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Abstract
In this paper, a nonlinearity-tolerant sphere shaping is proposed based on the parallel bisection structure. Compared to enumerative sphere shaping (ESS), the bisection-based sphere shaping (BS-SS) can achieve a lower average energy and rate loss at the same blocklength by selecting all preferential sequences. The nonlinearity-tolerant BS-SS (BS-SS-NLI) can mitigate the fiber nonlinearity by reducing the energy variation, which is demonstrated in the simulations with probabilistically shaped, single channel, dual-polarization 64-ary quadrature amplitude modulation (DP-64QAM) for a multi-span long-haul transmission over 2500 km of standard single-mode fiber (SSMF) applying digital subcarrier multiplexing (SCM), comparing with recent literature kurtosis-limited ESS (K-ESS) and list-encoding constant composition distribution matching (L-CCDM). The BS-SS-NLI can provide 0.25 dB effective signal-to-noise ratio (SNR) gain over ESS and 0.39 dB gain over L-CCDM. The achievable information rate (AIR) can be improved ∼0.1 bits/4D-symbol by BS-SS-NLI over ESS.
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1. Introduction
Probabilistic amplitude shaping (PAS) [1] can be used to approach the capacity of a linear additive white Gaussian noise (AWGN) channel by generating non-uniform symbols. An important structure of PAS is the amplitude shaper. Many different amplitude shapers have been investigated: constant composition distribution matching (CCDM) [2], product distribution matching (PDM) [3], and enumerative sphere shaping (ESS) [4], and Huffman-coded sphere shaping (HCSS) [5,6]. To further improve the linear performance, some modifications had been adopted to the above shapers. Multi-dimensional distribution matching (MDDM) [7] was proposed for high modulation format. Optimum ESS (OESS) [8] includes a reverse trellis to achieve the lowest average energy. In the nonlinear fiber channel, nonlinear interference (NLI) is a substantial limitation caused by the nonlinear interactions among the transmitted symbols. Digital subcarrier multiplexing (SCM) scheme has been experimentally demonstrated that it can improve the intra-channel fiber nonlinearity tolerance in [9]. In [10], it investigated the performance of utilizing both PAS and SCM to the long-haul transmission.
The Gaussian noise (GN) model derived from perturbation analysis in [11] can approximate NLI. The original GN model does not however predict the observed modulation format dependency of NLI, requiring an enhanced GN (EGN) model to be developed [12,13]. The EGN model indicates the NLI is dependent on the fourth order standardized moment, i.e., kurtosis, of the input distribution. This in turn has led to kurtosis-limited ESS (K-ESS) being proposed [14] which select sphere shaping sequences with the low accumulated kurtosis of the amplitude, i.e., $\sum _{i=1}^{n}a_{i}^{4}$, where $a$ is the amplitude. The approach is demonstrated to provide effective signal-to-noise ratio (SNR) gain over ESS in an extended single-span transmission.
Recently, numerous metrics have been proposed to characterise NLI in the fiber transmission for systems employing PAS. In [15], the simulation results demonstrate the effective SNR after transmission decreases with increasing shaping blocklength, and a heuristic metric called run ratio (RR) is proposed to evaluate this effect by calculating the number of consecutive symbols in a sequence. The windowed energy is used to model NLI for constant composition (CC) sequence in [16] with a new metric called energy dispersion index (EDI) proposed, which is an accurate metric to indicate the blocklength-dependent NLI. However, since the statistical correlation of the shaped symbols cannot be explained by the conventional EGN model, it has been proposed to use windowed moments in place of the conventional moments in [17,18]. Based on various metrics proposed, list-encoding CCDM (L-CCDM) was proposed to improve the NLI tolerance of conventional CCDM by reducing the EDI of transmitted symbols [19]. In this work, we take an alternative approach, focusing on the sphere shaping with short blocklength and nonlinearity mitigation inspired by the recently proposed metrics for NLI, which is demonstrated in a multi-span long-hual transmission.
The paper is organized as follows. In Section 2, the principle of BS-SS is discussed. Then, the nonlinearity-tolerant BS-SS was proposed based on NLI characterisation in Section 3. A brief introduction of other nonlinearity-tolerant shaping algorithms is in Section 4. Section 5. contains two parts of the simulation: AWGN channel and nonlinear fiber channel. Under nonlinear fiber channel simulation, both single channel and 5 channels long-haul transmissions are investigated. Conclusion is stated in Section 6.
2. Principle of BS-SS
In this paper, we propose a new encoding algorithm of sphere shaping, parallel bisection-based sphere shaping (BS-SS), based on divide-and-conquer principle [20]. This idea is inspired from the parallel bisection-based CCDM (BS-CCDM) proposed in [21]. BS-CCDM targets a constant composition but BS-SS targets multiple compositions with blocklength of $n$. BS-SS has the same shaping set with other sphere shaping approaches which can be defined as:
Given that $\mathcal {A}=\{1,3,\dots,2^{m}-1\}$ for integer $m\geq 1$, a composition with blocklength $n$ can be represented in a form of $(N_{1},N_{3},\dots,N_{2^{m}-1})$, where $N_{2^{m}-1}$ is the number of the amplitude $(2^{m}-1)$ in the composition and it satisfies $N_{1}+N_{3}+\cdots +N_{2^{m}-1}=n$. Each target composition with blocklength of $n$ can be divided into two sub-compositions with blocklength of $n/2$, where $n$ is a power of two. The sub-composition pair can be denoted as $\{C_i,\overline {C}_{i}\}$, where $\overline {C}_{i}$ is the counterpart of $C_i$. The total number of each amplitude in a sub-composition pair has to be equal to the one in its target composition. This process can recursively divide the target composition to $n$ blocks with a length of 1. Then, the target composition can be realised by concatenating all length of 1 blocks together. Similarly to the trellis of ESS, a look-up table (LUT) is also required in BS-SS which needs to store the number of permutations, $M(\cdot )$, and sub-composition pairs. Compositions can be separated by its blocklength where compositions with blocklength of $n$ is at the first layer and blocklength of one is at the last layer. The LUT can be constructed from the last layer with the number of permutations of the composition stored. In BS-CCDM, when constructing the LUT, all sub-compositions from the last to the second layer has to satisfy the condition that the number of each amplitude is less than or equal to the number of corresponding amplitude of the composition at the top layer. This will make sure that there will be only one composition at the top layer. However, in BS-SS, no such condition for sub-compositions and, eventually, there will be multiple compositions at the top layer. Given the number of permutations of two sub-compositions $M(C_{i})$ and $M(\overline {C}_{i})$, the total number of permutations of a sub-composition pair $\{C_i,\overline {C}_{i}\}$ can be calculated as
where $M(\cdot )$ represents the number of permutations of a composition and $i$ is the index of the sub-composition pair in the current layer. To ensure the one-to-one mapping, The total number of permutations of all compositions at the first layer should be equal to or greater than $2^{k}$, where $k$ is number of input bits. With an input bit sequence $B = \{b_{k},b_{k-1},\dots,b_{1}\}$, we can represent it to a decimal integer $U$ as At the first layer, a composition with blocklength of $n$ should be selected initially. The condition can be described as and the $l$ th composition will be selected. The input integer value for the next layer is From the second layer, the condition for selecting sub-composition pair and the update of the input integer are
Fig. 1. An example of the encoding process of BS-SS for $\mathcal {A}=\{1,3,5,7\}$, $k=10$, and $n=8$.
Compared to ESS, BS-SS can select all sequences from low to high energy. ESS encodes sequences lexicographically through its trellis and will miss some preferential sequences with low energy [22]. Compared to shell mapping and OESS which also select sequences from low energy to high, BS-SS can manage the energy difference of the sub-composition pairs at the second layer of the LUT. The sequences with low energy difference will be selected first resulting in a different one-to-one mapping of the input bits and output amplitudes.
3. Nonlinearity-tolerant BS-SS
Based on the recent research [16–18], windowed statistics are introduced to characterize NLI. Compared to the proposed accurate metrics, the variation of the energy can be considered as a simple metric to roughly evaluate the performance of transmitted symbols in the nonlinear fiber channel. In this section, we introduce the nonlinearity-tolerant BS-SS by reducing the energy variation of the output sequences. Firstly, given an amplitude sequence $A=\{a_{1},a_{2},\dots,a_{n}\}$ with blocklength $n$, we can change the sequence to a combination of blocks with short length $n_l$, $A=\{A_{1},\dots,A_{n/n_l}\}$, i.e., $A_{1}=\{a_{1},\dots,a_{n_l}\}$. Then, the block energy of $A_{i}$, $E(A_{i})=\sum _{j=1}^{n_l} a_{(i-1)\cdot n_l+j}^{2}$. In BS-SS, $\text {Var}[E]$ can be considered as the variance of sub-composition pairs with the same length $n_l$ shown as
where $\mu _E$ is the average block energy of BS-SS with the length as $n_l$. Therefore, we can calculate the amount of contribution from a sub-composition pair to $\text {Var}[E]$ as where $E(C_i)$ and $E(\overline {C}_i)$ are the energies of a sub-composition pair $\{C_i,\overline {C}_{i}\}$. As a composition can be composed by multiple sub-composition pairs, the one with large energy variation can be deleted from the LUT. Given $n_l$ and a threshold value of $V$, $V_{\text {thr}}$, when the LUT is constructed to the layer with length $2n_l$, any sub-composition pair that has $V > V_{\text {thr}}$, equivalently, all sequences that contain such sub-composition pairs will be eliminated. With a smaller number of permutations for some compositions, the number of compositions used at the first layer will be increased. We call this approach BS-SS-NLI in the following.The short length $n_l$ can be any power of 2 less than the blocklength $n$. If $n_l$ is small, for example $n_l=1$, any compositions with that amplitude will be deleted and the linear performance will have some degradation. If $n_l$ is large, for example $n/2$, only a small amount of sequences will be deleted and the nonlinear performance will not have much improvement. Given $n_l$, the value of $V_{\text {thr}}$ will affect $\text {Var}[E]$ of the output sequences from BS-SS-NLI. In Section 5.3, the relationship between the nonlinear performance and $\text {Var}[E]$ and $V_{\text {thr}}$ will be shown. An example of the probability of symbol sequences for BS-SS, BS-SS-NLI, and ESS with $\mathcal {A}=\{1,3,5,7\}$, $k=10$, and $n=8$ is shown in the Fig. 2. As shown in Fig. 2, BS-SS can achieve the lowest average symbol sequence energy and BS-SS-NLI can achieve the lowest variance of symbol sequence energy.
Fig. 2. The probability of symbol sequences with increasing energy selected by BS-SS, BS-SS-NLI, and ESS for $\mathcal {A}=\{1,3,5,7\}$, $k=10$, and $n=8$. Symbol sequence energy is defined as $S = \sum _{i=1}^{n} \lvert X_{i}\rvert ^{2} = \sum _{i=1}^{n} a_{I,i}^{2}+a_{Q,i}^{2}$. The inner figure shows the probability of each amplitude and the inner table shows $\mathbb {E}[S]$ and $\text {Var}[S]$ of all schemes.
4. Other nonlinearity-tolerant shaping algorithms
4.1 Kurtosis-limited ESS
Kurtosis-limited ESS (K-ESS) is a nonlinear-tolerant ESS by limiting the kurtosis of its outputs. The shaping set of K-ESS is defined as [14]:
4.2 List-encoding CCDM
List-encoding CCDM (L-CCDM) can improve the effective SNR of transmission over CCDM by using a recently proposed metric EDI. Given a sequence of symbols $[\dots,X_{i-1},X_{i},X_{i+1},\dots ]$ and a window size $W$, EDI can be calculated as:
where and $G_{i}^{W} = \sum _{j=i-W/2}^{i+W/2}|X_{j}|^{2}$ denoting the sum of $W + 1$ symbol energies centered around the symbol $X_{i}$.In L-CCDM, an EDI selector is added to the PAS transmitter after the CCDM encoder. The best sequence with the lowest EDI is selected within $2^{2v}$ number of sequences, which leads to an extra rate loss of $v/n$ bits/2D-symbol, where $n$ is the blocklength. The optimal window size depends on the transmission parameters.
5. Simulations
5.1 AWGN Channel
Firstly, the performance of BS-SS and BS-SS-NLI is validated in an additive white Gaussian noise (AWGN) channel by Monte Carlo simulations. We measure the achievable information rate (AIR) as the performance metric which can be calculated as [23]
where $X$ and $Y$ are the transmitted and received symbols respectively, $B_{i}$ represents the $i$ th bit of the binary label, $m$ is the bit length of the binary label, $H(\cdot )$ represents the entropy, and $R_{\text {loss}}$ denotes the rate loss between the entropy and spectral efficiency (SE) of the transmitted symbols, which has a significant impact for higher order modulation formats and short shaping blocklength.The shaping rate ($R_{s} = k/n$), 1.25 bits/amp, of 64 QAM BS-SS and BS-SS-NLI with $n=32$ are illustrated in this paper. The result is shown in Fig. 3 where the performance is compared to CCDM with $n=1024$ and $n=256$ and ESS with $n=32$. As shown in Fig. 3, BS-SS can achieve a $\sim$0.2 dB SNR gain over ESS but the performance is still not better than long blocklength CCDM ($n=1024$) which has a lower rate loss and can provide near-optimal shaping gain in AWGN channels [2].
The rate loss of BS-SS, ESS, and CCDM are investigated with different $R_{s}$ and blocklength in Fig. 4. With same short blocklength as BS-SS, CCDM has significantly larger rate loss. The rate loss gain from BS-SS over ESS varies with $R_{s}$. The largest gain achieved is 0.0527 bits/amp at $R_{s}=0.9375$ bits/amp and 0.0285 bits/amp at $R_{s}=1.125$ bits/amp for $n=16$ and $n=32$ respectively. In Fig. 5, the relationship between the rate loss, $\text {Var}[E]$ and $V_{\text {thr}}$ of BS-SS-NLI with $n=32$ and $n_l=4$ is investigated. Since the expression of $V$ in Eq. (9) is derived from the formula of $\text {Var}[E]$ in Eq. (8), a larger $V_{\text {thr}}$ allows the BS-SS-NLI select sequences with larger $\text {Var}[E]$, as shown in Fig. 5(a). Equivalently, the rate loss of BS-SS-NLI decreases as $V_{\text {thr}}$ increases, as fewer sequences are removed which is demonstrated in Fig. 5(b). Given an infinite $V_{\text {thr}}$, BS-SS-NLI has the same rate loss as BS-SS. A larger $V_{\text {thr}}$ is required to achieve a rate loss for a larger shaping rate. For each shaping rate, there will be a minimum $V_{\text {thr}}$ to ensure the one-to-one mapping. The impact of $V_{\text {thr}}$ and $\text {Var}[E]$ on nonlinear performance of BS-SS-NLI will be discussed in Section 5.3.
Fig. 5. (a) $\text {Var}[E]$ vs. $V_{\text {thr}}$ and (b) Rate loss vs. $V_{\text {thr}}$ for BS-SS-NLI with $n=32$ and $n_l=4$.
5.2 Digital subcarrier multiplexing
Symbol rate has been demonstrated by numerical simulations to have significant influences on the nonlinear performance of optical communication system in [24,25]. Therefore, the nonlinear performance can be improved by using multiple multiplexed low symbol rate subcarrier signals to replace a high symbol rate single carrier signal. To reduce the increasing number of lasers and modulators for each subcarrier, digital subcarrier multiplexing can be applied. The baud rate is different for different number of subcarriers when considering the central frequency in the coherent detection scheme. The roll-off factor and frequency spacing are chosen to prevent overlap of the spectrum of all subcarriers. In [10], the number of sub-carriers should ideally be powers of 2 and small subcarrier symbol rates should be avoided for reducing implementation complexity. As the short blocklength effect is not significant in long-haul transmission at a high symbol rate [17], the nonlinear performance of sphere shaping with short blocklength will be more significant in the SCM system. In the following work, we will investigate the performances of long-haul fiber transmission in the SCM scheme.
5.3 Nonlinear fiber channel
Secondly, we evaluate the performance of BS-SS and BS-SS-NLI in a nonlinear fiber simulation. The simulation setup is shown in the Fig. 6. It contains three parts: transmitter digital signal processing (DSP), fiber transmission, and receiver DSP. At the receiver, an ideal chromatic dispersion (CD) compensation and matched filtering is used. The carrier phase recovery is adopted by a single phase rotation. The simulation parameter is shown in Table 1.
Table 1. Simulation parameters
The optimal number of subcarriers and blocklength for different amplitude shapers based on the parameters shown in Table 1 and a total distance of 2500 km (25 spans) is demonstrated. Firstly, the result in Fig. 7(a) shows that the optimum blocklengths are $n=64$, 32, 64, 256 for BS-SS, BS-SS-NLI with optimized $V_{\text {thr}}$, ESS, and CCDM respectively with $R_s=1.25$ bits/amp. To make a fair comparison of BS-SS and BS-SS-NLI to ESS, the same blocklength needs to be selected. In this paper, we choose $n=32$ for BS-SS, BS-SS-NLI, and ESS. Other than sphere shaping, CCDM has only one composition. Therefore, we choose the optimum blocklength $n=256$ for CCDM to illustrate the potential of BS-SS and BS-SS-NLI to improve the simulation performance. Then, the result in Fig. 7(b) shows that the optimum number of subcarriers of BS-SS and BS-SS-NLI is 16 and 8 for ESS and CCDM. As mentioned in Section 5.2, the symbol rate has a significant effect on the fiber nonlinearity. For a fair comparison of the performance, the same number of subcarriers should be selected for all PAS shapers. In the following simulations, 16 subcarriers will be used.
The nonlinearity tolerance of BS-SS-NLI is first demonstrated. Using the same shaping rates, $R_s=0.875$, 1.25, and 1.5 bits/amp and $V_{\text {thr}}$ for each shaping rate as the one used in Fig. 5, the effective SNR after the transmission for each case is shown in Fig. 8. In Fig. 8(a), the result shows that if we reduce $V_{\text {thr}}$, the nonlinear performance of BS-SS-NLI can be improved. Correspondingly, the effective SNR decreases as $\text {Var}[E]$ increases as shown in Fig. 8(b). Therefore, there is a trade-off between rate loss and nonlinearity tolerance which needs to be considered when optimizing $V_{\text {thr}}$.
Fig. 8. Effective SNR vs. (a) $V_{\text {thr}}$ and (b) $\text {Var}[E]$ for BS-SS-NLI with $n=32$ and $n_l=4$.
Parameters in the recently proposed nonlinearity-tolerant shaping approaches are optimized under the given simulation parameters. $E_{\text {max}}$ and $K_{\text {max}}$ of K-ESS are 1440 and 2864 respectively with $n = 32$. In L-CCDM, optimal values of $W$ and $v$ are 10 and 2 with $n=256$. Then, the simulation of 2500 km and 16 subcarriers with a range of total launch power between 2 to 7 dBm is run and the result is provided in Fig. 9. The result in Fig. 9(a) shows that all nonlinearity-tolerant shaping approaches, BS-SS-NLI, L-CCDM, and K-ESS, can provide effective SNR gain over their conventional counterparts. BS-SS-NLI can achieve the highest effective SNR with a $\sim$0.59 dB and $\sim$0.28 dB gain over CCDM and ESS and a $\sim$0.39 dB and $\sim$0.11 dB gain over L-CCDM and K-ESS showing a better nonlinearity tolerance. There is also a small effective SNR gain from BS-SS over ESS which is caused by the sorting of sub-composition pairs on energy difference discussed in Section 2. In Fig. 9(b), BS-SS-NLI can achieve the highest AIR and provide AIR gain of $\sim$0.1 bits/4D-sym over ESS and $\sim$0.045 bits/4D-sym over K-ESS. The AIR improvement of BS-SS-NLI is relatively small since there is an AWGN performance degradation. The AIR gain from BS-SS over ESS is caused by the effective SNR gain discussed above and the selection of all preferential sequences. Even CCDM and L-CCDM can only achieve a relatively low effective SNR, it can provide closing AIR to short blocklength sphere shaping schemes due to their low rate loss at a longer blocklength. To further investigate the effective SNR gain from nonlinearity-tolerant PAS shapers, we listed the EDI of all schemes with optimized window size $W=20$ in Table 2. The smallest EDI achieved is by BS-SS-NLI which can also demonstrate that BS-SS-NLI can mitigate fiber nonlinearity.
Fig. 9. Simulation results of nonlinear fiber channel: (a) effective SNR vs. launch power, (b) AIR vs. launch power. The effective SNR gain of BS-SS-NLI over ESS in an extended 205 km transmission with 50 GBd symbol rate is $\sim$0.8 dB.
Table 2. Effective SNR and EDI of different PAS algorithms with $W=20$
In WDM system with SCM for each channel, these nonlinearity-tolerant shaping algorithms are investigated by a 5-channel, 1500 km simulation with parameters shown in Table 1 and 100 GHz channel spacing. The result is shown in Fig. 10. Figure 10(a) shows that BS-SS-NLI can still achieve the highest effective SNR, but the gain over K-ESS is small. The result in Fig. 10(b) shows that L-CCDM can achieve the highest AIR in multi-channel transmission. The fewer improvements on effective SNR and AIR by BS-SS-NLI and K-ESS are limited by a small channel spacing, i.e., channels are very close to each other so that there is a large inter-channel NLI from other channels which can be inferred by the approximate analytical GN-type model. As both BS-SS-NLI and K-ESS can be considered as deleting the sequences which contribute more to NLI, the AIR degradation is more obvious when NLI is not well mitigated. L-CCDM performs better in multi-channel SCM system as it has a low rate loss compared to other nonlinearity tolerant shaping algorithms, BS-SS-NLI and K-ESS.
Fig. 10. Simulation results of nonlinear fiber channel: (a) effective SNR vs. launch power, (b) AIR vs. launch power
6. Conclusion
In this paper, we propose two shaping techniques: BS-SS and its nonlinearity-tolerant scheme, BS-SS-NLI, by reducing the energy variation of the transmitted symbols. In the AWGN channel, BS-SS can achieve a better result than ESS with the same blocklength, and the gain will be varying as the shaping rate changes. BS-SS-NLI introduces a linear performance degradation to BS-SS as some sequences are deleted. In nonlinear fiber channel, short blocklength BS-SS-NLI can achieve the highest effective SNR and AIR in single channel long-hual transmission compared to the same blocklength ESS, K-ESS and longer blocklength CCDM, L-CCDM in SCM system. The performance of BS-SS-NLI in 5-channel WDM system is limited by the small channel spacing which also implies that BS-SS-NLI can effectively mitigate intra-channel NLI but not inter-channel NLI.
Acknowledgments
The authors thank BT and Huawei for financial support and useful discussions.
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. G. Böcherer, 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]
2. P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016). [CrossRef]
3. G. Böcherer, P. Schulte, and F. Steiner, “High throughput probabilistic shaping with product distribution matching,” arXiv, arXiv:1702.07510 (2017). [CrossRef]
4. A. Amari, S. Goossens, Y. C. Gültekin, O. Vassilieva, I. Kim, T. Ikeuchi, C. M. Okonkwo, F. M. J. Willems, and A. Alvarado, “Introducing enumerative sphere shaping for optical communication systems with short blocklengths,” J. Lightwave Technol. 37(23), 5926–5936 (2019). [CrossRef]
5. T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, K. Parsons, and H. Griesser, “Huffman-coded sphere shaping and distribution matching algorithms via lookup tables,” J. Lightwave Technol. 38(10), 2826–2834 (2020). [CrossRef]
6. P. Skvortcov, I. Phillips, W. Forysiak, T. Koike-Akino, K. Kojima, K. Parsons, and D. S. Millar, “Nonlinearity tolerant lut-based probabilistic shaping for extended-reach single-span links,” IEEE Photonics Technol. Lett. 32(16), 967–970 (2020). [CrossRef]
7. M. Fu, Q. Liu, Y. Xu, H. Jiang, X. Zeng, Y. Wu, L. Yi, W. Hu, and Q. Zhuge, “Multi-dimensional distribution matching with bit-level shaping for probabilistically shaped high order modulation formats,” J. Lightwave Technol. 40(9), 2870–2879 (2022). [CrossRef]
8. Y. Chen, J. Chen, W. Li, M. Zhang, D. Liu, and M. Tang, “On optimization and analysis of enumerative sphere shaping for short blocklengths,” Journal of Lightwave Technology pp. 1–14 (2022).
9. M. Qiu, Q. Zhuge, M. Chagnon, Y. Gao, X. Xu, M. Morsy-Osman, and D. V. Plant, “Digital subcarrier multiplexing for fiber nonlinearity mitigation in coherent optical communication systems,” Opt. Express 22(15), 18770–18777 (2014). [CrossRef]
10. H. Sun, M. Torbatian, M. Karimi, R. Maher, S. Thomson, M. Tehrani, Y. Gao, A. Kumpera, G. Soliman, A. Kakkar, M. Osman, Z. A. El-Sahn, C. Doggart, W. Hou, S. Sutarwala, Y. Wu, M. R. Chitgarha, V. Lal, H.-S. Tsai, S. Corzine, J. Zhang, J. Osenbach, S. Buggaveeti, Z. Morbi, M. I. Olmedo, I. Leung, X. Xu, P. Samra, V. Dominic, S. Sanders, M. Ziari, A. Napoli, B. Spinnler, K.-T. Wu, and P. Kandappan, “800g dsp asic design using probabilistic shaping and digital sub-carrier multiplexing,” J. Lightwave Technol. 38(17), 4744–4756 (2020). [CrossRef]
11. P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The gn-model of fiber non-linear propagation and its applications,” J. Lightwave Technol. 32(4), 694–721 (2014). [CrossRef]
12. A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “EGN model of non-linear fiber propagation,” Opt. Express 22(13), 16335–16362 (2014). [CrossRef]
13. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 22(12), 14199–14211 (2014). [CrossRef]
14. Y. C. Gültekin, A. Alvarado, O. Vassilieva, I. Kim, P. Palacharla, C. M. Okonkwo, and F. M. J. Willems, “Kurtosis-limited sphere shaping for nonlinear interference noise reduction in optical channels,” J. Lightwave Technol. 40(1), 101–112 (2022). [CrossRef]
15. T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, K. Parsons, and H. Griesser, “Analysis of nonlinear fiber interactions for finite-length constant-composition sequences,” J. Lightwave Technol. 38(2), 457–465 (2020). [CrossRef]
16. K. Wu, G. Liga, A. Sheikh, F. M. J. Willems, and A. Alvarado, “Temporal energy analysis of symbol sequences for fiber nonlinear interference modelling via energy dispersion index,” J. Lightwave Technol. 39(18), 5766–5782 (2021). [CrossRef]
17. J. Cho and R. Tkach, “On the kurtosis of modulation formats for characterizing the nonlinear fiber propagation,” J. Lightwave Technol. 40(12), 3739–3748 (2022). [CrossRef]
18. J. Cho, X. Chen, G. Raybon, D. Che, E. Burrows, S. Olsson, and R. Tkach, “Shaping lightwaves in time and frequency for optical fiber communication,” Nat. Commun. 13(1), 785 (2022). [CrossRef]
19. K. Wu, G. Liga, A. Sheikh, Y. C. Gültekin, F. M. J. Willems, and A. Alvarado, “List-encoding CCDM: A nonlinearity-tolerant shaper aided by energy dispersion index,” J. Lightwave Technol. 40(4), 1064–1071 (2022). [CrossRef]
20. A. Khandani and P. Kabal, “Shaping multidimensional signal spaces. i. optimum shaping, shell mapping,” IEEE Trans. Inf. Theory 39(6), 1799–1808 (1993). [CrossRef]
21. M. Fu, Q. Liu, H. Lun, H. Jiang, Y. Wu, X. Liu, Z. Yang, L. Yi, W. Hu, and Q. Zhuge, “Parallel bisection-based distribution matching for nonlinearity-tolerant probabilistic shaping in coherent optical communication systems,” J. Lightwave Technol. 39(20), 6459–6469 (2021). [CrossRef]
22. Y. C. Gültekin, W. J. van Houtum, A. G. C. Koppelaar, and F. M. J. Willems, “Comparison and optimization of enumerative coding techniques for amplitude shaping,” IEEE Commun. Lett. 25(4), 1231–1235 (2021). [CrossRef]
23. 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]
24. L. B. Du and A. J. Lowery, “Optimizing the subcarrier granularity of coherent optical communications systems,” Opt. Express 19(9), 8079–8084 (2011). [CrossRef]
25. A. Bononi, N. Rossi, and P. Serena, “Performance dependence on channel baud-rate of coherent single-carrier wdm systems,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), (2013), paper Th.1.D.5.
