ML-Enhanced Digital Backpropagation for Long-Reach Single-Span Systems

We propose a digital backpropagation method that employs machine-learning-aided joint optimization of dispersion step lengths and nonlinear phase rotation filters within an FFT-based enhanced split-step Fourier structure, achieving improved accuracy at low computational complexity.

💡 Research Summary

The paper introduces a novel low‑complexity digital back‑propagation (DBP) technique called Learned Enhanced Split‑Step Fourier Method (L‑ESSFM). Traditional DBP based on the split‑step Fourier method (SSFM) alternates linear dispersion and nonlinear Kerr steps, but achieving high accuracy typically requires hundreds of very short steps, leading to prohibitive computational load for real‑time systems. Two existing low‑complexity approaches have been pursued: the Enhanced SSFM (ESSFM), which refines the nonlinear step with a frequency‑domain filter while preserving the FFT/IFFT implementation, and Learned DBP (LDBP), which treats the SSFM as a deep neural network and jointly learns time‑domain FIR coefficients for all steps. ESSFM scales logarithmically with channel memory but often fixes step sizes and uses identical nonlinear filters across steps, limiting its adaptability. LDBP offers full joint optimization but its FIR‑based time‑domain implementation scales quadratically with symbol rate, making it unsuitable for very high‑speed links.

L‑ESSFM merges the strengths of both methods. It retains the FFT‑based ESSFM architecture but makes every linear step length (L_i) and every nonlinear phase‑rotation (NLPR) filter (c_i) trainable. The linear step is represented by the conventional group‑velocity‑dispersion transfer function (H_{\text{GVD}}(L_i)=\exp(-j2\pi^2\beta_2 f^2 L_i)), while the NLPR filter is a symmetric real‑valued impulse response converted to the frequency domain via a real FFT (RFFT). By allowing each step to have its own (L_i) and (c_i), the method captures the interaction between dispersion and nonlinearity more precisely than a globally fixed filter. The total number of trainable parameters grows linearly with the number of steps (N_s) and the filter length (N_c), yet the overall computational cost remains dominated by the FFT/IFFT pairs, preserving the favorable logarithmic scaling.

Training is performed offline using supervised learning. Large batches of i.i.d. complex Gaussian symbols are generated, propagated through a simulated 170 km SMF link, and corrupted by amplified spontaneous emission noise. The received samples constitute the neural‑network inputs, while the original transmitted symbols are the desired outputs. Initial values for (L_i) are set to equally divide the link, and the NLPR filters start as instantaneous nonlinearities. Standard back‑propagation with the Adam optimizer refines all parameters jointly. Once trained, the L‑ESSFM can be deployed in real time with the same FFT‑based processing pipeline as conventional ESSFM, but with per‑step optimized distances and filters.

Performance evaluation uses a 5‑channel, 93 GBd dual‑polarization 64‑QAM WDM system (100 GHz spacing) over a single‑span 170 km SMF link (0.2 dB/km loss, 17 ps/nm/km dispersion, 1.27 W⁻¹km⁻¹ Kerr coefficient). The receiver includes an EDF‑A pre‑amplifier (NF = 4.5 dB), channel demultiplexing, and either DBP or electronic dispersion compensation (EDC). Complexity is measured in real multiplications per complex symbol (RM/2D). L‑ESSFM with four steps ((N_s=4)) requires 172 RM/2D and delivers a 0.8 dB SNR gain over pure EDC. This gain is achieved with roughly one‑quarter the computational effort of the conventional ESSFM (761 RM/2D) and with an order‑of‑magnitude less effort than LDBP (≈2000 RM/2D for comparable performance). As the number of steps increases, L‑ESSFM asymptotically approaches the ideal infinite‑step SSFM performance while its complexity grows only modestly, confirming the efficiency of the joint optimization.

The authors conclude that L‑ESSFM provides a practical pathway to high‑accuracy, low‑complexity DBP for modern high‑speed optical links, especially those involving long‑reach, single‑span transmission such as data‑center interconnects. The method can be readily extended to multi‑span links with inline amplification and further enhanced by sub‑band processing. Future work is suggested on hardware‑friendly training, parameter quantization, and integration with adaptive modulation formats.