## 3 Results
### Simulation System and Result Analysis
#### 3.1.3 Computational Effort Analysis  
In this section, computational efforts of the conventional DBP and A-DBP are analyzed by complex multiplication (CM) per transmitted bit. Both of these two schemes employ a symmetric SSFM to compensate by linear and nonlinear steps. The linear step is implemented in the FD, using the overlap-save method as the FFT strategy. The nonlinear step is implemented in the TD, which needs IFFT to achieve the conversion from the FD to the TD. The CMs of the SSFM per overlap block can be obtained as [36; 37]:  
\[N_{\text{step}}(2M_{\text{FFT}}+2M_{D}+M_{N}) \tag{10}\]  
where \(N_{step}\) is the number of SSFM steps, \(M_{FFT}\) is the number of CMs for each FFT/IFFT and is calculated as \(M_{FFT}=N_{FFT}/2\text{log}_{2}(N_{FFT})\) CMs considering a standard Cooley-Tukey radix-2 implementation, in which \(N_{FFT}\) is the block size of FFT employing the overlap-save method. \(M_{D}\) and \(M_{N}\) represent the CMs for evaluating every linear and nonlinear step of each SSFM step, respectively. \(M_{D}\) equals \(N_{FFT}\) CMs per linear step. As for the part inside the exponential operator of the nonlinear step, \(2N_{FFT}\) real multiplications (RMs) are needed for the power operator and \(N_{FFT}\) RMs are required for the multiplication of the real nonlinear constant. There are \(6N_{FFT}\) RMs (\(3/2\)\(N_{FFT}\) CMs) as the total cost for solving the exponential operator by a 4 th order Taylor expansion [38]. \(N_{FFT}\) CMs are needed for the multiplication of \(A_{x,y}(z,t)\) outside the exponential operator which is described by the inverse Manakov equation. Hence, \(M_{N}\) equals \(5/2\)\(N_{FFT}\) CMs per nonlinear step. In addition, the factor of 2 associated with \(M_{FFT}\) is for the FFT/IFFT pair in one SSFM step and the factor of 2 connected with \(M_{D}\) is for two linear steps in each SSFM step. Considering the whole iterations of \(n\), extra FFT and IFFT located before and after SSFM should be considered as shown in Figure 1. The CMs of DBP/A-DBP per overlap block can be extended from Equation (10) as:  
\[n\times[N_{\text{step}}(2M_{\text{FFT}}+2M_{D}+M_{N})+2M_{\text{FFT}}] \tag{11}\]  
When Equation (11) is applied to DBP, \(n=1\). In A-DBP, \(n\) represents the times of iterations by gradient-descent algorithms. Due to the memory length \(N_{M}\) of DC in the overlap-save method which is related to the channel impulse response [36], only \(N_{FFT}-N_{M}+1\) samples are useful in each overlap block. The useful bits per overlap block are given as [36]:  
\[\frac{N_{\text{FFT}}-N_{M}+1}{n_{\text{sc}}}\times\text{log}_{2}(M) \tag{12}\]  
where \(M\) is the order of the modulation format, \(n_{\text{sc}}\) is the oversampling ratio. It is noted that there is only one calculation of the iterative formula for plentiful samples, so it is  
Figure 9: \(\text{log}_{10}(\text{BER})\) as the function of launching power for LE, the conventional DBP, and the proposed method in heterogeneous link of **(a)**: fiber length, **(b)**: fiber type. tiny enough to be ignored since few real multiplications generated from that formula are averaged into every sample. Therefore, the complexity \(N\) of conventional DBP and A-DBP equals the following expression:  
\[ \begin{array}{l}N=\frac{n\times n_{sc}\times[N_{\text{step}}(2M_{\text{TFT}}+2M _{\text{D}}+M_{N})+2M_{\text{FFT}}]}{(N_{\text{TFT}}-N_{\text{M}}+1)\times\log_ {2}(M)}\\ =\frac{n\times n_{sc}\times N_{\text{TFT}}\times[N_{\text{step}}(\log_{2}(N_{ \text{FFT}})+\frac{2}{2})+\log_{2}(N_{\text{FFT}})]}{(N_{\text{FFT}}-N_{\text {M}}+1)\times\log_{2}(M)}\\ \end{array}  \tag{13}\]  
We assess complexity efforts of the conventional DBP and our proposed method with six and one SSFM steps per span, respectively, based on the situation shown in Figure 7, showing that both methods reach the FEC threshold. Table 1 depicts the number of complex multiplications per transmitted bit calculated by (13).