## 4 Results and Discussion
### Results
The optimal block length is again determined by considering the trade-off between phase noise and amplitude noise. It can be found that the BER floors of the block-wise average carrier phase recovery algorithm is more robust than the BER floors of the block-wise average carrier phase recovery algorithm. Figure 10: BER floors versus transmission distances in the block-wise average carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, and the linewidth of both the Tx and the LO lasers are 1 MHz. Figure 9: BER floors versus laser linewidths in the block-wise average carrier phase recovery in the coherent optical transmission systems using different modulation formats. The block length is 11, and the indicated linewidth value is the 3-dB linewidth for both the Tx and the LO lasers. that the phase noise induced BER floors in the Viterbi-Viterbi carrier phase recovery algorithm are also deteriorated with the increment of block lengths. Similar to the block-wise average algorithm, the block length of \(N_{VV}=11\) is also selected as an example in the Viterbi-Viterbi carrier phase recovery to consider the mitigation of both the phase noise and the amplitude noise in practical applications. Based on Equation (15), the performance of the Viterbi-Viterbi carrier phase recovery in the coherent optical communication systems using different modulation formats is shown in Figure 11b, where the block length is 11. It is found in Figure 11b that the Viterbi-Viterbi carrier phase recovery algorithm is also very sensitive to the phase noise variance and the modulation formats, when phase noise variance is less than 0.15. Without considering EEPN (or for the back-to-back case), the BER floors versus laser linewidths in the Viterbi-Viterbi carrier phase recovery have been studied in Figure 12 based on the analyses in Equations (1) and (15), where the indicated linewidth value is again the 3-dB linewidth for both the transmitter laser and the LO laser. It can be found that the BER floors are also degraded significantly with the increment of laser linewidths, and this degradation is also more severe with the increment of modulation format levels. As shown in Figure 13, the BER floors of the Viterbi-Viterbi carrier phase recovery have also been investigated for different transmission distances considering the impact of EEPN, where the linewidths of the Tx and the LO lasers are both set to 1 MHz. It can be found that in the Viterbi-Viterbi carrier phase recovery, the EEPN influence will also increase with the increment of transmission  
Figure 11: BER floors versus phase noise variances in the Viterbi-Viterbi carrier phase recovery in the coherent optical transmission systems. (a) Different block lengths in the 8-PSK transmission system; (b) different modulation formats with the block length of 11.  
distances and modulation formats. The performance of the Viterbi-Viterbi algorithm is degraded significantly with the increment of transmission distances, and this effect is more serious for higher-level modulation formats. The comparisons of the one-tap normalized LMS, the block-wise average, and the Viterbi-Viterbi carrier phase recovery algorithms have also been investigated in detail. The BER floors versus different phase noise variances in the above three carrier phase recovery algorithms in the 8-PSK optical fiber communication system have been studied and are shown in Figure 14, where the block length in the block-wise average and the Viterbi-Viterbi algorithms varies from 5 to 17, in Figure 14a-c, respectively. It can be seen that the phase noise induced BER floors in the block-wise average and Viterbi-Viterbi algorithms are degraded with increment of the block length, and the one-tap normalized LMS algorithm keeps the same performance due to its optimized operation.