## 4 Simulation Results and Discussion
Figure 10 presents the NMSE curve comparison for the two algorithms under different average received optical power conditions, while Figure 10 shows the BER curve comparison for the two algorithms under the same conditions. From Figure 10, it can be observed that the proposed algorithm has a lower NMSE, indicating higher estimation accuracy and superior performance compared to the QPSK partitioning algorithm, consistent with the conclusions drawn under B2B conditions. Figure 10 compares the average receiver sensitivity required for both algorithms to reach the FEC decision threshold under weak turbulence conditions. The results reveal that the proposed scheme requires a lower average received optical power to achieve the FEC threshold compared to traditional QPSK partitioning, reducing the average received optical power by 0.76 dBm. Thus, the proposed algorithm is able to enhance receiver sensitivity even under weak turbulence conditions. Figure 9: Performance comparison between the proposed algorithm and traditional QPSK partitioning under B2B conditions: (**a**) NMSE curve under different average received optical power, (**b**) BER curve under different average received optical power. Figure 11 illustrates the performance comparison between the proposed algorithm and QPSK partitioning under strong turbulence conditions, utilizing four-branch reception. Figure 10(a) presents the NMSE curves for the two algorithms under various average received optical power conditions, while panel (b) illustrates the BER curves for the same conditions. As shown in Figure 10(a), the proposed algorithm exhibits a significant advantage under strong turbulence conditions compared to B2B and weak turbulence scenarios. The estimation accuracy of the proposed algorithm is notably higher at the same levels of received optical power. Specifically, under strong turbulence, the proposed algorithm maintains superior receiver sensitivity relative to QPSK partitioning. At the FEC threshold, the proposed algorithm achieves a reduction in average received optical power by 0.71 dBm. Next, we will assess the estimation accuracy and range of the algorithms under varying frequency offset conditions to validate their effectiveness. This evaluation will help  
Figure 11: Performance comparison between the proposed algorithm and traditional QPSK Partitioning under strong turbulence conditions: (**a**) NMSE curve under different average received optical power, (**b**) BER curve under different average received optical power. Figure 10: Performance comparison between the proposed algorithm and traditional QPSK partitioning under weak turbulence conditions: (**a**) NMSE curve under different average received optical power, (**b**) BER curve under different average received optical power. determine the robustness and adaptability of each algorithm in handling frequency offset variations, ensuring their performance remains robust in practical applications. We will compare the estimation accuracy and estimation range of the algorithms under different frequency offset conditions to validate their effectiveness. This comparison will help assess the robustness and adaptability of each algorithm in response to variations in frequency offset, ensuring that the algorithms can maintain good performance in practical applications. Figure 12 shows the NMSE performance curves of the proposed algorithm and QPSK partitioning under different frequency offsets in a four-branch received MRC scenario, for both weak and strong turbulence environments. Figure 12a shows a comparison of the NMSE curves under weak turbulence with an average received optical power of \(-16\) dBm, while Figure 12b illustrates the NMSE performance comparison under strong turbulence conditions, with an average received optical power of \(-18\) dBm. As illustrated in Figure 12a, the NMSE of the proposed algorithm consistently surpasses that of the traditional QPSK partitioning algorithm under various frequency offset conditions.