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This paper considers a massive random access problem in which a large numberof sporadically active devices wish to communicate with a base station (BS)equipped with massive multiple-input multiple-output (MIMO) antennas. Eachdevice is preassigned a unique signature sequence, and the BS identifies theactive devices by detecting which sequences are transmitted. This deviceactivity detection problem can be formulated as a maximum likelihood estimation(MLE) problem for which the sample covariance matrix of the received signal isa sufficient statistic. The goal of this paper is to characterize the feasibleset of problem parameters under which this covariance based approach is able tosuccessfully recover the device activities in the massive MIMO regime. Throughan analysis of the asymptotic behaviors of MLE via its associated Fisherinformation matrix, this paper derives a necessary and sufficient condition onthe Fisher information matrix to ensure a vanishing probability of detectionerror as the number of antennas goes to infinity, based on which a numericalphase transition analysis is obtained. This condition is also examined from aperspective of covariance matching, which relates the phase transition analysisto a recently derived scaling law. Further, we provide a characterization ofthe distribution of the estimation error in MLE, based on which the errorprobabilities in device activity detection can be accurately predicted.Finally, this paper studies a random access scheme with joint device activityand data detection and analyzes its performance in a similar way.

Phase Transition Analysis for Covariance Based Massive Random Access with Massive MIMO