This paper focuses on the covariance-based activity detection problem in a multi-cell massive multiple-input multiple-output (MIMO) system. In this system, active devices transmit their signature sequences to multiple base stations (BSs), and the BSs cooperatively detect the active devices based on the received signals. While the scaling law for the covariance-based activity detection in the single-cell scenario has been extensively analyzed in the literature, this paper aims to analyze the scaling law for the covariance-based activity detection in the multi-cell massive MIMO system. Specifically, this paper demonstrates 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. This finding shows that, in the multi-cell MIMO system, the maximum number of active devices that can be detected correctly 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). Moreover, in addition to analyzing the scaling law for the signature sequences randomly and uniformly distributed on a sphere, the paper also establishes the scaling law for signature sequences generated from a finite alphabet, which are easier to generate and store. Moreover, this paper proposes two efficient accelerated coordinate descent (CD) algorithms with a convergence guarantee for solving the device activity detection problem. The first algorithm reduces the complexity of CD by using an inexact coordinate update strategy. The second algorithm avoids unnecessary computations of CD by using an active set selection strategy. Simulation results show that the proposed algorithms exhibit excellent performance in terms of computational efficiency and detection error probability.

翻译：本文聚焦于多小区大规模多输入多输出(MIMO)系统中的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发送特征序列，基站协同根据接收信号检测活跃设备。尽管现有文献已深入分析了单小区场景下基于协方差的活动检测缩放律，本文旨在分析多小区大规模MIMO系统中该问题的缩放律。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，多小区系统存在二次缩放律。这一发现表明，在多小区MIMO系统中，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，并随小区数量呈对数递减（当天线数趋于无穷时）。此外，除了分析球面上随机均匀分布的特征序列的缩放律外，本文还建立了易于生成和存储的有限字母表特征序列的缩放律。同时，本文提出两种具有收敛保证的高效加速坐标下降算法以解决设备活动检测问题。第一种算法采用非精确坐标更新策略降低计算复杂度，第二种算法通过活动集选择策略避免不必要的计算。仿真结果表明，所提算法在计算效率和检测错误概率方面均表现出优异性能。