## 5 Discussion
For example, in [9], the authors propose an improved carrier phase estimation (CPE) method that employs dual pilot tones to suppress equalization-enhanced phase noise (EEPN) while mitigating cycle slips. Compared to the BPS algorithm, this approach introduces additional filtering and pilot tone processing, but eliminates the need for complex phase correction. Similarly, Cosine-RLOWESS inherently resists cycle slips, as discussed in the introduction of Section 4. In Section 4, we compared the computational complexity of the Cosine-RLOWESS and BPS algorithms. The blind phase search (BPS) algorithm employs a brute-force phase estimation approach, where multiple test phase angles are applied to the received symbols, and the optimal phase is selected. The primary computational components of BPS are as follows [1; 32]:  
* **Multiplications:** Each test phase requires a phase rotation (complex multiplication), leading to a complexity of \(\mathcal{O}(NB)\). * **Additions:** Computing the phase error metric in all test phases scales as \(\mathcal{O}(NB)\). * **Comparisons:** Selecting the best phase requires additional comparisons, adding a complexity of \(\mathcal{O}(B)\). Thus, the total computational complexity of BPS is given by \(\mathcal{O}(NB)\). In contrast, the Cosine-RLOWESS method reduces the computational complexity to \(\mathcal{O}(Nn)\), where \(n\) represents the pilot spacing. Overall, the proposed algorithm is computationally more efficient than BPS, particularly for high-order modulation formats, where BPS requires a large \(B\), making it computationally expensive. A comparison of Cosine-RLOWESS with other pilot-based algorithms is summarized in Table 1.  
Table 1 compares the computational complexity of various carrier phase estimation (CPE) algorithms, including methods based on Cosine-RLOWESS, BPS, and Kalman. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Algorithm** & **Multiplications (MULT)** & **Additions (ADD)** \\ \hline CMA+BPS [46] & \(16M+16+6 NB+4N\) & \(16M+4+6 NB-B+2N+2\) \\ AKF [46] & \(585\) & \(459\) \\
2-Stage BPS [27] & \(6 NBT+4N\) & \(5 NBT+2N+3\) \\ PCPE [27] & \(10N+8\) & \(6N+3\) \\ MPD-BPS [27] & \(12 NBT\) & \(21N\) \\ UKF [27] & \(56N\) & \(264N\) \\ EKF-PC [27] & \(56N\) & \(88N\) \\ Cosine-RLOWESS & \(\mathcal{O}(N\log n)\) & \(\mathcal{O}(N)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational complexity comparison of CPE algorithms based on pilot symbols. Compared to extended Kalman filtering (EKF-PC) and unscented Kalman filtering (UKF), Cosine-RLOWESS has a lower computational burden, particularly in high-order modulation formats, where Kalman-based methods require matrix operations. However, potential performance trade-offs should be considered, as Kalman filtering techniques may offer superior phase noise tracking under highly dynamic conditions. The AKF algorithm maintains a fixed complexity of \(\mathcal{O}(N)\), as stated in Table 1, since it processes symbols sequentially with a constant computational cost. In contrast, the general Cosine-RLOWESS has a complexity of \(\mathcal{O}(Nn)\), where \(n\) denotes the pilot spacing. Since \(n\) is typically a slot of \(N\), the computational cost of Cosine-RLOWESS is slightly higher than that of AKF, especially as the number of pilot symbols increases. However, for small values of \(n\), its complexity remains close to \(\mathcal{O}(N)\), making Cosine-RLOWESS a computationally viable alternative. Overall, while AKF is more efficient in terms of computational complexity, Cosine-RLOWESS may offer advantages in phase noise robustness, depending on the pilot symbol density and modulation order. Moreover, it achieves a superior spectral and power efficiencies compared to AKF approaches. Furthermore, Cosine-RLOWESS provides a computationally efficient alternative to BPS for phase estimation, particularly in high-order modulation formats, while maintaining a lower complexity compared to Kalman-based approaches.