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Fundamental Limits of Two-Hop MIMO Channels: An Asymptotic Approach

Zeyan Zhuang, Xin Zhang, Dongfang Xu, Shenghui Song

TL;DR

The paper addresses the fundamental limits of two-hop MIMO channels by deriving deterministic approximations for the mean and variance of MI using large-scale random matrix theory with Gaussian tools. It establishes a joint CLT for the MI terms, proves the asymptotic Gaussianity, and provides an iterative algorithm to compute the fixed-point equations that govern the deterministic equivalents. The results give O(1/N) convergence rates and explicit covariance structure, enabling rigorous outage and ergodic-capacity assessments for AF-relay and active-IRS aided MIMO systems. The framework generalizes to a broad class of two-hop (and product) channels and offers practical insight into the role of active elements and channel correlations in system performance.

Abstract

Multi-antenna relays and intelligent reflecting surfaces (IRSs) have been utilized to construct favorable channels to improve the performance of wireless systems. A common feature between relay systems and IRS-aided systems is the two-hop multiple-input multiple-output (MIMO) channel. As a result, the mutual information (MI) of two-hop MIMO channels has been widely investigated with very engaging results. However, a rigorous investigation on the fundamental limits of two-hop MIMO channels, i.e., the first and second-order analysis, is not yet available in the literature, due to the difficulties caused by the two-hop (product) channel and the noise introduced by the relay (active IRS). In this paper, we employ large-scale random matrix theory (RMT), specifically Gaussian tools, to derive the closed-form deterministic approximation for the mean and variance of the MI. Additionally, we determine the convergence rate for the mean, variance and the characteristic function of the MI, and prove the asymptotic Gaussianity. Furthermore, we also investigate the analytical properties of the fundamental equations that describe the closed-form approximation and prove the existence and uniqueness of the solution. An iterative algorithm is then proposed to obtain the solution for the fundamental equations. Numerical results validate the accuracy of the theoretical analysis.

Fundamental Limits of Two-Hop MIMO Channels: An Asymptotic Approach

TL;DR

The paper addresses the fundamental limits of two-hop MIMO channels by deriving deterministic approximations for the mean and variance of MI using large-scale random matrix theory with Gaussian tools. It establishes a joint CLT for the MI terms, proves the asymptotic Gaussianity, and provides an iterative algorithm to compute the fixed-point equations that govern the deterministic equivalents. The results give O(1/N) convergence rates and explicit covariance structure, enabling rigorous outage and ergodic-capacity assessments for AF-relay and active-IRS aided MIMO systems. The framework generalizes to a broad class of two-hop (and product) channels and offers practical insight into the role of active elements and channel correlations in system performance.

Abstract

Multi-antenna relays and intelligent reflecting surfaces (IRSs) have been utilized to construct favorable channels to improve the performance of wireless systems. A common feature between relay systems and IRS-aided systems is the two-hop multiple-input multiple-output (MIMO) channel. As a result, the mutual information (MI) of two-hop MIMO channels has been widely investigated with very engaging results. However, a rigorous investigation on the fundamental limits of two-hop MIMO channels, i.e., the first and second-order analysis, is not yet available in the literature, due to the difficulties caused by the two-hop (product) channel and the noise introduced by the relay (active IRS). In this paper, we employ large-scale random matrix theory (RMT), specifically Gaussian tools, to derive the closed-form deterministic approximation for the mean and variance of the MI. Additionally, we determine the convergence rate for the mean, variance and the characteristic function of the MI, and prove the asymptotic Gaussianity. Furthermore, we also investigate the analytical properties of the fundamental equations that describe the closed-form approximation and prove the existence and uniqueness of the solution. An iterative algorithm is then proposed to obtain the solution for the fundamental equations. Numerical results validate the accuracy of the theoretical analysis.
Paper Structure (70 sections, 25 theorems, 266 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 70 sections, 25 theorems, 266 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Two-Hop MIMO Channels
  3. Existing Works
  4. Challenges
  5. Contributions
  6. Paper Organization and Notations
  7. System Model and Problem Formulation
  8. System Model
  9. Problem Formulation
  10. Assumptions
  11. Useful Notations and the Main Results
  12. Useful Notations
  13. Fundamental System of Equations
  14. The Resolvent
  15. Deterministic Quantities
  16. ...and 55 more sections

Key Result

Proposition 1

Assuming that assumptions A-1-A-3 hold, the system of equations in DE_system_1 admit unique positive solutions $(\delta, \overline{\omega}, \underline{\omega}, \gamma)$ for $z > 0$, $\overline{s} \geq 0$.

Figures (6)

  • Figure 1: Diagram for the logic flow of this paper
  • Figure 2: ESD of $\mathbf{H}_1 \mathbf{H}_2 \mathbf{H}_2^H \mathbf{H}_1^H + \overline{s} \mathbf{H}_1\mathbf{H}_1^H$ versus LSD
  • Figure 3: Approximation Error versus $N$
  • Figure 4: i.i.d. Case
  • Figure 5: Chi-square plot
  • ...and 1 more figures

Theorems & Definitions (36)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Lemma 1
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Remark 4
  • Proposition 3
  • ...and 26 more