolve these components in delay, their coherent superposition induces structured perturbations in the carrier phase measurements. To address this, we develop a Gaussian process (GP)-based model that learns the quasi-periodic phase distortion caused by ground reflection from carrier phase measurements collected at a small number of training locations and uses it to generate high-quality synthetic samples. Building on these samples, we propose the Phase-Only Positioning Transformer (POPT), an encoder-only transformer that captures inter-antenna point (AP) phase relationships without solving the highly non-convex maximum-likelihood (ML) problem underlying model-based estimators. We also derive the fundamental position error bound (PEB) and develop maximum-likelihood estimation (MLE) for the considered multipath phase-only D-MIMO positioning problem. Numerical results show that, with only 50 GP training locations, the proposed method achieves near-PEB accuracy. The analytical MLE is highly sensitive to relative permittivity mismatch, whereas the proposed method mitigates this dependence by learning the phase perturbation directly from measurement data. Compared with MLE, the proposed approach also reduces floating-point operation (FLOP) complexity and inference time by about 1.7 and 3.6 orders of magnitude, respectively.

I. INTRODUCTION

The introduction motivates phase-only positioning as a route to high precision with limited bandwidth, while identifying unresolved multipath and data requirements in prior D-MIMO methods. It presents a two-ray, single-snapshot 3D framework combining shared Gaussian-process modeling, synthetic data, and transformer-based positioning.

I. INTRODUCTION

  • 6G use cases such as cooperative robots and digital twins require better than the roughly 10 cm accuracy available under favorable 5G-Advanced conditions.Time- and angle-based methods face bandwidth and antenna-aperture limits, respectively.Source: §I · p. 1

A. Related Works and CPP Prior Art

  • Prior phase-only D-MIMO methods are limited by LoS-only, 2D assumptions, while narrowband observations coherently superimpose unresolved reflections with LoS.Earlier approaches also face large labeled-data needs or deployment-dependent coverage; related SLAM work does not directly position the UE from phase alone.Source: A. Related Works and CPP Prior Art · p. 2

B. Contributions and Novelty

  • A shared GP models ground-reflection phase perturbations across APs from limited labeled locations and generates synthetic phase-only training samples.The framework targets quasi-periodic perturbations and reduces the need for dense labeled coverage.Source: B. Contributions and Novelty · p. 3
  • POPT uses an encoder-only transformer to capture inter-AP phase relationships, alongside a derived PEB and a phase-only MLE baseline.The MLE is described as theoretically grounded but computationally complex.Source: B. Contributions and Novelty · p. 3

II. SYSTEM MODEL, BOUNDS, AND MLE

The system model uses synchronized distributed APs to observe a narrowband uplink under two-ray propagation. Its phase measurements combine LoS and ground-reflected paths, with an isotropic antenna-gain assumption for analytical simplicity.

A. System Model

  • A single-antenna UE transmits a narrowband pilot to synchronized single-antenna APs, which forward measurements to a coordinating DLMF.The bandwidth is assumed too small to resolve LoS and NLoS components in delay.Source: A. System Model · p. 3
  • Under the two-ray model, the received signal includes a LoS path and a single-bounce ground-reflected path, whose coherent combination yields one carrier-phase observation per AP.The phase measurements are wrapped, and their model includes a common phase offset and measurement noise.Source: A. System Model · pp. 3–4
  • The analytical model assumes isotropic antenna gains, although directional patterns and device orientations may alter path amplitudes, effective phase, and phase-noise variance.Incorporating orientation-dependent antenna patterns is identified as future work.Source: A. System Model · p. 4

B. Problem Formulation

The positioning task estimates the unknown UE location from a single vector of wrapped carrier-phase observations. Wrapped phases, a shared unknown offset, and nonlinear two-ray propagation make the estimation problem challenging.

B. Problem Formulation

  • The objective is to estimate the UE’s unknown 3D position from the AP carrier-phase vector rather than the full received-signal vector.The framework targets low-complexity, high-accuracy single-snapshot positioning.Source: B. Problem Formulation · p. 4
  • Wrapped measurements create integer ambiguity, while the unknown common phase offset couples observations across APs.The offset is common to the AP network.Source: B. Problem Formulation · p. 4
  • The two-ray phase mapping is highly nonlinear and depends on reflection properties, making the likelihood non-convex and dense-grid MLE computationally costly.The proposed learning-based method does not require prior knowledge of relative permittivity or polarization; the analytical PEB and MLE do.Source: B. Problem Formulation · p. 4

C. Performance Bounds

The performance-bound derivation forms a Fisher information matrix for channel parameters, eliminates nuisance parameters to obtain information about phase, then maps it to position.

C. Performance Bounds

  • The Fisher information matrix is formed for the channel parameter vector containing the common phase offset, AP amplitudes, and AP phases.The derivation partitions the parameters of interest from nuisance parameters.Source: C. Performance Bounds · p. 4
  • The equivalent Fisher information for AP phases is obtained by applying a Schur complement to eliminate nuisance parameters.The nuisance parameters are the common phase offset and AP amplitudes.Source: C. Performance Bounds · p. 4
  • A Jacobian transforms phase information into UE-position information, from which the position error bound is computed.This bound presumes perfect knowledge of relative permittivity and polarization through the reflection coefficient and Jacobian.Source: C. Performance Bounds · p. 5

D. Maximum Likelihood Estimator (MLE)

The MLE removes the unknown common phase offset by estimating it from the likelihood, leaving a position-only optimization problem. Because its cost is highly non-convex, the estimator uses a least-squares grid search for initialization before gradient-descent refinement; both steps presume correct reflection-material knowledge.

D. Maximum Likelihood Estimator (MLE)

  • The likelihood uses a Gaussian approximation to the von Mises distribution under small zero-mean Gaussian measurement noise and independent measurements.These assumptions support the stated likelihood formulation.Source: D. Maximum Likelihood Estimator (MLE) · p. 5
  • Wrapped phase residuals are corrected for boundary crossings before evaluating the position-dependent likelihood.When the CPO is near ±π, uncorrected residuals can cross the wrapping boundary.Source: D. Maximum Likelihood Estimator (MLE) · p. 5
  • Eliminating the unknown CPO by maximizing the likelihood leaves an optimization problem that depends only on UE position.The CPO estimate is obtained by differentiating the log-likelihood with respect to the CPO, setting the derivative to zero, and solving.Source: D. Maximum Likelihood Estimator (MLE) · p. 5
  • The non-convex cost is sensitive to initialization, so the estimator first searches a predefined position grid using LS and then refines with gradient descent.The gradient-descent estimate is initialized from the grid-search result.Source: D. Maximum Likelihood Estimator (MLE) · p. 5
  • The MLE and LS estimator presume perfect knowledge of relative permittivity and polarization through the analytical phase model.The article presents this multipath phase-only D-MIMO MLE as not previously reported.Source: D. Maximum Likelihood Estimator (MLE) · p. 5

III. GAUSSIAN PROCESS CARRIER PHASE MODEL

The GP carrier-phase model uses limited measurements to learn a shared data-driven model, avoiding the need to know the material’s polarization type and relative permittivity. Its covariance is designed to reflect the wavelength-linked, quasi-periodic phase perturbation produced by the NLoS path.

III. GAUSSIAN PROCESS CARRIER PHASE MODEL

  • The GP model learns carrier-phase measurements from limited data instead of requiring known polarization type and relative permittivity in the theoretical phase model.The model bridges the propagation model and the learning-based positioning method.Source: §III · p. 5, A. Proposed GP Model · p. 5

A. Proposed GP Model

  • The phase model represents the observation as wrapped LoS carrier phase and CPO, an NLoS phase perturbation, and input-dependent measurement noise.A zero-mean GP prior is assigned to the unknown perturbation function because relative permittivity and polarization are assumed unknown.Source: A. Proposed GP Model · p. 5
  • A shared GP prior across APs supports sparse-data training under the assumptions of known reflection-surface normal and homogeneous dielectric properties.If those assumptions do not hold, each AP would need its own prior.Source: A. Proposed GP Model · pp. 5–6
  • The covariance uses path-distance difference and incidence angle as regressors, allowing wavelength-periodic phase fluctuations whose shape varies quasi-periodically.The periodic component has period set by the wavelength; the quasi-periodic shape can change with path-distance difference and incidence angle.Source: A. Proposed GP Model · p. 6

B. Training the GP Model

The GP model is trained from measurements across UE locations by optimizing its marginal likelihood, then predicts phase perturbations for test locations. Those predictions capture the quasi-periodicity accurately in the illustrated example and support synthetic training-data generation as well as online refinement.

B. Training the GP Model

  • GP hyperparameters and unknown CPOs are learned from the L-location measurement dataset by maximizing the log marginal likelihood, with a gradient-based optimizer available.The model uses measurements from the APs at the training UE locations.Source: B. Training the GP Model · p. 6

C. Gaussian Process Regression

  • For a known test location, GP regression predicts the unknown perturbation with a Gaussian predictive density whose mean and variance feed the modeled carrier-phase measurement.The phase-measurement prediction combines the modeled phase with measurement noise.Source: C. Gaussian Process Regression · p. 6
  • With L = 10 training locations, the GP captures quasi-periodic NLoS perturbations accurately, with example estimation error below one degree.The figure visualizes training points and compares predicted and theoretical phase perturbations and carrier-phase curves.Source: C. Gaussian Process Regression · p. 6, Figure 3 · p. 7
  • The trained GP generates synthetic phase measurements at arbitrary training locations, drawing an independent uniform CPO for each sample to promote invariance to unknown CPO.The GP model is also used with gradient descent to refine POPT’s coarse online position estimate.Source: C. Gaussian Process Regression · pp. 6–7

IV. PROPOSED POSITIONING APPROACH

POPT estimates a coarse 3D UE position directly from one carrier-phase snapshot, using an encoder-only transformer to capture inter-AP phase relationships. Its phase-token preprocessing and learnable AP embeddings feed a transformer encoder, whose output is aggregated for position regression.

IV. PROPOSED POSITIONING APPROACH

  • A single carrier-phase snapshot is processed by POPT for a coarse UE position estimate, which is subsequently refined using GD.The architecture targets phase relationships across APs despite phase wrapping and unknown CPO.Source: §IV · p. 7, A. Proposed POPT Architecture · p. 7

A. Proposed POPT Architecture

  • The model uses self-attention with learnable AP embeddings to learn inter-AP phase relationships for 3D UE positioning.Preprocessing, Transformer Encoder, and Output Head are the three named components.Source: A. Proposed POPT Architecture · p. 7, Figure 4 · p. 7

1) Preprocessing:

  • POPT represents each phase measurement as [sin(r_m), cos(r_m)] and adds a learnable AP-specific embedding to each projected token.The sine-cosine representation mitigates the phase-wrapping discontinuity; AP embeddings distinguish tokens because the encoder has no positional information.Source: 1) Preprocessing · pp. 7–8

2) Transformer Encoder:

  • The encoder stacks transformer blocks with layer normalization, multi-head attention, residual connections, and token-wise MLPs to process inter-AP relationships.The encoder output is normalized and globally averaged across AP tokens before the Output Head.Source: 2) Transformer Encoder · p. 8, 3) Output Head · p. 8

3) Output Head:

The Output Head aggregates encoder tokens to regress a 3D position, while the complexity analysis describes POPT’s scaling and the positioning benchmarks used for comparison.

3) Output Head:

  • The Output Head averages the encoder’s M token embeddings, then applies a two-layer MLP to produce a 3D coarse UE position estimate.The MLP uses a hidden dimension D_h and GELU activation.Source: 3) Output Head · p. 8

B. Benchmarks for Positioning

  • Benchmark MLP directly maps phase observations to position, while Benchmark HI predicts differential integer ambiguities before gradient-descent position estimation.HI’s differential distance constraints are hyperboloids in the considered 3D setting; the method is included despite its single-path assumptions not matching the two-ray model.Source: B. Benchmarks for Positioning · p. 8

V. COMPLEXITY ANALYSIS

  • The complexity analysis uses big-O scaling with AP count and FLOP counts to quantify computational burden relevant to real-time deployment.The fine-estimation refinement is excluded because it is common to the compared coarse estimators.Source: §V · p. 8

A. Proposed POPT

  • In practical settings where D_emb and D_ff greatly exceed M, terms linear in M dominate POPT’s processing overhead, supporting scalability in large networks.The analysis neglects preprocessing and Output Head costs, and focuses on dominant matrix multiplications.Source: A. Proposed POPT · pp. 8–9

B. EGS/MLE Complexity

The section characterizes exhaustive-grid and benchmark inference costs, then specifies the simulation setting used for localization experiments.

B. EGS/MLE Complexity

  • EGS evaluates its cost over a predefined search set whose grid-point count scales as V/δ_x^3, with approximately 57 FLOPs per AP per point.Its total FLOP count is approximately 57MV/δ_x^3.Source: B. EGS/MLE Complexity · p. 9

C. Complexity of Benchmark Methods

  • Benchmark HI has total inference cost approximately 7.5 × 10^4M + 256Q + 1.92 × 10^5 FLOPs, where Q is the ambiguity estimator’s output-class count and grows linearly with M, yielding O(M) complexity.Source: C. Complexity of Benchmark Methods · p. 9

VI. NUMERICAL RESULTS

  • The simulated 3D area is 10 m × 10 m × 4 m, with 17 ceiling-deployed APs and an 800 MHz carrier under the two-ray channel model.The nominal ground material is concrete unless otherwise stated.Source: §VI · p. 9
  • Localization experiments use 1280 Monte Carlo realizations, P = −10 dBm, L = 50 GP training locations, and an EGS grid spacing of λ/8.The test locations are sampled independently from the GP measurement and POPT training and validation locations; all coarse estimates receive GD refinement.Source: §VI · p. 9

A. Training of POPT

The GP model captures material-dependent, quasi-periodic phase perturbations and predicts carrier-phase measurements with accuracy that improves with transmit power and training data. Its performance approaches the theoretical model for low-reflectivity materials and with sufficient training locations.

A. Training of POPT

  • POPT training uses a synthetic dataset generated from 1.125 × 10^6 random UE positions, split into 1 × 10^6 training and 0.125 × 10^6 validation samples.Separate POPT models are trained for each transmit power and GP training size.Source: A. Training of POPT · p. 9

B. Evaluation Metrics

  • The modeling evaluation reports localization RMSE, modeling RMSE, and NLPD to assess positioning and GP prediction accuracy.The modeling RMSE uses wrapped phase residuals, while NLPD evaluates predictive uncertainty.Source: B. Evaluation Metrics · p. 9

C. Modeling Results

  • Metal produces the largest and most rapidly varying phase perturbations, while concrete and wood show smaller variations; all materials exhibit wavelength-period quasi-periodicity.The GP captures the quasi-periodic structure, with perturbation magnitude depending on reflecting material and incidence angle.Source: C. Modeling Results · p. 10, Figure 5 · p. 10
  • As transmit power and training samples increase, GP prediction accuracy improves; with L = 100 it nearly matches the theoretical model, and at high power L = 5 yields error below one degree.The GP jointly models all APs, and Figure 5 evaluates predictive mean and variance using RMSE and NLPD.Source: C. Modeling Results · p. 10, Figure 5 · p. 10
  • For wood, plasterboard, and concrete, GP RMSE and NLPD are nearly identical to the theoretical model, while both metrics increase for glass and metal.The comparison uses L = 100 UE training locations and P = −10 dBm; stronger reflections are harder to learn from limited data.Source: C. Modeling Results · p. 10, Table I · p. 10

D. Localization Results

With 50 GP training locations, POPT approaches the PEB and mitigates sensitivity to relative-permittivity mismatch. Its reported inference cost is substantially lower than MLE’s when MLE uses a fine grid.

D. Localization Results

  • The tested MLE mismatch factors are β ∈ {1.00, 1.01, 1.05, 1.10}; POPT curves vary by GP training size L, not β.The measurements are generated using one fixed relative permittivity, while MLE assumes the perturbed value ε(β) = βεr.Source: D. Localization Results · p. 10
  • With L=50 GP training locations, POPT approaches the PEB across transmit powers and outperforms benchmarks designed for 2D LoS-only propagation.POPT learns phase perturbations from measurements rather than requiring explicit knowledge of relative permittivity.Source: D. Localization Results · p. 11
  • The PEB decreases as transmit power rises; matched MLE follows it, but relative-permittivity mismatch degrades fine estimates, especially at high power.POPT accuracy improves with more GP training locations, while the EGS coarse estimates are almost unaffected by mismatch and GD refinement is sensitive to it.Source: D. Localization Results · p. 11
  • POPT’s 99% of positioning errors remain below 0.191 mm, with a maximum observed error of 0.352 mm at L=50.Increasing L improves both average accuracy and empirical tail behavior.Source: D. Localization Results · p. 11

E. Computational Complexity Results

  • At δx=λ/8, POPT uses 81.10 × 10^6 FLOPs and 0.95 ms versus MLE’s 3771.04 × 10^6 FLOPs and 3716.05 ms.The reported runtimes cover coarse inference and exclude the GD refinement stage, which is applied to all methods.Source: Table II · p. 11, E. Computational Complexity Results · p. 11
  • POPT’s runtime advantage over MLE is larger than its FLOP ratio because optimized dense matrix multiplications contrast with MLE’s candidate sweeps and memory overhead.The FLOP accounting also counts each trigonometric and square-root evaluation as one operation, although these can require several to tens of elementary operations.Source: E. Computational Complexity Results · p. 11

VII. CONCLUSION

The paper develops a GP-based learning pipeline and POPT for single-snapshot phase-only positioning under two-ray propagation. With 50 GP training locations, it reports near-PEB accuracy and substantially lower inference cost than MLE, while identifying broader propagation and dynamic scenarios for future work.

VII. CONCLUSION

  • The GP model learns quasi-periodic phase perturbations from ground reflections and generates synthetic samples for the encoder-only POPT positioning model.The paper also derives the PEB and develops MLE for the multipath phase-only positioning problem.Source: §VII · p. 12
  • With 50 training locations, the proposed approach achieves near-PEB accuracy and outperforms the two considered benchmarks.The analytical MLE is vulnerable to modest relative-permittivity errors, while the GP-based approach learns phase perturbations from measurements without explicitly relying on that parameter.Source: §VII · p. 12
  • Compared with exhaustive-grid MLE, POPT reduces FLOP complexity and inference time by approximately 1.7 and 3.6 orders of magnitude, respectively.The conclusion identifies computationally feasible, low-latency, high-accuracy positioning as the framework’s potential in future phase-coherent distributed wireless networks.Source: §VII · p. 12
  • Future work includes more realistic propagation, orientation-dependent antenna patterns, heterogeneous reflecting surfaces, multipath beyond two rays, and moving-UE scenarios.The paper identifies multi-snapshot processing and sequential learning-based tracking as possible approaches for dynamic scenarios.Source: §VII · p. 12
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