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Fundamental Limits of Optical Fiber MIMO Channels With Finite Blocklength

Xin Zhang, Dongfang Xu, Shenghui Song, Mérouane Debbah

TL;DR

The paper addresses finite-blocklength limits for optical-fiber MIMO channels modeled by the Jacobi ensemble, deriving a central limit theorem for information density and Gaussian bounds on the optimal error probability for rates near capacity. It develops a closed-form ergodic-capacity approximation and a CLT-based dispersion expression, showing how the ID fluctuations translate into explicit, near-capacity error-probability bounds in the finite-blocklength regime. The results reveal that Jacobi channels degenerate to Rayleigh channels as the number of available channels grows and that, at high SNR, more available channels can increase error probability; numerical experiments confirm the bounds’ accuracy and their closer alignment with practical LDPC performance than outage. This framework provides rigorous, actionable FBL performance metrics for optical-fiber SDM/MIMO systems, enabling better design under latency-restricted constraints.

Abstract

The multiple-input and multiple-output (MIMO) technique is regarded as a promising approach to boost the throughput and reliability of optical fiber communications. However, the fundamental limits of optical fiber MIMO systems with finite block-length (FBL) are not available in the literature. This paper studies the fundamental limits of optical fiber multicore/multimode systems in the FBL regime when the coding rate is a perturbation within $\mathcal{O}(\frac{1}{\sqrt{ML}})$ of the capacity, where M and L represent the number of transmit channels and blocklength, respectively. Considering the Jacobi MIMO channel, which was proposed to model the nearly lossless propagation and the crosstalks in optical fiber systems, we derive the upper and lower bounds for the optimal error probability. For that purpose, we first set up the central limit theorem for the information density in the asymptotic regime where the number of transmit, receive, available channels and the blocklength go to infinity at the same pace. The result is then utilized to derive the upper and lower bounds for the optimal average error probability with the concerned rate. The derived theoretical results reveal interesting physical insights for Jacobi MIMO channels with FBL. First, the derived bounds for Jacobi channels degenerate to those for Rayleigh channels when the number of available channels approaches infinity. Second, the high signal-to-noise (SNR) approximation indicates that a larger number of available channels results in a larger error probability. Numerical results validate the accuracy of the theoretical results and show that the derived bounds are closer to the performance of practical LDPC codes than outage probability.

Fundamental Limits of Optical Fiber MIMO Channels With Finite Blocklength

TL;DR

The paper addresses finite-blocklength limits for optical-fiber MIMO channels modeled by the Jacobi ensemble, deriving a central limit theorem for information density and Gaussian bounds on the optimal error probability for rates near capacity. It develops a closed-form ergodic-capacity approximation and a CLT-based dispersion expression, showing how the ID fluctuations translate into explicit, near-capacity error-probability bounds in the finite-blocklength regime. The results reveal that Jacobi channels degenerate to Rayleigh channels as the number of available channels grows and that, at high SNR, more available channels can increase error probability; numerical experiments confirm the bounds’ accuracy and their closer alignment with practical LDPC performance than outage. This framework provides rigorous, actionable FBL performance metrics for optical-fiber SDM/MIMO systems, enabling better design under latency-restricted constraints.

Abstract

The multiple-input and multiple-output (MIMO) technique is regarded as a promising approach to boost the throughput and reliability of optical fiber communications. However, the fundamental limits of optical fiber MIMO systems with finite block-length (FBL) are not available in the literature. This paper studies the fundamental limits of optical fiber multicore/multimode systems in the FBL regime when the coding rate is a perturbation within of the capacity, where M and L represent the number of transmit channels and blocklength, respectively. Considering the Jacobi MIMO channel, which was proposed to model the nearly lossless propagation and the crosstalks in optical fiber systems, we derive the upper and lower bounds for the optimal error probability. For that purpose, we first set up the central limit theorem for the information density in the asymptotic regime where the number of transmit, receive, available channels and the blocklength go to infinity at the same pace. The result is then utilized to derive the upper and lower bounds for the optimal average error probability with the concerned rate. The derived theoretical results reveal interesting physical insights for Jacobi MIMO channels with FBL. First, the derived bounds for Jacobi channels degenerate to those for Rayleigh channels when the number of available channels approaches infinity. Second, the high signal-to-noise (SNR) approximation indicates that a larger number of available channels results in a larger error probability. Numerical results validate the accuracy of the theoretical results and show that the derived bounds are closer to the performance of practical LDPC codes than outage probability.
Paper Structure (42 sections, 6 theorems, 314 equations, 4 figures)

This paper contains 42 sections, 6 theorems, 314 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. System Model
  4. Jacobi Model
  5. Optimal Average Error Probability
  6. Information Density (ID) and Error Bounds
  7. Problem Formulation
  8. Main Results
  9. Capacity Analysis
  10. CLT for ID
  11. Comparison with the CLT for Rayleigh Channels
  12. $N\le M$
  13. $N> M$
  14. Upper and Lower Bounds for Error Probability
  15. Numerical Results
  16. ...and 27 more sections

Key Result

Theorem 1

Given $0<\lim\inf\limits_{N \ge 1} y_1 \le y_1 \le \lim \sup\limits_{N \ge 1} y_1 <\infty$ and $0<\lim\inf\limits_{N \ge 1} y_2 \le y_2 \le \lim \sup\limits_{N \ge 1} y_2 <\infty$, the following evaluation for $C(\sigma^2)=\frac{1}{M}\log\det({\bold{I}}_N+\frac{1}{\sigma^2}{\bold{H}}{\bold{H}}^{ and When $N\le M$, $\overline{C}(\sigma^2)$ is given by where When $N > M$, $\overline{C}(\sigma

Figures (4)

  • Figure 1: Optical SDM MIMO systems with $n$ channels.
  • Figure 2: Approximation accuracy of the derived bounds.
  • Figure 3: Bounds with different $n$.
  • Figure 4: Bounds and LDPC codes ($n=16$).

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1
  • ...and 4 more