Phase Transition Analysis for Covariance Based Massive Random Access with Massive MIMO
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This paper considers the massive random access problem in which a large number of sporadically active devices wish to communicate with a base-station (BS) equipped with a large number of antennas. Each device is preassigned a unique signature sequence, and the BS identifies the active devices in the random access by detecting which sequences are transmitted. This device activity detection problem can be formulated as a maximum likelihood estimation (MLE) problem with the sample covariance matrix of the received signal being a sufficient statistic. The aim of this paper is to characterize the parameter space in which this covariance based approach would be able to successfully recover the device activities in the massive multiple-input multiple-output (MIMO) regime. Through an analysis of the asymptotic behaviors of MLE via its associated Fisher information matrix, this paper derives a necessary and sufficient condition on the Fisher information matrix to ensure a vanishing detection error rate as the number of antennas goes to infinity, based on which a numerical phase transition analysis is obtained. This condition is also examined from a perspective of covariance matching that relates the phase transition analysis in this paper to a recently derived scaling law. Furthermore, this paper provides a characterization of the distribution of the estimation error in MLE, based on which the error probabilities in device activity detection can be accurately predicted. Finally, this paper studies a random access scheme with joint device activity and data detection and analyzes its performance in a similar way. Simulation results validate the analysis.
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