This paper studies the covariance based activity detection problem in a multi-cell massive multiple-input multiple-output (MIMO) system, where the active devices transmit their signature sequences to multiple base stations (BSs), and the BSs cooperatively detect the active devices based on the received signals. The scaling law of covariance based activity detection in the single-cell scenario has been thoroughly analyzed in the literature. This paper aims to analyze the scaling law of covariance based activity detection in the multi-cell massive MIMO system. In particular, this paper shows a quadratic scaling law in the multi-cell system under the assumption that the exponent in the classical path-loss model is greater than 2, which demonstrates that in the multi-cell MIMO system the maximum number of active devices that can be correctly detected in each cell increases quadratically with the length of the signature sequence and decreases logarithmically with the number of cells (as the number of antennas tends to infinity). This paper also characterizes the distribution of the estimation error in the multi-cell scenario.

翻译：本文研究了多小区大规模多输入多输出（MIMO）系统中基于协方差的活动检测问题。在该系统中，活跃设备向多个基站（BS）发送其特征序列，基站基于接收信号协同检测活跃设备。已有文献对单小区场景下基于协方差的活动检测的标度律进行了深入分析。本文旨在分析多小区大规模MIMO系统中基于协方差的活动检测的标度律。具体而言，本文证明：在经典路径损耗模型指数大于2的假设下，多小区系统呈现二次标度律——当天线数趋于无穷时，每个小区可正确检测的最大活跃设备数量随特征序列长度呈二次增长，且随小区数量呈对数减少。本文还刻画了多小区场景下估计误差的分布特征。