## 3 Simulation Results
Among the algorithms considered, the linear interpolator exhibits the weakest performance but serves as a valuable reference for benchmarking system improvements. The theoretical curve represents the performance of BER in the presence of pure ASE noise, modeling the bit error rate (BER) for M-QAM modulation in an AWGN channel when phase noise is absent. Notably, the incorporation of a hard-decision filter enhances the linear interpolator's performance, particularly at SNR values below 13 dB, demonstrating the utility of straightforward implementations. However, as the SNR increases, the effectiveness of the hard-decision filter diminishes. This reduction in performance is likely due to the filter's  
Figure 8: Comparison of various CPR interpolator algorithms for BER versus SNR for 16 QAM modulation, \(\Delta v=100\,\mathrm{kHz}\times 2,\ \mathrm{Rs}=50\) Gbaud. Figure 7: Constellation after carrier recovery, SNR = 20 dB, 16 QAM: (**a**) CPR—Linear interpolation with hard-decision filter (HDF) and (**b**) CPR—Cosine Robust LOWESS (Cosine-RLOWESS). reliance on fixed thresholds, which are sensitive to variations in power levels, affecting noise power estimation and outlier detection. Consequently, these power fluctuations lead to variability in the BER. To address these issues and improve the reliability of the system, we transitioned from a hard-decision approach to a more sophisticated filtering technique. A cosine filter was selected because of its proven effectiveness in achieving smoother interpolation. The combination of a cosine filter and robust locally weighted regression yields consistently impressive results across the entire SNR spectrum, demonstrating excellent stability. The efficiency of this algorithm is further enhanced by its burst analysis approach, which utilizes pilot symbols and eliminates the need for complete signal analysis to achieve effective tracking. In particular, a BER of \(10^{-3}\) is achievable at approximately 17 dB SNR, as shown by the blue line in Figure 8, which is remarkably close to the theoretical value. This result is based on the analysis of a signal containing \(2^{18}\) symbols. In Figure 9, the phase noise increases to \(\Delta\
u=400\times 2\,\mathrm{kHz}\) with a symbol rate of \(50\,\mathrm{Gbaud}\). To prevent frequency filter interpolation from experiencing losses, as observed in Figure 9 as an increase in signal-to-noise ratio (SNR), which may indicate a stability issue, the passband width is expanded to \(150\,\mathrm{MHz}\). As a result, the filter continues to perform effectively within the defined linewidth of \(\Delta\
u=400\times 2\,\mathrm{kHz}\). Among the algorithms evaluated, the combination of the cosine filter and RLOWESS (Cosine-RLOWESS) consistently delivers the best performance across the entire range of SNR values. Although an increase in phase noise leads to greater losses, these losses are effectively mitigated by the advanced processing capabilities of the Cosine-RLOWESS algorithm. The results for \(\Delta\
u=100\times 2\,\mathrm{kHz}\), a symbol rate of \(50\,\mathrm{Gbaud}\), and a modulation order of \(64\,\mathrm{QAM}\) are illustrated in Figure 10. The analysis shows that the linear interpolation algorithm, when combined with a hard-decision filter, performs acceptably at lower SNR values. However, as the SNR increases, certain limitations of both hard-decision filters and frequency interpolators become apparent. In contrast, the system that demonstrates stability across a wide range of SNR values is the linear interpolator combined with the cosine and RLOWESS techniques. Figure 9: Comparison of various CPR algorithms for BER versus SNR for 16 QAM modulation, \(\Delta\
u=400\,\mathrm{kHz}\times 2,\,\mathrm{Rs}=50\,\mathrm{Gbaud}\).