This paper explores the fundamental limits of unsourced random access (URA) with a random and unknown number ${\rm{K}}_a$ of active users in MIMO quasi-static Rayleigh fading channels. First, we derive an upper bound on the probability of incorrectly estimating the number of active users. We prove that it exponentially decays with the number of receive antennas and eventually vanishes, whereas reaches a plateau as the power and blocklength increase. Then, we derive non-asymptotic achievability and converse bounds on the minimum energy-per-bit required by each active user to reliably transmit $J$ bits with blocklength $n$. Numerical results verify the tightness of our bounds, suggesting that they provide benchmarks to evaluate existing schemes. The extra required energy-per-bit due to the uncertainty of the number of active users decreases as $\mathbb{E}[{\rm{K}}_a]$ increases. Compared to random access with individual codebooks, the URA paradigm achieves higher spectral and energy efficiency. Moreover, using codewords distributed on a sphere is shown to outperform the Gaussian random coding scheme in the non-asymptotic regime.

翻译：本文探究了在MIMO准静态瑞利衰落信道中，具有随机且未知活跃用户数${\rm{K}}_a$的无源随机接入（URA）的基本极限。首先，我们推导了活跃用户数估计错误概率的上界。证明表明，该概率随接收天线数量呈指数衰减并最终消失，但随着功率和块长的增加而趋于平稳。随后，我们推导了每个活跃用户以块长$n$可靠传输$J$比特所需最小每比特能量的非渐近可达界与逆界。数值结果验证了所提界的紧致性，表明其可作为评估现有方案的基准。由活跃用户数不确定性引起的额外每比特能量需求随$\mathbb{E}[{\rm{K}}_a]$增大而减小。与采用独立码本的随机接入相比，URA范式能实现更高的频谱效率和能量效率。此外，与非渐近体制下的高斯随机编码方案相比，采用分布于球面上的码字表现出更优性能。