With the development of innovative applications that require high reliability and low latency, ultra-reliable and low latency communications become critical for wireless networks. In this paper, the second-order coding rate of the coherent quasi-static Rayleigh-product MIMO channel is investigated. We consider the coding rate within O(1/\sqrt(Mn)) of the capacity, where M and n denote the number of transmit antennas and the blocklength, respectively, and derive the closed-form upper and lower bounds for the optimal average error probability. This analysis is achieved by setting up a central limit theorem (CLT) for the mutual information density (MID) with the assumption that the block-length, the number of the scatterers, and the number of the antennas go to infinity with the same pace. To obtain more physical insights, the high and low SNR approximations for the upper and lower bounds are also given. One interesting observation is that rank-deficiency degrades the performance of MIMO systems with FBL and the fundamental limits of the Rayleigh-product channel approaches those of the single Rayleigh case when the number of scatterers approaches infinity. Finally, the fitness of the CLT and the gap between the derived bounds and the performance of practical LDPC coding are illustrated by simulations.

翻译：随着需要高可靠性与低延迟的创新应用的发展，超可靠低延迟通信成为无线网络的关键。本文研究了相干准静态瑞利乘积MIMO信道的二阶编码率。我们考虑容量附近O(1/√(Mn))量级的编码率，其中M和n分别表示发射天线数和块长，并推导了最优平均错误概率的闭式上下界。该分析通过设定互信息密度（MID）的中心极限定理（CLT）实现，假设块长、散射体数量与天线数量以相同速度趋于无穷。为获得更多物理洞察，还给出了高、低信噪比下上下界的近似。一个有趣的发现是：秩亏会降低有限块长（FBL）下MIMO系统的性能，且当散射体数量趋于无穷时，瑞利乘积信道的基本极限趋近于单瑞利情况。最后，通过仿真验证了CLT的适用性以及所推导界与实际LDPC编码性能之间的差距。