Low-Complexity Adaptive Digital Backpropagation Based on Clock-Tone Strength for Nyquist and Bandwidth-Limited Coherent Optical Systems

Digital backpropagation (DBP) compensates fiber Kerr nonlinearities by approximately performing inverse propagation based on the nonlinear Schrödinger equation (NLSE), typically implemented via a split-step Fourier method (SSFM). However, in practical links, the assumed transmission parameters, particularly nonlinear coefficientγ, often deviate from their true values, leading to model mismatch and reduced DBP effectiveness. To address this issue, we propose a low-complexity, clock-tone assisted adaptive DBP (CT-ADBP) scheme that is tolerant to bandwidth limitations and across modulation formats. The CT-ADBP utilize the CT strength computed by an interpolated-power (IP) Gardner timing-error detector (TED) as the feedback metric to update the empirical coefficientηvia a steepest descent algorithm. This enables fine-tuning of the nonlinear phase compensation without introducing auxiliary per-iteration digital signal processing blocks. Owing to the spectral excess-bandwidth tolerance of the IP-TED, the CT-ADBP operates robustly for both Nyquist shaped signals and narrowband pre-filtered signals with 3-dB bandwidth below the Nyquist bandwidth. We experimentally validate the CT-ADBP on a 1200 km recirculating fiber loop using 30-GBd Nyquist-shaped 16- quadrature amplitude modulation (16-QAM) and 64-QAM, and narrowband pre-filtered 16-QAM signal with a 3-dB bandwidth of 6 GHz. Across all three modulation settings, the CT-ADBP exhibits the most reliable convergence. In contrast, the existing intensity-noise-variance ADBP (INV-ADBP) shows a larger estimation bias for the narrowband pre-filtered signal, whereas the phase variance ADBP (PV-ADBP) converges reliably only for 16-QAM. When all the schemes converge to their respective optima with comparable estimation bias, they deliver essentially the same performance. However, CT-ADBP achieves a reduction in per-symbol computational complexity of 52.9% and 45.4% relative to PV-ADBP and INV-ADBP, respectively.