Open Access
Add to BibSonomy
Abstract
We present and validate a statistical method able to separate nonlinear interference noise (NLIN) into a residual Gaussian (ResN) and a phase noise (NLPN) component. We take into account the interaction of the NLIN with the receiver’s DSP, mainly through carrier phase recovery (CPR), by considering the amount of correlation of the NLPN component. This allows obtaining in a straightforward way an accurate prediction of the achievable post-DSP transmission performance. We apply our method on simulated data in different scenarios. For this purpose: (i) several different quadrature amplitude modulation (QAM) and probabilistically shaped (PS) formats are investigated and (ii) simulations with standard single mode fiber (SSMF) and dispersion shifted fiber (DSF) are performed. In all these cases we validate the results provided by our method through comparison with ideal data-aided CPR and a more practical blind phase search (BPS) algorithm. The results obtained are finally compared with the predictions of existing theoretical models and the differences with our approach are pointed out.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In modern wavelength division multiplexed (WDM) systems, the nonlinear Kerr effect is known to be the limiting factor for the achievable transmission capacity [1]. This phenomenon has been proven to be modulation format-dependent; significant research activity has been dedicated to analyze its impact on the system performance, both theoretically [2,3] and experimentally [4,5]. In the context of modulation formats it is nowadays well established how shaped constellations make it possible to achieve better sensitivity and reach compared to unshaped ones [6,7]. While this has been proven in the linear channel assumption, where amplified spontaneous emission (ASE) noise dominates, the performance in the nonlinear regime at a system level is not taken for granted.
For this purpose we present a novel analysis method applied on simulated signals, through which we are able to provide a deeper insight about the nature of nonlinear interference noise (NLIN) and in particular of its interaction with the DSP at the receiver. Our method separates NLIN into a residual Gaussian (ResN) and a nonlinear phase noise (NLPN) component, regardless of the carrier phase recovery (CPR) algorithms employed in a specific DSP chain.
The reason for naming the Gaussian component as residual is that in several transmission scenarios NLPN presents long temporal correlation and it has been demonstrated how CPR implemented in coherent receivers is able to mitigate it. The nonlinearity remaining on the post-DSP constellation is in this case ResN only. Such a condition has been predicted and observed in common non-dispersion-compensated links [8–10]. In this case the information about the total induced NLIN power is lost after coherent demodulation. Theoretical models on the other hand predict that (i) NLIN power variation as a function of format mainly concerns NLPN, and (ii) that shaped constellations are predicted to constitute a worst case in this sense, due to higher modulation dependent factors [2]. This makes probabilistically shaped (PS) constellations performance estimation an interesting scenario to apply our correlation-independent method and to compare the results with these theoretical predictions. In particular we compare quantitatively our method with the enhanced Gaussian noise model (EGN) [2,3] and point out the differences between our analysis and the approach in [11] based on the time-domain pulse-collision theory [12].
The ResN obtained with our method is compared to the results obtained by applying ideal data-aided CPR (as in the current approach of [13]) and a more realistically implementable feed-forward blind phase search (BPS) algorithm [14] (as in [5,15]). This operation is performed in order to validate our method and to show its applicability in realistic conditions. Compared to this conventional approach, our method provides results which are not influenced by the CPR implementation. Furthermore it has the advantage of not requiring parameter tuning (e.g. CPR filter optimization), and of being independent of the temporal correlation of the NLPN. This allows for it to be used as a tool for predicting the limitations of a specific CPR implementation in any transmission scenario.
2. Analyzed system scenarios
The analysis is performed on signals simulated using VPIphotonics Design Suite 10.0. Five 32 Gbaud dual polarization multiplexed channels, with a channel spacing of 32 GHz, form a Nyquist WDM superchannel. A raised cosine filter with 0 roll-off is used to perform pulse shaping in the transmitters, which generate sequences of random symbols modulated either as standard quadrature amplitude modulated (QAM) signals or probabilistically shaped following a Maxwell-Boltzmann probability mass function (PMF). Propagation of the resulting optical signal through an 80 km span of standard single mode fiber (SSMF) is then simulated through the solution of the Manakov equation. This fiber is characterized, at $\lambda _{c} = 1550 \textrm { nm}$, by attenuation coefficient $\alpha = 0.2 \textrm { dB/km}$, dispersion $D = 16.7 \textrm { ps/(nm} \cdot \textrm {km)}$, dispersion slope $S = 80 \textrm { s}/\textrm {m}^3$, effective nonlinear coefficient $\gamma = 1.3 \textrm { W}^{-1} \textrm {km}^{-1}$ and effective area $A_{eff}=80 \;\mu \textrm {m}^2$. Another set of simulations are performed considering a dispersion shifted fiber (DSF) fiber at the same wavelength; in this case the dispersion parameter is modified to $D = 0.08 \textrm { ps/(nm} \cdot \textrm {km)}$ while the other values are kept fixed. This scenario is simulated in order to verify the impact of dispersion-induced pulse broadening [12] on NLIN. After every fiber span lumped amplification is applied. Throughout this work the amplifier is modeled either as noiseless or including a noise figure (NF) parameter to take into account ASE generation at the amplification stages. Polarization mode dispersion (PMD) has been shown to have a negligible impact on NLIN generation and it is not taken into account [16]. After fiber propagation, ideal chromatic dispersion compensation (CDC) is performed and a matched filter is used to separate the channel under test (CUT) at the center of the WDM comb, from the other four interfering channels (INTs). This ideal compensation and the absence of PMD are necessary in order to observe a constellation impaired by NLIN only, which is the quantity that we aim to analyze by means of our method. Although PMD is not introduced in our simulation, in realistic polarization multiplexed systems a time-domain equalizer (TDE) is part of the DSP chain at the receiver. In the time-domain pulse-collision theory it has been shown how NLIN introduces polarization rotation noise (PolRotN) and equalization is expected to interact with this component [17]. However, in our simulation scenario, the application of a TDE did not show significant NLIN mitigation at that stage and therefore was not included. The effect of linear adaptive equalization on our analysis, in which NLIN is separated only in a Gaussian and a NLPN component, appears then to be negligible. A block diagram of the simulation setup is presented in Fig. 1.
3. Analysis method description
We now describe our analysis method. Section 3.1 provides information on the application of the method on the simulated impaired constellations. In sections 3.2 to 3.4 the main steps of the analysis are detailed. Finally, in Section 3.5 the error introduced by the application of our statistical method is estimated.
3.1 Application of the method
Our method is applied on a received constellation impaired by NLIN and eventually ASE noise. Extracting the Rician-distributed magnitude of the received symbols, it is possible to ideally remove the phase information. Applying on this set of data the Method of Moments (MoM), it is possible to estimate the signal to noise ratio (SNR) of the constellation impaired by the Gaussian components resulting from ResN and ASE only [18,19]. To estimate instead the total noise power given by the combination of ASE, NLPN and ResN, the Euclidean distance of the received constellation points to the ideal ones is considered. We refer to this operation as the EVM method. When both total NLIN and ResN have been estimated, the NLPN is finally obtained as the difference of the two. We apply this method on the results of the simulation setup in Fig. 1, where the full WDM comb is propagated, and on single-channel simulations with the INTs turned off. In this way we are able to isolate single-channel interference (SCI) from the total NLIN which includes also cross- and multi-channel interference (XMCI). The steps of the analysis just described are graphically represented by Fig. 2. In sections 3.2 to 3.4 the single steps are separately presented in detail.
3.2 Method of moments
To estimate the Gaussian component of the noise power, which includes ASE noise and ResN, the M2M4 SNR estimation method [20] is implemented. As the name suggests, this operation requires the calculation of the second and fourth order moments of the received symbols’ magnitude distribution. The estimator can be applied separately on all the amplitude rings of a generic QAM constellation. In this case care must be taken in appropriately normalizing the amplitude values to the signal’s average energy, yielding Eq. (1). Here $P_i$ represents the probability of a transmitted symbol belonging to the $i^{th}$ ring, $M_{2,i}$ and $M_{4,i}$ are the second and fourth order moments evaluated over the elements of the $i^{th}$ ring, and $A_i$ is the respective amplitude at the transmitter. The ResN + ASE power is then simply calculated as $P_{ch}/\textrm {SNR}$, where $P_{ch}$ is the channel power.
Where $N_i$ is the number of symbols belonging to the $i^{th}$ ring.
Fig. 3. (left) Received 64-QAM-PS constellation with target entropy of 5 bits/symbol, impaired only by NLIN, after transmission over 20 fiber spans. (right) Corresponding amplitude distributions for each ring of the constellation. $2^{16}$ transmitted symbols were simulated in this example.
3.3 EVM method for noise power estimation
In order to estimate the amount of noise present on a constellation impaired by both NLPN and Gaussian noise, the variance of the Euclidean distances of the impaired points to the transmitted one is properly normalized as shown in Eq. (3). Here $Y_{RX}$ and $Y_{TX}$ are respectively the received and transmitted complex signals on the IQ plane.
This method introduces an additive error when a phase noise component is present, in which case a more accurate way of estimating the noise power would be to evaluate the arc distance from the ideal point, not the Euclidean one. Nonetheless in all the conditions analyzed throughout this work, this error has not been limiting the applicability of the method. A deeper analysis about its magnitude is provided in Section 3.5.3.4 CPR implementations and CPR-based NLPN estimation
Two different CPR DSP algorithms have been implemented for this work: the first is a maximum-likelihood fully data-aided algorithm (referred to as DA-ML); the second CPR realization applied is instead a more realistic single stage feed-forward BPS CPR [14]. For this implementation, 64 test phases are used. DA-ML is based on a moving average filter, while BPS is block averaged. In both cases the filter length is optimized for every analyzed signal. The optimization is performed by minimization of the Pearson correlation coefficient calculated between the I and Q samples of the compensated constellation. A smaller absolute value of this coefficient indicates that the received symbols are closer to a circular Gaussian distribution in the I-Q plane.
The results obtained through DA-ML CPR are used as a reference to check the estimation provided by the MoM and represent a best case scenario to understand the upper limit regarding NLPN mitigation through CPR. In contrast, the analysis performed with BPS aims at comparing the results obtained with the MoM with a realistic algorithm commonly found in commercially available coherent receivers to explore practical performance estimation potentialities.
Once the chosen CPR is applied, the information about the estimated phase error for each received symbol is obtained. The NLPN power is then calculated from (4). Here $\langle \cdot \rangle$ denotes time averaging and $\theta _e$ is the sample by sample phase error estimated by the CPR.
3.5 Error analysis of EVM method and MoM
The EVM method and the MoM are employed to calculate the noise components through statistical analysis of discrete complex signals with a finite number of samples; the direct consequence is that a statistical inaccuracy introduced by the two methods must be taken into account. Furthermore, it has been pointed out in Section 3.3 how the EVM method is exact only when the noise analyzed is purely Gaussian. Before applying our methodology we need to understand the conditions in which it is sufficiently accurate; for this purpose we study the estimation error introduced by the presence of phase noise in the impaired constellations and the statistical inaccuracy of the MoM and of the EVM method.
In the tests performed in this section an ideal QPSK constellation is impaired with Gaussian and/or phase noise of known powers. Ten thousand different noise realizations are tested in each simulation. In the second and third columns of Table 1(a) the average values and standard deviations of the relative errors obtained when estimating the power of a purely Gaussian noise through the MoM with different number of symbols are reported. The same constellations are employed for testing in the same way the statistical error of the EVM method. The analysis of the error caused by this method can be observed in the last two columns of the same table. As expected, for both operations the error decreases with the number of symbols considered. The MoM generally introduces a higher inaccuracy but always of the same order of magnitude as the EVM method. These results are obtained setting SNR=10 dB and did not vary considerably when lower noise powers were tested; this behavior is evidence of the statistical nature of the error. Only when the noise power is comparable to the signal the inaccuracy introduced by the MoM proved to increase consistently. In this limiting case care must be taken to consider a sufficient number of symbols. During our tests, using $2^{16}$ samples provided an acceptable error, characterized by an average value of 1.67% and a standard deviation of 1.27% for SNR=0 dB.
Table 1. (a): Maximum statistical error observed for MoM and EVM method. SNR=10dB. (b): EVM method error when applied to pure phase noise. $2^{16}$ symbols have been simulated.
Finally in Table 1(b) the error introduced by the EVM method is shown, when the impairment is purely Gaussian-distributed phase noise. This situation is taken into account since, although not realistic, it represents a worst case for the EVM method; $2^{16}$ symbols have been used for this simulation. The maximum error observed is higher than the one obtained in the Gaussian noise case, but it is still acceptable for our purposes. The main difference is instead represented by the evident SNR dependence of the error of the EVM method, in particular for low SNR values.
When looking at these tables it must be emphasized that this behavior is a pure mathematical construction to show that phase noise does not considerably affect our method at any of the considered operating conditions. During our work situations when NLPN is much larger than the Gaussian component have never been observed; therefore, no noticeable error is found in the results. The statistical error of Table 1(a) thus provides the uncertainty observed in our analysis.
Two conclusions can be drawn from this analysis: (i) At least $2^{14}$ symbols must be taken into account to avoid increasing excessively the statistical uncertainty related to the use of the MoM and EVM method. This guideline has been followed in all the simulations performed in the following sections. (ii) The presence of NLPN does not impact significantly the results provided by the EVM method until very strong levels of phase noise are present. A situation which has never been observed when simulating the nonlinear channel’s impairments of interest for this work.
4. Simulation results
4.1 Validation of MoM based ResN estimation
To verify the results provided by the MoM, a set of simulations for different channel powers are performed. The WDM comb is placed around $\lambda _c$ and a noise figure $\textrm {NF} = 6 \textrm { dB}$ is set for the inline amplifiers. A probabilistically shaped 64-QAM constellation with target entropy equal to 4 is chosen for the simulation (64-QAM-PS4). This is chosen as a reference constellation, since it represents a worst case of the set of examined formats for our realistic CPR implementation. Penalties when applying BPS to shaped constellations have been reported in [21].
In Fig. 4(a) it is possible to observe how the results provided by the DA-ML CPR method and the MoM perfectly agree in a wide range of $P_{ch}$ values and transmission distances. The only conditions in which a difference is observed is either for $P_{ch}<-16 \textrm { dBm}$ or for $P_{ch} >4 \textrm { dBm}$ after 20 fiber spans. In these operation points, limited respectively by ASE and NLIN, in which the ResN power is comparable to the signal power, the best case CPR fails. The MoM, in contrast, still provides accurate results, following the expected SNR linear slope for low channel power. With this, the MoM has not only been validated, but we also demonstrated that it can be exploited as a performance estimator for any CPR realization.
Fig. 4. (a) SNR estimated with the MoM and with the EVM method after DA-ML CPR on a 64-QAM-PS4 constellation with $P_{ch}$ ranging from -20 to 10 dBm. (b) SNR for $P_{ch} = 0$ dBm and difference between the MoM and the results obtained applying DA-ML and BPS CPR.
From Fig. 4(a) it is also possible to conclude that $P_{ch} = 0 \textrm { dBm}$ represents a reasonable working point for the purpose of optimizing the system SNR over the variable number of spans considered. This channel power is set in the simulation to obtain the results in Fig. 4(b), where the agreement between both BPS and DA-ML CPR-based, and MoM-based methods is shown in detail.
Here we can see how the agreement of the MoM with DA-ML CPR slightly degrades with decreasing SNR. Nevertheless, the deviation between the results is limited to 0.2 dB after 20 spans, a value comparable to the intrinsic statistical error of the method. The BPS-based results, on the contrary, show a maximum inconsistency of 0.4 dB after the first span. This is due to the fact that at this transmission distance the correlation of the NLPN is still insufficient to achieve complete mitigation with our realistic CPR. When the number of spans is increased further, dispersion-induced pulse broadening takes place and stronger correlation is introduced. In this case we observe how the BPS performance approaches the one of the data-aided algorithm.
The results demonstrate how our correlation-independent MoM is always able to isolate the Gaussian component of the NLIN. This estimated quantity is shown to correspond to the achievable NLIN that ultimately impacts system performance, with ideal data-aided CPR, in meaningful working points. Moreover, in the dispersion-unmanaged link, the agreement with the results obtained using a realistic CPR algorithm is also remarkable. This demonstrates the potentialities of the MoM as an accurate system performance estimator in several transmission scenarios.
4.2 Modulation format dependence of NLIN induced system performance penalty
A scenario in which our method is able to provide interesting insights is the modulation format dependence of NLIN, and in particular its impact on the post-DSP achievable performance. The NLIN dependence on the modulation format has been theoretically predicted and the results have been corroborated with simulations and experiments, also when employing shaped constellations [15,22,23]. Nonetheless, to the best of our knowledge these results have always been obtained without taking into account the interaction of NLIN with DSP, as in the approach used for the EGN method [3], or by application of a specific DSP realization as in [11]. We then propose our method not only as a tool to provide a straightforward and accurate prediction of the post-DSP transmission performance, but also to validate these theoretical predictions from a different perspective.
For the simulations in this section we set once again $P_{ch} = 0 \textrm { dBm}$. First we compare the predictions of the EGN model to the results obtained through application of our method on the simulated signals; this comparison is shown for different modulation formats in Fig. 5, where the noise powers plotted are normalized by $P_{ch}^3$, yielding values which are independent of the channel power, since nonlinear noise has a cubic dependence on this parameter.
As can be seen, the EGN model provides an accurate prediction of the total NLIN introduced, but by treating it as a single Gaussian component it does not distinguish between ResN and NLPN, and thus does not take into account the mitigation of the correlated NLPN by means of CPR. The final result, as visible in Fig. 5, is that the EGN model provides an overestimation of the system performance penalty induced by NLIN which is highest for PS modulation, since in this case NLPN is considerably stronger. We can observe this better by looking at the simulation results shown in Fig. 6, where we analyzed with our MoM approach several standard and shaped QAM formats.
Fig. 6. Normalized NLIN power versus number of fiber spans for QPSK, 16-QAM, 32-QAM, 64-QAM and 64-QAM-PSX, where X is the transmitted constellation entropy. (a) shows the total NLIN, (b) the NLPN and (c) the ResN.
Figure 6(a) suggests once again that the use of PS constellations introduces a severe increase of the NLIN power. However, as expected [2,8], the main reason for such behavior is a large increase in NLPN, as revealed by Fig. 6(b). The ResN plots in Fig. 6(c), on the other hand, show only a small variation of less than 0.5 dB for all formats after 5 spans. If we assume that NLPN can be mitigated in normal operating conditions by CPR and thus the performance is limited by the ResN, the simulation shows that PS formats do not suffer any significant penalty compared to unshaped M-QAM in the nonlinear fiber channel for medium- to long-haul transmission.
These results have been obtained thanks to the possibility given by our method of separating NLIN into two components. While this is a substantial difference with respect to the EGN model, the analytical approach in [11] is able to account for the correlation properties of NLIN and for its deviation from Gaussianity. Nevertheless, the analytical estimation is performed for XMCI only. To obtain the total SCI+XMCI residual noise after mitigation, as in our approach, a specific equalizer realization must still be introduced.
Because of this difference, it is useful to understand the impact of SCI mitigation in our analyzed scenario. For this purpose we perform single-channel simulations and compare the different NLIN components with respect to the full WDM simulation in Fig. 7. The outcome of this analysis is that for high-order modulations the NLPN generated by single-channel effects is a significant component already from the first spans, constituting between 35 and 45 % of the total NLPN. It is only for QPSK and short transmission distances that slightly lower values are observed. Furthermore, it is important to remark that these results have been obtained in a superchannel scenario, where the relative impact of XMCI is highest because of the minimized channel spacing. This analysis suggests then that a purely analytical estimation of the NLIN mitigation through the model developed in [11] would still lead to an overestimation of the ResN after CPR; at least in scenarios where the NLPN is almost perfectly compensated by the receiver DSP.
Fig. 7. Fraction of the NLPN generated by SCI versus the number of fiber spans simulated for different modulation formats.
4.3 NLIN mitigation dependence on dispersion-induced pulse broadening
The last scenario in which we apply our method is when transmission takes place over a DSF. In this case we still position our WDM superchannel around $\lambda _{c} = 1550 \textrm { nm}$ but we set $D = 0.08 \textrm { ps/(nm} \cdot \textrm {km)}$. Moreover $P_{ch}$ is lowered to -10 dBm because of the excessive NLIN introduced in these conditions for a launch power of 0 dBm. 16-QAM is chosen as the modulation format.
This scenario is considered since it has been demonstrated how in situations in which dispersion-induced pulse broadening has a reduced impact on the pulse shape, CPR is not able to mitigate NLPN [15,23]. This is due to the absence of sufficient temporal correlation in the received signal. In this context it is important to remark that the relative pulse temporal overlap after a given transmission distance is not exclusively determined by the dispersion coefficient, but is also dependent on the symbol rate and the channel spacing. Ultimately what allows NLPN mitigation is the correlation induced by the interaction among different symbols, as extensively detailed in the time-domain pulse-collision theory. A longer symbol period will then have a comparable effect to the reduction of the dispersion coefficient considered here, as demonstrated in [15,24]. The impact of strongly reduced pulse-broadening can be observed in Fig. 8.
Fig. 8. Total NLIN and residual nonlinearity as estimated by the MoM and on the post-CPR constellations.
In the first spans BPS has no effect on the constellation and the estimated ResN is simply equal to the total NLIN, including the uncompensated NLPN. Only in the last spans, after a certain amount of dispersion is accumulated, BPS starts to lightly mitigate the phase impairment. On the contrary we recall how our MoM based method is correlation-independent. The same plot shows that the result obtained with MoM agrees with the one provided by ideal DA-ML CPR. The result of this analysis is that in conditions in which temporal correlation is very low, it is not possible to approximate the post-DSP nonlinearity with the ResN if standard CPR is considered for the equalization. Nevertheless, recently more complex adaptive equalizers able to follow the NLIN dynamics also when characterized by short correlation have been investigated [17]. In this context our correlation-independent method can be considered as a tool to estimate the ultimately achievable performance by means of NLPN equalization.
5. Conclusions
In this paper we have presented a statistical method able to separate NLIN into a Gaussian and a phase noise component. The method has been fully validated, and its main sources of error have been analyzed, in order to understand the limitations of the model and its applicability in different simulation scenarios. The method has been proven to be sufficiently accurate in a wide range of operating conditions while being faster and easier to apply than the standard CPR-based approach, which suffers from the need for filter length optimization. The method has also proved to be correlation-independent, making it possible to analyze scenarios where conventional phase recovery is not applicable, and allowing for its use as a performance estimator for CPR realizations and adaptive equalization.
The method has been tested by analyzing the impact of probabilistic shaping on the NLIN system performance penalty. The results obtained confirm that in standard transmission scenarios, there is no significant penalty in the performance of a transmission system when using PS-QAM signals. We also confirm that this behavior is strongly related to the amount of temporal correlation present in the received signal. These findings have been compared to predictions provided by state-of-the-art analytical models. In this context we have shown the potentialities of our method as a unique validation tool to assess the achievable transmission performance without being limited by considering a specific DSP implementation.
Funding
H2020 Marie Skłodowska-Curie Actions (814276); Bundesministerium für Bildung und Forschung (16KIS0993).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]
2. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013). [CrossRef]
3. 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]
4. L. Galdino, G. Liga, G. Saavedra, D. Ives, R. Maher, A. Alvarado, S. Savory, R. Killey, and P. Bayvel, “Experimental demonstration of modulation-dependent nonlinear interference in optical fibre communication,” in ECOC 2016; 42nd European Conference on Optical Communication (VDE, 2016), pp. 1–3.
5. Y. Fan, L. Dou, Z. Tao, L. Lei, S. Oda, T. Hoshida, and J. C. Rasmussen, “Modulation format dependent phase noise caused by intra-channel nonlinearity,” in 2012 38th European Conference and Exhibition on Optical Communications (IEEE, 2012), pp. 1–3.
6. Z. Qu and I. B. Djordjevic, “On the probabilistic shaping and geometric shaping in optical communication systems,” IEEE Access 7, 21454–21464 (2019). [CrossRef]
7. T. Fehenberger, A. Alvarado, G. Böcherer, and N. Hanik, “On probabilistic shaping of quadrature amplitude modulation for the nonlinear fiber channel,” J. Lightwave Technol. 34(21), 5063–5073 (2016). [CrossRef]
8. R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Opt. Lett. 39(2), 398–401 (2014). [CrossRef]
9. O. Golani, M. Feder, A. Mecozzi, and M. Shtaif, “Correlations and phase noise in nlin-modelling and system implications,” in Optical Fiber Communication Conference (Optical Society of America, 2016), pp. W3I–2.
10. O. Golani, D. Elson, D. Lavery, L. Galdino, R. Killey, P. Bayvel, and M. Shtaif, “Experimental characterization of the time correlation properties of nonlinear interference noise,” in 2017 European Conference on Optical Communication (ECOC) (IEEE, 2017), pp. 1–3.
11. O. Golani, R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Modeling the bit-error-rate performance of nonlinear fiber-optic systems,” J. Lightwave Technol. 34(15), 3482–3489 (2016). [CrossRef]
12. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Pulse collision picture of inter-channel nonlinear interference in fiber-optic communications,” J. Lightwave Technol. 34(2), 593–607 (2016). [CrossRef]
13. F. Frey, R. Elschner, C. Schmidt-Langhorst, R. Emmerich, C. Schubert, J. K. Fischer, and R. F. Fischer, “Investigation of nonlinear inter-channel interference in heterogeneous flexible optical networks,” in SCC 2017; 11th International ITG Conference on Systems, Communications and Coding (VDE, 2017), pp. 1–6.
14. T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for m-qam constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
15. D. Pilori, F. Forghieri, and G. Bosco, “Residual non-linear phase noise in probabilistically shaped 64-qam optical links,” in 2018 Optical Fiber Communications Conference and Exposition (OFC) (IEEE, 2018), pp. 1–3.
16. M. Cantono, D. Pilori, A. Ferrari, A. Carena, and V. Curri, “Observing the interaction of pmd with generation of nli in uncompensated amplified optical links,” in 2018 Optical Fiber Communications Conference and Exposition (OFC) (IEEE, 2018), pp. 1–3.
17. O. Golani, M. Feder, and M. Shtaif, “Kalman-mlse equalization for nlin mitigation,” J. Lightwave Technol. 36(12), 2541–2550 (2018).
18. S. Dris, C. Spatharakis, P. Bakopoulos, I. Lazarou, and H. Avramopoulos, “Blind snr estimation for qam constellations based on the signal magnitude statistics,” in Next-Generation Optical Communication: Components, Sub-Systems, and Systems III, vol. 9009 (International Society for Optics and Photonics, 2014), p. 900905.
19. S. Dris, H. Louchet, and A. Richter, “Analysis of nonlinear interference noise in flexible optical networks,” in 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC) (IEEE, 2017), pp. 1–3.
20. R. Matzner and F. Englberger, “An snr estimation algorithm using fourth-order moments,” in Proceedings of 1994 IEEE International Symposium on Information Theory (IEEE, 1994), p. 119.
21. D. 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).
22. T. Fehenberger, A. Alvarado, G. Böcherer, and N. Hanik, “On the impact of probabilistic shaping on snr and information rates in multi-span wdm systems,” in Optical Fiber Communication Conference (Optical Society of America, 2017), pp. M3C–4.
23. D. Pilori, A. Nespola, F. Forghieri, and G. Bosco, “Non-linear phase noise mitigation over systems using constellation shaping,” J. Lightwave Technol. 37(14), 3475–3482 (2019). [CrossRef]
24. F. Guiomar, A. Carena, G. Bosco, L. Bertignono, A. Nespola, and P. Poggiolini, “Nonlinear mitigation on subcarrier-multiplexed pm-16qam optical systems,” Opt. Express 25(4), 4298–4311 (2017). [CrossRef]
