Carrier Phase Differential GNSS (CDGNSS)

• Carrier Phase Differential GNSS (CDGNSS) is a high-precision positioning method that leverages differential carrier-phase observations for centimeter-level accuracy.
• It addresses the integer ambiguity challenge by first estimating float ambiguities and then applying integer least-squares methods, such as LAMBDA, to refine solutions.
• CDGNSS integrates various estimation architectures—ranging from least-squares and Kalman filters to factor graphs—to robustly fuse dynamics, geometry, and measurement uncertainties.

Searching arXiv for recent and foundational papers on Carrier Phase Differential GNSS (CDGNSS). Carrier Phase Differential GNSS (CDGNSS) denotes the family of high-precision GNSS techniques that use carrier-phase observables, typically in differential form, to estimate position, velocity, baseline, or attitude with much higher precision than code-only methods. In recent GNSS estimation literature, CDGNSS problems are commonly formulated with state variables that include position, velocity, receiver clock terms, and carrier-phase ambiguities, while real-world vehicular experiments in dense urban conditions have reported 17-cm-accurate 3D urban positioning (95% probability) with solution availability greater than 87% without inertial, electro-optical, or odometry aiding (Suzuki, 12 Feb 2025, Humphreys et al., 2019).

1. Differential carrier-phase observation model

The defining feature of CDGNSS is the use of carrier phase rather than only pseudorange. For a receiver $k$ and satellite $p$, one low-cost interferometric formulation writes the carrier phase as

$$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$

where the measurement contains an unknown integer ambiguity $N_{p,k}$, receiver and satellite clock terms, propagation effects, and noise (Slutsker et al., 2017).

Differencing is central because it suppresses common-mode errors. A single difference between two receivers $k$ and $m$ for satellite $p$,

$SD_p = \Phi_{p,k}(t) - \Phi_{p,m}(t),$

eliminates satellite clock bias and reduces atmospheric effects to first order. A double difference between satellites $p$ and $q$,

$p$0

further eliminates receiver clock terms and leaves a baseline-geometry term, an integer ambiguity term, and residual noise (Slutsker et al., 2017).

A recent factor-graph-oriented CDGNSS model uses the double-differenced carrier phase

$p$1

where $p$2 is the carrier wavelength, $p$3 is the double-differenced observation between satellites $p$4 and reference $p$5, $p$6 is the double-differenced geometric range, $p$7 is the integer ambiguity, and $p$8 is measurement noise (Suzuki, 12 Feb 2025). In this formulation the ambiguity is estimated as a float value inside the estimator and fixed afterward.

This differential structure also appears in related formulations based on time rather than space. Time-differential carrier phase and time-relative RTK-GNSS treat the displacement between epochs as an RTK-like baseline, preserving the same carrier-phase precision while changing the differencing topology (Suzuki, 2023, Suzuki, 2023).

2. Integer ambiguity as the central estimation problem

The principal algorithmic difficulty in CDGNSS is integer ambiguity resolution. In one recent implementation, the carrier-phase factor for CDGNSS is written as

$p$9

so that both the position error state and the ambiguity state appear explicitly in the residual (Suzuki, 12 Feb 2025). The associated CDGNSS objective for position and ambiguity estimation combines double-differenced pseudorange and carrier-phase factors, and an extended model can also include Doppler and motion factors (Suzuki, 12 Feb 2025).

A standard workflow is to estimate float ambiguities first and then apply integer least-squares. In gtsam_gnss, integer least-squares methods such as LAMBDA are applied after optimization using the float ambiguity estimate, its covariance, and a ratio test, with the threshold typically set at 2.0 (Suzuki, 12 Feb 2025). In a visual-inertial EKF formulation, the ambiguity vector is likewise fixed after a float solution, with MLAMBDA used for the integer search and a ratio test that accepts a fixed solution if the ratio exceeds a threshold, typically 3 (Dong et al., 2023).

Alternative formulations address the same integer structure differently. For ultra-short-baseline attitude determination, a quaternion-based model

$$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$0

is combined with Monte Carlo sampling to build an empirical ambiguity probability density and then with LAMBDA plus candidate screening; experiments reported a 100% ambiguity resolution success rate for ultra-short baselines, including cases with as few as 4 satellites and without code measurements or prior attitude information (Sun et al., 2017). In another attitude-oriented line of work, constrained wrapped least squares estimates attitude parameters directly from ambiguous carrier-phase observations, without requiring prior ambiguity fixing, and is reported to outperform the ambiguity function method, constrained LAMBDA, and multivariate constrained LAMBDA (Liu et al., 2021).

Across these formulations, the recurring pattern is the same: CDGNSS accuracy depends on how sharply the float solution and its covariance restrict the admissible integer lattice. Richer kinematic or geometric constraints generally improve that restriction.

3. Estimation architectures and software realizations

CDGNSS has been implemented in least-squares, Kalman, batch-optimization, and factor-graph frameworks. A recent open-source package, gtsam_gnss, uses GTSAM as its backend and defines pseudorange, Doppler, and carrier-phase observations as graph factors. Its design separates preprocessing from optimization: satellite position computation, tropospheric and ionospheric corrections, measurement differencing, and line-of-sight calculations are performed outside the graph, while the graph receives preprocessed residuals and Jacobians as generic inputs (Suzuki, 12 Feb 2025). This separation is intended to simplify the transition from ordinary least-squares positioning to factor graph optimization and to make user-specific GNSS research easier to implement (Suzuki, 12 Feb 2025).

Tightly coupled filtering remains prominent. A visual-inertial fusion system uses a loosely coupled EKF in which VIO incremental pose and covariance drive the prediction step and double-differenced GNSS carrier-phase and pseudorange equations are used in the update step; the same system also solves an antenna-to-IMU extrinsic calibration problem to align VIO to the ECEF frame (Dong et al., 2023). For deep-urban multi-antenna navigation, a matrix-Lie-group unscented Kalman filter tightly couples CDGNSS with a low-cost MEMS IMU and vehicle dynamics constraints; its unscented linearization is designed to support integer least-squares ambiguity resolution while implicitly enforcing known-baseline-length constraints and exploiting inter-baseline correlations (Yoder et al., 2022).

Sliding-window formulations have also been developed. SRI-GVINS deeply fuses pseudorange, Doppler shift, single-differenced pseudorange, and double-differenced carrier phase with visual-inertial measurements inside a square-root inverse sliding window filter, while also performing online GNSS-IMU extrinsic calibration (Hu et al., 2024). For particle methods, the Multiple Update Particle Filter addresses the sharp-peaked likelihoods created by carrier-phase observables by applying likelihoods sequentially from broad to narrow and resampling between updates; in static testing it reported 1.64 cm 3D RMS error and 100% fixed rate after 20 epochs, and in an urban vehicle test it achieved 80.3\% of epoch errors within 0.5 m versus 63.9\% for RTKLIB (Suzuki, 2024).

These architectures differ in numerical strategy, but they converge on a common CDGNSS design principle: the carrier phase is most effective when processed jointly with dynamics, geometry, and carefully modeled uncertainty rather than as an isolated measurement stream.

4. Attitude determination and single-receiver relative formulations

A major branch of CDGNSS concerns attitude determination. In multi-antenna settings, the baseline geometry is known in the body frame and the carrier-phase observations constrain the vehicle rotation. The quaternion-based model

$$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$1

treats $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$2 as the attitude matrix and $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$3 as the integer ambiguity matrix, with ambiguity particles generated by Monte Carlo sampling and screened after LAMBDA search through consistency with the GNSS-attitude model (Sun et al., 2017). For low-cost hardware, a two-receiver “GNSS Differential Interferometer” based on double-difference processing reported azimuth standard deviations as low as 0.57 degrees for a 2 m baseline, with one field test reporting mean 47.18° and standard deviation 0.5752° (Slutsker et al., 2017).

Another important extension replaces spatial differencing between a base and a rover with temporal differencing at a single receiver. Time-Relative RTK-GNSS uses time-differential carrier phase between current and past epochs to create loop-closure constraints in a pose graph, and kinematic UAV tests reported trajectory estimation with approximately 3 cm accuracy using only a stand-alone GNSS receiver (Suzuki, 2023). A related “GNSS Odometry” approach uses time differences of carrier phase together with explicit cycle-slip estimation in factor graph optimization; by estimating and correcting slip integers, UAV experiments reported trajectory accuracy of 5 to 30 cm using only a single GNSS receiver and no external sensors (Suzuki, 2023).

Windowed carrier-phase methods generalize the two-epoch TDCP idea. Instead of differencing only neighboring epochs, the Window Carrier-Phase formulation stacks carrier-phase measurements across a window that shares a common ambiguity and applies a left null space matrix to eliminate that ambiguity, thereby correlating multiple receiver states inside a factor graph (Bai et al., 2021). In two urban canyons in Hong Kong, this approach achieved mean positioning errors of 1.76 meters and 2.96 meters, respectively, with an automobile-level GNSS receiver (Bai et al., 2021). Although these results are not differential RTK fixes in the conventional short-baseline sense, they show that carrier-phase temporal structure can be used as a robust relative constraint even without a reference receiver.

5. Application domains and reported performance

Reported CDGNSS performance varies strongly with geometry, aiding, and environment. Representative results from the cited literature are summarized below.

Domain	Reported result	Source
Unaided deep-urban vehicle positioning	17-cm-accurate 3D positioning (95%) with availability greater than 87%	(Humphreys et al., 2019)
FGO CDGNSS with Doppler/velocity	Fix rate 40.8% $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$4 51.4%	(Suzuki, 12 Feb 2025)
VIO-aided carrier-phase GNSS, light-blocked urban canyon	RMSE 0.033 m, FSR 92.9%	(Dong et al., 2023)
Deep-urban multi-antenna CDGNSS + consumer/industrial IMU	96.6% / 97.5% integer fix availability; 12.0 cm / 10.1 cm 95th-percentile horizontal error	(Yoder et al., 2022)
SRI-GVINS with Ddcp	0.143 m mean position RMSE	(Hu et al., 2024)
Distributed space systems post-IAR relative navigation	1.79 mm RMS relative position; 0.040–0.041 mm/s RMS relative velocity	(Low et al., 25 Aug 2025)

In urban vehicular CDGNSS without aiding, performance sensitivity is dominated by signal tracking quality and infrastructure. A detailed Austin, Texas study found that navigation data bit wipeoff for fully modulated GNSS signals and a dense reference network are key to high-performance urban RTK positioning (Humphreys et al., 2019). Disabling data bit prediction caused a “catastrophic drop,” with $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$5 falling from 84.8% to 54.4% and $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$6 rising from 2.4% to 25%; increasing the reference baseline to 15 km reduced $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$7 to 78.9% and increased $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$8 to 7.7% (Humphreys et al., 2019).

Sensor fusion changes both accuracy and fix behavior. In gtsam_gnss, adding Doppler and velocity factors to the CDGNSS graph improved fix rate from 40.8% to 51.4%, raised ratio-test values, and reduced 3D position error through better float ambiguity estimation (Suzuki, 12 Feb 2025). In real-world urban canyons, a VIO-aided carrier-phase GNSS system improved fixed solution rate from 39.4% to 50.8% in one environment and from 8.1% to 36.5% in another, while also outperforming RTKLIB in simulated light-blocked conditions with 0.033 m RMSE versus 0.78 m for RTKLIB (Dong et al., 2023). For deep-urban multi-antenna navigation, tight inertial coupling with consumer-grade and industrial-grade IMUs achieved 96.6% and 97.5% integer fix availability, respectively, with 12.0 cm and 10.1 cm overall 95th-percentile horizontal error (Yoder et al., 2022).

CDGNSS is not restricted to terrestrial vehicles. On UAV datasets, tightly coupled double-differenced carrier phase in SRI-GVINS yielded 0.143 m mean position RMSE, versus 0.410 m for VIO+Dopp+Dpsr and 0.595 m for VIO+Dopp+Psr (Hu et al., 2024). For distributed space systems, a flight-ready software architecture using float ambiguities with integer resolution where possible reported 1.79 mm RMS post-IAR relative position accuracy and 0.040–0.041 mm/s RMS relative velocity on VISORS-like tests (Low et al., 25 Aug 2025). This broad application range shows that CDGNSS is best viewed as a measurement and estimation paradigm rather than a single terrestrial RTK workflow.

6. Integrity, spoofing, and relation to emerging carrier-phase systems

Because CDGNSS resolves integers and enforces strong dynamical consistency, its internal residuals are also useful for integrity monitoring. One tightly coupled CDGNSS-IMU spoofing detector defines a carrier-phase fixed-ambiguity residual cost $$\Phi_{p,k}(t) = \Phi_p^{(t)} - \Phi_{k}^{(t)} + N_{p,k} + S_{p,k} + f_p t_p + f_k t_k - B_{\text{iono}} + \text{Strop},$$9 and a windowed sum

$N_{p,k}$0

called the Windowed Fixed-Ambiguity Residual Cost (WFARC) (Clements et al., 2022). Using vehicle data from Austin, Texas, the study reported that artificial worst-case spoofing attacks were detected within two seconds, and in shallow urban conditions the most subtle attacks were detected within 1 second with an industrial-grade IMU and within two seconds with a consumer-grade IMU (Clements et al., 2022).

A second line of work uses short-time inertial tracking of deliberate high-frequency antenna motion and checks whether the carrier phases from different satellites exhibit the spatial diversity expected from real transmitters rather than the collapsed geometry of a single-antenna spoofer. In laboratory and Jammertest 2024 evaluations, this method reported accuracy of up to 90% in correctly identifying spoofing, or the lack of it, using mass-production-grade INS typical for mobile phones and without modification to the receiver structure (Johansson et al., 6 Feb 2025).

Related carrier-phase positioning systems have begun to appear outside GNSS. In 3GPP Release-18, carrier phase positioning is standardized for the first time in 5G NR, with differential bias-cancellation logic that the literature explicitly compares to CDGNSS; the basic measurement is written as

$N_{p,k}$1

and differential processing with a Positioning Reference Unit is used to cancel hardware phase biases (Cha et al., 2024). In LEO-based NR-NTN research, joint delay-and-carrier-phase positioning has been compared directly with GNSS: simulations reported cm-level accuracy with convergence times on the order of a few seconds for LEO, whereas GNSS remained limited to meter-level accuracy over comparable short observation windows (Dureppagari et al., 18 Mar 2026). These developments do not replace CDGNSS in the cited literature, but they show that integer ambiguity resolution, differencing, and phase-continuity management are becoming general positioning principles across multiple radio-navigation domains.