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Precise Analysis of Covariance Identifiability for Activity Detection in Grant-Free Random Access

Shengsong Luo, Junjie Ma, Chongbin Xu, Xin Wang

TL;DR

The paper addresses identifiability of covariance-based activity detection in grant-free massive MIMO by analyzing when the maximum-likelihood estimator can uniquely recover the active-user pattern from second-order statistics. It introduces a semi-random surrogate for the Khatri–Rao-structured mapping and leverages Tropp's convex geometry framework to derive a sharp phase-transition boundary, depending on the active fraction $\,\epsilon = K/N\,$ and a rank parameter $\,\alpha\,$ related to the signature structure. The main result characterizes the boundary via $\delta_*(\epsilon)$ and $\mu_*(\epsilon)$, with $\delta_*(\epsilon) = 1 - (1-\epsilon) \Phi(\mu_*(\epsilon))$ and $(1-\epsilon)[\mu(1-\Phi(\mu)) - \phi(\mu)] + \epsilon \mu = 0$, establishing high-probability identifiability when $\alpha > \delta_*(\epsilon)$ and non-identifiability otherwise; the sparse-limit behavior $\delta_*(\epsilon) \sim 2\epsilon \log(1/\epsilon)$ recovers compressed-sensing scalings. Simulations across Gaussian, Rademacher, and sub-sampled Hadamard signatures corroborate the theory, and the work motivates a potential full rigor via spectral universality.

Abstract

We consider the identifiability issue of maximum likelihood based activity detection in massive MIMO based grant-free random access. A prior work by Chen et al. indicates that the identifiability undergoes a phase transition for commonly-used random signatures. In this paper, we provide an analytical characterization of the boundary of the phase transition curve. Our theoretical results agree well with the numerical experiments.

Precise Analysis of Covariance Identifiability for Activity Detection in Grant-Free Random Access

TL;DR

The paper addresses identifiability of covariance-based activity detection in grant-free massive MIMO by analyzing when the maximum-likelihood estimator can uniquely recover the active-user pattern from second-order statistics. It introduces a semi-random surrogate for the Khatri–Rao-structured mapping and leverages Tropp's convex geometry framework to derive a sharp phase-transition boundary, depending on the active fraction and a rank parameter related to the signature structure. The main result characterizes the boundary via and , with and , establishing high-probability identifiability when and non-identifiability otherwise; the sparse-limit behavior recovers compressed-sensing scalings. Simulations across Gaussian, Rademacher, and sub-sampled Hadamard signatures corroborate the theory, and the work motivates a potential full rigor via spectral universality.

Abstract

We consider the identifiability issue of maximum likelihood based activity detection in massive MIMO based grant-free random access. A prior work by Chen et al. indicates that the identifiability undergoes a phase transition for commonly-used random signatures. In this paper, we provide an analytical characterization of the boundary of the phase transition curve. Our theoretical results agree well with the numerical experiments.
Paper Structure (9 sections, 4 theorems, 36 equations, 1 figure)

This paper contains 9 sections, 4 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Semi-Random Model: Phase Transition Analysis
  4. Semi-Random Model
  5. Phase Transition Analysis
  6. Behavior of the Statistical Dimension in the Sparse Limit
  7. Simulation Results
  8. Conclusion
  9. Proof of Theorem \ref{['The:main']}

Key Result

Proposition 1

Tropp: Let $\bm{V}\in\mathbb{R}^{N\times N}$ be a Haar distributed random orthogonal matrix. Let $\mathcal{D},\mathcal{K}$ be two closed convex cones in $\mathbb{R}^N$. The following holds for any $\eta \in (0,1)$: where $\xi_{\eta} := \sqrt{8 \log(4/\eta)}$, and $\delta(\mathcal{D})$ (similarly for $\delta(\mathcal{K})$) denotes the statistical dimension of $\mathcal{D}$: Here, ${\Pi}_{\mathcal

Figures (1)

  • Figure 1: Empirical phase transition of the identifiability of MLE under various signature matrices. $d:=\frac{L^2-L}{2}$ where $L$ is length of user signatures. $K$ is the number of active users and $N$ is the number of total users.

Theorems & Definitions (4)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Lemma 1