Four-Dimensional Coded Modulation with Bit-wise Decoders for Future Optical Communications

TL;DR

This work analyzes four-dimensional coded modulation for coherent optical communications with bit-wise decoders. It demonstrates that generalized mutual information ($I^{\text{gmi}}$) is the robust metric for predicting post-FEC BER in BW demapping, and shows that constellations transmitting independently across dimensions (PM-QPSK, PM-16QAM, PM-64QAM) outperform 4D constellations optimized for uncoded performance when used with BW decoders. Through extensive simulations across $M=16$, $256$, and $4096$ points, the authors reveal that $I^{\text{gmi}}$ aligns well with practical LDPC-coded performance, with PM constellations providing favorable trade-offs between complexity and reliability. The study emphasizes the limitation of MI for BW decoders in finite-SNR regimes and highlights labeling importance, suggesting that per-dimension PM formats are preferable for future optical systems employing BW FEC decoders.

Abstract

Coded modulation (CM) is the combination of forward error correction (FEC) and multilevel constellations. Coherent optical communication systems result in a four-dimensional (4D) signal space, which naturally leads to 4D-CM transceivers. A practically attractive design paradigm is to use a bit-wise decoder, where the detection process is (suboptimally) separated into two steps: soft-decision demapping followed by binary decoding. In this paper, bit-wise decoders are studied from an information-theoretic viewpoint. 4D constellations with up to 4096 constellation points are considered. Metrics to predict the post-FEC bit-error rate (BER) of bit-wise decoders are analyzed. The mutual information is shown to fail at predicting the post-FEC BER of bit-wise decoders and the so-called generalized mutual information is shown to be a much more robust metric. For the suboptimal scheme under consideration, it is also shown that constellations that transmit and receive information in each polarization and quadrature independently (e.g., PM-QPSK, PM-16QAM, and PM-64QAM) outperform the best 4D constellations designed for uncoded transmission. Theoretical gains are as high as 4 dB, which are then validated via numerical simulations of low-density parity check codes.

Figures (11)

• Figure 1: CM structure under consideration. The CM encoder is a concatenation of a rate-$R_\mathrm{c}$ binary FEC encoder and a memoryless mapper. The CM decoder is either an ML decoder or a BW decoder (see Fig. \ref{['receivers']}).
• Figure 2: Two implementations of the CM decoder in Fig. \ref{['model']}: Optimum (ML) decoder (top) and BW decoder (bottom).
• Figure 3: Post-FEC BER (${\textnormal{BER}}_{{\textnormal{pos}}}$) for different code rates $R_\mathrm{c}$ and constellations as a function of the SNR $\gamma$. The constellations are PM-QPSK (squares), PM-16QAM (circles), PM-64QAM (triangles), and PM-256QAM (stars).
• Figure 4: Post-FEC BER (${\textnormal{BER}}_{{\textnormal{pos}}}$) as a function of pre-FEC BER (${\textnormal{BER}}_{{\textnormal{pre}}}$) for the $24$ cases in Fig. \ref{['BERout_vs_SNR']}. The same markers are used.
• Figure 5: Post-FEC BER (${\textnormal{BER}}_{{\textnormal{pos}}}$) as a function of the normalized MI ($I(\boldsymbol{X};\boldsymbol{Y})/m$) for the $24$ cases in Fig. \ref{['BERout_vs_SNR']} (same markers).