# Post-FEC Performance of Pilot-Aided Carrier Phase Estimation over Cycle Slip
## 3 Experimental Results and Discussion  
Figure 3a-c show phase errors between the phase values estimated by CPE and the referenced phase values. Errors between CPE + PAPU and the referenced phase values are shown in Figure 3e-f. The traces in each figure indicate the errors under different CPE filter lengths (CPE filter length = 4, 8, 16, 20, 32, 48). To obtain the referenced phase values, we firstly calculate the phase error between original symbol and received symbol, then usual PU (2 pi) and an appropriate interpolation filter are used to remove the residual noise. The outputs of interpolation filter are used as referenced phase values. As shown in Figure 3, frequent CS is clearly observed for the VVPE scheme, producing \(\pm 90^{\lx@math@degree}\) phase error jumps under the condition of lower OSNR (which is equivalent to 10.5 dB or 12.1 dB) and small CPE filter length (which is less than 20). Even if OSNR is equivalent to 15.4 dB, the CPE scheme still shows frequent CS when CPE filter length is 4. Conversely, even when OSNR is equivalent to 10.5 dB and CPE filter length is less than 8, the VVPE + PAPU scheme just shows a few discrete CS, which are induced by additive white Gaussian noise (AWGN) and can be corrected by FEC. The results indicate that the CPE scheme seems to be more sensitive to CPE filter length and OSNR, compared to the CPE + PAPU scheme. Probability of cycle slips as a function of OSNR under different CPE filter lengths are shown in Figure 4a. With the increasing of OSNR, the cycle slips probability of both schemes is decreasing and the difference between the VVPE scheme and the VVPE + PAPU scheme is becoming less obvious. The probability of CS is less than \(1\times 10^{-7}\) when OSNR is large than 17.5 dB, which indicates that the CS problem can be avoided under high OSNR. However, with OSNR less than 12 dB, the VVPE scheme exhibits a high CS-rate with CPE filter length less than 48, compared with the VVPE + PAPU scheme. The reason is that frequent continuous CS occurs in the CPE scheme with lower OSNR, while only some discrete CS exists in the CPE + PAPU scheme. The results in Figure 4a indicate that the CPE + PAPU scheme can effectively mitigate the continuous CS due to the improper PU of the CPE. The discrete CS on account of AWGN cannot be corrected by the CPE + PAPU scheme, but these independent CS can be effectively corrected by FEC. Figure 4b shows the relationship between PRE-FEC BER and OSNR under different CPE filter lengths for the CPE scheme and the CPE + PAPU scheme. The AWGN  
Figure 3: Phase errors for (**a-c**) the carrier phase estimation (CPE) scheme, (**e–f**) the CPE + pilot-aided-phase-unwrap (PAPU) scheme. PRE-FEC BER is presented by gray circle. There is a similar trend between Figure 4a,b. When the CPE filter length is less than 48, the sudden degrade of PRE-FEC BER is consistent with the probability of CS for the CPE scheme. Figure 5a depicts the probability of CS versus CPE filter length under OSNR of 10.5 dB, 12.1 dB, and 15.4 dB for the CPE scheme and the CPE + PAPU scheme. As shown in Figure 5a, Under OSNR of 10.5 dB, 12.1 dB, and 15.4 dB, the CPE scheme exhibits high probability of CS with CPE filter length less than 32, 16, and 8, respectively. As shown in Figure 5b, with OSNR equivalent to 10.5 dB, 12.1 dB, 15.4 dB, to achieve the same PRE-FEC BER (respectively about \(2\times 10^{-2}\), \(1\times 10^{-2}\), \(9.6\times 10^{-4}\)), the required CPE filter length of the CPE scheme and the CPE + PAPU scheme are 32 and 8, 16 and 6, and 8 and 8, respectively. With the OSNR equivalent to 10.5 dB, the PRE-FEC BER of CPE scheme sharply degrade from \(2\times 10^{-2}\) to 0.5 with CPE filter length less than 32, moreover, the tendency of PRE-FEC BER agree with the CS probability in Figure 5a. When OSNR is increasing from 12.1 dB to 15.4 dB, the min CPE filter length to avoid the sharply degrading of PRE-FEC BER performance, decreases from 16 to 8. Figure 6a shows the comparison of PRE-FEC BER and POST-FEC BER versus CPE filter lengths with different OSNR.