A Simultaneous Clustering and Tracking Algorithm for Capturing Cluster-Level Spatial Consistency in 6G Wireless Channels

Technical Summary: A Simultaneous Clustering and Tracking Algorithm for Capturing Cluster-Level Spatial Consistency in 6G Wireless Channels

Problem Statement
Spatial consistency is a critical physical property for realistic channel modeling in 6G systems, particularly for technologies like massive MIMO, integrated sensing and communication, and terahertz (THz) communication. At the cluster level, this property requires that multipath components (MPCs) with similar characteristics remain grouped into the same clusters as transceivers move, ensuring smooth evolution of channel parameters.

Existing methods for extracting cluster-level spatial consistency generally fall into two categories: "tracking-after-clustering" and "joint clustering-and-tracking."

• Tracking-after-clustering decouples the processes, performing clustering on individual snapshots (e.g., using KPowerMeans or DBSCAN) and then associating clusters across snapshots. This often leads to fragmented trajectories when cluster compositions change between snapshots.
• Joint clustering-and-tracking attempts to use information from previous snapshots (typically cluster centroids or Kalman filter predictions) to guide current clustering. However, these methods often rely heavily on simple centroid matching and do not fully exploit the joint distribution of MPCs across delay, angular, and spatial domains.

Consequently, existing metrics (such as Davies–Bouldin or Calinski–Harabasz indices for clustering, and Gradient Change Rate for tracking) evaluate these aspects separately, failing to jointly reflect clustering quality and tracking consistency.

Methodology: MD-SCT Algorithm
The authors propose a Mahalanobis-distance-based Simultaneous Clustering and Tracking (MD-SCT) algorithm. Unlike Euclidean-based approaches, MD-SCT utilizes the Mahalanobis distance to measure the distance between a newly observed MPC and the parameter distribution of existing clusters. This approach inherently accounts for the covariance structure among MPC parameters (delay, azimuth/zenith angles of departure and arrival, and transceiver locations).

The algorithm operates as follows:

1. Initialization: In a dense spatial region, conventional clustering methods (e.g., KPowerMeans or DBSCAN) are applied to an initial set of MPCs to establish $K_1$ initial clusters. This ensures robust estimation of the initial mean vectors and covariance matrices.
2. Simultaneous Assignment: For subsequent snapshots, each new MPC is assigned to an existing cluster $k$ if its Mahalanobis distance ($D_{Mah}$) to the cluster's parameter distribution is below a threshold ($D_{th}$). This step inherently performs tracking by associating new observations with historical distributions.
3. Outlier Handling: MPCs that do not fit existing clusters are marked as outliers. These are subsequently re-clustered using conventional methods to form new clusters (representing cluster birth).
4. Cluster Lifecycle: Clusters with no assigned MPCs over consecutive snapshots are considered inactive (cluster death). The algorithm also incorporates a memory mechanism to retain dormant clusters during temporary observation gaps, improving robustness against blockage.
5. Complexity Optimization: To manage computational cost, the inverse covariance matrix is updated iteratively using the Woodbury matrix inversion lemma, reducing complexity from $O(N_{par}^3)$ to $O(N_{par}^2)$.

Key Contributions

1. Algorithm Design: The proposal of the MD-SCT algorithm, which improves cluster-level spatial consistency by exploiting the joint distribution structure of delay, angle, and transceiver position formed by previously assigned MPCs.
2. Application: The application of this algorithm to sub-THz channels at 132 GHz, generated via measurement-based deterministic ray tracing in an industrial Internet of Things (IIoT) scenario.
3. Evaluation Metric: The introduction of the Mean Square Successive Difference (MSSD) as a novel metric to quantify the spatial consistency of clustering statistics (e.g., DB index, CH index, parameter standard deviations) across successive snapshots.

Experimental Results
The algorithm was validated against a multidimensional feature metric-based (MFM) tracking method using channel data from a 132 GHz IIoT scenario.

• Cluster Continuity: The MD-SCT algorithm produced significantly fewer but longer and more continuous cluster trajectories compared to MFM. Specifically, MD-SCT identified 17 clusters with an average tracked length of 11.894 m, whereas MFM produced 104 clusters with an average length of only 1.457 m.
• Tracking Consistency: The MD-SCT algorithm achieved a Gradient Change Rate (GCR) of 0.028, significantly lower than the MFM method's 0.561, indicating much smoother cluster-center evolution.
• Statistical Stability: The MSSD values for clustering statistics (DB, CH) and parameter standard deviations (delay, angles) were consistently lower for MD-SCT. For instance, the MSSD for the Davies–Bouldin index was 0.003 for MD-SCT versus 0.007 for MFM.
• Clustering Quality: Despite the focus on tracking, MD-SCT maintained comparable cluster compactness and separation (Avg. DB: 0.369; Avg. CH: 146.072) to the snapshot-optimized MFM method (Avg. DB: 0.402; Avg. CH: 134.825).

Significance
The paper claims that the MD-SCT algorithm yields smoother cluster evolution by inherently tracking while clustering, thereby improving the reliability of clustered channels for spatial consistency modeling in 6G. The authors state that these findings support spatial-consistency-aware channel modeling, which is essential for accurate beam tracking, environment-aware sensing, and the design of massive MIMO and sub-THz communication systems. The work emphasizes that by capturing the joint distribution of MPCs, the proposed method addresses the limitations of decoupled or centroid-dependent tracking approaches.