Mutual Information Density of Massive MIMO Systems over Rayleigh-Product Channels

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.