## 2 Principle
### GS-Based CD Pre-Compensation Algorithm at the Transmitter  
Figure 1 shows the principle of the iterative GS-based CD pre-compensation algorithm. Firstly, the ideal signal \(E(t)\) is assigned to \(A(t)\) for initialization at the transmitter. The dispersion transfer module is then executed to simulate the transmission of data in the fiber. At the receiver, the phase information of the data is retained. The amplitude information is replaced by the amplitude of the ideal signal \(E(t)\) as a receiver constraint. Next, the updated data are transmitted to the transmitter by the inverse dispersion transfer function. At the transmitter, we only need to obtain the amplitude information of the data and ignore the phase information as a transmitter constraint. At this point, one whole iteration is completed. The next iteration initiates by following the steps above. It is worth mentioning that \(A(t)\) usually takes the square root of \(E(t)\). After the iteration, the GS algorithm is completed, and \(A(t)\) is squared to obtain the transmitted data.  
The influence of CD can be expressed as \(H_{CD}(f)=\cos(D\lambda^{2}\omega^{2}L/4\pi c)\), where \(\lambda=1550.12\,\mathrm{nm}\), \(c=3\times 10^{8}\) m/s, and \(D=17\times 10^{-6}\) ps/nm/km. To simplify the description, we define \(\theta=D\lambda^{2}\omega^{2}L/4\pi c\). According to Figure 1, the iterative GS-based CD pre-compensation algorithm process can be described by the followed expression. At the 1 st iteration, the transmitted data are the ideal intensity signal without phase information. It passes through the dispersion transmission module and can be described as follows:  
\[\begin{bmatrix}B^{1}\\ \phi^{1}\end{bmatrix}=\begin{bmatrix}\cos\theta&-2P_{0}\sin\theta\\ \sin\theta/2P_{0}&\cos\theta\end{bmatrix}\begin{bmatrix}E\\ 0\end{bmatrix} \tag{1}\]  
Then, the amplitude of \(B^{1}\) is replaced by \(E\) as the receiver constraint. The phase of \(\phi^{1}\) is retained. It subsequently passes through the inverse dispersion transmission module and becomes the following:  
\[\begin{bmatrix}A^{1}\\ \psi^{1}\end{bmatrix}=\begin{bmatrix}\cos\theta&2P_{0}\sin\theta\\ -\sin\theta/2P_{0}&\cos\theta\end{bmatrix}\begin{bmatrix}E\\ \phi^{1}\end{bmatrix} \tag{2}\]  
where \(A^{1}=\left[\cos\theta+\sin^{2}\theta\right]E\) is the desired transmitted signal at the 1 st iteration. In the similar way, \(A^{2}=\left[\cos\theta+\sin^{2}\theta\left[\cos\theta+\sin^{2}\theta\right] \right]E\) is the desired transmitted signal at the 2 nd iteration. It follows that the desired transmitted signal at the \(n\)th iteration is \(A^{n}=\left[\cos\theta+\sin^{2}\theta\left[\cos\theta+\sin^{2}\theta\left[ \ldots\right]\right]\right]E\). That is to say, the desired transmitted intensity signal can be achieved by multiplying the ideal signal and the function \(f(n)\). Through constructing geometric series and further deduction, the \(f(n)\) can be expressed:  
\[f(n)=\sin^{2n}\theta+\frac{1-\sin^{2n}\theta}{\cos\theta} \tag{3}\]  
where \(\theta\) is fixed when the transmission distance is confirmed. The only variable in \(f(n)\) is iteration number. However, the expression is the non-iterative version, and thus, the complexity is reduced.  
The role of the GS-based CD pre-compensation algorithm can further be observed through the power spectrum. As shown by the blue line in the Figure 2, the transmitted signal passes through the dispersion channel and generates the spectrum nulls. The green line in the Figure 2 corresponds to the effect of \(f(n)\). It elevates the spectrum on either side of spectrum nulls. At the receiver, the received signal undergoes the combined action of dispersion and \(f(n)\). Obviously, the effect of spectrum nulls on the received signal is weakened, as shown by the orange line in the Figure 2.  
Figure 1: Scheme diagram of GS-based CD pre-compensation algorithm.