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键孔效应引起的秩亏缺问题。然而，现有文献中尚未出现针对瑞利乘积信道的有限块长分析。本文将刻画互信息密度（MID）并进行有限块长分析，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、天线数量和块长以相同速率趋于无穷的渐近机制下，建立了瑞利乘积MIMO信道上互信息密度的中心极限定理。然后利用该中心极限定理推导了数据包错误概率的上下界，进而得到高信噪比和低信噪比机制下的近似表达式，以阐明秩亏缺的影响。一个有趣的发现是：秩亏缺会降低采用有限块长的MIMO系统性能，而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至瑞利情形。