Mutual information density (MID) plays an important role in the analysis of MIMO systems with finite-blocklength (FBL). However, with the increasing antenna size, e.g., massive MIMO, the full-rank channel condition may no longer hold, but the MID analysis for rank-deficient MIMO channels is not available in the literature, making it difficult to unveil the performance loss due to rank-deficiency. In this paper, we will characterize the MID of Rayleigh-product channels, which are able to model both the full-rank and rank-deficient cases, and perform the FBL analysis to reveal the impact of rank-deficiency. To this end, we first set up a central limit theorem for the MID in the asymptotic regime where the number of scatterers, numbers of antennas, and blocklength go to infinity at the same pace. Then, we utilize the CLT to obtain the upper and lower bounds for the packet error probability, whose approximations in the high and low signal to noise ratio regimes are then derived to illustrate the impact of rank-deficiency. One interesting observation is that rank-deficiency degrades the performance of MIMO systems with FBL and the fundamental limits of Rayleigh-product channels degenerate to those of the Rayleigh case when the number of scatterers approaches infinity.

翻译：互信息密度（MID）在有限块长（FBL）MIMO系统的分析中扮演着重要角色。然而，随着天线规模的增大（例如大规模MIMO），满秩信道条件可能不再成立，但文献中尚缺乏对秩亏MIMO信道进行MID分析的工作，这使得揭示秩亏导致的性能损失变得困难。本文将对能够同时建模满秩和秩亏情况的瑞利-乘积信道的MID进行刻画，并通过FBL分析揭示秩亏的影响。为此，我们首先在散射体数量、天线数量及块长以相同速度趋于无穷的渐进域中，建立了MID的中心极限定理（CLT）。随后，利用该CLT得到了分组错误概率的上界和下界，并推导了高信噪比和低信噪比下的近似表达式，以阐明秩亏的影响。一个有趣的观测是：秩亏会降低FBL 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退化为瑞利信道的情形。