Results and Discussions In the dual-carrier PDM-16QAM FTN system, the Δf·T_s is set to 1×10^-4. With the decrease of accelerating factor named α, the artificially introduced ISI will become more serious. The PA-Viterbi-ML algorithm can overcome the effect of ISI introduced by FTN when α does not exceed the Mazo limit, and effectively estimate the phase noise caused by the laser line-width (Fig. 7). Finally, the performance of the proposed algorithm and the traditional PA-CPR is compared. When the Δf·T_s is small, the OSNR penalties of both algorithms are almost the same. As the Δf·T_s increases, the OSNR penalty of the traditional PA-CPR is relatively higher (Fig. 9). This means that the linewidth tolerance of the proposed PA-Viterbi-ML algorithm is larger for the same OSNR penalty, and the OSNR penalty of the proposed algorithm is less for the same Δf·T_s. In addition, with the decrease of α, the performance advantage of the PA-Viterbi-ML algorithm is more significant. When α=0.850, Δf·T_s is 1×10^-4, the OSNR penalty of the PA-Viterbi-ML algorithm is 0.02 dB lower than that of the PA-CPR algorithm. When α=0.833, Δf·T_s is 1×10^-4, the OSNR penalty of the PA-Viterbi-ML algorithm is 0.28 dB lower than that of the PA-CPR algorithm. Compared with the traditional algorithm, the performance advantage of the PA-Viterbi-ML algorithm will become larger when the smaller value of α and the larger value of Δf·T_s are taken (Table 1).

The large-scale commercialization of 5G networks promotes the development of fiber optical communication, cloud computing, and Internet of Things technologies. To realize the optical communication systems with high capacity and high spectrum efficiency, higher-order modulation formats are required. However, the higher-order modulation formats require high signal-to-noise ratios (SNR) to ensure communication quality, which will limit the transmission distance. Within the same amount of time, the faster-than-Nyquist (FTN) technology can transmit more signals than the Nyquist system with the same modulation format. Thus, this technology becomes a key technology for the next-generation optical communication networks with the advantages of high spectral efficiency and large capacity. The inter-symbol interference is artificially introduced into the FTN system to make the symbol interval between two adjacent pulses much smaller than the corresponding Nyquist symbol period. Thus, high-speed digital signal processing (DSP) unit, which can equalize and compensate the signal impairments efficiently in electrical domain, is a key module in FTN systems. It can improve the signal quality and support the optical fiber communication systems with ultra-high capacity. Pilot-aided carrier phase recovery (PA-CPR) is an important DSP algorithm for optical receiver, which is used to compensate the frequency deviation between the lasers on both sides of the transceiver and the phase noise generated by the laser linewidth. In present study, we report a two-stage electric-domain pilot-aided carrier phase estimation algorithm, named PA-Viterbi-ML, in which the PA-CPR algorithm is combined with the Viterbi-based maximum likelihood (ML) estimation algorithm. The simulation results verify that the proposed two-stage PA-Viterbi-ML algorithm can effectively track the phase noise when the Mazo limit is not exceeded in the FTN-16QAM system.

The proposed PA-Viterbi-ML, which combines the PA-CPR algorithm with the Viterbi-based maximum likelihood estimation algorithm, can effectively overcome the intersymbol interference (ISI) introduced by FTN technology. The first stage of the algorithm can estimate and compensate most of the phase noise in FTN-16QAM system. In order to compensate the residual phase deviation, the ML phase estimation is used as the second stage of the phase recovery algorithm to obtain a more refined phase estimate value. However, the ML phase estimation will also fail under the influence of the ISI introduced by FTN system, so the Viterbi algorithm is cascaded with it to remove the influence of ISI, and the estimated value of the phase noise close to the real value is obtained. During the simulation, the pilot-signal-ratio (PSR) and the bandwidth of the low-pass filter (B_LPF), two important parameters of the PA-Viterbi-ML algorithm, are optimized first to achieve the optimal system performance. Then, the performance of the proposed algorithm for tracking system phase noise within the Mazo limit is shown. Finally, the maximum linewidth tolerance of the algorithm is determined at the threshold of the bit error rate.

In the present study, a two-stage electric-domain pilot-aided carrier phase estimation algorithm, named PA-Viterbi-ML, is proposed, in which the PA-CPR algorithm is combined with the Viterbi-based maximum likelihood estimation algorithm. The PA-Viterbi-ML algorithm will occupy about 1.7% of the bandwidth, which can effectively overcome the ISI introduced by the FTN technology. The simulation is taken to verify that the proposed two-stage PA-Viterbi-ML algorithm can effectively track the phase noise when the Mazo limit is not exceeded in the FTN-16QAM system. The simulation results show that the OSNR penalty requirements of the proposed algorithm are smaller than those of the traditional PA-CPR when the linewidth tolerance values are the same, which means the performance of the linewidth tolerance of the proposed algorithm is better. The maximum linewidth tolerance value is defined when the BER and the OSNR penalty are equal to 2×10^-2 and 1 dB, respectively. When the accelerating factor α is as low as 0.833, the maximum linewidth tolerance value is about 5×10^-4 for the PA-Viterbi-ML algorithm, and the corresponding value for the traditional PA-CPR algorithm is about 1×10^-4.