## 3 Simulation Results  
In this section, we analyze the simulations generated in our study. Figure 5 presents the constellation results for 16 QAM with phase rotation. Furthermore, we applied our analysis to other modulation schemes, including QPSK, 64 QAM, and 256 QAM. For this experiment, phase noise was defined as \(\Delta\
u=100\,\mathrm{kHz}\times 2\), accounting for contributions from both the transmitter and the receiver. A linewidth of \(100\,\mathrm{kHz}\) is commonly used as a benchmark in phase compensation studies and is considered low enough that phase-locked loop (PLL) techniques become ineffective [1]. To achieve lower laser linewidths, below \(100\,\mathrm{kHz}\), external cavity lasers (ECLs) are often employed. These devices are typically compact and suitable for small form factor transceivers and modules [38]. Therefore, instead of using PLL techniques, a feedforward carrier recovery method is recommended. The power level was set high, with \(\mathrm{SNR}=20\,\mathrm{dB}\), to produce a cleaner constellation where the rotation effect due to phase noise is clearly visible. Figure 4: Comparison between the data points with noise and outliers, and the fitting result: (**a**) LOESS smoothing, (**b**) Robust LOESS smoothing, (**c**) LOWESS smoothing, and (**d**) Robust LOWESS smoothing. In Figure 6, we present the phase tracking analysis. This figure highlights the effectiveness of the hard-decision filter, as depicted in Figure 6b, in reducing peak values during the tracking process. Compared to the results produced by the pure linear interpolator in Figure 6a, the application of this hard-decision filter significantly attenuates the peaks. However, the tracking is not entirely smooth. In contrast, the results obtained using the robust interpolator, shown in Figure 6c, show much smoother phase tracking. This comparison clearly demonstrates the superiority of the RLOWESS method over other interpolation techniques for phase tracking. In Figure 7, the results of the 16 QAM constellation after phase recovery are illustrated, comparing two interpolation methods. Figure 7, on the left, shows the constellation after phase recovery using the linear method combined with a hard-decision filter. In particular, the symbols at the outer edges of the constellation exhibit significant elongation, indicating distortion in the recovered symbols. In contrast, Figure 7, on the right, presents the scenario in which pilot symbols have been removed and carrier recovery has been performed using a cosine filter in conjunction with the robust locally weighted smoothing interpolation method (Cosine-RLOWESS). Here, the external symbols appear in a Gaussian format, demonstrating that smoother phase recovery is crucial for effective carrier phase recovery. This improvement will have a significant impact on the subsequent analysis of the bit error rate. Figure 5: Constellation rotating due to phase noise effect, for 16 QAM modulation, \(\Delta\
u=100\,\mathrm{kHz}\times 2\), \(\mathrm{SNR}=20\,\mathrm{dB}\). Figure 6: Phase noise tracking, 16 QAM modulation, \(\Delta\
u=100\,\mathrm{kHz}\times 2\), \(\mathrm{SNR}=25\,\mathrm{dB}\): (a) CPR—Linear interpolation, (b) CPR—Linear interpolation with hard-decision filter (HDF), and (c) CPR—Cosine Robust Smooth LOWESS (Cosine-RLOWESS). A key conclusion from the constellation analysis is that the hard-decision filter-based interpolator exhibits a notable disadvantage: it causes elongation in the outer symbols. However, this limitation can be alleviated by using probabilistic shaping techniques. These techniques reduce the probability of symbols occupying these outer regions, while leveraging the simplicity and efficiency of linear interpolation. In the BER vs. SNR analysis, shown in Figure 8, we evaluate the performance of various CPR algorithms with interpolation for 16 QAM modulation. The system operates with a linewidth of \(\Delta v=100\times 2\,\mathrm{kHz}\) and a symbol rate of 50 Gbaud.