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Statistical Framework for Clustering MU-MIMO Wireless via Second Order Statistics

Roberto Pereira, Xavier Mestre

TL;DR

This work addresses clustering of users in MU-MIMO systems by exploiting second-order channel statistics on the SPD manifold. It introduces a log-Euclidean distance $d_M$ between covariance matrices and a consistent estimator $\hat{d}_M$, proving a central limit theorem for $M(\hat{d}_M-d_M)$ and extending the result to a multivariate setting for multiple SCM comparisons. The authors derive the explicit form of the consistent estimator, establish assumptions ensuring consistency, and demonstrate via numerical validation that the consistent estimator markedly improves clustering performance over the traditional plug-in distance, with accurate asymptotic predictions even at moderate system dimensions. The framework provides a practical, predictive tool for assessing clustering quality in realistic MU-MIMO deployments, guiding channel estimation budgets and interference management strategies. Key contributions include the CLT for the distance estimator, the high-dimensional clustering analysis, and robust numerical validation across varying sample regimes.

Abstract

This work explores the clustering of wireless users by examining the distances between their channel covariance matrices, which reside on the Riemannian manifold of positive definite matrices. Specifically, we consider an estimator of the Log-Euclidean distance between multiple sample covariance matrices (SCMs) consistent when the number of samples and the observation size grow unbounded at the same rate. Within the context of multi-user MIMO (MU-MIMO) wireless communication systems, we develop a statistical framework that allows to accurate predictions of the clustering algorithm's performance under realistic conditions. Specifically, we present a central limit theorem that establishes the asymptotic Gaussianity of the consistent estimator of the log-Euclidean distance computed over two sample covariance matrices.

Statistical Framework for Clustering MU-MIMO Wireless via Second Order Statistics

TL;DR

This work addresses clustering of users in MU-MIMO systems by exploiting second-order channel statistics on the SPD manifold. It introduces a log-Euclidean distance between covariance matrices and a consistent estimator , proving a central limit theorem for and extending the result to a multivariate setting for multiple SCM comparisons. The authors derive the explicit form of the consistent estimator, establish assumptions ensuring consistency, and demonstrate via numerical validation that the consistent estimator markedly improves clustering performance over the traditional plug-in distance, with accurate asymptotic predictions even at moderate system dimensions. The framework provides a practical, predictive tool for assessing clustering quality in realistic MU-MIMO deployments, guiding channel estimation budgets and interference management strategies. Key contributions include the CLT for the distance estimator, the high-dimensional clustering analysis, and robust numerical validation across varying sample regimes.

Abstract

This work explores the clustering of wireless users by examining the distances between their channel covariance matrices, which reside on the Riemannian manifold of positive definite matrices. Specifically, we consider an estimator of the Log-Euclidean distance between multiple sample covariance matrices (SCMs) consistent when the number of samples and the observation size grow unbounded at the same rate. Within the context of multi-user MIMO (MU-MIMO) wireless communication systems, we develop a statistical framework that allows to accurate predictions of the clustering algorithm's performance under realistic conditions. Specifically, we present a central limit theorem that establishes the asymptotic Gaussianity of the consistent estimator of the log-Euclidean distance computed over two sample covariance matrices.
Paper Structure (7 sections, 1 theorem, 18 equations, 4 figures)

This paper contains 7 sections, 1 theorem, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimators of log-Euclidean Distance
  4. Asymptotic Fluctuation of $\hat{d}_M$
  5. Numerical Validation
  6. Clustering of Multiple UEs
  7. Conclusion

Key Result

Theorem 3.1

In addition to (As1)-(As2), assume that the observations are complex circularly symmetric Gaussian distributed and that the minimum eigenvalue of $\bar{\boldsymbol{\Sigma}}_M$ is bounded away from zero. Then, the random vector converges in law to a multivariate standard Gaussian.

Figures (4)

  • Figure 1: Illustrative example of the simulation scenario with $G=3$ groups (blue, red, and green), $K=21$ UEs and two different values of $\tau= 1,2$ (represented by the circles and the cross). From the BS perspective (located at the origin), the parameter $\tau$ controls the cluster separability.
  • Figure 2: Comparison between empirical distribution of $\hat{d}_M$ and its asymptotic according to Theorem \ref{['th:cltEstimators']} under different scenarios and $N_1 = N_2 = 24$.
  • Figure 3: Probability of correct clustering (y-axis) $K$ UEs into three groups, different ratios $M/N^{(g)}, g=1,2,3$ and growing $\tau$ (x-axis).
  • Figure 4: Probability of correct clustering (y-axis) $K$ UEs into three groups for fixed $M = 8$, $\tau=1.5$ and growing $N^{(g)}$.

Theorems & Definitions (1)

  • Theorem 3.1