Electromagnetic information theory (EIT) is an interdisciplinary subject that serves to integrate deterministic electromagnetic theory with stochastic Shannon's information theory. Existing EIT analysis operates in the continuous space domain, which is not aligned with the practical algorithms working in the discrete space domain. This mismatch leads to a significant difficulty in application of EIT methodologies to practical discrete space systems, which is called as the discrete-continuous gap in this paper. To bridge this gap, we establish the discrete-continuous correspondence with a prolate spheroidal wave function (PSWF)-based ergodic capacity analysis framework. Specifically, we state and prove some discrete-continuous correspondence lemmas to establish a firm theoretical connection between discrete information-theoretic quantities to their continuous counterparts. With these lemmas, we apply the PSWF ergodic capacity bound to advanced MIMO architectures such as continuous-aperture MIMO (CAP-MIMO) and extremely large-scale MIMO (XL-MIMO). From this PSWF capacity bound, we discover the capacity saturation phenomenon both theoretically and empirically. Although the growth of MIMO performance is fundamentally limited in this EIT-based analysis framework, we reveal new opportunities in MIMO channel estimation by exploiting the EIT knowledge about the channel. Inspired by the PSWF capacity bound, we utilize continuous PSWFs to improve the pilot design of discrete MIMO channel estimators, which is called as the PSWF channel estimator (PSWF-CE). Simulation results demonstrate improved performances of the proposed PSWF-CE, compared to traditional minimum mean squared error (MMSE) and compressed sensing-based estimators.

翻译：电磁信息论（EIT）是一门旨在将确定性电磁理论与随机性香农信息论相融合的交叉学科。现有的EIT分析在连续空间域中进行，这与实际在离散空间域中运行的算法并不一致。这种不匹配导致EIT方法在实际离散空间系统中的应用存在显著困难，本文称之为离散-连续鸿沟。为弥合此鸿沟，我们建立了一种基于扁球面波函数（PSWF）的遍历容量分析框架的离散-连续对应关系。具体而言，我们陈述并证明了一些离散-连续对应引理，以在离散信息论量与它们的连续对应量之间建立坚实的理论联系。利用这些引理，我们将PSWF遍历容量界应用于先进的MIMO架构，如连续孔径MIMO（CAP-MIMO）和超大规模MIMO（XL-MIMO）。基于此PSWF容量界，我们从理论和实证两方面发现了容量饱和现象。尽管在此基于EIT的分析框架中，MIMO性能的增长受到根本性限制，但我们通过利用关于信道的EIT知识揭示了MIMO信道估计的新机遇。受PSWF容量界的启发，我们利用连续PSWF改进离散MIMO信道估计器的导频设计，称之为PSWF信道估计器（PSWF-CE）。仿真结果表明，与传统的基于最小均方误差（MMSE）和压缩感知的估计器相比，所提出的PSWF-CE具有更优的性能。