Open Access
Add to BibSonomy
Abstract
In this paper, a low-complexity optimized detection scheme consisting of a post filter with weight sharing (PF-WS) and cluster-assisted log-maximum a posteriori estimation (CA-Log-MAP) is proposed. Besides, a modified equal-width discrete (MEWD) clustering algorithm is proposed to eliminate the training process during clustering. After channel equalization, optimized detection schemes improve performance by suppressing the in-band noise raised by the equalizers. The proposed optimized detection scheme was experimentally performed in a C-band 64-Gb/s on-off keying (OOK) transmission system over 100-km standard single-mode fiber (SSMF) transmission. Compared with the optimized detection scheme with the lowest complexity, the proposed method saves 69.23% required number of real-valued multiplications per symbol (RNRM) at 7% hard-decision forward error correction (HD-FEC). In addition, when the detection performance reaches saturation, the proposed CA-Log-MAP with MEWD saves 82.93% RNRM. Compared with the classic k-means clustering algorithm, the proposed MEWD has the same performance without a training process. To the best of our knowledge, this is the first time clustering algorithms have been applied to optimize decision schemes.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Driven by the rapid development of high-definition video streaming, virtual/augmented reality, and cloud network, high-speed data center interconnects (DCIs) are in ongoing demand [1–3]. To meet low cost, low power consumption and small footprint, DCIs are inclined to adopt intensity modulation and direct detection (IM/DD) optical system [4], which the P802.3cn Task Force recommends for 400G Ethernet. However, in the C-band IM/DD optical system, the combination of square-law detection and chromatic dispersion (CD) causes spectral nulls, which is the main obstacle limiting the achievable capacity-distance product [5]. With the capacity-distance product increasing, the CD becomes severe and results in the degradation of transmission performance [6]. O-band transmission is the simplest method, which can fundamentally avoid CD [7]. However, compared with C-band transmission, O-band transmission has higher fiber loss limiting its transmission distance [8,9]. Therefore, how to compensate for CD-caused spectral nulls is an urgent issue for the C-band IM/DD optical system.
Recently, a large number of CD compensation methods have been proposed. For optical dispersion compensation, dispersion compensation fiber (DCF) can solve the C-band CD problem, but DCF has high insertion loss and implementation costs [10]. Some schemes that change the structure of traditional IM/DD optical systems can effectively compensate for CD-caused distortions, including single sideband or vestigial sideband modulation (SSB/VSB) [11–13], CD pre-compensation [14], and Kramers-Kronig receiver [15]. However, these solutions include complex system structures and additional devices, which greatly increase the cost and difficulty of commercialization. There are other schemes that use advanced digital signal processing (DSP) technology at low cost to improve the dispersion tolerance of the system in the electrical domain without changing the traditional IM/DD structure. Popular DSP techniques include feed-forward equalizer (FFE), decision feedback equalizer (DFE) [16], and Tomlinson-Harashima precoding (THP) [17]. However, These schemes not only compensate for signal fading but also raise the noise in the spectral nulls. In order to suppress the raised noise, our previous work proposed an adaptive channel-matched detection (ACMD) [18–20]. This work consists of two parts, channel equalization and optimization detection including maximum likelihood sequence estimation (MLSE) and log-maximum a posteriori estimation with a fixed number of surviving states (fixed-state Log-MAP). However, the required number of real-valued multiplications per symbol (RNRM) of the optimization detection is ~1300, while achieving a 64-Gb/s on-off keying (OOK) transmission over 100-km standard single-mode fiber (SSMF). In the part of channel equalization, low-complexity absolute-term-based nonlinear FFE combined with a DFE with weight sharing (AT-NLE-WS) is proposed [21] to reduce the RNRM. Nevertheless, the used k-means clustering algorithm, which needs to be iterated continuously according to the real-time changes of DFE taps, makes the AT-NLE-WS difficult to be encapsulated into hardware. Therefore, simplifying the clustering process can make this algorithm more applicable. In the part of optimization detection, due to the high complexity of existing schemes, it is worthwhile to investigate low-complexity optimal detection schemes.
In this work, we proposed a low-complexity optimized detection scheme consisting of a post filter with weight sharing (PF-WS) and cluster-assisted log-maximum a posteriori estimation (CA-Log-MAP), which has a fixed number of surviving states. Besides, a modified equal-width discrete (MEWD) clustering algorithm is proposed to eliminate the training process during clustering. Thanks to the proposed MEWD, we have the same clustering performance as the k-means clustering algorithm without the training process. Based on the best performance of fixed-state Log-MAP, CA-Log-MAP with different numbers of clusters and different clustering algorithms are investigated to measure its detection performance and RNRM in a C-band 64-Gb/s OOK transmission system over a 100-km SSMF. The experimental results show that the RNRM of CA-Log-MAP is only 128 for a bit error ratio (BER) under 7% hard-decision forward error correction (HD-FEC) threshold of $~3.8\times 10^{-3}$. Compared with the fixed-state Log-MAP, the proposed CA-Log-MAP saves 69.23% RNRM at 7% HD-FEC. In addition, when the detection performance reaches saturation, the proposed CA-Log-MAP saves 82.93% RNRM (i.e., 224). This study represents the first application of clustering algorithms to optimize decision schemes for the reduction of calculation complexity.
2. Principle of the proposed optimal detection
2.1 PF-WS and MEWD
It has been proved that the optimized detection scheme performs best when the channel noise is additive white Gaussian noise (AWGN) [22]. However, the in-band noise is enhanced in the spectral nulls by the equalizers and converted to colored noise. Therefore, the noise-whitening post filter (PF) is used to whiten the colored in-band noise. The output $v(k)$ of $(L+1)$-tap noise-whitening PF is given by
where $x(k-i)$ is the output of equalizers with the colored noise, and $h(i)$ is the tap coefficient. The tap coefficients can be obtained adaptively based on Yule-Walker equations for the autoregressive (AR) coefficients extraction [23]. PF introduces the known inter-symbol interference (ISI) with memory length $L$. As for PF-WS, it is given by where $w(i)$ and $L_c+1$ are centroids (i.e., the new weight of PF-WS) and the number of clusters obtained by clustering algorithms, respectively. Here, $x_c(k-i)$ is the sum of outputs from channel equalization belonging to the same cluster. Then, the number of taps of PF changes from $L+1$ to $L_c+1$, which affects the complexity of CA-Log-MAP. In this paper, we investigate two clustering algorithms, the k-means clustering algorithm and the proposed MEWD.The process of the k-means clustering algorithm is as follows [24,25]: (1) $L_c+1$ centroids are initially chosen from the $L+1$ estimated the weights of PF at random; (2) the Euclidean distance is calculated between each tap weight and each centroid and then it is assigned to the nearest cluster; (3) the centroids are updated by calculating the mean of each cluster; (4) finally, steps (2) and (3) are repeated until the standard metric function converges. This k-means clustering process results in extra $O\left [(L+1) \cdot (L_c +1)\cdot t\right ]$ computational complexity in the whole training process, dependent on $L+1$, $L_c+1$ and the number of iterations $t$.
In our research, we found that the tap coefficients of PF-WS have the following distribution characteristics: except for the first two taps, the weight distribution of the rest shows the law of equal width dispersion. Besides, they are distributed around zero and have more positive values. In order to confirm this, we obtained the post-filter coefficients with the number of taps from 3 to 50 and arranged them from small to large in Fig. 1. It is worth noting that all the coefficients of these PFs are sorted in Fig. 1(a), while the tap coefficients except for the first two taps are sorted in Fig. 1(b). Thanks to the distribution characteristics, the use of MEWD eliminates the training process of traditional clustering algorithms. Figure 2 shows the principle of MEWD. The process of the MEWD clustering algorithm is as follows: (1) 2 centroids are chosen from the first two estimated tap weights of PF; (2) the rest of the estimated tap weights are divided into two parts depending on whether they are less than zero; (3) for each part, the tap weights are uniformly quantized into $(L_c-1)/2$ values; (4) the rest of centroids are chosen form the $L_c-1$ values; (5) for each tap weight, the Euclidean distance is calculated between it and each centroid and then it is assigned to the nearest cluster; (6) finally, the centroids are updated by calculating the mean of each cluster. Since there are more positive values in tap weights, when $L_c-1$ cannot be divided by 2, the positive value part will be assigned an additional cluster. This MEWD clustering process results in extra $O\left [(L+1) \cdot (L_c +1)\right ]$ computational complexity in the whole training process. Compared with k-means clustering algorithms, the complexity of MEWD is only $1/t$.
Fig. 1. Distribution of post-filter coefficients: (a). Overall distribution; (b). Distribution except for the first two taps
2.2 CA-Log-MAP
The output of CA-Log-MAP detection is the log-likelihood ratio (LLR). For an OOK system, the output can be expressed as [19]
where $u_k$ is the transmitted symbol with the elements of $\{-1,1\}$. $P(u_k|z)$ is the probability of transmitted symbol $u$ at time $k$ in the condition of the received sequence $z$. $P(u_k|z)$ is derived asFinally, the LLR is sent to a hard decision module for detection output. $LLR(u_k)\geq 0$ means that the probability of $u_k=+1$ is greater than $u_k=-1$. Therefore, the hard-decision output is $\hat {u}_k=+1$ at $LLR(u_k)\geq 0$. Conversely, the hard-decision output is $\hat {u}_k=-1$. The output of the hard-decision module is expressed as
Then, the computational complexity of the algorithms is analyzed. As for fixed-state Log-MAP, we reserve $M$ surviving states and each of them has 2 branches. The calculation procedure includes four steps, branch transition probability, forward probability, backward probability, and LLR calculation. Among them, only the branch transition probability involves $2 M \cdot (L+2)$ real-valued multiplications. Therefore, the RNRM per symbol of the fixed-state Log-MAP is $2 M \cdot (L+2)$. Although the complexity of the optimized detection has been greatly reduced due to the fixed states, its complexity is still an order of magnitude larger than that of channel equalization. Compared with the fixed-state Log-MAP, we reduced the computational complexity while calculating the branch transition probability $\gamma _k(s,s^{\prime })$ in the CA-Log-MAP. Thanks to this optimization, the RNRM of CA-Log-MAP is $2 M \cdot (L_c+2)$, which is affected by the number of clusters $L_c+1$ but not the number of the weights of PF $L+1$. More essentially, the CA-Log-MAP performs the addition operations first to reduce the RNRM, which is similar to the associative law of multiplication in principle. Figure 3 shows an example of 4-cluster CA-Log-MAP based on 6-tap fixed-state Log-MAP while calculating $v_k^{\prime }$. Figure 3(a) is the process of calculating $v_k^{\prime }$ before clustering. For each symbol, when the memory length is 5, each calculation of $v_k^{\prime }$ requires 6 times multiplication. Figure 3(b) is the process of calculating $v_k^{\prime }$ after clustering, in which we assume that $h(2)$ and $h(4)$, $h(3)$ and $h(5)$ can be clustered, respectively. Therefore, the number of centroids is 4. For each symbol, although the memory length is not changed, each calculation of $v_k^{\prime }$ requires only 4 times multiplication.
Fig. 3. An example of 4-cluster CA-Log-MAP based on 6-tap fixed-state Log-MAP while calculating $v_k^{\prime }$: (a). Before clustering; (b). After clustering
3. Experimental setup
The experimental setup of a C-band 64-Gb/s IM/DD optical OOK system over a 100-km SSMF is illustrated in Fig. 4. First of all, we mapped a pseudo-random bit sequence (PRBS) into OOK symbols by offline processing. In one digital OOK frame, the first 5000 symbols served as training symbols and the rest 77240 symbols were payload symbols. Afterward, we used a root-raised-cosine digital filter for pulse shaping and resampled the digital OOK frame to match the sampling rate of the digital-to-analog converter (DAC). After that, the generated offline data was sent to a 90-GSa/s DAC with 16-GHz 3-dB bandwidth and 8-bit resolution. The amplitude of the offline-generated signal was adjusted by an electrical amplifier (EA, Centellax OA4SMM4) followed by a 3-dB attenuator (ATT) to avoid the modulation nonlinearity, and then fed into a 40-GHz Mach-Zehnder modulator (MZM, Fujitsu FTM7937EZ) with +2V DC bias for the electrical-optical conversion. An external cavity laser (ECL) with a central wavelength of 1550.116 nm was applied to generate an optical source. Next, the optical signal was launched into a 100-km SSMF without any dispersion compensation. The loss of SSMF is about 0.19 dB/km, and the total loss of transmission link is around 20 dB. At the receiver, a variable optical attenuator (VOA) was employed to adjust the received optical power (ROP). Then an Erbium-doped fiber amplifier (EDFA) was applied to boost the power up to −4 dBm. The optical signal was converted into an electrical signal by a 31-GHz PIN with a trans-impedance amplifier (TIA). Then, the detected electrical signal was fed into a 80-GSa/s real-time oscilloscope (RTO) with 36-GHz 3-dB bandwidth to be digitized. Finally, the offline DSP procedures were performed, including resampling, matched filter, time synchronization, channel equalization, optimize detection consisting of PF-WS and CA-Log-MAP, and bit error ratio (BER) counting.
Fig. 4. Experimental setup of C-band 64-Gb/s IM/DD optical OOK system over a 100-km SSMF: (a). Tx-DSP; (b). The electrical spectrum of received signal; (c). Rx-DSP.
The 64-Gb/s OOK signal with 32-GHz baseband bandwidth attenuates rapidly at the high-frequency regions. Figure 4(b) shows the normalized power spectrum of the received signal after 100-km SSMF transmission. For a 100-km dispersion-uncompensated link, CD leads to 14 spectral nulls, and the limited system bandwidth causes severe attenuation of the high-frequency spectrum.
4. Experimental results and analysis
We firstly optimized the required memory length $L$ and the number of surviving states $M$ of classic fixed-state Log-MAP in terms of the BER performance. $L$ is determined by the number of taps of the PF, $L+1$. The corresponding BER performance at the maximal ROP of −14dBm after 100-km SSMF transmission is depicted in Fig. 5. Note that $L=0$ represents that only channel equalization [19] is used. One can see that the BER performance of the algorithm is improved as the number of surviving states $M$ or the memory length $L$ increases but the improvement becomes negligible when $L \geq 49$ and $M \geq 16$. Based on BER results, we choose $L=49$ and $M=16$ here and also for the following experimental results. It is observed that all BER results are lower than 7% HD-FEC BER threshold of $3.8\times 10^{-3}$ satisfying $L \geq 11$. Furthermore, the weight distributions of PF are presented in Fig. 6(a), which contains 50 unique weights ranging from approximately $-0.1$ to $+0.1$ except for the first two weights.
Fig. 5. BER versus memory length $L$ under different number of surviving states $M$ for fixed-state Log-MAP at a ROP of −14 dBm after 100-km SSMF transmission.
Fig. 6. Some experimental results: (a). The weight distributions of PF; (b). BER versus the number of clusters with different clustering algorithms
We then compared the detection performance of different fixed-state Log-MAP and CA-Log-MAPs. Different from the MEWD-based CA-Log-MAP, the training process of k-means-based CA-Log-MAP ensures stable performance when the number of centroids is small. The measured BER performance of CA-Log-MAP with different clustering algorithms versus the number of clusters is analyzed and shown in Fig. 6(b) by clustering the estimated weights of PF. One can see that both the BERs of CA-Log-MAPs decrease as the number of clusters increases. In this case, when the number of clusters is 6, the saturated BER ($\sim 1.5\times 10^{-3}$) arrives and equals to that of the optimized fixed-state Log-MAP. In addition, the BER performance of MEWD-based CA-Log-MAP is close to that of k-means-based CA-Log-MAP.
Following the above discussions, the BER as a function of RNRM using the algorithm mentioned above at a ROP of −14dBm is shown in Fig. 7. The RNRM is altered by varying the memory length $L$ of fixed-state Log-MAP or the number of taps $L_c+1$ of CA-Log-MAP. Obviously, fixed-state Log-MAP and CA-Log-MAPs have the close saturated BER of $\sim 1.5\times 10^{-3}$. However, the former one has an approximately order-of-magnitude bigger RNRM of 1312 while the latter two only have the RNRM of 224. The RNRM of CA-Log-MAP is even smaller than the used channel equalization. Considering to reach the 7% HD-FEC threshold, the proposed 3-cluster CA-Log-MAP only requires the RNRM of 128, which is approximately 69.23% lower than the RNRM of 416 required by the optimized fixed-state Log-MAP.
Fig. 7. BER versus RNRM using different algorithms at a ROP of −14dBm after 100-km SSMF transmission
Finally, the BER performance versus different ROP after 100-km SSMF transmission is evaluated, using only channel equalization I: PNLE (291,81,41); only channel equalization II: PNLE (291,81,41) and DFE (71,61); channel equalization II and optimize detection: fixed-state Log-MAP ($L=49, M=16$); channel equalization II and optimize detection: 6-cluster CA-Log-MAP ($M=16$) with two clustering algorithms, and the results are shown in Fig. 8. It can be summarized that: (1) except for only using channel equalization, the other three cases can reach the 7% HD-FEC limit of $~3.8\times 10^{-3}$ under certain ROPs; (2) the in-band noise raised by the optimized channel equalization makes it difficult for low-cost systems to transmit high-speed signals over a long distance; (3) optimized detection can further improve BER performance after the optimized channel equalization; (4) compared with the optimized fixed-state Log-MAP, an approximate 0.2dB penalty on the receiver sensitivity can be achieved by the MEDW based CA-Log-MAP under the 7% HD-FEC threshold, but CA-Log-MAP reduces the RNRM by a factor of approximately 82.93% from 1312 to 224.
5. Conclusion
We have adopted a low-complexity CA-Log-MAP detection combined with MEWD to mitigate the noise rise caused by channel equalization for IM/DD optical systems over dispersion-uncompensated links. Compared with the k-means algorithm, the MEWD algorithm does not need a training process but performs the same. We experimentally demonstrated a C-band 64-Gb/s IM/DD OOK system over a 100-km SSMF with BER below the 7% HD-FEC threshold of $3.8\times 10^{-3}$. Under the 7% HD-FEC limit, the CA-Log-MAP with a small number of weight clusters has the RNRM much reduced to 128 and saves up to 69.23% complexity. In addition, under the saturated BER of $\sim 1.5\times 10^{-3}$, 6-cluster CA-Log-MAP saves 82.9% RNRM. Therefore, the CA-Log-MAP detection achieves better transmission performance under low computational complexity. Overall, the proposed optimized detection scheme shows promising results in improving performance while maintaining low complexity. The experimental results demonstrate the effectiveness of the proposed scheme in suppressing noise and reducing computational complexity.
Funding
Fund of State Key Laboratory of Information Photonics and Optical Communications (IPOC2022ZT06); National Natural Science Foundation of China (62271080).
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. J. Zhou, H. Wang, J. Wei, L. Liu, X. Huang, S. Gao, W. Liu, J. Li, C. Yu, and Z. Li, “Adaptive moment estimation for polynomial nonlinear equalizer in pam8-based optical interconnects,” Opt. Express 27(22), 32210–32216 (2019). [CrossRef]
2. Q. Cheng, M. Bahadori, M. Glick, S. Rumley, and K. Bergman, “Recent advances in optical technologies for data centers: a review,” Optica 5(11), 1354–1370 (2018). [CrossRef]
3. K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. P. T. Lau, “Digital signal processing for short-reach optical communications: A review of current technologies and future trends,” J. Lightwave Technol. 36(2), 377–400 (2018). [CrossRef]
4. M. Chagnon, “Direct-detection technologies for intra- and inter-data center optical links,” in 2019 Optical Fiber Communications Conference and Exhibition (OFC).
5. S. M. Ranzini, R. Dischler, F. Da Ros, H. Buelow, and D. Zibar, “Experimental demonstration of optoelectronic equalization for short-reach transmission with reservoir computing,” (2020).
6. 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,” Journal of Lightwave Technology PP, 1–1 (2020).
7. S. Yamamoto, H. Taniguchi, M. Nakamura, and Y. Kisaka, “O-band transmission of 255-gb/s pam8 signal through 10-km smf using mlse based on nonlinear channel estimation,” in 2019 24th OptoElectronics and Communications Conference (OECC) and 2019 International Conference on Photonics in Switching and Computing (PSC), (2019).
8. P. Torres-Ferrera, H. Wang, V. Ferrero, and R. Gaudino, “100 gbps/λ pon downstream o-and c-band alternatives using direct-detection and linear-impairment equalization,” J. Opt. Commun. Netw. 13(2), A111–A123 (2021). [CrossRef]
9. D. Zou, F. Li, W. Wang, Z. Li, and Z. Li, “Amplifier-less transmission of beyond 100-gbit/s/λ signal for 40-km dci-edge with 10g-class o-band dml,” J. Lightwave Technol. 38(20), 5649–5655 (2020). [CrossRef]
10. T. A. Eriksson, M. Chagnon, F. Buchali, K. Schuh, S. ten Brink, and L. Schmalen, “56 gbaud probabilistically shaped pam8 for data center interconnects,” in 2017 European Conference on Optical Communication (ECOC), (IEEE, 2017), pp. 1–3.
11. S. T. Le, V. Aref, K. Schuh, and H. N. Tan, “30 gbaud 128 qam ssb direct detection transmission over 80 km with clipped iterative ssbi cancellation,” in Optical Fiber Communication Conference, (Optica Publishing Group, 2020), pp. M4F–2.
12. Y. Zhu, L. Li, X. Miao, and W. Hu, “Time skew-based filter-free vsb nyquist pam-4/6 generation and 80km ssmf transmission with direct detection,” in 2020 European Conference on Optical Communications (ECOC), (IEEE, 2020), pp. 1–5.
13. W. Wang, D. Zou, Z. Li, Q. Sui, Z. Cao, C. Lu, F. Li, and Z. Li, “Optical single sideband signal reconstruction based on time-domain iteration,” J. Lightwave Technol. 39(8), 2319–2326 (2021). [CrossRef]
14. Q. Zhang, N. Stojanovic, C. Xie, C. Prodaniuc, and P. Laskowski, “Transmission of single lane 128 gbit/s pam-4 signals over an 80 km ssmf link, enabled by ddmzm aided dispersion pre-compensation,” Opt. Express 24(21), 24580–24591 (2016). [CrossRef]
15. X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers–kronig receivers for 100-km datacenter interconnects,” J. Lightwave Technol. 36(1), 79–89 (2018). [CrossRef]
16. 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]
17. Q. Hu, M. Chagnon, K. Schuh, F. Buchali, and H. Bülow, “Im/dd beyond bandwidth limitation for data center optical interconnects,” J. Lightwave Technol. 37(19), 4940–4946 (2019). [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. S. Liu, J. Zhou, H. Wang, X. Tang, M. Guo, Y. Lu, and Y. Qiao, “Fixed-state log-map detection for intensity-modulation and direct-detection systems over dispersion-uncompensated links,” IEEE Photonics J. 13(3), 1–10 (2021). [CrossRef]
20. J. Zhou, H. Wang, Y. Feng, W. Liu, and Z. Li, “Processing for dispersive intensity-modulation and direct-detection fiber-optic communications,” Opt. Lett. 46(1), 138–141 (2021). [CrossRef]
21. X. Wu, J. Zhang, A. P. T. Lau, and C. Lu, “Low-complexity absolute-term based nonlinear equalizer with weight sharing for c-band 85-gbaud ook transmission over a 100-km ssmf,” Opt. Lett. 47(6), 1565–1568 (2022). [CrossRef]
22. J. G. Proakis and M. Salehi, Digital communications, vol. 4 (McGraw-hill New York, 2001).
23. L. Liu, L. Li, and Y. Lu, “Detection of 56gbaud pdm-qpsk generated by commercial cmos dac with 11ghz analog bandwidth,” in 2014 The European Conference on Optical Communication (ECOC), (IEEE, 2014), pp. 1–3.
24. J. Zhang, W. Chen, M. Gao, and G. Shen, “K-means-clustering-based fiber nonlinearity equalization techniques for 64-qam coherent optical communication system,” Opt. Express 25(22), 27570–27580 (2017). [CrossRef]
25. N. Aggarwal and K. Aggarwal, An Improved K-means Clustering Algorithm for Data Mining (LAP LAMBERT Academic Publishing, 2012).
