## 3 Numerical Analysis and Performance Evaluation  
The nonlinear SchrOdinger equation (NLSE) of Equation (1) is only known as the expression for the propagation of the optical signal in the silica fiber such as SMF and DCF [17].  
\[\frac{\partial A_{j}}{\partial z}=-\frac{\alpha}{2}A_{j}-\frac{i}{2}\beta_{2j }\frac{\partial^{2}A_{j}}{\partial T^{2}}+\frac{1}{6}\beta_{3j}\frac{ \partial^{3}A_{j}}{\partial T^{3}}+i\gamma_{j}\big{|}A_{j}\big{|}^{2}A_{j}+2i \gamma_{j}\big{|}A_{k}\big{|}^{2}A_{j}, \tag{1}\]  
In Equation (1), \(j\), \(k\) = 1, 2, \(\ldots\), 24 (\(j\
eq k\)), \(A_{j}\) represents the complex amplitude of the optical signal of the \(j\)-th channel, \(z\) is the propagation distance, \(\beta_{2j}\) is the group velocity dispersion (GVD), \(\beta_{3j}\) is dispersion slope, \(\gamma_{j}\) is nonlinear coefficient, and \(T\) = \(t-z/v_{j}\) is the time measured in a retarded frame. The last two terms of (1) express the effect of nonlinearity, including self-phase modulation (SPM) and cross-phase modulation (XPM). It has been reported that the effect of XPM on WDM signals decreases as the dispersion of optical fiber increases [18]. The WDM signal may be affected by XPM due to the relatively high power of WDM signals, but since SMF with a large dispersion coefficient was used as the basic fiber constituting the optical link, we excluded the effect of XPM based on the inference that the effect of XPM would be lower than that of SPM.  
The most common approach for solving (1) is the split-step Fourier method [17]. The split-step Fourier method divides the transmission medium dominated by NLSE into small steps and processes linear calculations in the frequency domain and nonlinear calculations in the time domain separately and continuously through the Fourier transform and inverse Fourier transform for each step [17].  
The performance evaluation in this study is based on an an eye diagram. An eye diagram can simultaneously determine the level (amplitude) characteristics and timing characteristics of the signal. We use the eye-opening penalty (EOP) and timing jitter(TJ) to assess the compensated optical signals as the level characteristics and the timing characteristics, respectively.  
\[\text{EOP}[\text{dB}]=10\text{$\text{$\text{$\text{$\text{$\text{$\text{$ \text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$}}$}$}$}$}}$}}}}$}$}} \frac{EO_{\text{rec}}}{EO_{\text{btb}}}, \tag{2}\]  
The EOP is quantified (in dB) as Equation (2), where \(EO_{\text{rec}}\) and \(EO_{\text{btb}}\) are the eye opening of receiving optical signals and the back-to-back optical signals (i.e., the reference signals), respectively. The eye opening is defined as twice the average power of all of the received signals divided by the smallest power level of the 'one' optical signal minus the largest power level of the 'zero' optical signal.  
The compensation performance criterion is 1-dB EOP, which is equivalent to the pulse broadening (the ratio of root mean square (RMS) width of the received pulse to RMS width of the initial pulse) of 1.25 and corresponds to a bit error rate (BER) of \(10^{-12}\)[19]. We also use 2 ps, which is 8% of the duration of 40 Gb/s, as a performance criterion for timing jitter.  
In the 'random-inverse' schemed dispersion map examined in this paper, the RDPS of each optical fiber span has different values, but their total sum is 0 ps/nm, i.e., NRD is 0 ps/nm. The simplest general way to form the DM link with NRD of 0 ps/nm is to fix the SMF length at 80 km and the DCF length at 16 km. We call this method a 'non-random fixed' scheme. For the DM link with the 'random-inverse' dispersion map to be used for distortion compensation of WDM signals, the compensation performance must at least be better than the 'non-random fixed' scheme. For this reason, we also analyze the compensation characteristics of the WDM signal in the DM link designed with a 'non-random fixed' scheme, and the EOP and TJ of the worst channel obtained from this link will be used for comparison.