An arXiv preprint reports that a piecewise-linear way of calculating the reliability values used by a soft-decision decoder matched an exact, relative-intensity-noise-aware calculation in bit-error-rate tests. The approach, called the zero-crossing approximation (ZCA), is also described as asymptotically less complex.
A shortcut for a noisy link
Log-likelihood ratios, or LLRs, turn each received signal into a graded indication of which bit values it supports. The analysis compares exact LLRs with two lower-complexity alternatives: signal-independent AWGN-like LLRs based on averaged noise, and the ZCA's piecewise-linear approximation.
The modeled channel uses a Gaussian distribution whose variance depends on the transmitted symbol. After equalization, residual intersymbol interference is treated as negligible and the channel is considered memoryless.
The ZCA omits logarithm and exponential operations and has asymptotic complexity O(M), compared with O(M log M) for the exact, AWGN-like and max-log methods.
Where the methods diverged
In PAM-4, the AWGN-like calculation differed from exact LLRs by about 0.07 dB in optical modulation amplitude (OMA) at 400 Gb/s, 0.2 dB at 600 Gb/s and 0.63 dB at 800 Gb/s.
At high OMA, the AWGN-like method's error floor was 7.2 times higher at 600 Gb/s and 3.5 times higher at 800 Gb/s. The error floor is the residual bit-error rate left in the comparison after the signal level rises.
Under the reported RIN condition, PAM-8 post-forward-error-correction BER with the extended-Hamming code did not reach the KP4-FEC threshold. The ZCA matched exact LLRs, while the AWGN-like calculation showed a small penalty gap.
Noise still sets an information ceiling
The study's normalized generalized mutual information (GMI), an information-rate measure, showed the same advantage for the ZCA: it reached the same maximum normalized GMI as exact LLRs. The AWGN-like method was below the maximum by 0.0081 in one comparison, with a gap of up to 0.028 for PAM-8 as RIN increased.
At a symbol rate of 238.13 GBd and under strong RIN, PAM-8 normalized GMI saturated around 0.93. The reported maximum achievable information rate was capped at 664.38 Gb/s rather than 714.39 Gb/s.
The comparison also found that GMI was not a good predictor of extended-Hamming BER.
A second coding test
A separate test used DVB-S2 low-density parity-check (LDPC) codes at rates 2/3, 3/4, 5/6 and 8/9, with 64,800-bit blocks and a fixed symbol rate of 238.13 GBd. The reported setup included PAM-4 at 400 Gb/s and PAM-8 at 600 Gb/s.
For the LDPC codes, the three LLR calculations produced essentially identical post-FEC BER when plotted against NGMI, or normalized GMI. Relative to exact LLRs, the AWGN-like method had OMA gaps of 0.1 dB at code rate 3/4, 0.36 dB at 5/6 and 2.41 dB at 8/9.
What the preprint leaves open
The supplied document identifies itself as an arXiv preprint, version v1. Its channel model treats residual intersymbol interference after equalization as negligible and assumes a memoryless channel; the authors list experimental validation, PAM-6 and constellation shaping as future work.
The work was funded by Holland High Tech | TKI HSTM through the PPS allowance scheme for public-private partnerships as part of the COCOLI project.
Paper data and sources
Original title: Low-complexity Soft-decision LLR Calculations for Next-generation IM-DD Systems with RIN
Authors: Felipe Villenas, Yunus Can Gültekin, Alex Alvarado
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text