## 4 Computational Complexities
Assuming that \(m\) and \(n\) are independent of \(N\) and each other, and since \(m\) is small and can be treated as a constant, the complexity depends primarily on how \(n\) scales with respect to \(N\). Consequently, the overall complexity can still be simplified to \(\mathcal{O}(N)\) when considering practical scenarios. \[\mathcal{O}(Nn). \tag{22}\]  
Therefore, the overall computational complexity for running both processes one after the other is as follows:  
\[\mathcal{O}(N)+\mathcal{O}(Nn)=\mathcal{O}(Nn). \tag{23}\]  
This means that, for large values of \(N\), the term \((Nn)\) dominates, and thus represents the overall growth rate of the computational workload. Figure 18 compares the run times of the Cosine-RLOWESS and BPS algorithms, providing information on their relative performance under identical conditions. Although run times are influenced by factors such as processor speed and memory, this analysis ensures a fair assessment by standardizing parameters. The evaluation focuses on the impact of varying the number of input symbols while maintaining consistent phase noise and modulation settings. Specifically, 64 QAM modulation is used with a linewidth of \(\Delta\
u=800\,\mathrm{kHz}\times 2\) and a symbol rate of \(R_{s}=50\,\mathrm{Gbaud}\). The input symbol count ranges from \(2^{16}\) to \(2^{20}\), with the number of test phases for the BPS algorithm set to 64 and the filter length set to 12. The results indicate that the Cosine-RLOWESS algorithm consistently achieves shorter run times across all input sizes compared to the BPS algorithm. Moreover, the Cosine-RLOWESS algorithm demonstrates a computational complexity increase that is approximately \(50\%\) lower than that of the BPS algorithm, making it a more efficient choice for managing phase noise as computational demands grow. Figure 19 presents a comparative analysis of the run times of the BPS and Cosine-RLOWESS algorithms under varying conditions. For the BPS algorithm, the filter length is fixed at 12, while the number of test phases is varied. As anticipated, increasing the  
number of test phases improves the precision of the results but also leads to a corresponding increase in run time, as shown in the figure. For the Cosine-RLOWESS algorithm, we evaluated the impact of varying pilot symbol lengths, ranging from short lengths (8 symbols) to longer lengths (64 symbols). Unlike the BPS algorithm, reducing the pilot length increases precision, but at the cost of spectral  
Figure 19: Comparison of run time for the carrier phase recovery (CPR) algorithms with 64 QAM modulation, considering different system parameters. For the BPS algorithm, the number of test phases is varied, while for the Cosine-RLOWESS algorithm the pilot length is adjusted. The impact of these variations on computational time is analyzed. Figure 18: Comparison of run time for the carrier phase recovery (CPR) algorithms using 64 QAM modulation with a frequency offset of \(\Delta\
u=800\;\mathrm{kHz}\times 2\). The performance of the Cosine-RLOWESS and BPS algorithms is evaluated as a function of the number of input symbols. efficiency. A pilot length of 32 symbols strikes a favorable balance, delivering satisfactory precision with only a minimal increase in run-time, thus having a negligible impact on the algorithm's overall efficiency. Although a pilot length of 16 symbols improves performance, it results in a 6.6% loss in spectral efficiency, which is prohibitive. Therefore, we conclude that the Cosine-RLOWESS algorithm offers significant advantages in computational efficiency and run time, making a pilot length of 32 symbols an optimal choice. The run-time analysis indicates that integrating the BPS algorithm with pilot burst mode transmission in optical communications is not only feasible but also shows considerable potential to improve performance in specific scenarios.