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

We consider the identifiability issue of maximum-likelihood based activity detection in massive MIMO-based grant-free random access. An intriguing observation by (Chen et al., 2022) indicates that the identifiability undergoes a phase transition for commonly-used random user signatures as $L^{2}$ , $N$ and $K$ tend to infinity with fixed ratios, where $L$ , $N$ and $K$ denote the user signature length, the total number of users, and the number of active users, respectively. In this letter, we provide a precise analytical characterization of the phase transition based on a spectral universality conjecture. Numerical results demonstrate excellent agreement between our theoretical predictions and the empirical phase transitions.