Open Access
Add to BibSonomy
Abstract
Maximum likelihood sequence estimation (MLSE) is the optimal signal sequence detection that can remove the inter-symbol interference (ISI). However, we find that the MLSE causes burst consecutive errors alternating between +2 and –2 in M-ary pulse amplitude modulation (PAM-M) IM/DD systems with large ISI. In this paper, we propose to use precoding to suppress the burst consecutive errors resulted from MLSE. A 2 M modulo operation is employed to guarantee that the probability distribution as well as the peak-to-average power ratio (PAPR) of encoded signal remain unchanged. After the receiver-side MLSE, the decoding process that involves adding the current MLSE output to the previous one and applying a 2 M modulo is implemented to break the burst consecutive errors. We conduct experiments to transmit 112/150-Gb/s PAM-4 or beyond 200-Gb/s PAM-8 signals at C-band to investigate the performance of the proposed MLSE integrated with precoding. The results show that the precoding can break burst errors effectively. For 201-Gb/s PAM-8 signal transmission, the precoding MLSE can achieve 1.4-dB receiver sensitivity gain and reduce the maximum length of burst consecutive errors from 16 to 3.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Driven by various emerging internet applications such as 5 G communication, cloud computing and video streaming, there is an exponential increasing requirement of the data traffic in data center interconnects [1]. In recent years, the data traffic of intra-data center interconnects (Intra-DCIs) is typically greater than that of the traffic of inter-data center interconnects [2]. Intra-DCIs that cover 2-km or 10-km transmission distance are much sensitive to the cost and power consumption. Intensity modulation and direct-detection (IM/DD) is preferred to be a cost-effective solution due to the advantages like low power consumption, simpler transceiver design as well as smaller footprint [3]. IM/DD transmissions with beyond 100 Gb/s data rates per-wavelength have been extensively studied with different modulation formats, such as discrete multi-tone (DMT) [4], carrier-less amplitude phase (CAP) [5], and M-ary pulse amplitude modulation (PAM-M) [6]. Among various modulation formats, PAM-4 has been chosen as a standard format by IEEE P802.3bs Task Force for 400-G Ethernet due to its simpler architecture and lower energy consumption [7,8]. As data traffic increasing, larger than 200 Gb/s per lane is expected to scale up to 800 Gb/s or 1.6 Tb/s, which can reduce the requirement of more optical lanes and the complexity of integration for the next-generation Ethernet [9–11]. In this regard, continually adopting PAM-4 format will strongly rely on optical transceiver’s bandwidth and more complex receiver-side digital signal processing (DSP). Therefore, the implementation of PAM-8 for higher-order modulation format has been put on the agenda. The large inter-symbol interference (ISI) caused by severe bandwidth limitation and chromatic dispersion (CD), has become a critical issue in high-speed IM/DD transmissions [12,13]. Several methods have been reported to remove ISI without noise enhancement.
The decision feedback equalizer (DFE) combined with feed-forward equalizer (FFE) can compensate for linear and nonlinear distortions [14]. The transmitter-side DFE, Tomlinson-Harashima precoding (THP), has shown excellent performance in combating bandwidth limitation but requires prior transfer-function information which is unrealistic in a practical system [15]. However, the DFE is susceptible to error propagation. The suppression of burst consecutive errors resulted from DFE has been widely investigated [16,17]. The maximum likelihood sequence estimation (MLSE) has been proven to be the optimal signal detection for removing ISI in systems with additive white Gaussian noise (AWGN) [18]. In our previous work, we proposed an advanced trellis-compressed MLSE (TC-MLSE) that reduces the complexity by 98% without performance degradation compared to conventional MLSE [19]. Whereas, the burst consecutive errors are neglected in MLSE. We find that when subject to large ISI, MLSE compensates for the distortions, meantime, it causes burst consecutive errors. This degrades the performance of sequence detection with MLSE and increases the requirement for the subsequent forward error correction (FEC) [20,21].
In this paper, we propose and experimentally demonstrate the TC-MLSE integrated with simple precoding to suppress burst consecutive errors in bandwidth-limited PAM-M IM/DD transmission systems at C-band. The precoding correlates two adjacent PAM symbols. A 2 M modulo operation is employed to restrict the level number, so that the probability distribution as well as the peak-to-average power ratio (PAPR) of encoded signal are the same as original PAM signal, avoiding changing the signal’s features like partial response (PR) signaling or THP. The precoding removal is implemented after receiver-side TC-MLSE to break the consecutive burst errors. The precoding and decoding require only addition and 2 M modulo operation, resulting in negligible increase in complexity. We transmit 112/150-Gb/s PAM-4 or 201-Gb/s PAM-8 signal with FFE, individual TC-MLSE and the proposed precoding TC-MLSE. The experimental results show that the precoding can efficiently break down long burst errors resulted from TC-MLSE and convert them into individual errors at the beginning and end. For PAM-8 signal transmission, employing precoding can reduce the maximum length of burst errors from 16 to 3, resulting in a 1.4-dB receive optical power (ROP) gain.
2. Principle of TC-MLSE integrated with precoding
For high-speed optical IM/DD systems, larger ISI introduced by bandwidth-limited effect and the CD limits the practical transmission performance. Facing large ISI, the MLSE compensates for distortions meantime it causes burst consecutive errors. In this work, the TC-MLSE we employed is based on look-up-table (LUT) [19,22]. The LUT is indexed by the pattern of multiple symbols, and its entries record the distorted samples corresponding to the indexing symbol patterns. The transition metrics to different states are the Euclidean distances between the received sample and candidate LUT entries. When the memory length L of MLSE is set to one, the MLSE state or the indexing symbol pattern of the LUT is noted as (Sk-1, Sk), where S and subscript k represent the candidate symbol and the time slot, respectively.
2.1 Consecutive errors in ‘+2, –2’ zig-zag pattern
The FFE is usually used as a pre-equalizer for MLSE to eliminate part of ISI. Then, a post filter (PF) with transfer function of H_PF(z) = 1+α·z−1 is used to whiten the colored noise caused by FFE [23–25]. For simplicity, we take L equal to one as an example. It is assumed that the received signal is severely distorted (positive polarity noise) at time k and k–1 simultaneously, and the received signal yk contains larger positive noise. Thus, the Euclidean distance between the received signal and state (Sk-1, Sk + 2) is smaller than the Euclidean distance between it with the true state (Sk-1, Sk), so is the transition metric. This triggers a competitive path to Sk + 2 with lower accumulative metric. Because there is significant negative noise in yk relative to Sk + 2, the competitive path will transfer to (Sk + 2, Sk + 1–2) at time k + 1 to minimize the accumulative metric. Like this, the error propagates along the competitive path in ‘+2, –2’ zigzag pattern. The burst consecutive errors terminate when several consecutive symbols with minor error cause the path to re-converge to the true path. Note that if another burst error occurs before re-convergence, the length of consecutive errors will be extended. When tracking back, competitive path with lower accumulative metric will be chosen as the survival path, whose estimated errors alternating between +2 and –2. In this paper, the traceback depth is set to five times as long as the memory length of MLSE.
Table 1 shows an experimental example of 67-Gbaud PAM-8 signal transmission over 3-km standard single mode fiber (SSMF), where the tap coefficient of PF is set to 0.8. Here, we analyze the patterns consisted of ‘True symbol’ and ‘MLSE output’ to investigate how errors occur and propagate. In Table 1, after the PF, the noise in symbol 3 is –1.07 (0.8× (0.3–1) + (6.49–7) = –1.07). Thus, for symbol 3, the transition metric to level 5 is lower than it to level 7, indicating that the path transferred to level 5 becomes the competitive path and triggers the burst errors. Along this competitive path, the noise in symbol 3 relative to level 5 changes to 1.49 (6.49–5 = 1.49), leading the transition metric to (5, 7) lower than it to (5, 5). Like this, the error propagates along the competitive path with ‘+2, –2’ pattern. After symbol 11 and symbol 12 whose noise are small, the burst error propagation terminates and re-converges to the true path. Figure 1 depicts the cumulative metrics of the estimated path and the true path corresponding to Table 1. In Fig. 1, the accumulative metric of estimated path is lower than the true path before re-convergence (symbol 12) despite the transition metrics of true path are lower than that of estimated path at some nodes like symbol 6 and symbol 9. In conclusion, severe distortion of a received signal causes the path mismatch of the Viterbi algorithm, where the estimated errors alternate in ‘+2, –2’ zigzag pattern. Note that the burst consecutive errors also exist in traditional MLSE, not introduced by the operation of trellis compression or precoding.
Table 1. Example of 67-Gbaud PAM-8 signal transmission over 3-km SSMF with MLSE
2.2 Principle of precoding
Figure 2 depicts the block diagram of precoding and decoding of PAM-M signal, where hT(t) is the channel. The pre-coder is composed of a delay-feedback circuit, a sub-tractor and a 2 M modulo operation. Its output can be expressed as,
where ak is the original PAM-M symbol. The 2 M modulo operation is introduced to confine the level number of bk being consistent with original PAM-M symbols (belongs to {±1, ± 3, …, ±(M–1)}), where m = 1 when modulo circuit performs rounding down, m = –1 when rounding up, and otherwise m = 0. Thus, the precoding does not change the PAPR or frequency-domain power distribution of transmitted signal. At the receiver side, the received sequence is processed by FFE, PF and the TC-MLSE. Subsequently, the precoding removal is implemented by adding the TC-MLSE output to the previous one and then implementing the 2 M modulo operation. Combining with Eq. (1), the output of the decoder dk can be expressed as,Table 2. Example of how the precoding work in PAM-8 signal transmission
3. Experimental setup
Figure 3(a) shows the experimental setups of PAM signal transmission over dispersion-uncompensated link at C-band. At the transmitter, a pseudo-random bit sequence (PRBS) with 219 bits is used for PAM-4/PAM-8 symbol mapping then shaped by a root raised cosine (RRC) filter with 0.01 roll-off factor. Then, 67-Gbaud PAM-8 or 56/75-Gbaud PAM-4 signal with 33.84- or 28.28/37.88-GHz bandwidth is generated via fractional sampling where the 32-GHz bandwidth arbitrary waveform generator (AWG) is operating at 92 GSa/s. After amplified by an electrical amplifier (EA), a single-drive mode Mach-Zehnder modulator (MZM) with 40 GHz is used for electro/optic conversion. A 1.89-V DC bias is applied on the MZM. Then, a continuous-wave optical carrier at 1549.5 nm with 12.9-dBm optical power is launched into the MZM and the output power of the MZM is 3.9 dBm. Next, the generated optical PAM-4/PAM-8 signal is fed into 0/3/6-km SSMF with a fiber loss of 0.2-dB/km. At the receiver side, a variable optical attenuator (VOA) is employed to adjust the ROP. Given that our receiver employs a photodiode (PD) without a trans-impedance amplifier (TIA), we apply an Erbium doped fiber amplifier (EDFA) to boost the optical signal. Then the optical signal is directly detected via a TIA-free single-ended PD with 3-dB bandwidth of 40 GHz and captured by a digital phosphor oscilloscope (DPO) operating at 200 GSa/s. Subsequently, the received signal is processed by offline DSP, including resampling, matched filter, equalization, decoding, PAM de-mapping and bit error ratio (BER) calculation. As for equalization, we compare the performance of FFE, individual TC-MLSE and TC-MLSE with precoding. Figure 3(b) shows the frequency response of 67-Gbaud IM/DD PAM-8 optical back to back (OBTB) transmission at a ROP of 8 dBm. Figure 3(c) depicts the optical spectra of transmitted 67-Gbaud PAM-8 signal. The 10-dB bandwidth of the system is approximately 25.5 GHz. Moreover, the frequency response fades rapidly when the frequency is beyond 29 GHz, which leads to a great degradation of transmission performance. To eliminate severe ISI, TC-MLSE and TC-MLSE with precoding have been employed and compared.
Fig. 3. (a) The experimental setups of C-band IM/DD PAM signal transmission system. (b) The frequency response of 67-Gbaud IM/DD PAM-8 OBTB transmission system at a ROP of 8 dBm. (c) The optical spectra of transmitted 67-Gbaud PAM-8 signal.
4. Experimental results and discussions
We first evaluate the effectiveness of the proposed precoding TC-MLSE in suppressing burst consecutive errors in a 112-Gb/s PAM-4 signal transmission over 6-km SSMF system. Figure 4(a) shows the frequency spectrum of the received signal. Due to relatively low spectral bandwidth of the 112-Gb/s PAM-4 signal with respect to the transceiver’s bandwidth, the signal suffers less from the bandwidth limitation. While, after 6-km SSMF transmission, CD leads to one spectral null at around 25 GHz as shown in Fig. 4(a). This power fading introduces large ISI to the received signal. For comparisons, we employ 71-tap FFE, TC-MLSE without precoding and TC-MLSE integrated with precoding for equalization. Figure 4(b) shows the frequency spectrum of the signal processed by 71-tap FFE, in which the 25-GHz spectral null cannot be compensated. Figure 4(c) depicts the frequency response of the PF with α=0.8. Figure 4(d) shows the noise power spectral density (PSD) without and with noise whitening PF. In Fig. 4(d), the noise power after FFE equalization is enhanced in higher frequency region. With the help of 2-tap PF whose coefficient is set to 0.8, the fluctuation of noise PSD is greatly suppressed and the noise after FFE + PF can be approximated as white noise but introducing ISI. Figure 5(a) shows the BER performance of the 112-Gb/s PAM-4 signal after 6-km SSMF transmission. In Fig. 5, the 71-tap FFE fails to reach the 20% overhead soft-decision FEC (SD-FEC) threshold due to noise enhancement. Figure 5(b) depicts the probability distribution function (PDF) of signal after 71-tap FFE, in which the signal cannot be distinguished. Therefore, we reserve three levels with the highest probability of each symbol to compress the trellis graph in TC-MLSE. Affected by the burst consecutive errors, individual TC-MLSE with L = 2 cannot reach the 7% overhead HD- FEC threshold. When we increase the memory length of TC-MLSE to three, the BER performance is improved and achieves a BER of 2.65 × 10−3 at a ROP of 8 dBm, which is lower than the 7% overhead HD-FEC threshold. While, the proposed TC-MLSE (L = 2) with precoding outperforms individual TC-MLSE (L = 3), because the precoding can suppress burst consecutive errors. The precoding TC-MLSE (L = 2) achieves 3.7-dB receiver sensitivity gain considering the HD-FEC threshold of 3.8 × 10−3 compared with TC-MLSE (L = 3).
Fig. 4. The frequency spectrum of signal (a) after 6-km SSMF transmission, (b) processed by 71-tap FFE. (c) The frequency response of PF with α=0.8. (d) The noise PSD without or with PF.
Fig. 5. (a) BER performance of 112-Gb/s PAM-4 signal transmission over 6-km SSMF. (b) The PDF of signal after 71-tap FFE at a ROP of 10 dBm.
Next, we increase the bit rate of PAM-4 signal to 150-Gb/s to demonstrate the performance of the precoding in the scenario with more severe bandwidth limitation. To achieve optimal performance, we have optimized the tap number of FFE and the tap coefficient of following PF as 95 and 0.86, respectively. For TC-MLSE, its memory length L is increased to three to combat large ISI. Figure 6(a) shows the BER performance versus ROP of 150-Gb/s PAM-4 signal transmission over 3-km SSMF. Note that the performance of TC-MLSE is close to that of traditional MLSE, which has been proven in our previous work [19]. In Fig. 6(a), the 95-tap FFE and even TC-MLSE (L = 3) without precoding fail to reach the 7% overhead HD-FEC limit. While, the proposed precoding TC-MLSE (L = 2) shows a BER advantage than individual TC-MLSE (L = 3). Moreover, the TC-MLSE with precoding (L = 3) achieves a BER of 3.04 × 10−3 at a ROP of 10 dBm. The improvement in performance is benefited from the suppression of burst consecutive errors. Figure 6(b) shows the burst consecutive errors distribution of 150-Gb/s PAM-4 signal after processing by traditional MLSE, TC-MLSE (L = 3) with and without precoding at a ROP of 10 dBm. In Fig. 6(b), the traditional MLSE suffers from the same consecutive errors as the TC-MLSE. For TC-MLSE without precoding, the maximum length of error block can be as long as 15, which seriously degrades its performance. However, the maximum length of burst consecutive errors is compressed to only 3 with the help of precoding. This can relieve the requirement and consumption of the follow-up FEC decoding.
Fig. 6. (a) BER performance, (b) distribution of burst consecutive errors using traditional MLSE, TC-MLSE with and without precoding for 150-Gb/s PAM-4 signal transmission over 3-km SSMF.
We further introduce the precoding TC-MLSE into PAM-8 signal transmission to study its performance. Similarly, for TC-MLSE, the reserved level is set to three to compress the trellis graph. Figure 7(a) depicts the BER performance of 67-Gbaud PAM-8 signal transmission over 3-km SSMF. In Fig. 7(a), all the equalization schemes cannot reach the HD-FEC limit, because PAM-8 signaling has poor noise immunity compared to PAM-4. While, the precoding TC-MLSE (L = 3) shows BER advantage than other equalizers. By utilizing precoding, the receiver sensitivity can be improved by approximately 1.4 dB when considering 20% overhead soft-decision FEC, compared to individual TC-MLSE processing. Figure 7(b) shows the burst consecutive errors distribution of PAM-8 signal processed by traditional MLSE, TC-MLSE with and without precoding. In Fig. 7(b), the maximum length of error block for TC-MLSE without precoding is 16. The traditional MLSE also suffers from the consecutive errors as the TC-MLSE. After implementing precoding, long error blocks are broken and converted into individual errors. Thus, the maximum length of burst consecutive errors is compressed to only 3. Therefore, the proposed precoding TC-MLSE has excellent performance on suppression of burst consecutive errors, achieving better BER performance at the cost of negligible complexity increasing.
Fig. 7. (a) BER performance, (b) distribution of burst consecutive errors using traditional MLSE, TC-MLSE with and without precoding for 201-Gb/s PAM-8 signal transmission over 3-km SSMF.
5. Conclusion
In this paper, we have proposed and experimentally demonstrated that the precoding can suppress burst consecutive errors resulted from MLSE in high-speed PAM signal transmission IM/DD systems with severe bandwidth limitation. The experimental results show that the precoding can effectively break long burst consecutive errors and convert them into individual error at the head and the tail of the error block. In beyond 200-Gb/s PAM-8 signal transmission system, utilizing TC-MLSE with precoding reduces the maximum length of burst errors from 16 to 3, and achieves 1.4-dB receiver sensitivity gain.
Funding
National Natural Science Foundation of China (61871082, 62111530150); Open Fund of IPOC (IPOC2020A011); Science and Technology Commission of Shanghai Municipality (SKLSFO2021-01); Fundamental Research Funds for the Central Universities (ZYGX2019J008, ZYGX2020ZB043).
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 may be obtained from the authors upon reasonable request.
References
1. T. Marc, P. Maniotis, and T. Asser, “Optics enabled networks and architectures for data center cost and power efficiency [Invited],” J. Opt. Commun. Netw. 14(1), A41–A49 (2022). [CrossRef]
2. S. J. Ben Yoo, “Prospects and challenges of photonic switching in data centers and computing systems,” J. Lightwave Technol. 40(8), 2214–2243 (2022). [CrossRef]
3. J. Cheng, C. Xie, Y. Chen, X. Chen, M. Tang, and S. Fu, “Comparison of Coherent and IMDD Transceivers for Intra Datacenter Optical Interconnects,” in Optical Fiber Communication Conference (OFC) 2019, OSA Technical Digest (Optical Society of America, 2019), paper W1F.2.
4. S. Liu, J. Zhou, H. Wang, X. Tang, M. Guo, H. Cui, Y. Lu, and Y. Qiao, “Non-orthogonal DMT enabled by iterative ICI cancellation algorithm for bandwidth-limited IM/DD Optical Systems,” J. Lightwave Technol. 40(9), 2799–2806 (2022). [CrossRef]
5. J. Chen, J. Shi, J. Hu, C. Shen, and N. Chi, “DC-balanced even-dimensional CAP modulation for visible light communication,” J. Lightwave Technol. 40(15), 5041–5051 (2022). [CrossRef]
6. D. Che and X. Chen, “Higher-order modulation vs faster-than-nyquist PAM-4 for datacenter IM-DD optics: an AIR comparison under practical bandwidth limits,” J. Lightwave Technol. 40(10), 3347–3357 (2022). [CrossRef]
7. X. Tang, S. Liu, Z. Sun, H. Cui, X. Xu, J. Qi, M. Guo, Y. Lu, and Y. Qiao, “C-band 56-Gb/s PAM4 transmission over 80-km SSMF with electrical equalization at receiver,” Opt. Express 27(18), 25708–25717 (2019). [CrossRef]
8. M. Guo, Y. Qiao, X. Tang, S. Liu, Z. Sun, H. Cui, X. Xu, and L. A. Rusch, “112-Gb/s PAM4 with Joint Pre- and Post-Equalization for Data Center Interconnects,” Asia Communications and Photonics Conference (ACPC) 2019, OSA Technical Digest (Optical Society of America, 2019), paper T2G.2.
9. B. Shariati, L. Velasco, J.-J. Pedreno-Manresa, et al., “Demonstration of latency-aware 5 G network slicing on optical metro networks,” J. Opt. Commun. Netw. 14(1), A81–A90 (2022). [CrossRef]
10. X. Guo, X. Xue, F. Yan, B. Pan, G. Exarchakos, and N. Calabretta, “DACON: a reconfigurable application-centric optical network for disaggregated data center infrastructures [Invited],” J. Opt. Commun. Netw. 14(1), A69–A80 (2022). [CrossRef]
11. H. Mardoyan, M. A. Mestre, R.-M. Rafael, A. Konczykowska, J. Renaudier, F. Jorge, B. Duval, J.-Y. Dupuy, A. Ghazisaeidi, P. Jennevé, M. Achouche, and S. Bigo, “Single carrier 168-Gb/s line-rate PAM direct detection transmission using high-speed selector power DAC for optical interconnects,” J. Lightwave Technol. 34(7), 1593–1598 (2016). [CrossRef]
12. M. Xiang, S. Fu, O. Xu, J. Li, D. Peng, Z. Gao, Y. Wang, and Y. Qin, “Advanced DSP enabled C-band 112 Gbit/s/λ PAM-4 transmissions with severe bandwidth-constraint,” J. Lightwave Technol. 40(4), 987–996 (2022). [CrossRef]
13. S. Hu, J. Zhang, J. Tang, T. Jin, W. Jin, Q. Liu, Z. Zhong, R. Giddings, Y. Hong, B. Xu, X. Yi, and K. Qiu, “Multi-constraint Gerchberg-Saxton iteration algorithms for linearizing IM/DD transmission systems,” Opt. Express 30(6), 10019–10031 (2022). [CrossRef]
14. J. Zhang, X. Wu, L. Sun, J. Liu, A. P. T. Lau, C. Guo, S. Yu, and C. Lu, “C-band 120-Gb/s PAM-4 transmission over 50-km SSMF with improved weighted decision-feedback equalizer,” Opt. Express 29(25), 41622–41633 (2021). [CrossRef]
15. H. Xin, K. Zhang, D. Kong, Q. Zhuge, Y. Fu, S. Jia, W. Hu, and H. Hu, “Nonlinear Tomlinson-Harashima precoding for direct-detected double sideband PAM-4 transmission without dispersion compensation,” Opt. Express 27(14), 19156–19167 (2019). [CrossRef]
16. J. Zhou, C. Yang, D. Wang, Q. Sui, H. Wang, S. Gao, Y. Feng, W. Liu, Y. Yan, J. Li, C. Yu, and Z. Li, “Burst-error-propagation suppression for decision-feedback equalizer in field-trial submarine fiber-optic communications,” J. Lightwave Technol. 39(14), 4601–4606 (2021). [CrossRef]
17. X. Tang, Y. Qiao, Y.-W. Chen, Y. Lu, and G.-K. Chang, “Digital pre- and post-equalization for C-band 112-Gb/s PAM4 short-reach transport systems,” J. Lightwave Technol. 38(17), 4683–4690 (2020). [CrossRef]
18. 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]
19. J. Zhou, J. Zhang, X. Zhao, T. Jin, S. Hu, H. Lin, Z. Yu, Q. Zhang, B. Xu, and K. Qiu, “Advanced nonlinearity equalizer with TC-NL-MLSE for transmitting beyond 200-Gb/s PAM-8 in IM/DD systems,” Opt. Express 30(21), 37416–37425 (2022). [CrossRef]
20. H. Song and J. R. Cruz, “Reduced-complexity decoding of q-ary LDPC codes for magnetic recording,” IEEE Trans. Magn. 39(2), 1081–1087 (2003). [CrossRef]
21. N. Xie, T. Zhang, and E. F. Haratsch, “Improving burst error tolerance of LDPC-centric coding systems in read channel,” IEEE Trans. Magn. 46(3), 933–941 (2010). [CrossRef]
22. Md. Khairuzzaman, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Equalization of nonlinear transmission impairments by maximum-likelihood-sequence estimation in digital coherent receivers,” Opt. Express 18(5), 4776–4782 (2010). [CrossRef]
23. D. Li, H. Song, W. Cheng, M. Cheng, S. Fu, M. Tang, D. Liu, and L. Deng, “Low-complexity equalization scheme for suppressing FFE-enhanced in-band noise and ISI in 100 Gbps PAM4 optical IMDD system,” Opt. Lett. 45(9), 2555–2558 (2020). [CrossRef]
24. K. Zhong, X. Zhou, Y. Gao, W. Chen, J. Man, L. Zeng, A. P. T. Lau, and C. Lu, “140-Gb/s 20-km Transmission of PAM-4 Signal at 1.3 µm for Short Reach Communications,” IEEE Photonics Technol. Lett. 27(16), 1757–1760 (2015). [CrossRef]
25. X. Tang, Y. Qiao, J. Zhou, M. Guo, J. Qi, S. Liu, X. Xu, and Y. Lu, “Equalization scheme of C-band PAM4 signal for optical amplified 50-Gb/s PON,” Opt. Express 26(25), 33418–33427 (2018). [CrossRef]
