- Email: [email protected]
Nonlinear distortion estimation in dual-carrier PM-QPSK signals for uncompensated fiber links
Optical Fiber Technology 52 (2019) 101943
Contents lists available at ScienceDirect
Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
Nonlinear distortion estimation in dual-carrier PM-QPSK signals for uncompensated fiber links
T
Abdullah S. Karar College of Engineering and Technology, American University of the Middle East, Kuwait
A R T I C LE I N FO
A B S T R A C T
Keywords: Fiber nonlinearity Coherent optical communications Kerr nonlinearity Numerical methods
Kerr nonlinearity is particularly detrimental in coherent optical communications over long-haul, sub-sea and transatlantic links. The correct and accurate determination of the penalty due to fiber nonlinearity is critical for estimating the link capacity and optical signal to noise ratio margins. Two distinct approaches for calculating the power in the nonlinear field are proposed. The first approach utilizes the nonlinear variance at the decision gate after coherent detection, while the second attempts to extract the electrical field of the nonlinear interference noise. Subsequently, the power spectral density of the nonlinear interference noise is integrated to estimate the nonlinear power within a pre-defined signal-to-nose ratio criterion. Based on the assumptions in this study, a temporal resolution of 16 samples per symbol or (6.25 ps) is required to achieve a signal-to-noise ratio sensitivity of 40 dB on the nonlinear power estimate for a 10 Gbaud signal. Furthermore, a minimum pattern length of 128 symbols is required to obtain an adequate estimate of the nonlinear noise power within 30 dB signal-to-noise ratio for two adjacent wavelength-division-multiplexed (WDM) channels. The influence of the pulse shaping rolloff factor and number of WDM channels on the required pattern length and the nonlinear power estimation is also investigated.
1. Introduction The capacity of transmission in optical fiber communication systems are not only limited by amplified spontaneous emission (ASE), but also by the nonlinear Shannon limit [1]. Kerr nonlinearity is particularly detrimental in coherent optical communications over long-haul, sub-sea and transatlantic links. The correct and accurate determination of the penalty due to fiber nonlinearity is critical for estimating the link capacity and optical signal to noise ratio (OSNR) margins. The primary objective of this study is to numerically characterize the nonlinear distortion in a 10 Gbaud dual-carrier polarization multiplexed quadrature phase shift keying (PM-QPSK) signal transmitted over 2000 km of uncompensated fiber in the absence of the amplified spontaneous emission (ASE) noise. A numerical study of methods to calculate the power in the nonlinear optical field are examined in this work. Long-haul transmission is fundamentally limited by two distinct phenomena: amplified spontaneous emission (ASE) noise and the nonlinear interference (NLI) due to the Kerr effect. This study illustrates the statistical distribution of the NLI after propagation in an uncompensated fiber link in the absence of ASE noise to isolate the effect of signal propagation alone. The simulation at hand models the propagation of two 10 Gbaud polarization multiplexed quadrature-phase-shift keying (PM-QPSK)
signals separated by 50 GHz.The number of WDM channels will be later increased to 9 channels, to estimate the influence of the WDM set-up on the nonlinear interference noise and simulation pattern length. The optical link consists of 20 spans of standard single mode fiber with 100 km per span. The fiber parameters used are listed below: Attenuation Dispersion Nonlinear coefficient
α D γ
Wavelength
λ
0.2 dB/km 17 ps/nm/km 1.3 W−1/km 1.55 μm
The Tx performed root raised cosine (RRC) digital pulse shaping with a roll-off factor of 1.0 and a time domain impulse response window of 64 symbols. A coherent Rx was employed with matched filtering and standard digital signal processing (DSP) algorithms [1]. Chromatic dispersion was compensated for at the coherent Rx using a fixed frequency domain equalizer. Lasers were assumed ideal (zero linewidth) and exhibited perfect tuning (no frequency offset). The state of polarization (SOP) was scrambled randomly prior to fiber propagation, while the effect of polarization mode dispersion was not included. The optical fiber was modeled using the coupled nonlinear Schrödinger equation (CNLSE) [2]. The split-step Fourier method (SSFM) was employed to numerically solve the CNLSE with a variable
E-mail address: [email protected]. https://doi.org/10.1016/j.yofte.2019.101943 Received 13 March 2019; Received in revised form 18 May 2019; Accepted 28 May 2019 Available online 11 June 2019 1068-5200/ © 2019 Elsevier Inc. All rights reserved.
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
Fig. 1. (a) Eye-diagram of the drive signal for channel-1 IX (32 samples per symbol). (b) PSD of the optical field at back-to-back (c) PSD of the optical field after transmission at a per channel launch power of +3 dBm (d) Recovered constellation diagram of channel-1 IX after coherent detection and DSP.
random patterns. The random number generator used exhibits a periodicity much larger than the collective number of symbols simulated.
2. Simulation test case To verify the modelling approach an initial simulation was performed with a pattern length of 214 completely random symbols and 32 temporal samples per symbol. The per channel launch power was set to +3 dBm to operate in the nonlinear region. Fig. 1 illustrates the eyediagram of the drive signal of channel-1 IX, the power spectral density (PSD) before and after transmission, and the recovered constellation diagram of channel-1 IX at 1 sample per symbol after coherent detection and DSP. The recovered constellation diagram shows that the NLI distortion is manifested as Gaussian distributed noise about the ideal constellation point even in the absence of ASE noise [3,4]. In other words, the effect of nonlinear propagation results in an introduction of additive Gaussian noise (nonlinear noise). The Gaussian statistics for the nonlinear phase noise is expected for uncompensated links similar to the one implemented in this study. On the other hand, optically compensated links will manifest a strong phase rotation to the constellation points at high launch powers. A simple method to quantify the dependence of the NLI on launch power is through calculating the variance of the nonlinear 2 noise (σNLI ) at the decision gate stage [5]. This dependence is shown in Fig. 2 for both linear and nonlinear propagation cases. This result is consistent with previous published work [5] and testifies to the validity of the simulation model at hand. In the linear propagation case the value of 10log10 (σ 2) is around −34 dB ± 0.1 dB. This is expected as it would be independent of the launch power. However, the value indicates a distortion contribution from the Rx down-conversion and subsequent DSP equalization alone. This effect should be accounted for if the variance of the nonlinear noise 2 (σNLI ) is used in estimating the power in the nonlinear field.
Fig. 2. The dependence of the average variance of the QPSK signal constellation points after coherent detection and DSP equalization on the per channel launch power in both nonlinear and linear propagation cases.
step size exhibiting a maximum nonlinear phase rotation of 3 × 10−3 rad. The wavelength division multiplexed (WDM) channels were propagated as a unique (single) optical field, offering a complete account of self-phase modulation (SPM), cross-phase modulation (XPM) and fourwave mixing (FWM). A single 2n deBruijn bit sequence (DBBS) is decimated into the Inphase (I) and Quadrature (Q) data contents of the both WDM channels in both X and Y polarization with appropriate de-correlation lengths. To investigate the effects of the pattern length irrespective of the data content an option has been added to facilitate generating completely
2
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
The estimation of the power in the nonlinear field PNLI and the OSNR (0.1 nm) can be readily obtained from ENLI and ETX .The mean nonlinear phase rotation is obtained numerically through direct access to the time-domain applied phase rotation for the nonlinear section of the split-step algorithm [3].
3. Calculating the power in the nonlinear field In this work, the power in the nonlinear field (PNLI ) will be defined as the power of the nonlinear noise resulting from Kerr nonlinearity. Two distinct approaches for calculating the power in the nonlinear field are considered in this work. The former approach utilizes the nonlinear variance at the decision gate, while the latter attempts to extract the electrical field of the nonlinear interference.
3.3. Comparison To compare both approaches used to estimate PNLI and OSNR (0.1 nm) a bench-mark simulation is performed with 32 temporal samples per symbol and a total of 214 completely random symbols. The dependence of PNLI and OSNR (0.1 nm) on per channel launch power is shown in Fig. 4 for both approaches. It is evident that both approaches are consistent within ± 1 dB. A diagram showing the PSD of the nonlinear noise superimposed on the PSD of the transmitted signal at a lunch power of 0 dBm and the OBPF is shown in Fig. 5 for illustration.
3.1. Estimating PNLI using the nonlinear variance at the decision gate Due to the quasi-Gaussian nature of the statistics of the nonlinear interference in dispersion unmanaged links, it is possible to calculate the optical signal-to-noise ratio (OSNR) using the SNR at the decision gate. In the absence of ASE noise, the electrical SNR due to Kerr nonlinearity can be expressed as:
SNR =
A2 − σR2x
2 σNLI
(1) 4. On the precision of PNLI estimation
where σRx is the variance of the extra receiver noise arising from backto-back cross talk and intersymbol interference (ISI) [6] (Here 10log10 (σR2x ) = −34 dB). The term ( A 2 ) denotes the average squared distance of the noise-less signal constellation point from the origin. The OSNR in 0.1 nm (12.48 GHz reference bandwidth at 1550 nm) can be related to the SNR through the following expression:
R OSNRdB = 10log10 (SNR) + 10log10 ⎡ s ⎤ ⎣B⎦
The results generated thus far assume the nonlinear noise power of interest falls within the first null-to-null optical bandwidth of the signal. This assumption allows the use of coherent detection and DSP to estimate the nonlinear noise within the match-filter bandwidth. Furthermore, the simulations performed were computationally expensive employing 32 temporal samples per symbol and a total of 214 symbols. To facilitate using lower temporal resolution and a short sequence length, the nonlinear noise power could be estimated from the calculated PSD. In this section two fundamental assumptions are made which differ from the simulation performed in earlier sections of this work:
(2)
where the symbol rate is Rs = 10 Gbaud and the reference bandwidth B = 12.48 GHz. Subsequently, through knowledge of the transmitted average power and the OSNR a rough estimate of the power in the nonlinear field PNLI within the bandwidth of the coherent receiver assuming matched filtering and quasi-Gaussian statistics can be made. 2 Utilizing σNLI at the decision gate has the ability to estimate PNLI with random and non-deterministic effects in the fiber transmission such as polarization mode dispersion. However, the accuracy of the estimate is in question, as it relies on only one sample per symbol, is influenced by the DSP algorithms employed, requires a large number of symbols for adaptive equalization, and assumes strictly Gaussian statistics.
• The definition of the nonlinear noise P •
3.2. Estimating PNLI using the NLI electrical field
The estimation of the total power from the PSD is simply frequency domain integration through calculating the signal and noise power over frequency bins. For this purpose the SNR will be defined as the max signal power divided by the power in the nonlinear noise bin. According to Parceval’s theorem, the total signal power computed in the time and frequency domain should coincide. However, as we do not have an infinite simulation time window, the PSD will be estimated using a pwelch function, which divides the sample sequence into shorter blocks and computes the windowed periodogram of each block producing an average PSD estimate. Fig. 6, shows an example of noise bandwidth selection at a launch power of 0 dBm per channel and a SNRs = 40 dB. It is evident that the out-of-band nonlinear noise is captured. A simulation was performed with a total of 210 completely random symbols and a temporal resolution of 16 samples per symbol. The dependence of PNLI and OSNR (0.1 nm) estimated through integrating the power spectral density (SNRs = 40 dB) on the per channel launch power is shown in Fig. 7. The total nonlinear noise estimated from the PSD is consistent with earlier calculations. The nonlinear noise power presented in Fig. 7(a) is expected to be at least 3 dB higher than the per-channel PNLI reported in Fig. 4(a). However, the difference is more than 3 dB at higher launch powers as the selection criterion accepts more out-of-band nonlinear noise.
A more rigorous approach in estimating PNLI would require extracting the complex NLI electrical field ENLI within a specified optical bandwidth and calculating its average power. A simple block diagram describing the procedure to estimate PNLI is illustrated in Fig. 3. A perfectly square optical bandpass filter (OBPF) is used to extract both channels of the transmitted electrical field ETX within a null-tonull optical bandwidth of 10 GHz. The original dual-carrier QPSK signal is transmitted over 20 spans of 100 km long SMF with both linear and nonlinear effects included and ideal amplification (no ASE noise). The accumulated dispersion in the link is Dc . To facilitate isolating the effect of Kerr nonlinearity, the signal is propagated through a single span of linear post-compensating fiber with the negative accumulated dispersion (− Dc ), no attenuation and no nonlinearity. The same OBPF is used to extract both channels after dispersion post-compensation producing the received electrical field ERX . Simply subtracting ETX from ERX will not yield the correct NLI electrical field ENLI particularly under high lunch powers as the average nonlinear phase rotation induced by the link must be taken into consideration. The nonlinear phase rotation applied at each step of the numerical SSFM when solving the CNLSE was accumulated during simulation and the average nonlinear phase rotation ϕ was determined. The NLI electrical field ENLI is calculated as:
ENLI = ERX × exp (−jϕ) − ETX
NLI will be re-defined to include the spectral content present over the entire simulation frequency window, which meets a pre-defined signal-to-noise ratio selection criterion (SNRs ). The value of SNRs allows capturing out-of band nonlinear noise. The PNLI will be estimated from integrating the PSD as oppose to calculating the average power in the time domain as in earlier sections of this work.
(3) 3
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
Fig. 3. The procedure for extracting the NLI electrical field. OBPF: optical bandpass filter; ETX : electrical field before transmission. SMF: single mode fiber with both linear and nonlinear propagation. Dc : cumulative dispersion; ERX : electrical field after linear post-compensation of dispersion. ϕ : average nonlinear phase rotation induced in the link.
Fig. 4. Comparison between both approaches used to estimate the nonlinear power PNLI and OSNR. The dependence of the per channel PNLI and OSNR (0.1 nm) on the per channel launch power PTX is shown in (a) and (b), respectively.
Fig. 5. The power spectral density of the transmitted optical field and nonlinear noise filed at 0 dBm launch power.
Fig. 6. Example of estimating PNLI using the power spectral density of the transmitted optical field, nonlinear noise field and selected nonlinear noise field with SNRs = 40 dB at 0 dBm launch power.
4
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
Fig. 7. The dependence of the total channel PNLI and OSNR (0.1 nm) on the per channel launch power PTX is shown in (a) and (b), respectively. These results are estimated through integrating the nonlinear power spectral density (SNRs = 40 dB).
Fig. 8. PSD of the transmitted signal and nonlinear noise with a temporal resolution of (a) 8 samples/symbol and (b) 16 samples/symbol. (Simulation with 210 completely random symbols at 3 dBm).
Fig. 9. Plot of the estimated OSNR (0.1 nm) obtained using the integration of the noise PSD (with SNRs = 30 dB) as a function of pattern length at 3 dBm launch power. Fig. 10. The power spectral density of the transmitted optical field and nonlinear noise filed at −3 dBm launch power for 9 WDM channels and a roll-off factor of 1.0.
4.1. The effect of temporal resolution on PNLI estimation (two channels) According to the interpretation of the question and subsequent treatment of PNLI estimation, the temporal resolution (samples per symbol) appears to play a critical role in determining the precision. The PSD of the nonlinear noise when simulating 210 completely random symbols with 8 samples per symbol at a per channel launch power of 3 dBm is shown in Fig. 8(a), while increasing the temporal resolution to 16 samples per symbol results in the PSD shown in Fig. 8(b).
Observing Fig. 8(a), a so called noise floor above SNRs = 40 dB is indicated, which is absent when the temporal resolution is increased. To achieve a sensitivity of SNRs < 40 dB, the number of samples per symbol should be greater than 16 or equivalently a time resolution of 6.25 ps for a 10 Gbaud signal. 5
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
The power of the nonlinear field was not found to depend on the roll-off factor, as the method of integrating the PSD accounts for the null-to-null bandwidth variation of the nonlinear noise irrespective of the pulse shaping used. Furthermore, the required pattern length for 9 WDM channels was found to follow a similar dependency as for two channels shown in Fig. 9. This is expected as the pattern length has an effect on the resolution (smoothness) of the PSD required for integration purposes. It is worth noting that the estimate of PNLI obtained using the nonlinear variation at the decision gate was found to depend on both the pattern length and modulation format used. However, the primary objective of this work is to present a generalized method for estimating the total power in the nonlinear field, which does not depend on modulation format or roll-off factor. Once the PNLI is estimated using the PSD integration method, it can be used to estimate the BER penalty for each modulation format separately.
Fig. 11. The power spectral density of the transmitted optical field and nonlinear noise filed at −3 dBm launch power for 9 WDM channels and a roll-off factor of 0.1.
6. Conclusion 4.2. The effect of pattern length on PNLI estimation (two channels)
Two distinct approaches for calculating the power in the nonlinear field were considered in this study. The first approach utilizes the nonlinear variance at the decision gate after coherent detection, while the second attempts to extract the electrical field of the nonlinear interference noise. Subsequently, the power spectral density of the nonlinear interference noise was integrated to estimate the nonlinear power within a pre-defined SNR criterion. Based on the assumptions in this study, a temporal resolution of 16 samples per symbol or (6.25 ps) is required to achieve a SNR sensitivity of 40 dB on the nonlinear power estimate for a 10 Gbaud signal. Furthermore, a minimum pattern length of 128 symbols is required to obtain an adequate estimate of the nonlinear noise power within 30 dB SNR for two adjacent WDM channels. The influence of the pulse shaping roll-off factor and number of WDM channels on the required pattern length and the nonlinear power estimation was presented.
To quantify the effect of the pattern length, the number of samples per symbol was fixed at 16 and the SNRs was set to 30 dB. For each pattern length a completely random sequence was generated as to avoid data dependence and each simulation was run 10 times per pattern length. A plot of the estimated OSNR (0.1 nm) obtained using the integration of the noise PSD as a function of pattern length at 3 dBm launch power is illustrated in Fig. 9. A pattern length greater than 128 symbols is of interest as it would allow obtaining similar nonlinear noise power (and consequently OSNR) for each run. Using a long sequence length in each simulation run is imperative given the long correlation time of nonlinear interference noise. Longer sequences will also eliminate numerical artifacts arising from signal periodicity resulting from discrete FFT operations. If time permitted a more comprehensive simulation would be needed with more than 10 runs per pattern length.
References 5. On the effect of the roll-off factor and the number of WDM channels on the PSD integration method
[1] A.D. Ellis, J. Zhao, D. Cotter, Approaching the non-linear Shannon limit, J. Lightwave Technol. 28 (4) (2010) 423–433. [2] J.H. Ke, K.P. Zhong, Y. Gao, J.C. Cartledge, A.S. Karar, M.A. Rezania, Linewidthtolerant and low-complexity two-stage carrier phase estimation for dual-polarization 16-QAM coherent optical fiber communications, J. Lightwave Technol. 30 (24) (2012) 3987–3992. [3] P. Serena, M. Bertolini, A. Vannucci, Optilux Toolbox, available atwww.optilux. sourceforge.net, 2009. [4] A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links, J. Lightwave Technol. 30 (10) (2012) 1524–1539. [5] P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, Analytical modeling of nonlinear propagation in uncompensated optical transmission links, IEEE Photon. Tech. Lett. 23 (11) (2011) 742–744. [6] A. Carena, G. Bosco, V. Curri, P. Poggiolini, M.T. Taiba, F. Forghieri, Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links, ECOC 2010, pp. 19–23, Torino, Italy, 2010.
The number of WDM channels was increased to 9 adjacent channels, were the power in the nonlinear field was estimated for the middle channel. Furthermore, the roll-off factor was varied between the values of one to zero. The roll-off factor of one, is indicative of NRZ modulation from binary pulse pattern generators, while a roll-off factor of zero, is the limit Nyquist pulse shaping case. The number of samples per symbol was increased to 64 samples to avoid aliasing between the frequency spectrum of the WDM channels. The launch power was maintained at −3 dBm per channel. An example of estimating PNLI using the power spectral density of the transmitted optical field is shown in Fig. 10 for a roll-off factor of 1.0 andFig. 11for a roll-off factor of 0.1.
6
Contents lists available at ScienceDirect
Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
Nonlinear distortion estimation in dual-carrier PM-QPSK signals for uncompensated fiber links
T
Abdullah S. Karar College of Engineering and Technology, American University of the Middle East, Kuwait
A R T I C LE I N FO
A B S T R A C T
Keywords: Fiber nonlinearity Coherent optical communications Kerr nonlinearity Numerical methods
Kerr nonlinearity is particularly detrimental in coherent optical communications over long-haul, sub-sea and transatlantic links. The correct and accurate determination of the penalty due to fiber nonlinearity is critical for estimating the link capacity and optical signal to noise ratio margins. Two distinct approaches for calculating the power in the nonlinear field are proposed. The first approach utilizes the nonlinear variance at the decision gate after coherent detection, while the second attempts to extract the electrical field of the nonlinear interference noise. Subsequently, the power spectral density of the nonlinear interference noise is integrated to estimate the nonlinear power within a pre-defined signal-to-nose ratio criterion. Based on the assumptions in this study, a temporal resolution of 16 samples per symbol or (6.25 ps) is required to achieve a signal-to-noise ratio sensitivity of 40 dB on the nonlinear power estimate for a 10 Gbaud signal. Furthermore, a minimum pattern length of 128 symbols is required to obtain an adequate estimate of the nonlinear noise power within 30 dB signal-to-noise ratio for two adjacent wavelength-division-multiplexed (WDM) channels. The influence of the pulse shaping rolloff factor and number of WDM channels on the required pattern length and the nonlinear power estimation is also investigated.
1. Introduction The capacity of transmission in optical fiber communication systems are not only limited by amplified spontaneous emission (ASE), but also by the nonlinear Shannon limit [1]. Kerr nonlinearity is particularly detrimental in coherent optical communications over long-haul, sub-sea and transatlantic links. The correct and accurate determination of the penalty due to fiber nonlinearity is critical for estimating the link capacity and optical signal to noise ratio (OSNR) margins. The primary objective of this study is to numerically characterize the nonlinear distortion in a 10 Gbaud dual-carrier polarization multiplexed quadrature phase shift keying (PM-QPSK) signal transmitted over 2000 km of uncompensated fiber in the absence of the amplified spontaneous emission (ASE) noise. A numerical study of methods to calculate the power in the nonlinear optical field are examined in this work. Long-haul transmission is fundamentally limited by two distinct phenomena: amplified spontaneous emission (ASE) noise and the nonlinear interference (NLI) due to the Kerr effect. This study illustrates the statistical distribution of the NLI after propagation in an uncompensated fiber link in the absence of ASE noise to isolate the effect of signal propagation alone. The simulation at hand models the propagation of two 10 Gbaud polarization multiplexed quadrature-phase-shift keying (PM-QPSK)
signals separated by 50 GHz.The number of WDM channels will be later increased to 9 channels, to estimate the influence of the WDM set-up on the nonlinear interference noise and simulation pattern length. The optical link consists of 20 spans of standard single mode fiber with 100 km per span. The fiber parameters used are listed below: Attenuation Dispersion Nonlinear coefficient
α D γ
Wavelength
λ
0.2 dB/km 17 ps/nm/km 1.3 W−1/km 1.55 μm
The Tx performed root raised cosine (RRC) digital pulse shaping with a roll-off factor of 1.0 and a time domain impulse response window of 64 symbols. A coherent Rx was employed with matched filtering and standard digital signal processing (DSP) algorithms [1]. Chromatic dispersion was compensated for at the coherent Rx using a fixed frequency domain equalizer. Lasers were assumed ideal (zero linewidth) and exhibited perfect tuning (no frequency offset). The state of polarization (SOP) was scrambled randomly prior to fiber propagation, while the effect of polarization mode dispersion was not included. The optical fiber was modeled using the coupled nonlinear Schrödinger equation (CNLSE) [2]. The split-step Fourier method (SSFM) was employed to numerically solve the CNLSE with a variable
E-mail address: [email protected]. https://doi.org/10.1016/j.yofte.2019.101943 Received 13 March 2019; Received in revised form 18 May 2019; Accepted 28 May 2019 Available online 11 June 2019 1068-5200/ © 2019 Elsevier Inc. All rights reserved.
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
Fig. 1. (a) Eye-diagram of the drive signal for channel-1 IX (32 samples per symbol). (b) PSD of the optical field at back-to-back (c) PSD of the optical field after transmission at a per channel launch power of +3 dBm (d) Recovered constellation diagram of channel-1 IX after coherent detection and DSP.
random patterns. The random number generator used exhibits a periodicity much larger than the collective number of symbols simulated.
2. Simulation test case To verify the modelling approach an initial simulation was performed with a pattern length of 214 completely random symbols and 32 temporal samples per symbol. The per channel launch power was set to +3 dBm to operate in the nonlinear region. Fig. 1 illustrates the eyediagram of the drive signal of channel-1 IX, the power spectral density (PSD) before and after transmission, and the recovered constellation diagram of channel-1 IX at 1 sample per symbol after coherent detection and DSP. The recovered constellation diagram shows that the NLI distortion is manifested as Gaussian distributed noise about the ideal constellation point even in the absence of ASE noise [3,4]. In other words, the effect of nonlinear propagation results in an introduction of additive Gaussian noise (nonlinear noise). The Gaussian statistics for the nonlinear phase noise is expected for uncompensated links similar to the one implemented in this study. On the other hand, optically compensated links will manifest a strong phase rotation to the constellation points at high launch powers. A simple method to quantify the dependence of the NLI on launch power is through calculating the variance of the nonlinear 2 noise (σNLI ) at the decision gate stage [5]. This dependence is shown in Fig. 2 for both linear and nonlinear propagation cases. This result is consistent with previous published work [5] and testifies to the validity of the simulation model at hand. In the linear propagation case the value of 10log10 (σ 2) is around −34 dB ± 0.1 dB. This is expected as it would be independent of the launch power. However, the value indicates a distortion contribution from the Rx down-conversion and subsequent DSP equalization alone. This effect should be accounted for if the variance of the nonlinear noise 2 (σNLI ) is used in estimating the power in the nonlinear field.
Fig. 2. The dependence of the average variance of the QPSK signal constellation points after coherent detection and DSP equalization on the per channel launch power in both nonlinear and linear propagation cases.
step size exhibiting a maximum nonlinear phase rotation of 3 × 10−3 rad. The wavelength division multiplexed (WDM) channels were propagated as a unique (single) optical field, offering a complete account of self-phase modulation (SPM), cross-phase modulation (XPM) and fourwave mixing (FWM). A single 2n deBruijn bit sequence (DBBS) is decimated into the Inphase (I) and Quadrature (Q) data contents of the both WDM channels in both X and Y polarization with appropriate de-correlation lengths. To investigate the effects of the pattern length irrespective of the data content an option has been added to facilitate generating completely
2
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
The estimation of the power in the nonlinear field PNLI and the OSNR (0.1 nm) can be readily obtained from ENLI and ETX .The mean nonlinear phase rotation is obtained numerically through direct access to the time-domain applied phase rotation for the nonlinear section of the split-step algorithm [3].
3. Calculating the power in the nonlinear field In this work, the power in the nonlinear field (PNLI ) will be defined as the power of the nonlinear noise resulting from Kerr nonlinearity. Two distinct approaches for calculating the power in the nonlinear field are considered in this work. The former approach utilizes the nonlinear variance at the decision gate, while the latter attempts to extract the electrical field of the nonlinear interference.
3.3. Comparison To compare both approaches used to estimate PNLI and OSNR (0.1 nm) a bench-mark simulation is performed with 32 temporal samples per symbol and a total of 214 completely random symbols. The dependence of PNLI and OSNR (0.1 nm) on per channel launch power is shown in Fig. 4 for both approaches. It is evident that both approaches are consistent within ± 1 dB. A diagram showing the PSD of the nonlinear noise superimposed on the PSD of the transmitted signal at a lunch power of 0 dBm and the OBPF is shown in Fig. 5 for illustration.
3.1. Estimating PNLI using the nonlinear variance at the decision gate Due to the quasi-Gaussian nature of the statistics of the nonlinear interference in dispersion unmanaged links, it is possible to calculate the optical signal-to-noise ratio (OSNR) using the SNR at the decision gate. In the absence of ASE noise, the electrical SNR due to Kerr nonlinearity can be expressed as:
SNR =
A2 − σR2x
2 σNLI
(1) 4. On the precision of PNLI estimation
where σRx is the variance of the extra receiver noise arising from backto-back cross talk and intersymbol interference (ISI) [6] (Here 10log10 (σR2x ) = −34 dB). The term ( A 2 ) denotes the average squared distance of the noise-less signal constellation point from the origin. The OSNR in 0.1 nm (12.48 GHz reference bandwidth at 1550 nm) can be related to the SNR through the following expression:
R OSNRdB = 10log10 (SNR) + 10log10 ⎡ s ⎤ ⎣B⎦
The results generated thus far assume the nonlinear noise power of interest falls within the first null-to-null optical bandwidth of the signal. This assumption allows the use of coherent detection and DSP to estimate the nonlinear noise within the match-filter bandwidth. Furthermore, the simulations performed were computationally expensive employing 32 temporal samples per symbol and a total of 214 symbols. To facilitate using lower temporal resolution and a short sequence length, the nonlinear noise power could be estimated from the calculated PSD. In this section two fundamental assumptions are made which differ from the simulation performed in earlier sections of this work:
(2)
where the symbol rate is Rs = 10 Gbaud and the reference bandwidth B = 12.48 GHz. Subsequently, through knowledge of the transmitted average power and the OSNR a rough estimate of the power in the nonlinear field PNLI within the bandwidth of the coherent receiver assuming matched filtering and quasi-Gaussian statistics can be made. 2 Utilizing σNLI at the decision gate has the ability to estimate PNLI with random and non-deterministic effects in the fiber transmission such as polarization mode dispersion. However, the accuracy of the estimate is in question, as it relies on only one sample per symbol, is influenced by the DSP algorithms employed, requires a large number of symbols for adaptive equalization, and assumes strictly Gaussian statistics.
• The definition of the nonlinear noise P •
3.2. Estimating PNLI using the NLI electrical field
The estimation of the total power from the PSD is simply frequency domain integration through calculating the signal and noise power over frequency bins. For this purpose the SNR will be defined as the max signal power divided by the power in the nonlinear noise bin. According to Parceval’s theorem, the total signal power computed in the time and frequency domain should coincide. However, as we do not have an infinite simulation time window, the PSD will be estimated using a pwelch function, which divides the sample sequence into shorter blocks and computes the windowed periodogram of each block producing an average PSD estimate. Fig. 6, shows an example of noise bandwidth selection at a launch power of 0 dBm per channel and a SNRs = 40 dB. It is evident that the out-of-band nonlinear noise is captured. A simulation was performed with a total of 210 completely random symbols and a temporal resolution of 16 samples per symbol. The dependence of PNLI and OSNR (0.1 nm) estimated through integrating the power spectral density (SNRs = 40 dB) on the per channel launch power is shown in Fig. 7. The total nonlinear noise estimated from the PSD is consistent with earlier calculations. The nonlinear noise power presented in Fig. 7(a) is expected to be at least 3 dB higher than the per-channel PNLI reported in Fig. 4(a). However, the difference is more than 3 dB at higher launch powers as the selection criterion accepts more out-of-band nonlinear noise.
A more rigorous approach in estimating PNLI would require extracting the complex NLI electrical field ENLI within a specified optical bandwidth and calculating its average power. A simple block diagram describing the procedure to estimate PNLI is illustrated in Fig. 3. A perfectly square optical bandpass filter (OBPF) is used to extract both channels of the transmitted electrical field ETX within a null-tonull optical bandwidth of 10 GHz. The original dual-carrier QPSK signal is transmitted over 20 spans of 100 km long SMF with both linear and nonlinear effects included and ideal amplification (no ASE noise). The accumulated dispersion in the link is Dc . To facilitate isolating the effect of Kerr nonlinearity, the signal is propagated through a single span of linear post-compensating fiber with the negative accumulated dispersion (− Dc ), no attenuation and no nonlinearity. The same OBPF is used to extract both channels after dispersion post-compensation producing the received electrical field ERX . Simply subtracting ETX from ERX will not yield the correct NLI electrical field ENLI particularly under high lunch powers as the average nonlinear phase rotation induced by the link must be taken into consideration. The nonlinear phase rotation applied at each step of the numerical SSFM when solving the CNLSE was accumulated during simulation and the average nonlinear phase rotation ϕ was determined. The NLI electrical field ENLI is calculated as:
ENLI = ERX × exp (−jϕ) − ETX
NLI will be re-defined to include the spectral content present over the entire simulation frequency window, which meets a pre-defined signal-to-noise ratio selection criterion (SNRs ). The value of SNRs allows capturing out-of band nonlinear noise. The PNLI will be estimated from integrating the PSD as oppose to calculating the average power in the time domain as in earlier sections of this work.
(3) 3
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
Fig. 3. The procedure for extracting the NLI electrical field. OBPF: optical bandpass filter; ETX : electrical field before transmission. SMF: single mode fiber with both linear and nonlinear propagation. Dc : cumulative dispersion; ERX : electrical field after linear post-compensation of dispersion. ϕ : average nonlinear phase rotation induced in the link.
Fig. 4. Comparison between both approaches used to estimate the nonlinear power PNLI and OSNR. The dependence of the per channel PNLI and OSNR (0.1 nm) on the per channel launch power PTX is shown in (a) and (b), respectively.
Fig. 5. The power spectral density of the transmitted optical field and nonlinear noise filed at 0 dBm launch power.
Fig. 6. Example of estimating PNLI using the power spectral density of the transmitted optical field, nonlinear noise field and selected nonlinear noise field with SNRs = 40 dB at 0 dBm launch power.
4
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
Fig. 7. The dependence of the total channel PNLI and OSNR (0.1 nm) on the per channel launch power PTX is shown in (a) and (b), respectively. These results are estimated through integrating the nonlinear power spectral density (SNRs = 40 dB).
Fig. 8. PSD of the transmitted signal and nonlinear noise with a temporal resolution of (a) 8 samples/symbol and (b) 16 samples/symbol. (Simulation with 210 completely random symbols at 3 dBm).
Fig. 9. Plot of the estimated OSNR (0.1 nm) obtained using the integration of the noise PSD (with SNRs = 30 dB) as a function of pattern length at 3 dBm launch power. Fig. 10. The power spectral density of the transmitted optical field and nonlinear noise filed at −3 dBm launch power for 9 WDM channels and a roll-off factor of 1.0.
4.1. The effect of temporal resolution on PNLI estimation (two channels) According to the interpretation of the question and subsequent treatment of PNLI estimation, the temporal resolution (samples per symbol) appears to play a critical role in determining the precision. The PSD of the nonlinear noise when simulating 210 completely random symbols with 8 samples per symbol at a per channel launch power of 3 dBm is shown in Fig. 8(a), while increasing the temporal resolution to 16 samples per symbol results in the PSD shown in Fig. 8(b).
Observing Fig. 8(a), a so called noise floor above SNRs = 40 dB is indicated, which is absent when the temporal resolution is increased. To achieve a sensitivity of SNRs < 40 dB, the number of samples per symbol should be greater than 16 or equivalently a time resolution of 6.25 ps for a 10 Gbaud signal. 5
Optical Fiber Technology 52 (2019) 101943
A.S. Karar
The power of the nonlinear field was not found to depend on the roll-off factor, as the method of integrating the PSD accounts for the null-to-null bandwidth variation of the nonlinear noise irrespective of the pulse shaping used. Furthermore, the required pattern length for 9 WDM channels was found to follow a similar dependency as for two channels shown in Fig. 9. This is expected as the pattern length has an effect on the resolution (smoothness) of the PSD required for integration purposes. It is worth noting that the estimate of PNLI obtained using the nonlinear variation at the decision gate was found to depend on both the pattern length and modulation format used. However, the primary objective of this work is to present a generalized method for estimating the total power in the nonlinear field, which does not depend on modulation format or roll-off factor. Once the PNLI is estimated using the PSD integration method, it can be used to estimate the BER penalty for each modulation format separately.
Fig. 11. The power spectral density of the transmitted optical field and nonlinear noise filed at −3 dBm launch power for 9 WDM channels and a roll-off factor of 0.1.
6. Conclusion 4.2. The effect of pattern length on PNLI estimation (two channels)
Two distinct approaches for calculating the power in the nonlinear field were considered in this study. The first approach utilizes the nonlinear variance at the decision gate after coherent detection, while the second attempts to extract the electrical field of the nonlinear interference noise. Subsequently, the power spectral density of the nonlinear interference noise was integrated to estimate the nonlinear power within a pre-defined SNR criterion. Based on the assumptions in this study, a temporal resolution of 16 samples per symbol or (6.25 ps) is required to achieve a SNR sensitivity of 40 dB on the nonlinear power estimate for a 10 Gbaud signal. Furthermore, a minimum pattern length of 128 symbols is required to obtain an adequate estimate of the nonlinear noise power within 30 dB SNR for two adjacent WDM channels. The influence of the pulse shaping roll-off factor and number of WDM channels on the required pattern length and the nonlinear power estimation was presented.
To quantify the effect of the pattern length, the number of samples per symbol was fixed at 16 and the SNRs was set to 30 dB. For each pattern length a completely random sequence was generated as to avoid data dependence and each simulation was run 10 times per pattern length. A plot of the estimated OSNR (0.1 nm) obtained using the integration of the noise PSD as a function of pattern length at 3 dBm launch power is illustrated in Fig. 9. A pattern length greater than 128 symbols is of interest as it would allow obtaining similar nonlinear noise power (and consequently OSNR) for each run. Using a long sequence length in each simulation run is imperative given the long correlation time of nonlinear interference noise. Longer sequences will also eliminate numerical artifacts arising from signal periodicity resulting from discrete FFT operations. If time permitted a more comprehensive simulation would be needed with more than 10 runs per pattern length.
References 5. On the effect of the roll-off factor and the number of WDM channels on the PSD integration method
[1] A.D. Ellis, J. Zhao, D. Cotter, Approaching the non-linear Shannon limit, J. Lightwave Technol. 28 (4) (2010) 423–433. [2] J.H. Ke, K.P. Zhong, Y. Gao, J.C. Cartledge, A.S. Karar, M.A. Rezania, Linewidthtolerant and low-complexity two-stage carrier phase estimation for dual-polarization 16-QAM coherent optical fiber communications, J. Lightwave Technol. 30 (24) (2012) 3987–3992. [3] P. Serena, M. Bertolini, A. Vannucci, Optilux Toolbox, available atwww.optilux. sourceforge.net, 2009. [4] A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links, J. Lightwave Technol. 30 (10) (2012) 1524–1539. [5] P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, Analytical modeling of nonlinear propagation in uncompensated optical transmission links, IEEE Photon. Tech. Lett. 23 (11) (2011) 742–744. [6] A. Carena, G. Bosco, V. Curri, P. Poggiolini, M.T. Taiba, F. Forghieri, Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links, ECOC 2010, pp. 19–23, Torino, Italy, 2010.
The number of WDM channels was increased to 9 adjacent channels, were the power in the nonlinear field was estimated for the middle channel. Furthermore, the roll-off factor was varied between the values of one to zero. The roll-off factor of one, is indicative of NRZ modulation from binary pulse pattern generators, while a roll-off factor of zero, is the limit Nyquist pulse shaping case. The number of samples per symbol was increased to 64 samples to avoid aliasing between the frequency spectrum of the WDM channels. The launch power was maintained at −3 dBm per channel. An example of estimating PNLI using the power spectral density of the transmitted optical field is shown in Fig. 10 for a roll-off factor of 1.0 andFig. 11for a roll-off factor of 0.1.
6