## 3 Results
### Simulation System and Result Analysis
#### 3.1.2 Analysis and Discussion of Simulation Results  
Figure 4 compares EVM performances of different algorithms versus the number of iterations in A-DBP with a five-step SSFM at 5 dBm launching power. We initialize \(\gamma=3\) /W/km for all algorithms and set \(\mu=1.1\)\(\times\)\(10^{-5}\), \(8\)\(\times\)\(10^{-6}\), \(1\)\(\times\)\(10^{-5}\), \(2.2\)\(\times\)\(10^{-4}\) for the GDA, GDAM, ANAGDA, A-DBP-LTM, respectively. A-DBP-LTM is implemented by RMSProp, AdaGrad, Adam, or AdaDelta. We can see that the CF of the GDA has fallen into a local minimum, while that of the GDAM fluctuates violently because of overdependence on the previous gradient. For ANAGDA, the introduction of the predictive term doubles the number of iterations, which prevents the low-complexity nonlinearity compensation. Among these algorithms, the A-DBP-LTM implemented by RMSProp simultaneously indicates the best compensation performance and the fastest and the most stable convergence process even compared with AdaGrad, Adam, and AdaDelta. Thus, we choose RMSProp to carry out the later numerical simulation and experiments to further validate the effectiveness of our proposed A-DBP-LTM method. Figure 4: EVM versus number of iterations for different A-DBP with 5-step SSFM at 5 dBm launching power. Figure 3: The heterogeneous links of length and nonlinear coefficient in the simulation: (**a**) the first hybrid link A; (**b**) the second hybrid link B. Figure 5 depicts the number of iterations for different A-DBP methods under variations of \(\Delta\mu\) and \(k\) at the optimal launching power with five SSFM steps. To prevent fluctuations and ensure good convergence, the initial \(k\) is set within [\(2\times 10^{-3}\), \(4\times 10^{-3}\)], exceeding the transmission link's default value. Based on the optimal \(\mu\) and \(m\) (for the GDAM) at the middle value of the range of \(k\), \(\mu\) is swept across [\(-1\times 10^{-5}\), \(1\times 10^{-5}\)], with \(\Delta\mu\) denoting the change of \(\mu\). It is noted that the iteration process converges when \(\Delta\mu<0\), and a number of iterations greater than 20, shown as the same color as 20 in Figure 5, is caused by the low convergence speed. For \(\Delta\mu>0\), fluctuations cause more iterations or divergence, also shaded as 20 in Figure 5. Figure 5b shows the number of iterations of A-DBP based on the GDAM is reduced compared to A-DBP based on the GDA shown in Figure 5a because of the additional momentum term when \(\Delta\mu<0\). However, when \(\Delta\mu>0\), the momentum term \(m\) intensifies fluctuations, leading to more iterations or divergence compared to A-DBP based on the GDA. Note that the performance of A-DBP based on the GDAM relies on \(m\) and fewer iterations are realizable by adjusting \(m\). But it cannot show the superiority owing to the hyperparameter \(m\). Conversely, our proposed method demonstrates strong robustness on hyperparameters by adapting \(\mu\) and consistently maintains a low number of iterations compared to A-DBP based on the GDA and A-DBP based on the GDAM. In order to verify the compensation performance in the situations of deviated \(\gamma\), the conventional DBP and the proposed method are implemented by employing different \(\gamma\) (0.6, 1.31, 2/W/km). It should be noted that, because the nonlinear coefficient is fixed  
Figure 5: The number of iterations under different changes of \(\Delta\mu\) and initial \(k\) using (**a**) A-DBP based on GDA; (**b**) A-DBP based on GDAM; (**c**) the proposed method at the best launching. as 1.31/W/km for each fiber span in the simulation system, the performance of the conventional DBP algorithm has been optimized with \(\gamma\) of 1.31/W/km, which ensures a fair comparison with the proposed A-DBP-LTM implemented by RMSProp. The FEC threshold is 3 \(\times\) 10\({}^{-2}\). Figure 6 compares the power-dependent log10(BER) performance of linear equalizer (LE), the conventional DBP, and the proposed method, all with five SSFM steps.