## 6 Conclusions  
An efficient, low-complexity, pilot-assisted phase estimation technique has been presented for coherent optical receivers. This method is based on a robust interpolation technique designed for quasioptimal operation, offering improved phase estimation accuracy. It effectively addresses the unique challenges posed by new scenarios in optical communication systems. We conducted comprehensive simulations to demonstrate the performance of the method compared to conventional techniques, highlighting the improvements in the curves of BER as a function of SNR and in the penalties in SNR at the pre-FEC BER thresholds of 1.0 \(\times\) 10\({}^{-3}\) and 2.0 \(\times\) 10\({}^{-2}\).  
The proposed Cosine-RLOWESS interpolation outperforms all other pilot-assisted interpolation techniques studied across entire ranges of SNRs. In fact, the BERxSNR curves for the Cosine-RLOWESS method are very close to the optimal AWGN channel curves. The Cosine-RLOWESS method surpasses BPS in all cases throughout the SNR range, except in the higher range of SNR > 28 dB for 256 QAM, where both algorithms performed equally well. We show the sensibility penalties incurred by the BPS and Cosine-RLOWESS methods at a pre-FEC BER of 1.0 \(\times\) 10\({}^{-3}\), respectively. First, we observe that the breakdown thresholds for \(\Delta vT_{s}\), above which there is an exponential increase in the sensitivity penalties, are approximately the same for all the respective modulation formats (QPSK, 16 QAM, 64 QAM, and 256 QAM) for both the BPS and Cosine-RLOWESS methods. However, in the flat working region for \(\Delta vT_{s}\) below the breakdown thresholds for the four modulation formats, the Cosine-RLOWESS sensibility penalties are approximately 0.42 dB (QPSK), 0.4 dB (16 QAM), 0.3 dB (64 QAM), and 0.3 dB (256 QAM) lower than the corresponding penalties for the BPS algorithm. For a pre-FEC BER of 2.0 \(\times\) 10\({}^{-2}\), there is an even greater improvement in the Cosine-RLOWESS sensibility penalties.  
In summary, the simulation results show a significant performance improvement of the Cosine-RLOWESS method compared to other linear interpolation methods and the BPS algorithm. Moreover, the proposed pilot-assisted method avoids cycle slips, presents low computational complexity, and can be applied to probabilistic shaping modulation, where the BPS fails at optimal operational distributions.  
**Author Contributions:** All the authors have contributed to various degrees to ensuring the quality of this work: system conceptualization, C.C.B.; weighted interpolation conceptualization, C.C.B. and K.S.M.; methodology, C.C.B. and V.R.M.; formal analysis, C.C.B., V.R.M., K.S.M., J.A.S. and D.S.A.; writing--original draft preparation, C.C.B., V.R.M. and K.S.M.; theorems, corollaries, and proofs, C.C.B. and K.S.M.; writing--review and editing, C.C.B., V.R.M., K.S.M., J.A.S. and D.S.A.; supervision, D.S.A. All authors have read and agreed to the published version of the manuscript.  
**Funding:** Carlos Carrion Betancourt is supported in part by the Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior, Brasil (CAPES), with Finance Code 001, and the Secretary of Science and Technology of Ecuador (SENESCYT), with Contract Number 2015-AR7L7856.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.