The Rayleigh-product channel model is utilized to characterize the rank deficiency caused by keyhole effects. However, the finite blocklength analysis for Rayleigh-product channels is not available in the literature. In this paper, we will characterize the mutual information density (MID) and perform the FBL analysis to reveal the impact of rank-deficiency in Rayleigh-product channels. To this end, we first set up a central limit theorem for the MID over Rayleigh-product MIMO channels in the asymptotic regime where the number of scatterers, number 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：瑞利-乘积信道模型用于表征由密钥孔效应导致的秩亏缺问题。然而，文献中尚未见针对瑞利-乘积信道的有限分组长度分析。本文将刻画互信息密度（MID）并进行FBL分析，以揭示瑞利-乘积信道中秩亏缺的影响。为此，我们首先在散射体数量、天线数量及分组长度同步趋于无穷的渐近框架下，建立了瑞利-乘积MIMO信道上互信息密度的中心极限定理（CLT）。进而利用此CLT推导分组错误概率的上下界，并分别在信噪比高、低区域导出其近似表达式以阐释秩亏缺的影响。一个有趣的发现是：在FBL条件下，秩亏缺会劣化MIMO系统性能；当散射体数量趋于无穷时，瑞利-乘积信道的基本极限将退化为经典瑞利信道的情形。