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Terminated and Tailbiting Spatially-Coupled Codes with Optimized Bit Mappings for Spectrally Efficient Fiber-Optical Systems

Christian Häger, Alexandre Graell i Amat, Fredrik Brännström, Alex Alvarado, Erik Agrell

TL;DR

The paper investigates how unequal protection across modulation bits can be exploited by optimizing the bit mapper for spectrally efficient fiber-optic systems using SC-LDPC and SC-GLDPC codes. It combines density-evolution/P-EXIT analyses with differential-evolution optimization to tailor bit allocations for terminated and tailbiting configurations, showing tailbiting codes can closely match terminated performance at lower FEC overhead. Simulation results on linear and nonlinear channels confirm the analytical predictions, with tailbiting achieving up to about $0.55$ dB improvement at $10^{-5}$ and an estimated reach gain of around 13% in some scenarios. Overall, optimized bit mappers enable tailbiting SC codes to approach capacity more closely, while terminated codes experience diminishing returns at long spatial lengths.

Abstract

We study the design of spectrally efficient fiber-optical communication systems based on different spatially coupled (SC) forward error correction (FEC) schemes. In particular, we optimize the allocation of the coded bits from the FEC encoder to the modulation bits of the signal constellation. Two SC code classes are considered. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes which are decoded using iterative hard-decision decoding. For both code classes, the bit allocation is optimized for the terminated and tailbiting SC cases based on a density evolution analysis. An optimized bit allocation can significantly improve the performance of tailbiting SC codes codes over the baseline sequential allocation, up to the point where they have a comparable gap to capacity as their terminated counterparts, at a lower FEC overhead. For the considered terminated SC codes, the optimization only results in marginal performance improvements, suggesting that in this case a sequential allocation is close to optimal.

Terminated and Tailbiting Spatially-Coupled Codes with Optimized Bit Mappings for Spectrally Efficient Fiber-Optical Systems

TL;DR

The paper investigates how unequal protection across modulation bits can be exploited by optimizing the bit mapper for spectrally efficient fiber-optic systems using SC-LDPC and SC-GLDPC codes. It combines density-evolution/P-EXIT analyses with differential-evolution optimization to tailor bit allocations for terminated and tailbiting configurations, showing tailbiting codes can closely match terminated performance at lower FEC overhead. Simulation results on linear and nonlinear channels confirm the analytical predictions, with tailbiting achieving up to about dB improvement at and an estimated reach gain of around 13% in some scenarios. Overall, optimized bit mappers enable tailbiting SC codes to approach capacity more closely, while terminated codes experience diminishing returns at long spatial lengths.

Abstract

We study the design of spectrally efficient fiber-optical communication systems based on different spatially coupled (SC) forward error correction (FEC) schemes. In particular, we optimize the allocation of the coded bits from the FEC encoder to the modulation bits of the signal constellation. Two SC code classes are considered. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes which are decoded using iterative hard-decision decoding. For both code classes, the bit allocation is optimized for the terminated and tailbiting SC cases based on a density evolution analysis. An optimized bit allocation can significantly improve the performance of tailbiting SC codes codes over the baseline sequential allocation, up to the point where they have a comparable gap to capacity as their terminated counterparts, at a lower FEC overhead. For the considered terminated SC codes, the optimization only results in marginal performance improvements, suggesting that in this case a sequential allocation is close to optimal.

Paper Structure

This paper contains 16 sections, 10 equations, 11 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Protograph-Based SC-LDPC Codes
  4. Code Construction
  5. Soft-Decision Decoding and Density Evolution
  6. SC-GLDPC Codes with BCH Component Codes
  7. Code Construction
  8. Hard-Decision Decoding and Density Evolution
  9. Bit Mapper Optimization
  10. Results and Discussion
  11. Structure of the Optimized Bit Mapper for Tailbiting Codes
  12. Optimization Gain
  13. Gap to Capacity
  14. Simulation Results
  15. Conclusions
  16. ...and 1 more sections

Figures (11)

  • Figure 1: Block diagram of the considered PM transmission system.
  • Figure 2: Illustration of the Bit Mapper.
  • Figure 3: Base matrices $\mathbf{P}$ for protograph-based SC-LDPC codes.
  • Figure 4: Protographs for the code with ${T} = 5$ in Example \ref{['ex:regldpc']}. The first step of the lifting procedure to obtain the Tanner graph (i.e., copying the protograph $M$ times) is indicated in light gray.
  • Figure 5: Predicted (solid lines) and finite-length (dashed lines) performance for the codes in Examples \ref{['ex:exitldpc']} (left) and \ref{['ex:gldpc']} (right). The codes have lengths $80\,000$ (a, crosses), $400\,000$ (a, dots), $168\,000$ (b, crosses), and $1\,260\,000$ (b, dots).
  • ...and 6 more figures

Theorems & Definitions (6)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6