## 4 Results and Discussion  
### Results  
In this section, the performance of carrier phase recovery in the long-haul high-speed \(n\)-PSK coherent optical transmission systems using the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi algorithms are investigated based on the above theoretical studies, which can be regarded as extensions of the QPSK results in previously reported work [26; 37; 51]. The comparisons of the three carrier phase recovery approaches have also been carried out in detail. In all these analyses and discussions, the standard single mode fiber (SSMF) has been employed in the \(n\)-PSK coherent optical transmission systems, where the fiber dispersion is 17 ps/nm/km, the central wavelengths of both the transmitter laser and the LO laser are 1550 nm, and the signal symbol rate is 32-Gbaud. The fiber attenuation, the PMD, and the nonlinear effects are all neglected. Based on Equation (9), the performance of the one-tap normalized LMS carrier phase recovery in the coherent optical fiber communication systems using different modulation formats is shown in Figure 5. The optimization of the one-tap normalized LMS carrier phase recovery has been investigated and discussed in detail in Reference [27; 37], where the step size varying from 0.01 to 1 has been applied. A smaller step size will degrade the BER floor due to the fast phase changing occurring in the long effective symbol average-span, while a larger step size will degrade the signal-to-noise ratio (SNR) sensitivity in the one-tap normalized LMS carrier phase recovery. The optimal step sizes for different effective linewidths have been studied, with a resolution of 0.005 used in the optimization process [37]. In this section we assume that all the operations of the one-tap normalized LMS algorithm have been optimized. It can be found in Figure 5 that the one-tap normalized LMS carrier phase recovery algorithm is very sensitive to the phase noise variance and the modulation formats, especially when phase noise variance is less than 0.1. In the case of back-to-back (BtB) or without considering EEPN, the performance of BER floors versus laser linewidths in the one-tap normalized LMS carrier phase estimation has been studied in Figure 6 based on the analysis in Equations (1) and (9). The indicated linewidth value in Figure 6 is the 3-dB linewidth for both the transmitter laser and the LO laser. It can be seen that the BER floors of the one-tap normalized LMS carrier phase recovery are deteriorated with the increment of the modulation format levels and the laser linewidths. The degradations due to the laser phase noise (laser linewidths) are more drastic for higher-level modulation formats. Considering the impact of EEPN, the effective phase noise variance will increase with the increment of the transmission distance and the laser linewidth. As shown in Figure 7, the BER floors  
Figure 5: Bit-error-rate (BER) floors versus phase noise variance in the one-tap normalized LMS carrier phase recovery in the coherent optical transmission systems using different modulation formats. of the one-tap normalized LMS carrier phase recovery has been investigated for different transmission distances, when the linewidths of both the Tx and the LO lasers are set to 1 MHz. It can be found that the performance of the one-tap normalized LMS carrier phase recovery is degraded significantly with the increment of transmission distances, and this effect is more severe for higher-level modulation formats due to less tolerance to laser phase noise and EEPN. The performance of the block-wise average carrier phase recovery approach has also been investigated, as shown in Figures 8-10. The BER floors versus different phase noise variances (or effective phase noise variance) in the block-wise average carrier phase recovery is described in Figure 8. In Figure 8a, the performance of the block-wise average algorithm is studied in terms of different block lengths in the 8-PSK optical transmission system.