## 3 Simulation Results
This indicates that employing advanced laser oscillators with finely tuned linewidths could offer significant advantages [38]. Furthermore, in the design of low-margin optical networks, adjusting the data rate according to demand can help to maintain \(\Delta vT_{s}\) within an optimal range. As the modulation order and linewidth increase, the performance of Cosine-RLOWESS converges with that of BPS. For higher values of \(\Delta vT_{s}\)--such as when the symbol period  
Figure 5:2 Comparative analysis of Cosine-RLOWESS and BPS algorithms in BER performance vs. SNR for 256 QAM modulation at a linewidth of \(\Delta v=2\times 200\,\mathrm{kHz}\). Figure 5:3 Receiver tolerance against phase noise for different square QAM constellations trough BPS algorithm and Cosine-RLOWESS for BER = \(1\times 10^{-3}\). decreases or the laser linewidth increases to several megahertz--BPS demonstrates better tolerance. However, the Cosine-RLOWESS algorithm offers the advantage of a lower complexity compared to BPS, as reflected in the reduced processing time discussed in Section 4. Advances in forward error correction (FEC) algorithms allow for more flexible bit error rate (BER) thresholds, particularly when FEC is not employed. Specifically, new approaches are shifting from traditional hard-decision methods to soft-decision methods, or a combination of both, to improve overall system performance. As noted in [40], setting a fixed BER threshold should be avoided, especially when using soft-decision decoding. Instead of relying on the BER threshold, it is recommended to use generalized mutual information (GMI) as a more accurate reference metric. However, when hard-decision decoding is used, setting a BER threshold remains a valid approach. Traditionally, this threshold has been set to \(3.8\times 10^{-3}\). According to [41], the hard-decision FEC thresholds can reach \(2.24\times 10^{-2}\), but achieving a post-FEC BER of \(10^{-15}\) requires an overhead of 33.33%. Based on these findings, we set the sensitivity to a minimum BER of \(2\times 10^{-2}\). Figure 16 presents the sensitivity results for the blind phase search algorithm (BPS) and the Cosine-RLOWESS method. The Cosine-RLOWESS consistently outperforms the BPS algorithm, even at a sensitivity level corresponding to a BER of \(1\times 10^{-3}\). Specifically, across all constellations compared, the Cosine-RLOWESS algorithm exhibits superior performance when the product \(\Delta vT_{s}\) is very low or extremely high. This advantage enables the system to support higher laser linewidth values more effectively. As mentioned above, using these hard-decision decoding values requires an increase in overhead. Therefore, it is important to consider the trade-off between the minimum required BER and the maximum allowable overhead in practical scenarios. In Figure 16, with a BER of \(2\times 10^{-2}\), the Cosine-RLOWESS algorithm consistently outperforms the BPS algorithm. For example, with a linewidth of \(\Delta v=2500\,\mathrm{kHz}\) and a symbol rate of \(R_{S}=50\,\mathrm{Gbaud}\), corresponding to \(\Delta vT_{s}=1\times 10^{-4}\), the performance difference is 0.56 dB for 16 QAM and increases to 0.6 dB for 64 QAM. As the sensitivity curves indicate, this performance gap becomes more pronounced as the laser linewidth narrows. Figure 17a analyzes the residual phase noise, which is calculated as the difference between the phase noise and the output of a tracking algorithm using the Cosine-RLOWESS method. The residual noise is influenced primarily by two factors: the variance of the phase  
Figure 5:4 Receiver tolerance against phase noise for different square QAM constellations trough BPS algorithm and Cosine-RLOWESS for BER = \(2\times 10^{-2}\). noise and the signal-to-noise ratio (SNR). Occasionally, spurious values appear that are not directly caused by phase noise errors but resemble Gaussian channel noise, particularly near the quadrant boundaries in the constellation diagram.