Open Access
Add to BibSonomy
Abstract
In this paper, for the first time, a probability-aided maximum-likelihood sequence detector (PMLSD) is experimentally investigated through a 64-GBaud probabilistic shaped 16-ary quadrature amplitude modulation (PS-16QAM) transmission experiment. In order to relax the impacts of PS technology on the decision module, a PMLSD decision scheme is investigated by modifying the decision criterion of maximum-likelihood sequence detector (MLSD) correctly. Meanwhile, a symbol-wise probability-aided maximum a posteriori probability (PMAP) scheme is also demonstrated for comparison. The results show that the PMLSD scheme outperforms the direct decision scheme about 1.0-dB optical signal to noise ratio (OSNR) sensitivity. Compared with symbol-wise PMAP scheme, PMLSD scheme can effectively relax the impacts of PS technology on the decision module and a more than 0.8-dB improvement in terms of OSNR sensitivity in back-to-back (B2B) case is obtained. Finally, we successfully transmit the PS-16QAM signals over a 2400-km fiber link with a bit error ratio (BER) lower than 1.00×10−3 by adopting the PMLSD scheme.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid development of the Internet of Things (IoT), cloud services and backbone networks, probabilistic shaped (PS) technology has received more attention in recent years owing to its flexibility in rate adaption and spectral efficiency (SE). The advantages introduced by PS technology directly promote the first development stage of PS technology that developing various forms of PS schemes to obtain better shaping gains with low complexity as possible, such as many-to-one (MTO) mapping based PS schemes [1,2] and distribution matcher-based PS schemes [3–8]. Numerous PS schemes are deployed to achieve various PS systems with different entropies for different transmission requirements respectively, and they have already widely used in coherent optical communication system. As for coherent optical transmission system, the classical receiver-side digital signal processing (DSP) chain mainly consists of chromatic dispersion compensation (DC) module, clock recovery (CR) module, equalizer module, frequency offset estimation (FOE) module, carrier phase recovery (CPR) module, decision module and forward error correction decoder (FEC-Decoder) module [9,10]. When coherent-optical-based DSP algorithms with blind adaptation are deployed for unshaped or slightly shaped constellations, the classical DSP algorithms could work as expected and contribute to obvious gains, whereas for stronger shaped formats, parts of these algorithms were not applicable without modifications [5]. It means that there are several efforts needed to be attached in this field due to parts of the classis DSP algorithms may be suboptimal for stronger probabilistic shaped system. Researchers have noticed this character and promote the second development stage of PS technology in which several algorithms are modified correctly. For instance, the sensitivity of Gardner-based CR technology is strongly affected by the combination of PS magnitude and pulse roll-off factor. The results motivate the research of other CR schemes oriented to PS system [11]. Besides, Yan et al. have investigated the applications of traditional FOE algorithms in PS-16QAM systems, and observed significant performance degradations when the shaping degree increases [12]. Furthermore, researchers have pointed out that the constellations on the corners of each square ring are more important to CPR than other constellations in 16-QAM systems, and the outer the constellations are located in, the more correct the estimation of phase noise will be [13]. However, the outer constellations have a lower probability compared to the inner constellations in PS-16QAM. Therefore, PS technology may introduce a performance penalty induced by the imperfect estimation of phase noise and some schemes are reported to illustrate how to eliminate this incompatibility between CPR technology and PS technology [14–16]. Fortunately, numerous literatures are demonstrated to show that common equalizers can be deployed directly in PS systems, such as [17,18].
As for the decision module, there are two prevailing decision schemes deployed in coherent optical systems, i.e. symbol-wise maximum a posteriori probability (MAP) scheme and maximum likelihood sequence detector (MLSD) scheme [19–22]. The symbol-wise MAP is a point-to-point-based estimation scheme and it makes symbol decisions on each received symbol individually. MLSD is a channel-information-based scheme and it makes symbol decisions on several received symbols simultaneously. Recently, a hot topic on PS system is how to alleviate the impacts of PS technology on decision module, which will influence the optimal decision threshold as illustrated in Sec.2 and will be aggravated in strongly shaped PS systems. To address this question, a symbol-wise probability-aided MAP (PMAP) decision scheme is investigated experimentally in PS system and results show that the implementation of symbol-wise PMAP decision is beneficial for PS-based optical fiber transmissions [19]. Although symbol-wise PMAP scheme has some contributions on combating the impacts of PS technology on decision module, there is still a performance penalty can be observed because symbol-wise PMAP scheme only utilizes the noise variance of channel and it could not utilize the magnitude response of the channel, which is vital and significant to signal recovery and can be obtained by partial response filter [20–22]. As for MLSD, it is an effective tool in direct-detection systems and coherent systems, which takes advantage of channel information to execute symbol decisions [20–22]. Thus, in order to investigate the impacts of PS technology on MLSD decision scheme, a probability-aided MLSD (PMLSD) decision scheme is investigated from theoretical deductions aspect and experiment aspect in this paper. This is the motivation of this paper. For clarifying conveniently, original MLSD (OMLSD) is utilized to represent the traditional MLSD scheme.
In this paper, aiming to analyze and alleviate the impacts of PS technology on MLSD scheme, a novel probability-aided maximum likelihood sequence detector (PMLSD) is experimentally investigated through a 64-GBaud PS-16QAM transmission experiment for the first time. Comparing to direct decision scheme, about 0.8-dB and 1.0-dB improvements of OSNR sensitivity under optical back-to-back (OBTB) case are realized aided by OMLSD and PMLSD respectively. Besides, symbol-wise PMAP scheme is also demonstrated for comparison and results show that comparing to symbol-wise PMAP scheme, about 0.8-dB OSNR sensitivity improvement can be achieved aided by PMLSD. Finally, the PS-16QAM signal is successfully transmitted over 2400-km fiber link aided by PMLSD scheme with a BER lower than 1.00×10−3.
2. Principle
2.1 Principle of PS-16QAM
It is known that Maxwell-Boltzmann distribution can maximize the entropy in linear AWGN channel under the constraint of signal power [5]. As demonstrated in [3], constant composition distribution matching (CCDM) can transform independent and Bernoulli (1/2) distributed input bits into a sequence of symbols with the desired probability distribution, including Maxwell-Boltzmann distribution. The Maxwell-Boltzmann distribution can be expressed as Eq. (1)
2.2 Principle of OMLSD and PMLSD
OMLSD is a powerful tool to reconstruct signal by the prior-information of channel [20], and deploying a partial-response filter (PF) HPF is a practical way to utilize OMLSD in practical system. The partial-response filter can shape the channel response into an intermediate truncated one with a short memory, and thus aided by a less-state OMLSD to eliminate the known inter-symbol interference (ISI) induced by the HPF. The HPF utilized in this paper can be expressed as Eq. (2).
where α is a coefficient to adjust the response of HPF and z means that this function is expressed by its Z-transform format. OMLSD is a Viterbi-based scheme and the target of OMLSD is to find a survival path with the minimum accumulative metric (AM), which is the sum of distance metric (DM) and can be expressed as Eq. (3).In Eq. (3), AM has a prerequisite that the distribution of candidate symbol X should be uniform distributed (UD) [20]. However, the distribution of candidate symbol X follows Maxwell-Boltzmann distribution in PS system and it violates the prerequisite. Thus, there will be a gap between the optimal decision threshold of PS signal and optimal decision threshold of UD signal, which is illustrated in Fig. 3.
Fig. 3. The probability density function of I-path PAM4 symbols of UD-16QAM and PS-16QAM and the corresponding optimal decision threshold when the noise variance is 0.2504.
Figure 3 shows the probability density function (PDF) of I-path PAM4 signal of UD signal and PS signal respectively. The red dashed lines represent the optimal decision threshold (VPS) of PS signal and the blue dashed lines indicate the optimal decision threshold (VUD) of UD signal. The green marked area represents the error-prone area in which PS symbols are easy to be decided incorrectly. For example, if a received PS symbol is located at the left-side error-prone area, it should be decided to be symbol ‘−1’ correctly when VPS-based decision scheme is deployed; however, it will be incorrectly decided to be symbol ‘−3’ and contribute to BER if the VUD-based decision scheme is deployed. This is a demonstration of the impacts of PS technology on decision module. Thus, an offset metric (OM), which can be expressed as Eq. (4), should be introduced in Eq. (3) to address this question, as well as alleviating the impacts of PS technology.
In Eq. (4), ${\sigma ^2}$ represents the estimated noise variance, p(Xk) represents the probability of candidate symbol Xk and ln(.) represents natural logarithm operation. The noise variance ${\sigma ^2}$ can be evaluated by adding training sequences with the method mentioned in our previous work [23]. At the beginning of the transmitted data, we send a training sequence T with a length of Lt and the noise variance can be estimated as Eq. (5).
Finally, the optimal HPF could be obtained as the performance of which can be adjusted by the coefficient α. There are many ways to obtain an optimal parameter α, such as Yule-Walker equations [25] or scanning in a given range [26]. The frequency response of HPF with different coefficient α is illustrated in Fig. 4(a). It can be found that the frequency response of HPF becomes steeper with the increase of coefficient α. However, with the increase of α, more severe ISI is also introduced by the HPF and there should be a trade-off between the coefficient α and the performance, which can be confirmed by Fig. 4(b). After 400-km transmission, PS signal without MLSD contributes to BER = 5.93×10−5 (α = 0), while the BER can reaches to 5.60×10−6 and 4.31×10−6 with OMLSD and PMLSD in the case of α is 0.3 and 0.4 respectively.
Fig. 4. The frequency response of HPF with different coefficient α and its influence on performance under 400 km transmission of PS signal @ Launch Power =−3dBm.
3. Experimental setup
To investigate the performance of PMLSD, experiments have been carried out directly as illustrated in Fig. 5. At the transmitter side, two pseudo random bit sequences (PRBS) of length 213−1 and 214−1 are generated for distribution matcher to generate two PS-PAM4 sequences following the designed probability distribution for in-phase (I-path) and quadrature (Q-path) components respectively. Then, I-path and Q-path PS-PAM4 symbols are combined into PS-16QAM signals and then up-sampled to 2 samples per symbol and shaped by a root raised cosine (RRC) filter with a roll-off factor of 0.6. The probability distribution of PS-16QAM and its optical spectrum are given in inset (a) and inset (b) of Fig. 5, and the entropy of PS-16QAM is 3.4088 bit/symbol for matching the channel character. Finally, the offline signals are ported to an arbitrary waveform generator (AWG, Keysight M8196A) with 92-GSa/s sampling rate and the output 64-GBaud signals are amplified by a pair of electrical amplifiers (EA, SHF S804B). The laser source at 193.4 THz from an external cavity laser (ECL) with a linewidth less than 100 kHz is fed into the I/Q modulator (Fujitsu FTM7977 HQA) with a 3-dB bandwidth of 23-GHz. Before coupling the signal into the recirculating loop, an erbium-doped fiber amplifier (EDFA) is deployed to amplify the optical signal to make it suitable for transmission. Then an attenuator is utilized to adjust the launch power. Afterwards, the signals are launched into the recirculating loop, consisting of eight spans of 50-km ultra large effective area fiber (ULAF, G.654.E). In the recirculating loop, we use a backward-pumped Raman amplifier (RFA) to make up power loss of each span. Besides, a wavelength selective switch (WSS) is also utilized to flatten the gain slope in the recirculating loop.
Fig. 5. Experimental setup. The distribution of (a) PS-16QAM and (b) its optical spectrum in this experiment. PRBS, pseudo random bit sequences; AWG, arbitrary waveform generator; EA, electrical amplifier; ECL: external cavity laser; EDFA, erbium-doped fiber amplifier; ATT, attenuator; OS, optical switch; ULAF, ultra large effective area fiber; RFA, Raman fiber amplifier; WSS, wavelength selective switch; TOF, tunable optical filter; LO, local oscillator; EDC, electrical chromatic dispersion compensation; CR, clock recovery; CMMA, cascaded multi-modulus algorithm, FOE, frequency offset estimation; CPR, carrier phase recovery;
At the receiver side, the combination of EDFA and tunable optical filter (TOF) is inserted before coherent receiver to amplify the signal and remove the out-of-band noise. Subsequently, the filtered signals are captured by an 80-GSa/s digital sampling oscilloscope with a 3-dB bandwidth of 36-GHz (Lecroy LabMaster 10-36Zi-A), and the DSP is carried out offline. The Gram–Schmidt orthogonalization procedure (GSOP)-based IQ quadrature imbalance compensation is utilized first and then electrical chromatic dispersion compensation (EDC) based on the frequency-domain compensation is carried out to compensate accumulated chromatic dispersion [27]. Afterwards, Gardner-algorithm-based clock recovery [28] is deployed to eliminate the sampling clock offset, and linear equalization based on the cascaded multi-modulus algorithm (CMMA) [29] is used to compensate the linear impairments. After then, Fast Fourier Transform (FFT) based frequency offset estimation (FOE) [30] and Viterbi-Viterbi phase estimation (VVPE) based carrier phase estimation (CPE) [31] are performed for carrier recovery. Before QAM de-mapping and bit error ratio (BER) calculation, different decision schemes are executed for comparison, which have been illustrated in Fig. 5. Finally, about 10 million QAM symbols are utilized for performance comparisons, which can guarantee the reliability of results. Besides, a 54.5-GBaud UD-16QAM signal is also demonstrated for comparison, which has the same net date rates as PS-16QAM.
4. Results and discussions
In this section, signals after carrier phase recovery (CPR) will be processed by different decision schemes separately for comparison, which have been demonstrated in Fig. 5. The three schemes are labeled as without MLSD (w/o MLSD), with OMLSD (w/ OMLSD) and with PMLSD (w/ PMLSD) to represent direct decision, recovered by OMLSD and recovered by PMLSD respectively. Besides, the performance of symbol-wise PMAP scheme is also demonstrated for comparison. Firstly, Fig. 6 demonstrates the experimental results of BER versus OSNR with different decision schemes under OBTB case. It should be pointed out that the results illustrated in this paper are aided by optimal coefficient α [25]. The performance of UD signal with and without MLSD are also demonstrated for comparison.
Fig. 6. BER versus OSNR under OBTB case. (a) The constellation of PS signal without MLSD @ OSNR = 16-dB and (b) UD signal without MLSD @ OSNR = 16-dB. (c) The enlarged figure of the black-dashed-line-marked area.
As expected, about 0.8-dB OSNR gain @BER = 1.00×10−3 is obtained when OMLSD is deployed and more than 0.2-dB OSNR gain can be further achieved aided by PMLSD for PS signal. For UD signal, the OSNR gain becomes 0.8-dB. The reasons why the PMLSD could contribute to better performance than OMLSD can be explained as follows. The decision threshold of OMLSD is suboptimal for PS signal as illustrated in Fig. 3, it means that symbols located at the error-prone area are decided incorrectly and contribute to performance degradation. Aided by the OM term in Eq. (6), symbols located at error-prone area could be decided correctly and contribute to 0.2-dB OSNR gain compared with OMLSD. Besides, about 0.8-dB OSNR gain is obtained when PMLSD is deployed compared with symbol-wise PMAP decision scheme owing to the abilities to combat the enhanced in-band noise [21,26].
Then, fiber transmission experiments are carried out using different transmission distances. Launch power is a key parameter for fiber transmission. Low launch power will degrade the performance of signal because of the low OSNR, while higher launch power will introduce fiber nonlinearity and further decrease the system sensitivity. Therefore, it is necessary to find an optimal launch power to obtain optimal fiber transmission performance. For example, the results of the optimization of launch power over 2400-km transmission (6 loops) are shown in Fig. 7. After 2400-km transmission, the BER of PS-16QAM reaches to 1.65×10−3 when direct decision scheme is deployed, and the BER can be lower to 1.40×10−3, 1.22×10−3 and 9.97×10−4 with symbol-wise PMAP, OMLSD and PMLSD at the optimal launch power respectively.
Fig. 7. BER versus launch power using different schemes after 2400-km transmission of PS signal. (a) The constellation of PS signal without MLSD after 2400-km.
Finally, the BER performance under different transmission distance is analyzed in Fig. 8 to investigate the transmission performance of PMLSD. When the transmission distance is up to 1350-km, the BER of UD-16QAM without MLSD reaches to 1.00×10−3 and with the help of OMLSD, the transmission distance can further reach to 1600-km for UD-16QAM. When the transmission distance is further increased to 2400-km, the BER of PS-16QAM without MLSD is 1.65×10−3, while the BER will be lower than 1.00×10−3 and reach to 9.97×10−4 aided by PMLSD. Experiment results show that 2400-km transmission with a BER lower than 1.00×10−3 is achieved aided by PMLSD.
Fig. 8. BER performance versus transmission distance using different schemes at optimal launch power. The constellation of PS-16QAM signal after (a) 2000-km and (b) 2400-km transmission without MLSD.
5. Conclusion
In this paper, we have investigated the probability-aided maximum likelihood sequence detector (PMLSD) decision scheme through a 64-GBaud PS-16QAM transmission experiment over a 2400-km G.654.E fiber link. The PMLSD scheme shows a superior ability to relax the impacts of PS technology on decision module, thus an enhanced BER performance is obtained when symbols located at error-prone area. Besides, other decision schemes are also demonstrated for comparison. Results show that PMLSD shows a superior ability to execute optimal symbol decisions under optical back-to-back (OBTB) case and about 1.0-dB optical signal-to-noise-ratio (OSNR) gain is obtained for PS-16QAM compared with direct decision. In case of fiber transmission, we successfully transmit the 64-GBaud PS-16QAM signal over 2400-km G.654.E fiber link with a BER lower than 1.00×10−3.
Funding
National Key Research and Development Program of China (2019YFB1803601); National Natural Science Foundation of China (61675034, 61875019, 62021005).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
References
1. Z. Cao, Y. Chen, X. Zhao, L. Zhou, and Z. Dong, “Probabilistic Shaping 44QAM Based on Many-to-One Mapping in DMT-WDM-PON,” IEEE Photonics Technol. Lett. 32(11), 639–642 (2020). [CrossRef]
2. H. Zhou, Y. Li, X. Jia, C. Gao, Y. Liu, J. Qiu, X. Hong, H. Guo, Y. Zuo, and J. Wu, “Polar coded probabilistic shaping PAM8 based on many-to-one mapping for short-reach optical interconnection,” Opt. Express 29(7), 10209–10220 (2021). [CrossRef]
3. P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016). [CrossRef]
4. J. Cho and P. J. Winzer, “Probabilistic Constellation Shaping for Optical Fiber Communications,” J. Lightwave Technol. 37(6), 1590–1607 (2019). [CrossRef]
5. F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Isdler, “Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016). [CrossRef]
6. M. Fu, Q. Liu, X. Zeng, Y. Wu, L. Yi, W. Hu, and Q. Zhuge, “Parallel bisection-based distribution matching for probabilistic shaping,” in Proceedings of Optical Fiber Communication Conference (2020), paper Th1G.2.
7. T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Multiset-partition distribution matching,” IEEE Trans. Commun. 67(3), 1885–1893 (2019). [CrossRef]
8. G. Bocherer, F. Steiner, and P. Schulte, “Bandwidth Efficient and Rate-Matched Low-Density Parity-Check Coded Modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015). [CrossRef]
9. X. Zhou, L. E. Nelson, P. Magill, R. Isaac, B. Zhu, D. W. Peckham, P. I. Borel, and K. Carlson, “High Spectral Efficiency 400 Gb/s Transmission and Training-Assisted Carrier Recovery,” J. Lightwave Technol. 31(7), 999–1005 (2013). [CrossRef]
10. J. Yu, K. Wang, J. Zhang, B. Zhu, S. Dzioba, X. Li, H. C. Chien, X. Xiao, Y. Cai, J. Shi, Y. Chen, S. Shi, and Y. Xia, “8×506-Gb/s 16QAM WDM signal coherent transmission over 6000-km enabled by PS and HB-CDM,” in Proceedings of Optical Fiber Communication Conference (2018), paper M2C.3.
11. F. A. Barbosa, S. M. Rossi, and D. A. A. Mello, “Clock recovery limitations in probabilistically shaped transmission,” in Proceedings of Optical Fiber Communication Conference (2020), paper M4J.4.
12. X. Wang, Q. Zhang, X. Xin, R. Gao, K. Lv, Q. Tian, F. Tian, C. Wang, X. Pan, Y. Wang, and L. Yang, “Robust and Low-Complexity Principal Component-Based Phase Estimation Algorithm for Probabilistically Shaped Square-QAM Systems,” J. Lightwave Technol. 38(22), 6153–6162 (2020). [CrossRef]
13. X. Wang, Q. Zhang, J. Yu, X. Xin, K. Lv, R. Gao, J. Ren, F. Tian, Q. Tian, C. Wang, X. Pan, Y. Wang, D. Guo, and L. Yang, “Carrier phase recovery friendly probabilistic shaping scheme based on a quasi-Maxwell–Boltzmann distribution model,” Opt. Lett. 45(17), 4883 (2020). [CrossRef]
14. D. A. A. Mello, F. A. Barbosa, and J. D. Reis, “Interplay of probabilistic shaping and the blind phase search algorithm,” J. Lightwave Technol. 36(22), 5096–5105 (2018). [CrossRef]
15. F. A. Barbosa and D. A. A. Mello, “Shaping factor detuning for optimized phase recovery in probabilistically-shaped systems,” in Proceedings of Optical Fiber Communication Conference (2019), paper W1D.4.
16. N. Zhang, X. Zhang, N. Cui, X. Li, L. Xi, and W. Zhang, “A Kalman Filter Based Carrier Phase Recovery Scheme for Probabilistic Shaping M-QAM System,” in Proceedings of European Conference on Optical Communications (2020).
17. M. Kong, K. Wang, J. Ding, J. Zhang, W. Li, J. Shi, F. Wang, L. Zhao, C. Liu, Y. Wang, W. Zhou, and J. Yu, “640-Gbps/Carrier WDM Transmission over 6,400 km Based on PS-16QAM at 106 Gbaud Employing Advanced DSP,” J. Lightwave Technol. 39(1), 55–63 (2021). [CrossRef]
18. M. Kong, C. Liu, B. Sang, K. Wang, J. Ding, J. Shi, L. Zhao, W. Zhou, X. Xin, B. Liu, B. Ye, W. Chen, and J. Yu, “Demonstration of 800-Gbit/s/carrier TPS-64QAM WDM Transmission over 2, 000 km Using MIMO Volterra Equalization,” in Proceedings of Optical Fiber Communication Conference (2021), paper W1I.4.
19. S. Hu, W. Zhang, X. Yi, Z. Li, F. Li, X. Huang, M. Zhu, J. Zhang, and K. Qiu, “Map detection of probabilistically shaped constellations in optical fiber transmissions,” in Proceedings of Optical Fiber Communication Conference (2019), paper W1D.3.
20. G. D. Forney, “Maximum-Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference,” IEEE Trans. Inf. Theory 18(3), 363–378 (1972). [CrossRef]
21. J. Li, E. Tipsuwannakul, T. Eriksson, M. Karlsson, and P. A. Andrekson, “Approaching nyquist limit in WDM systems by low-complexity receiver-side duobinary shaping,” J. Lightwave Technol. 30(11), 1664–1676 (2012). [CrossRef]
22. O. Golani, M. Feder, and M. Shtaif, “Kalman-MLSE Equalization for NLIN Mitigation,” J. Lightwave Technol. 36(12), 2541–2550 (2018). [CrossRef]
23. H. Zhou, Y. Li, T. Dong, J. Qiu, X. Hong, H. Guo, Y. Zuo, J. Yin, Y. Su, and J. Wu, “Improved polar decoding for optical PAM transmission via non-identical Gaussian distribution based LLR estimation,” Opt. Express 28(26), 38456 (2020). [CrossRef]
24. W. Zhang, S. Hu, X. Yi, J. Zhang, W. Zhou, X. Gao, and K. Qiu, “On the Optimum Decision Threshold for Probabilistically Shaped PAM Constellation,” in Proceedings of Conference on Lasers and Electro-Optics (2018).
25. L. Liu, L. Li, and Y. Lu, “Detection of 56GBaud PDM-QPSK generated by commercial CMOS DAC with 11 GHz analog bandwidth,” in Proceedings of European Conference on Optical Communications (2014).
26. H. Wang, J. Zhou, D. Guo, Y. Feng, W. Liu, C. Yu, and Z. Li, “Adaptive Channel-Matched Detection for C-Band 64-Gbit/s Optical OOK System over 100-km Dispersion-Uncompensated Link,” J. Lightwave Technol. 38(18), 5048–5055 (2020). [CrossRef]
27. Y. Liu, Y. Li, J. Song, L. Yue, H. Zhou, M. Luo, Z. He, J. Qiu, Y. Zuo, W. Li, X. Hong, H. Guo, and J. Wu, “Transmission of a 112-Gbit/s 16-QAM over a 1440-km SSMF with parallel Kramers-Kronig receivers enabled by an overlap approach and bandwidth compensation,” Opt. Express 29(6), 8117 (2021). [CrossRef]
28. F. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986). [CrossRef]
29. J. Yang, J. J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Select. Areas Commun. 20(5), 997–1015 (2002). [CrossRef]
30. M. Selmi, Y. Jaouen, and P. Ciblat, “Accurate digital frequency offset estimator for coherent PolMux QAM transmission systems,” in Proceedings of European Conference on Optical Communications (2009).
31. S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010). [CrossRef]
