# Low-Complexity Chromatic Dispersion Equalization FIR Digital Filter for Coherent Receiver
## 4 Discussion
### Comparision of FD-CDE and CQWO-TD-CDE  
From Section 3, we know that the computational complexity of FD-CDE is proportional to the size of the FFT block \(N_{\text{FFT}}\) In the optical communication system of long-distance transmission from cost, FD-CDE is far superior to TD-CDE. Therefore, CQWO-TD-CDE must be compared with FD-CDE to evaluate the replaceability. In the overlap preservation method, the \(N_{\text{FFT}}\) size signal block is transformed into the FD through FFT, then multiplied by the transmission function of the dispersion equalization filter, and then transformed into the TD through IFFT. An FFT operation includes \(N_{\text{FFT}}/2\cdot\log_{2}N_{\text{FFT}}\) times of complex number multiplication operations and \(N_{\text{FFT}}\cdot\log_{2}N_{\text{FFT}}\) times of complex number addition operations. It can be calculated that in FD-CDE, the real number multiplication and real number addition required for each, equalized, are  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**L/km** & \multicolumn{2}{c}{**500**} & \multicolumn{4}{c}{**1000**} & \multicolumn{4}{c}{**1500**} \\ \hline
**N** & \multicolumn{2}{c}{**207**} & \multicolumn{4}{c}{**415**} & \multicolumn{4}{c}{**623**} \\ \hline M & 8 & 12 & 16 & 8 & 12 & 16 & 8 & 12 & 16 \\ \(\eta_{\text{mul}}\) & 97.4 & 96.1 & 94.8 & 98.7 & 98.1 & 97.4 & 99.1 & 98.7 & 98.3 \\ \(\eta_{\text{add}}\) & 41.9 & 41.4 & 40.8 & 42.4 & 42.1 & 41.9 & 42.6 & 42.2 & 42.2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The degree of correlation between the calculation amount of real number multiplication \(\eta_{\text{mul}}\) and real number addition \(\eta_{\text{add}}\) and quantization stages M.  
\[\begin{split}\text{n}_{\text{R}_{\text{rand}}}&=3\cdot(N_{ \text{FFT}}\cdot\log_{2}N_{\text{FFT}}+N_{\text{FFT}})\\ \text{n}_{\text{R}_{\text{add}}}&=9\cdot N_{\text{FFT }}\cdot\log_{2}N_{\text{FFT}}+5\cdot N_{\text{FFT}}\end{split} \tag{26}\]  
The overlap value in FD-CDE is the key parameter determined by CD. As long as an integer is greater than the necessary overlap, it can theoretically be used as the \(N_{\text{FFT}}\). However, for different CD values, there is an optimal \(N_{\text{FFT}}\) to achieve the optimal calculation amount of FD-CDE, and as the CD value increases, the optimal \(N_{\text{FFT}}\) also increases. The optimal \(N_{\text{FFT}}\) used in the experiment comes from [22] The overlap size is specified as half of the \(N_{\text{FFT}}\) to simplify the calculation. Considering that the multiplication module is the most important indicator of the complexity in the digital signal processing module, we mainly compare the multiplication operations in FD-CDE and CQWO-TD-CDE, as shown in Figure 8.  
The results in Figure 8 prove that in long-distance optical fiber communication, the computational complexity of CQWO-TD-CDE is still better than that of FD-CDE. With the increase in optical fiber distance, this advantage presents an upward trend. The lower the quantization level M, the better performance optimization. When M = 8 and the optical fiber distance is 4000 km, the calculation complexity optimization reaches more than 60%; even in the short-distance optical fiber communication, such as 200 km, the performance index is also close to 35%. If weighting the performance requirements, such as setting the quantization level M = 16, when the fiber distance is 4000 km, the calculation complexity is also close to 30% optimization.