## 4 Simulation Results and Discussion  
Under the conditions of four-branch received MRC, the relationship between the normalized mean square error (NMSE) and received optical power across different symbol block lengths was investigated on the B2B platform. The NMSE is defined as \(E\Big{[}\left|f_{est}\cdot T-f_{tru}\cdot T\right|^{2}\Big{]}\), where \(f_{est}\) is the estimated frequency offset, \(f_{tru}\) is the true frequency offset, and \(T\) is the symbol duration. Figure 8 illustrates that, with constant received optical power, the normalized mean square error (NMSE) decreases as the symbol block length increases. However, the reduction in NMSE is not substantial when increasing the symbol block length from 1024 to 1536. The NMSE curves for symbol block lengths of 1024 and 1536 show minimal differences, and as the received optical power increases, these two curves exhibit similar trends. Theoretically, longer symbol blocks can enhance estimation accuracy. However, excessively long symbol blocks may lead to increased resource usage and delays. Shorter symbol blocks exhibit limited performance in smoothing additive noise, but offer advantages such as faster frequency offset tracking and reduced resource consumption and latency. This algorithm aims to provide offline design and validation for hardware implementation; \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameter Name** & **Parameter Value** & **Unit** \\ \hline \multirow{3}{*}{transmitter parameters} & laser center wavelength & 1550 & nm \\  & laser linewidth & 50 & KHz \\  & symbol rate & 10 & GBaud \\ \hline \multirow{3}{*}{atmospheric channel parameters} & atmospheric structure constant & \(1\times 10^{-14}/1\times 10^{-16}\) & m\({}^{-2/3}\) \\  & transmission distance & 10 & km \\  & outer scale factor & inf & m \\  & inner scale factor & 0 & m \\ \hline \multirow{2}{*}{receiver parameters} & local oscillator optical power & 7.5 & dBm \\  & local oscillator laser linewidth & 50 & KHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation system parameters. Figure 8: The relationship between the Normalized Mean Square Error (NMSE) and received optical power at different symbol block lengths under B2B conditions. therefore, considering both algorithm performance and complexity, a symbol block length of 1024 was selected. Figure 9 illustrates the performance comparison between the proposed algorithm and traditional QPSK partitioning under B2B conditions, utilizing four-branch reception. The comparison of algorithm performance was comprehensively conducted from both NMSE and BER perspectives. Figure 9 presents the NMSE curve comparison for the two algorithms under different average received optical power conditions, while Figure 9 shows the BER curve comparison for the two algorithms under the same conditions. From Figure 9, it can be observed that, under the same average received optical power, the proposed algorithm exhibits a lower NMSE, indicating higher estimation accuracy. As the average received optical power increases, the estimation accuracy continues to improve, with the NMSE reaching the order of \(10^{-8}\). In contrast, the traditional QPSK partitioning was only able to achieve an NMSE to the order of \(10^{-7}\), indicating a need for improvement in estimation accuracy. Figure 9 shows that as the average received optical power increases, the system's bit error rate (BER) decreases. When the BER reaches \(3.8\times 10^{-3}\) (the hard decision threshold with 7% FEC overhead), the proposed scheme requires a lower average received optical power compared to traditional QPSK partitioning, reducing it by 0.67 dBm and thereby improving receiver sensitivity. Figure 10 illustrates the performance comparison between the proposed algorithm and QPSK partitioning under weak turbulence conditions, utilizing four-branch reception. The algorithm performance is comprehensively compared from both NMSE and BER perspectives.