Phase-Only Positioning: Methods & Challenges

• Phase-only positioning is a localization technique that relies solely on carrier phase measurements, achieving sub-meter to centimeter accuracy despite inherent cycle ambiguities.
• It employs differential models, hyperbola intersections, and estimators—from classical least-squares to deep learning—to address the integer ambiguity challenge.
• Applications include distributed MIMO, 5G-Advanced NR, and large-constellation GNSS, utilizing multi-band and impairment-aware methods for robust performance.

Phase-only positioning denotes localization from phase observations alone, typically carrier-phase measurements, without time-of-arrival or other channel measurements. In current wireless formulations it appears in uplink carrier phase positioning for cell-free or distributed antenna systems, in distributed MIMO indoor localization, in 5G-Advanced NR carrier-phase positioning, and in large-constellation GNSS. Its appeal is that carrier phase can support sub-meter to centimeter-level accuracy, sometimes with minimal bandwidth requirements, but the observation is periodic with respect to wavelength, so each measurement is ambiguous by an integer number of cycles. The resulting integer ambiguity problem is the defining technical obstacle, and much of the literature can be read as a sequence of increasingly specialized strategies for either resolving, marginalizing, or circumventing that ambiguity (Ayten et al., 9 Jun 2025, Fouda et al., 2022, Niesen et al., 2018).

1. Measurement models and differential observables

A standard phase-only observation model in distributed antenna systems writes the carrier phase at the $i$-th AP as

$$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$

where $d_i$ is the transmitter-AP distance, $\theta$ is an unknown phase offset, $z_i\in\mathbb{Z}$ is the integer ambiguity, and $n_i$ is noise. Using one AP as reference, the corresponding differential measurement is

$$\delta_m = -\frac{\lambda}{2\pi}(r_m - r_0) = \Delta_m + k_m \lambda + w_m$$

with $\Delta_m=d_m-d_0$, $k_m=z_0-z_m$, and $w_m$ the noise term. This differential form removes the common phase and converts the problem from absolute phase recovery to differential distance recovery modulo the wavelength (Ayten et al., 9 Jun 2025).

In 5G-Advanced NR, the carrier-phase measurement model is commonly written as

$$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$0

where $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$1 and $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$2 are transmitter and receiver clock biases, $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$3 is the unknown integer ambiguity, and $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$4 is the measurement error. Single differencing between gNBs removes the UE clock bias, while double differencing between a target UE and a reference UE removes both transmitter and receiver clock biases. These differencing operations are central because practical cellular deployments do not assume perfect synchronization at all nodes (Fouda et al., 2022).

A geometric interpretation emerges once differential phases are mapped to differential distances. When both APs are operational, each differential phase measurement defines a hyperbola with foci at the two APs, parameterized by the corresponding integer ambiguity. Intersecting multiple such hyperbolas ideally yields a unique UE position. This hyperbola-intersection principle appears repeatedly in recent phase-only indoor and D-MIMO work because it converts wrapped phase information into a geometric consistency problem (Ayten et al., 20 Aug 2025).

2. Integer ambiguity as the central identifiability problem

The integer ambiguity problem arises because carrier phase is periodic with respect to the wavelength: path lengths separated by integer multiples of $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$5 produce indistinguishable phases. In the differential model, the nuisance parameters become differential ambiguities $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$6, but the problem remains combinatorial. With only phase and no absolute ranging observable, exhaustive search or fine-grid maximum likelihood quickly becomes computationally prohibitive as the number of APs grows (Ayten et al., 9 Jun 2025).

This difficulty is not confined to terrestrial networks. In large-constellation GNSS, asymptotic analysis shows that the standard carrier-phase positioning approach based on resolving the carrier-phase integer ambiguities fails in the large-satellite regime. Even with the true receiver position and clock bias known, the probability of correctly resolving all ambiguities decays exponentially to zero as the number of satellites increases, unless carrier-phase noise is infinitesimal. The alternative proposed there is Bayesian: ambiguities are treated as noise terms and not explicitly estimated. The resulting estimator is consistent, its error scales as $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$7 with the number of satellites, and carrier-phase measurements remain useful even when traditional ambiguity fixing becomes impossible (Niesen et al., 2018).

A common misconception is that phase-only positioning is synonymous with exact ambiguity fixing. The large-constellation GNSS result shows that this is not generally true: explicit ambiguity resolution is one strategy, but not the only principled one. A second misconception is that phase-only observations are intrinsically too ambiguous to be useful. The distributed MIMO and GNSS results jointly show the opposite: the difficulty is not lack of information, but how that information is represented and regularized (Niesen et al., 2018, Ayten et al., 9 Jun 2025).

3. Classical, geometric, and search-reduced estimators

The classical toolbox includes least-squares, maximum-likelihood, and weighted least-squares estimators built on differenced carrier phases. In 5G-Advanced NR, the phase-based LS position update is obtained from a Taylor-series linearization and a geometric design matrix, after single or double differencing has removed clock biases. The same literature assumes ideal integer ambiguity fixing, for example through LAMBDA or time-based assistance, and evaluates sensitivity to non-ideal ambiguity fixing through an ambiguity error model. Under indoor factory simulations, this carrier-phase approach yields horizontal 90% error of approximately $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$8 cm, vertical 90% error of approximately $$r_i = -\frac{2\pi}{\lambda}d_i + \theta + 2\pi z_i + n_i$$9 cm, and 3D 90% error below $d_i$0 cm, whereas the DL-TDOA baseline remains at meter level (Fouda et al., 2022).

Maximum-likelihood formulations are more direct but typically produce highly non-convex or highly spiky cost surfaces. In cell-free or distributed MIMO phase-only positioning, the maximum-likelihood baseline is often implemented through dense grid search, which is accurate but expensive. This motivates algorithms that reduce the search space before invoking ML evaluation (Ayten et al., 9 Jun 2025, Dey et al., 10 Dec 2025).

The POLO framework is an explicit example of search reduction. POLO-I selects three APs and uses two differential carrier-phase measurements to form hyperbola intersections in closed form, thereby generating a compact candidate set. POLO-II selects four APs, forms two AP pairs, and uses an alternative intersection strategy that improves coverage at marginally higher runtime. In both cases the ML cost function is evaluated only at candidate points rather than over a dense grid. Analytical and simulation results characterize a coverage-complexity tradeoff: exhaustive grid search has complexity $d_i$1, whereas POLO scales as $d_i$2, reducing cost by orders of magnitude with only marginal loss in coverage (Dey et al., 10 Dec 2025).

Hyperbola-intersection methods also appear in impairment-aware indoor localization. There, the ambiguities are estimated first, differential distances are reconstructed as

$d_i$3

and the UE location is refined through gradient descent on the squared mismatch between estimated and geometry-induced differential distances. This separates ambiguity estimation from geometric consistency and makes failure detection possible through the final optimization cost (Ayten et al., 20 Aug 2025).

4. Deep learning for phase-only positioning in cell-free and distributed MIMO

A major recent shift is the use of deep neural networks to learn either the position directly from ambiguous phase data or the ambiguity structure that makes geometric estimation tractable. In cell-free carrier phase positioning with phase-only observations, two architectures were proposed. The first is a direct MLP that maps the differential phase vector $d_i$4 to the UE coordinates. It uses one input layer, seven hidden layers with widths $d_i$5, ReLU activations, and a linear two-dimensional output. The second is ambiguity-aided: an MLP with shared layers and parallel softmax branches estimates each $d_i$6, and a CNN then combines $d_i$7 and $d_i$8 in an input of shape $d_i$9 for final position estimation. Training is supervised on simulated UE locations in a $\theta$0 area, with up to 20 APs, two frequencies of 800 MHz and 1.8 GHz, and post-training pruning to reduce FLOPs (Ayten et al., 9 Jun 2025).

Both networks achieve centimeter-level RMSE, and the ambiguity-aided model reaches 1 cm RMSE at 800 MHz and 0 dBm transmit power. The per-dimension ambiguity-estimation accuracy exceeds 98.9% at high SNR for 800 MHz. Compared with the maximum-likelihood baseline, the learned estimators reduce inference complexity by factors between 100 and 1800, i.e. two to three orders of magnitude, while often improving accuracy under equal-FLOP constraints (Ayten et al., 9 Jun 2025).

Configuration	FLOPs	RMSE (at 0 dBm)
MLP-based (Direct), 800 MHz, 50% pruning	1,376,000	~2 cm
CNN-based (Amb.-aided), 800 MHz, 75% pruning	1,148,000	1 cm
MLP-based (Direct), 1.8 GHz, 50% pruning	1,376,000	~4 cm
CNN-based (Amb.-aided), 1.8 GHz, 75% pruning	2,330,000	~3 cm

The significance of these results is not merely lower runtime. They show that the mapping from wrapped differential phases to location can be learned as a structured inference problem rather than treated only as integer search. This suggests that, in sufficiently regular deployment geometries, ambiguity resolution and localization can be partially amortized into the model parameters (Ayten et al., 9 Jun 2025).

5. Robustness to failures and phase impairments

Early deep phase-only localization work assumed ideal hardware. More recent results explicitly target AP failures and phase synchronization errors. In failure-tolerant indoor positioning, the phase observation model includes a binary AP status $\theta$1, so failed APs effectively zero their contribution. The proposed pipeline uses an MLP ambiguity estimator with hidden size $\theta$2 and total output classes $\theta$3 for $\theta$4 APs, trained with cross-entropy on simulated data containing both fault-free and antenna-failure cases. Once ambiguities are estimated, a gradient-descent hyperbola-intersection solver minimizes

$\theta$5

over the UE location. If the terminal cost exceeds a threshold $\theta$6, the estimate is flagged as likely faulty. Numerically, the ambiguity estimation accuracy is $\theta$7 at 0 dBm for $\theta$8 and $\theta$9 for $z_i\in\mathbb{Z}$0; the 95th-percentile position error is $z_i\in\mathbb{Z}$1 cm at 0 dBm and $z_i\in\mathbb{Z}$2 cm at $z_i\in\mathbb{Z}$3 dBm; the same percentile becomes $z_i\in\mathbb{Z}$4 cm for $z_i\in\mathbb{Z}$5; failure detection exceeds 99% reliability; and inference complexity is $z_i\in\mathbb{Z}$6 FLOPs (Ayten et al., 20 Aug 2025).

Phase synchronization errors introduce a different impairment class. In distributed MIMO, each AP may carry a random phase perturbation $z_i\in\mathbb{Z}$7, so the differential measurement becomes

$z_i\in\mathbb{Z}$8

Impairment-aware training is decisive here. When the ambiguity estimator is trained and tested on data with phase perturbation, the overall ambiguity estimation accuracy remains above 99%, and the 95th-percentile positioning error stays below 0.5 cm even for moderate phase noise such as $z_i\in\mathbb{Z}$9. If the same model is trained without phase errors and tested with them, performance degrades greatly. To exploit this robustness more efficiently, a deep AP-selection network is trained on measurement-based and geometry-based features concatenated into $n_i$0, with five fully connected hidden layers of sizes $n_i$1 and output size $n_i$2. Its purpose is to select the AP pair that minimizes positioning error under phase impairments, reducing total complexity by approximately 19.7% (Ayten et al., 4 Feb 2026).

At $n_i$3, the reported 95th-percentile errors are 0.14 cm for the best achievable pair, 0.26 cm for the proposed selector, 0.30 cm for no selection, 509.8 cm for Max-SNR, and 551.1 cm for Random. These results formalize a practical lesson: in phase-only positioning, AP choice is part of the estimator, not merely a preprocessing detail (Ayten et al., 4 Feb 2026).

6. Multi-band, large-constellation, and theoretical extensions

Single-band phase-only positioning is dominated by ambiguity structure. Multi-band carrier phase positioning mitigates that structure through frequency diversity. A recent terrestrial 6G formulation considers intra- and inter-band carrier aggregation across FR1, mmWave-FR2, and emerging FR3, with unknown UE clock bias, mutual clock imperfections between network nodes, and band-dependent or band-independent phase offsets. The central performance metric is a mixed-integer Cramér-Rao bound, obtained by relaxing integer ambiguities, mapping them back through integer least squares, and averaging the error covariance over ambiguity errors. The same work proposes a two-stage practical estimator based on an initial ToA-based estimate, local linearization, joint WLS for real parameters and relaxed ambiguities, ILS ambiguity resolution, and optional search-based refinement for narrowband IoT. Its main empirical conclusion is that only two carriers suffice to substantially facilitate resolving the integer ambiguity problem while also largely enhancing robustness against network-side clock imperfections and multipath propagation (Shourezari et al., 8 Jan 2026).

The large-constellation GNSS literature provides a complementary asymptotic extension. Rather than increase frequency diversity, it increases spatial redundancy through many satellites and then replaces ambiguity fixing by Bayesian marginalization. The resulting estimator exploits carrier phase even when explicit ambiguity resolution is asymptotically doomed, which broadens the conceptual design space of phase-only positioning beyond “fix then localize” pipelines (Niesen et al., 2018).

At a more abstract level, phase-only compressed sensing studies the recovery of structured signals from the phases of complex measurements. Two key results are that the nonlinear phase-only problem can be recast as a linear compressed sensing problem, and that basis pursuit or basis pursuit denoising then yields strong guarantees: perfect directional recovery with high probability for sparse signals under complex Gaussian measurements, and uniform instance-optimal recovery with robustness to small dense disturbances and sparse corruptions. These works are not positioning algorithms, but they directly support the broader proposition that phase-only measurements need not be information-poor if the inference model is chosen appropriately (Jacques et al., 2020, Chen et al., 2024).

Taken together, these strands define phase-only positioning as a family of localization problems in which the measurement budget is intentionally reduced to phase, while the inferential burden is shifted to geometry, statistical modeling, optimization, frequency diversity, or learned structure. The recurring theme is not that phase eliminates difficulty, but that it relocates difficulty into ambiguity management. The current literature shows several viable resolutions: explicit ambiguity estimation, Bayesian marginalization, multi-band diversity, candidate-set construction from hyperbola geometry, and deep learning trained on the relevant impairment distribution (Ayten et al., 9 Jun 2025, Shourezari et al., 8 Jan 2026).