## 5 Discussion  
In this section, we analyze the results of our study and compare them with related work in the field. Most existing carrier phase estimation (CPE) algorithms rely on pilot symbols for assistance, which can slightly increase system complexity and, for most practical applications, only slightly reduce the overall information rate [27]. In contrast, blind phase estimation methods, such as the blind phase search (BPS) algorithm [1] and the Viterbi-Viterbi phase estimator (VV-PE) [22], eliminate the need for pilot symbols. However, VV-PE requires adaptation for higher-order modulation formats, while BPS, despite being modulation-format-independent, suffers from high computational complexity. Moreover, both methods are vulnerable to severe burst errors caused by cycle slips, which can significantly degrade the system performance [25]. In our research, we analyze the advantages of eliminating outliers in phase tracking, as their presence degrades the accuracy of the interpolator, thereby affecting phase recovery. This effect is evident in Figure 6. Among the interpolation methods considered, RLOESS and RLOWESS provide the best fit, the latter being selected because of its ability to introduce greater flexibility in modifying the weights. This flexibility offers more control over the smoothing process, allowing for the fine-tuning of phase correction, which is particularly beneficial in more challenging scenarios. Figure 6 illustrates the benefits of incorporating a filter into the interpolator. The process begins with the hard-decision filter, shown in Figure 6b, and culminates in a smooth interpolation achieved through the combination of the cosine function and the RLOWESS interpolator, as depicted in Figure 6c. Furthermore, Figure 8 presents a comparison of different interpolators with the theoretical curve, which represents the performance of a communication system without phase noise disturbances. In this scenario, the only impairment is amplified spontaneous emission (ASE) noise, modeled as additive white Gaussian noise (AWGN) in the channel. Although the frequency domain interpolator shows good performance, as discussed in [25], its implementation complexity is significantly high. Computing the FFT requires a substantial portion of the signal, making this approach less practical for real-time applications. For instance, when the modulation order increases or when phase noise is exacerbated by an increase in the laser linewidth to \(\Delta v=2\times 400\) kHz, an adjustment in the bandwidth of the frequency filter becomes necessary to maintain acceptable performance. Specifically, the filter bandwidth must be increased from 150 MHz to 450 MHz to compensate for the increased phase noise. The impact of this adjustment on system performance is illustrated in Figure 11.  
In contrast, the Cosine-RLOWESS method provides a more efficient solution for phase noise tracking while maintaining lower computational complexity, making it suitable for real-time implementations. In this approach, the tracking is performed based only on a slot portion of the signal, unlike the frequency-filter approach. In our system, the spacing of the pilot symbols was set to a ratio of 1:32, corresponding to 3% of the transmitted symbols. It is interesting to compare this pilot symbol configuration with related studies, such as [PERSON] et al. [46], which uses the aiding Kalman filter (AKF) as the carrier phase estimation (CPE) algorithm. The ratios of pilot symbols used in their study are relatively very high (1:1, 1:3, and 1:7), thus limiting the power and spectral efficiencies, while the Cosine-RLOWESS method maintains acceptable power and spectral efficiencies for pilot-based systems, within the recommended maximum overhead of up to 5% [42; 43; 47]. In general, studies that use pilot symbols exhibit significantly lower computational complexity, as reported in [9; 27; 46]. Another key advantage is the ability to mitigate cycle slips without requiring differential coding.