3D Gaussian Distribution and RGB Reconstruction (3D Gaussian-RGB) based GNSS Multipath Detection — Part I: Derivation and Analysis of the 3D Gaussian ellipsoid Modelling

Technical Summary: 3D Gaussian Distribution and RGB Reconstruction (3D Gaussian-RGB) based GNSS Multipath Detection — Part I: Derivation and Analysis of the 3D Gaussian ellipsoid Modelling

Problem Statement
Global Navigation Satellite Systems (GNSS) face significant challenges in positioning accuracy and robustness within complex environments, such as urban canyons, due to multipath effects. While existing detection methods utilize measurement-domain features (e.g., pseudorange, C/N0) or correlation-domain distortions, they often suffer from limitations. Measurement-domain approaches can obscure original signal distortions due to receiver filtering and estimation processes. Conversely, existing correlation-domain methods frequently rely on high-resolution correlators or fail to sufficiently explore the nonlinear relationships and intrinsic connections between multipath features and correlation parameters. There is a need for a theoretical model that balances detection capability with computational complexity while accounting for the statistical dependencies among Early (E), Prompt (P), and Late (L) correlator outputs.

Methodology
This study proposes a theoretical correlation-domain model based on the statistical analysis of E, P, and L correlation values from the quadrature (Q) branch of GNSS signals. The Q-branch is selected because it is more sensitive to multipath-induced code and phase distortions than the in-phase (I) branch, particularly when the tracking loop is locked.

The methodology proceeds through the following steps:

1. Signal Modeling: Mathematical models are established for LOS, Non-Line-of-Sight (NLOS), and multipath signals. The multipath signal is modeled as a synthesis of LOS and NLOS components, introducing parameters for relative delay ($\delta\tau_N$), fading ($\alpha$), Doppler shift, and phase rotation ($\delta\phi_N$).
2. Normalization and Linearization: The E/P/L correlation values in the Q-branch are normalized relative to the Prompt correlator output to mitigate time-varying signal strength inconsistencies. A first-order multivariate Taylor expansion is applied to linearize these values around the noise vector, assuming noise is small relative to signal strength.
3. 3D Joint Gaussian Distribution: The linearized E/P/L values are shown to follow a 3D joint Gaussian distribution. The mean vector ($\mu_R$) and covariance matrix ($\Sigma_R$) are derived, accounting for noise coherency (via Cholesky decomposition) and phase jitter propagated through the tracking loop.
4. Equiprobability Ellipsoid Modeling: The joint distribution is geometrically represented as a 3D equiprobability ellipsoid. Spectral decomposition of the covariance matrix provides an orthogonal basis for the ellipsoid. The model characterizes signal conditions through three geometric transformations:
   • Translation: Governed by the mean vector $\mu_R$, primarily affecting multipath signals due to phase and delay differences.
   • Scaling: Determined by the eigenvalues of the covariance matrix, influenced by noise variance and signal amplitude (C/N0).
   • Rotation: Defined by the eigenvectors of the covariance matrix, dependent on the Early-Late correlator spacing ($\Delta$).
5. Detection Threshold Derivation: Using the Neyman-Pearson (NP) criterion, a detection boundary is derived. The hypothesis testing problem is framed as distinguishing between LOS (Null Hypothesis $H_0$) and multipath (Alternative Hypothesis $H_1$). The detection threshold $\gamma$ corresponds to the boundary of the equiprobability ellipsoid for a specified false-alarm probability ($P_{fa}$).

Key Contributions
The paper makes four primary contributions:

1. Derivation of 3D Joint Gaussian Distribution: The authors derive the 3D Gaussian joint distribution of E-P-L values, explicitly accounting for the interdependency among these values within the same autocorrelation function (ACF).
2. Equiprobability Ellipsoid Model: An ellipsoid model is established where the geometric properties (translation, scaling, rotation) are analytically linked to multipath parameters. This allows for the quantification of how multipath parameters (delay, fading, phase) deform the statistical distribution.
3. Theoretical Detection Threshold: A detection threshold is derived using the NP criterion, utilizing the equiprobability ellipsoid as the decision boundary. This provides a theoretical basis for quantifying detection probability ($P_d$) under varying multipath conditions.
4. Validation via Simulation: The theoretical models are validated through Monte Carlo simulations, demonstrating the effectiveness of the 3D Gaussian models for multipath detection.

Results
Simulation results using a semi-analytical model confirm the theoretical derivations:

• Gaussianity: For signal strengths (C/N0) exceeding 30 dB-Hz, the simulated E-P-L correlation values exhibit strong Gaussianity, aligning closely with the theoretical joint distribution. Below this threshold, deviations occur, particularly in multipath scenarios, due to the non-negligible influence of higher-order noise terms.
• Geometric Consistency: The simulated equiprobability ellipsoids align well with the theoretical models at C/N0 > 30 dB-Hz. Specifically, multipath signals exhibit a distinct translation effect along the X and Z axes, whereas LOS and NLOS signals show negligible translation. Scaling and rotation properties remain consistent across signal types when C/N0 is fixed, as they are primarily determined by the correlator spacing.
• Detection Performance: Receiver Operating Characteristic (ROC) analysis shows that detection performance improves with higher C/N0 and lower energy attenuation ($\alpha$). Performance is non-monotonic with respect to relative code delay ($\delta\tau_N$) and phase rotation ($\delta\phi_N$), peaking at intermediate values (approx. 0.6 chips and 0.6$\pi$, respectively). The method exhibits detection blind spots for multipath signals with extremely short/long delays or phases near 0 or $\pi$.
• Threshold Accuracy: The theoretical detection threshold $\gamma$ matches simulated results closely when C/N0 > 30 dB-Hz. Discrepancies increase at lower signal strengths due to degraded distribution Gaussianity.

Significance and Claims
The paper claims that this work provides the statistical-domain foundation for a broader multipath detection framework. By establishing a rigorous 3D Gaussian ellipsoid model, the study offers a geometrically interpretable method to characterize multipath-induced signal distortions. The authors state that this statistical model serves as the prerequisite for Part II of their research, where the normalized E/P/L vector is projected into RGB color space. The current work (Part I) focuses on deriving the theoretical thresholds and analyzing the statistical properties, while the companion paper will utilize these thresholds to interpret color features.

The study emphasizes that while the method effectively distinguishes between LOS and multipath signals based on distribution translation, distinguishing LOS from NLOS signals remains challenging under similar C/N0 conditions due to overlapping ellipsoid scaling characteristics. The authors maintain a modest scope, noting that the zero-mean Gaussian assumption for the Prompt correlator in multipath scenarios is an approximation justified by empirical validation within practical error margins, despite potential phase synchronization perturbations in real-world environments.