## 4 Simulation Result Analysis and Compensation Mechanism
Regarding situations where the dispersion has not been completely compensated, the insertion of pilot symbols was used to conduct optic fiber channel estimation and compensate for the phase shift. First, the received signal was assumed to be as expressed in Equation (9). \[r_{ik}=x_{ik}\otimes h_{k}+n_{ik}, \tag{9}\]  
where \(i\) represents the \(i\)th data symbol, \(k\) is the \(k\)th subcarrier, \(r_{ik}\) is the received signal, \(x_{ik}\) is the transmitted signal, \(h_{k}\) is the fiber channel effect, and \(n_{ik}\) is random noise. Subsequently, a least-square algorithm was used to estimate the frequency response in the fiber channel, as shown in Equation (10), where \(\hat{H}_{ik}\) equals FFT\(\{h_{k}\}\). \[\hat{H}_{ik}=X_{ik}^{-1}\cdot R_{ik}=X_{ik}^{-1}\cdot(X_{ik}\cdot H_{ik}+N_{ik}) =H_{ik}+X_{ik}^{-1}\cdot N_{ik}, \tag{10}\]  
Regarding the pilot symbol arrangement, a comb-type arrangement was adopted; pilot symbols were periodically inserted on the subcarriers and then removed at the receiver after they had been subjected to channel interference, and the calculation was performed. For the subcarriers without pilot symbols, the interpolation method was used to estimate the channel's frequency response. This study primarily adopted a linear interpolation method to conduct channel estimation. The schematic diagram is shown in Figure 11 and the interpolation value can be obtained using Equation (11). Figure 11: Schematic diagram of the linear interpolation method. Figure 10: Receiver constellation diagrams after DCF compensation (DCF length is set to 4 km). \[f_{1}(x)=f(x_{1})+\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}} \left(x-x_{1}\right) \tag{11}\]  
In Equation (11), the point slope form is utilized to calculate the estimated numerical value between two points. Furthermore, the estimation results are typically more accurate the smaller the interval between two data points is in situations when the linear interpolation method is used. Figure 12 shows a receiver constellation diagram after 10 km of SMF transmission and 4 km of DCF transmission, with no optical amplifier added and after electrical equalization. One pilot symbol is inserted at intervals of 64 subcarriers. In addition, a cyclic prefix with a length of 1/32 of the IFFT amount is added to suppress ISI. A comparison of Figures 10 and 12 indicates that after compensation through linear interpolation equalization, the cluster points in the constellation diagram became relatively converged and exhibited no phase shift. In addition to adopting linear interpolation technology, this study also investigated three common equalizer technologies, namely piecewise cubic spline interpolation, piecewise cubic Hermite interpolation, and FFT interpolation. On the other hand, the constellation diagrams using the linear, piecewise cubic spline, and piecewise cubic Hermite interpolation methods for channel estimation are highly similar after equalization, and they possess identical effectiveness, as shown by Figure 12a-c.  
Conversely, the constellation diagram obtained by using the FFT interpolation method is relatively diverged, as indicated by Figure 12d. The main reason for these results is that signals are subjected to  
Figure 12: Constellation diagrams after estimation and equalization of fiber channels using different interpolation methods: (**a**) linear interpolation; (**b**) piecewise cubic spline interpolation; (**c**) piecewise cubic Hermite interpolation; and (**d**) fast Fourier transform (FFT) interpolation. influence from different factors in different transmission channels. For example, in a wireless channel, the signal is affected by environmental factors and interference from barriers, while a signal transmitted in an optic fiber is influenced by interference from dispersion and other factors.