L-Moment-Based LOS and NLOS Channel Characterization via Four-parameter Kappa Distribution for AoA BLE CTE Measurements

Technical Summary: L-Moment-Based LOS/NLOS Channel Characterization via Four-parameter Kappa Distribution for AoA BLE CTE Measurements

Problem Statement
Bluetooth Low Energy (BLE) direction-finding systems rely on In-phase and Quadrature (IQ) samples collected during Constant Tone Extension (CTE) transmissions. The statistical properties of these samples are heavily influenced by the propagation regime: Line-of-Sight (LOS) conditions typically yield stable statistics, while Non-Line-of-Sight (NLOS) conditions introduce multipath distortions, shadowing, and phase-dependent superposition that degrade Angle-of-Arrival (AoA) estimation.

Existing datasets and models suffer from three primary limitations:

1. Lack of Paired Geometry: Most datasets do not provide tightly controlled, paired LOS/NLOS measurements under identical anchor-tag geometries, making it difficult to isolate propagation effects from geometric changes.
2. Inadequate Modeling: Common simulation frameworks rely on additive white Gaussian noise or simple flat-fading models (e.g., Rayleigh, Rician), which fail to capture the heavy-tailed and skewed amplitude statistics often observed in cluttered indoor NLOS environments.
3. Statistical Rigor: There is a lack of rigorously validated, IQ-level LOS/NLOS labels suitable for large-scale machine learning (ML) applications. Furthermore, classical product-moment statistics are sensitive to outliers and heavy tails, potentially leading to unreliable channel characterization.

Methodology
The authors propose a comprehensive framework combining a controlled measurement campaign with robust statistical modeling using L-moments.

1. Controlled Measurement Campaign:

   • Hardware: An off-the-shelf u-blox XPLR-AOA-3 development kit (ANT-B10 anchors and C209 tags) was used in a 630 × 385 cm indoor laboratory.
   • Protocol: A strictly paired geometry was enforced where LOS and NLOS measurements were taken at identical anchor-tag positions and orientations. NLOS conditions were created by inserting a blocker (graphene-coated paper stack) that obstructed the direct path while allowing reflections.
   • Data Volume: The campaign collected 132,000 labeled CTE packets (1000 packets per event across various configurations of tag location, orientation, and access points).

2. Robust Preprocessing:

   • A multi-stage outlier detection pipeline was applied to remove anomalous CTE packets. This combined three methods:
     • Inter-Quantile Range (IQR): For univariate filtering of random fluctuations.
     • Median Absolute Deviation (MAD): A robust, distribution-free scale estimator for detecting extreme deviations in marginal distributions.
     • Minimum Covariance Determinant (MCD): A multivariate method to identify points anomalous in their joint feature space.
   • This process reduced the dataset to 12,007 high-quality samples while retaining the central statistical tendency.

3. Statistical Validation:

   • Hypothesis Testing: Feature-wise Welch's t-tests and Levene's tests were applied to the 64-dimensional IQ-derived power features. These tests confirmed that mean and variance differences between LOS and NLOS subsets were statistically significant ($\alpha \le 0.01$, adjusted $P \le 10^{-6}$) across nearly all dimensions.

4. L-Moment Analysis and Kappa Modeling:

   • L-Moments: Instead of conventional moments, the authors utilized L-moments (linear combinations of order statistics), which are less sensitive to outliers and heavy tails.
   • L-Moment Ratio Diagram (LMRD): Empirical L-moment ratios (LMRs), specifically L-skewness ($\tau_3$) and L-kurtosis ($\tau_4$), were plotted. The analysis revealed distinct clusters for LOS and NLOS, with NLOS exhibiting heavier tails and stronger asymmetry.
   • Kappa Distribution: A four-parameter Kappa distribution family was fitted to the LMRs. This flexible family encompasses Generalized Extreme Value (GEV), Generalized Logistic (GLO), and Generalized Pareto (GPA) distributions as special cases, allowing it to better match the empirical data than three-parameter models.
   • Goodness-of-Fit (GoF): The Kappa models demonstrated significantly lower discrepancy scores ($D_{NLOS} = 3.20 \times 10^{-4}$) and near-zero standardized L-kurtosis deviations compared to traditional Rayleigh, Rice, or GEV models.

5. ML Application (Clustering):

   • The authors applied DBSCAN clustering to compare the separability of data using conventional product moments versus L-moment ratios. L-moment features yielded two distinct, well-separated clusters (LOS and NLOS), whereas product moments collapsed into a single cluster with high noise.

Key Contributions

• Paired BLE IQ Dataset: The creation of a large-scale, class-labeled dataset where LOS and NLOS captures are recorded under strictly identical geometric conditions, facilitating rigorous ML and channel characterization.
• Statistical Validation of Separability: Demonstration through robust outlier removal and hypothesis testing that LOS and NLOS conditions produce statistically distinct first- and second-order statistics across the IQ-derived power feature space.
• L-Moment Channel Characterization: The first application of L-moment analysis to BLE AoA CTE measurements. The study shows that NLOS channels exhibit heavy-tailed behavior that is poorly captured by classical models but well-modeled by the four-parameter Kappa distribution.
• Improved ML Representations: Evidence that L-moment statistics provide more compact, robust, and separable features for LOS/NLOS detection and channel simulation compared to conventional product moments, as demonstrated by superior DBSCAN clustering performance.

Results

• Statistical Significance: All mean differences between LOS and NLOS were statistically significant, with 92% of feature-wise variance differences also being significant.
• Model Fit: The Kappa distribution provided the best fit for the empirical data. Specifically, for NLOS data, the Kappa model achieved a discrepancy ($D_{NLOS}$) of $3.20 \times 10^{-4}$, the smallest in the LMRD, with a standardized L-kurtosis deviation ($Z_{NLOS}$) of $1.18 \times 10^{-2}$. In contrast, standard models like Rayleigh and Rice showed significantly larger errors.
• Clustering: L-moment-based representations successfully separated LOS and NLOS data into two clusters with a silhouette score of 0.018, while product-moment representations failed to form distinct clusters.

Significance and Claims
The paper claims to be the first to apply L-moment analysis to BLE AoA CTE measurements. Its significance lies in providing a robust statistical framework for characterizing indoor wireless channels that are heavy-tailed and asymmetric. The authors argue that:

• Classical Gaussian or Rayleigh assumptions are often inadequate for modeling NLOS power statistics in cluttered environments.
• L-moment descriptors offer a more reliable alternative for preprocessing, LOS/NLOS detection, and realistic channel simulation.
• The proposed Kappa-based modeling and L-moment features can be integrated into ML-assisted AoA estimation pipelines to improve robustness under heavy-tailed propagation conditions.

The authors remain modest regarding the scope, noting that the current analysis is limited to a single indoor environment and fixed hardware configuration. They suggest that future work should evaluate the transferability of these L-moment/Kappa relationships across diverse environments and dynamic scenarios involving moving objects.