## Abstract

Perturbation based nonlinearity pre-compensation has been performed for a 128 Gbit/s single-carrier dual-polarization 16-ary quadrature-amplitude-modulation (DP 16-QAM) signal. Without any performance degradation, a complexity reduction factor of 6.8 has been demonstrated for a transmission distance of 3600 km by combining symmetric electronic dispersion compensation and root-raised-cosine pulse shaping with a roll-off factor of 0.1. Transmission over 4200 km of standard single-mode fiber with EDFA amplification was achieved for the 128 Gbit/s DP 16-QAM signals with a forward error correction (FEC) threshold of 2 × 10^{−2}.

© 2014 Optical Society of America

## 1. Introduction

Fiber nonlinearities are major impairments in coherent transmission systems that limit the achievable transmission distance [1]. Compared to dispersion compensated links, dispersion uncompensated links have a better nonlinear impairment tolerance with and without nonlinear compensation [2]. However, the compensation complexity for dispersion uncompensated links increases quickly with accumulated fiber dispersion along the transmission link [3, 4]. Digital back propagation is an effective intra-channel non-linear compensation technique that has received considerable attention [3–7]. It normally requires multiple computation steps for long un-compensated fiber lengths and at least two samples per symbol, which lead to a high complexity [7]. Alternatively, the perturbation-based nonlinear pre-compensation technique compensates the accumulated nonlinearities with only one computation step and can be implemented with one sample per symbol [8]. However, calculation of the nonlinear perturbation involves single and double summations with the required number of terms depending on the accumulated fiber dispersion along the transmission link.

In this paper, we demonstrate that the number of terms in the summations for the perturbation-based nonlinear pre-compensation can be reduced by a factor up to 6.8 without degrading the system performance [9]. This reduction is achieved by combining symmetric electronic dispersion compensation (SEDC) and root-raised-cosine (RRC) pulse shaping with a roll-off factor of 0.1. SEDC has recently been shown to result in a beneficial nonlinear noise squeezing effect that only occurs for real-valued signals (e.g., BPSK) [10] or conjugated twin waves in the polarization domain [11, 12]. Here it is shown that for dual polarization (DP) complex-valued signals, SEDC can be used to reduce the complexity of perturbation-based nonlinear pre-compensation. The nonlinear perturbation coefficients in previous work have been obtained assuming a Gaussian pulse shape for return-to-zero (RZ), non-return-to-zero (NRZ) and root-raised-cosine (RRC) pulse shapes [8, 13–15]. With this assumption the coefficients can be obtained from analytical expressions which involve the exponential integral function [8, 16]. Using numerical calculations, the assumption about the pulse shape can be matched to the system configuration and the perturbation coefficients calculated accordingly [1]. It is shown that the use of a RRC pulse shape with a roll-off factor of 0.1 can also be used to reduce the complexity of perturbation-based nonlinear pre-compensation. In an experimental demonstration of this technique for a DP 16-ary quadrature-amplitude-modulation (16-QAM) signal, a distance of 4200 km is achieved for a 128 Gbit/s signal on a dispersion uncompensated link comprised of standard single mode fiber and EDFAs [9].

## 2. SEDC and RRC pulse shaping

The perturbation-based pre-compensation of a DP 16-QAM signal includes self-phase modulation, intra-channel cross phase modulation (IXPM) and intra-channel four-wave-mixing (IFWM) [13]. Without electronic pre-dispersion compensation, the optical field for the current symbol (at time 0) after nonlinear pre-compensation is:

*L*, ${P}_{0}$ is the optical launch power, $\left\{{A}_{n,x/y}\right\}$ is the sequence of complex transmitted symbols for the

*x*- and

*y*-polarization signals with zero dispersion, and

*E*denotes expectation [8, 9, 13]. The coefficients ${C}_{m,0}$ are imaginary-valued. The complexity of the algorithm is primarily determined by the second terms in Eqs. (5) and (6) for IXPM and Eqs. (7) and (8) for IFWM. The summations are truncated in practice based on the values of $\left|{C}_{m,n}\right|$ being larger than a specified criterion.

A benefit of nonlinearity pre-compensation in the transmitter is that the symbol values ${A}_{n,x/y}$ are known. For a QPSK signal, ${A}_{n,x/y}$ and the multiplication results for three ${A}_{n,x/y}$ values take the four values: 1, *i*, –1, –*i*. As a result, the multiplication ${A}_{n,x}{A}_{m,x}{A}^{\ast}{}_{m+n,x}$ can be easily realized by logical operations [8]. For a 16-QAM signal, the ${A}_{n,x/y}$ values can be formulated as a combination of two QPSK signals. The first QPSK signal is chosen from 2, 2*i*, –2, or –2*i*. The second QPSK signal is chosen from 1, *i*, –1, or –*i*. The multiplication of ${A}_{n,x}{A}_{m,x}{A}^{\ast}{}_{m+n,x}$ can be realized by adding the eight QPSK signals with moduli of 8, 4, 2 and 1 from the distributive rule of multiplication. By taking advantage of the known symbol sequence at the transmitter, high resolution complex multipliers can be avoided.

Assuming that the transmitted optical pulses have a Gaussian shape, analytical expressions in terms of the exponential integral function exist for the nonlinear coefficients ${C}_{m,n}$ [1, 8, 16]. This assumption has been used frequently in previous contributions for transmission systems with RZ, NRZ and RRC pulse shapes [8, 13–15]. For a Gaussian pulse shape, the amplitudes of the real and imaginary parts of ${C}_{m,n}(L)$ are plotted in Fig. 1(a) for a transmission distance of 3600 km without electronic pre-dispersion compensation. Since ${C}_{m,n}$ is complex-valued (*m*≠0, *n*≠0), both $\mathrm{Re}[{C}_{m,n}]$ and $\mathrm{Im}[{C}_{m,n}]$ are needed to calculate ${A}_{0,x/y}^{out}$. For comparison with what follows, each complex-valued ${C}_{m,n}$ term is regarded as contributing two terms to the summations. Using the criterion $20{\mathrm{log}}_{10}\left|{C}_{m,n}/{C}_{0,0}\right|$>-35 dB to determine the number of terms in the truncated summations, 8193 terms are required for a fiber length of 3600 km based on Fig. 1(a). The conventional nonlinearity pre-compensation algorithm with a Gaussian pulse shape assumption and post-compensation for dispersion is denoted as Gaussian.

For a 50% dispersion pre-compensated link of length *L*, the optical signal is dispersion free at the midpoint of the transmission link. Denoting the beginning, midpoint, and end of the link by $z=-L/2,0andL/2,$, respectively, the *x*-polarization IFWM perturbation for the link with SEDC is:

*x*-polarization signal. According to the analytical expression for Gaussian pulses [8], the relationship ${C}_{m,n}(-L/2)=-{C}_{{}_{m,n}}^{\ast}(L/2)$ is valid, which is equivalent to:

*y*-polarization signal is obtained by exchanging the subscripts

*x*and

*y*. Similarly, the phase perturbation for the

*x*-polarization signal with SEDC is,

*y*-polarization signal is also obtained by exchanging the subscripts

*x*and

*y*. It’s important to note that for a DP signal there are cross-polarization contributions in Eqs. (13) and (14). The perturbation for the

*x*-polarization signal depends on the transmitted symbol sequences for both the

*x*- and

*y*-polarization signals.

With SEDC, two simplifications result: 1) all the $\mathrm{Re}[{C}_{m,n}]$ terms are eliminated and 2) the ${C}_{m,n}$ coefficients are calculated based on half of the link length $L/2$. This reduces the dispersion induced pulse spreading and hence the required number of terms in the truncated summations. For a fiber length of 3600 km, $\left|\mathrm{Im}[{C}_{m,n}(L/2)]\right|$ is plotted in Fig. 1(b) with SEDC. For a selection criterion $20{\mathrm{log}}_{10}\left|{C}_{m,n}/{C}_{0,0}\right|$>-35 dB, with this SEDC-based simplification alone, the number of summation terms is reduced from 8193 to 2397. The simplification using SEDC and the Gaussian pulse assumption is denoted as Gaussian-SEDC.

The ${C}_{m,n}$ coefficients are fixed for a given transmission spectrum and fiber length. For a RRC pulse shape with a roll-off factor of 0.1, the coefficients are calculated numerically as an analytical solution is not known [1]. ${C}_{m,n}$ for a RRC pulse shape with a roll-off factor of 0.1 and matched filtering was calculated as [9]

*k*is a scaling factor, ${L}_{span}$ is the span length, ${f}_{pd}\left(z\right)$ is the power distribution profile along the link,

*T*is the symbol period, ${T}_{m}=mT$, ${u}_{0}(0,t)$ is the pulse shape with zero accumulated dispersion (Z = 0), and ${u}_{0}(z,t)$ is the dispersed pulse shape corresponding to a fiber length

*z*which is calculated according to ${u}_{0}(z,t)=ifft\{fft[{u}_{0}(0,t)]\times \mathrm{exp}[-i{\beta}_{2}{(2\pi f)}^{2}z/2]\}$, where (

*i*)

*fft*denotes the (inverse) Fourier transform,

*f*is frequency, and ${\beta}_{2}$ is the first order group velocity dispersion [1].

In a practical implementation, the ${C}_{m,n}$ values would be pre-calculated and stored in a look-up-table so the calculation requirements for ${C}_{m,n}$ are not of significant consequence. The coefficients still satisfy ${C}_{m,n}(-L/2)=-{C}_{{}_{m,n}}^{\ast}(L/2)$ With the RRC pulse shape and SEDC, $\left|\mathrm{Im}[{C}_{m,n}(L/2)]\right|$ is plotted in Fig. 1(c). The RRC pulse shape with a roll-off factor of 0.1 has a smaller bandwidth than the Gaussian pulse shape and RRC pulse shapes with larger roll-off factors. This yields a smaller dispersion induced pulse spreading and hence a reduction in the number of terms in truncated approximations to Eqs. (13) and (14) that are based on the values of *m* and *n* for which $20{\mathrm{log}}_{10}\left|{C}_{m,n}/{C}_{0,0}\right|$ is larger than a specified criterion. By combining SEDC and RRC pulse shaping, the number of summation terms is reduced from 8193 to 1201. The simplification using SEDC and a RRC pulse shape with a roll-off factor of 0.1 is denoted as RRC-SEDC.

## 3. Experimental set-up and back to back measurement

The experimental setup is shown in Fig. 2. The laser was operated at a wavelength of 1557.36 nm. A 2^{19} deBruijn bit sequence was mapped to symbols to generate a 128 Gbit/s DP 16-QAM signal. Each perturbation-based pre-compensation algorithm was implemented with one sample per symbol, RRC pulse shaping with a roll-off factor of 0.1, and SEDC as applicable. The generated waveforms were loaded into the programmable memory of a Ciena Wavelogic 3 transceiver which interfaced with 4 synchronized digital-to-analog converters (DACs) with a sampling rate of 39.4 GSa/s. The four synchronized DACs are used here since the nonlinear pre-compensations for the *x*-polarization and *y*-polarization signals both depend on the *x*- and *y*-polarization symbols. The output signals from the DACs were applied to a DP in-phase and quadrature (IQ) modulator. The pre-compensated signal was launched into a recirculating loop with four spans and a loop synchronous polarization scrambler (LSPS). Each span consisted of 75 km of standard single mode fiber (SSMF), an erbium doped fiber amplifier (EDFA), and a tunable optical band-pass filter (OBPF). The OBPFs were used in the loop to prevent the EDFAs from being saturated by ASE noise. The sample values obtained after coherent detection were recorded with two real-time sampling oscilloscopes operating at 80 GSa/s and processed off-line.

The off-line signal processing included (i) matched filtering for a RRC pulse shape with a roll-off factor of 0.1, (ii) quadrature imbalance compensation [17], (iii) re-sampling to two samples per symbol, (iv) fixed frequency domain equalization for estimated chromatic dispersion, (v) digital square and filter clock recovery [18], (vi) polarization recovery and residual distortion compensation using 11-tap adaptive equalizers in a butterfly configuration, (vii) carrier frequency offset recovery using a spectral domain algorithm [19], and (viii) carrier phase recovery using a sliding window two-stage simplified QPSK partitioning and QPSK constellation transformation algorithm [20]. The adaptive equalizer used a constant modulus algorithm for pre-convergence followed by a radius directed algorithm [21]. The BER was obtained by direct bit error counting using rectilinear decision boundaries.

The back-to-back signal generated with a RRC pulse shape with a roll-off factor of 0.1 was measured and analyzed. Without any dispersion compensation and amplified spontaneous emission (ASE) noise loading, the recovered constellation diagrams after the off-line signal processing are shown in Fig. 3 for one recorded data file. The average EVM based on 5 data files is 6.74%. The measured optical spectrum with a resolution of 0.38 pm is shown in Fig. 4.

The measured back-to-back BER as a function of the optical signal to noise ratio (OSNR) is shown in Fig. 5 for the 128 Gbit/s DP 16-QAM signal. For a forward error correction (FEC) threshold BER of 2 × 10^{−2}, the implementation penalty for the required OSNR is 1.8 dB relative to the theoretical performance. For a BER of 1 × 10^{−3}, the implementation penalty is 2.3 dB.

## 4. Transmission results and discussion

For a fiber length of 3600 km, the dependence of the BER on the launch power is shown in Fig. 6 for five different algorithms: linear post-compensation for dispersion (LC); Gaussian; symmetric (pre and post) linear compensation for dispersion (LC-SEDC); Gaussian-SEDC; and RRC-SEDC. The number of terms used in the truncated summations was based on $20{\mathrm{log}}_{10}\left|{C}_{m,n}/{C}_{0,0}\right|$>-35 dB. In the nonlinear region, the RRC-SEDC algorithm has a slightly lower BER than the Gaussian and Gaussian-SEDC algorithms. This is attributed to the calculation of the ${C}_{m,n}$ coefficients being based on the actual pulse shape. Without a degradation in the BER performance, the Gaussian-SEDC and RRC-SEDC algorithms reduced the number of summation terms by factors of 3.4 and 6.8, respectively.

The dependence of the BER at optimum launch power on fiber length for the five algorithms is shown in Fig. 7 and the corresponding complexity is shown in Fig. 8. The performances of the three nonlinear pre-compensation algorithms for a selection criterion $20{\mathrm{log}}_{10}\left|{C}_{m,n}/{C}_{0,0}\right|$>-35 dB are similar. However, the RRC-SEDC algorithm reduces the number of summation terms by factors from 6.1 to 7.8 compared to the Gaussian algorithm as the fiber length decreases from 4800 km to 2700 km. For the FEC threshold BER of 2 × 10^{−2}, transmission over 4200 km of fiber was achieved with a reduction in the number of summation terms by a factor of 6.4. Despite the near-linear dependence exhibited by the resultsin Fig. 8, since the number of terms is based on Eq. (15) and the selection criterion $20{\mathrm{log}}_{10}\left|{C}_{m,n}/{C}_{0,0}\right|$>-35 dB, an explicit formula for the complexity is not known.

The selection criterion for ${C}_{m,n}$ may be relaxed at the expense of an increase in the BER. For a fiber length of 3600 km, the pre-compensation algorithms have similar performances for different selection criteria, as shown in Fig. 9. For each ${C}_{m,n}$ selection criterion, the BER is shown at the optimum launch power for each nonlinear pre-compensation algorithm. Compared to the Gaussian algorithm, the reduction in complexity offered by the Gaussian-SEDC algorithm decreases with an increase in the BER and ranges from 3.4 to 2.2 for selection criteria of $20{\mathrm{log}}_{10}\left|{C}_{m,n}/{C}_{0,0}\right|$ from −35 to −10 dB. The corresponding reduction offered by the RRC-SEDC algorithm ranges from 6.8 to 2.2. The numbers of summation terms for a selection criterion of −10 dB, which are difficult to discern in Fig. 9, are 73 for the Gaussian algorithm and 33 for both the Gaussian-SEDC and RRC-SEDC algorithms.

For the RRC-SEDC algorithm, the dependence of the BER and required number of summation terms on the ${C}_{m,n}$ selection criterion is shown in Fig. 10 for fiber lengths of 3000 km, 3600 km and 4200 km. For each ${C}_{m,n}$ selection criterion, the BER is shown at the optimum launch power for each fiber length. For a given ${C}_{m,n}$ selection criterion, the BER increases with the fiber length, which is attributed to the accumulated linear noise and nonlinear impairments. For a given ${C}_{m,n}$ selection criterion, the required number of summation terms increases with the fiber length, which is attributed to the increased dispersion. The numbers of summation terms for a selection criterion of −10 dB, which are difficult to discern in Fig. 10, are 33 for both 3000 km and 3600 km, and 37 for 4200 km.

## 5. Conclusion

Without degrading the BER performance, the complexity of the perturbation-based pre-compensation algorithm has been reduced by a factor of 6.8 for transmission of a 128 Gbit/s DP 16-QAM signal over 3600 km, which is realized by two relatively simple modifications (SEDC and RRC pulse shaping). For all the perturbation based nonlinearity pre-compensation algorithms with and without simplification, the transmission distance of a single channel 128 Gbit/s DP 16-QAM signal has been extended from 3300 km to 4200 km by applying nonlinearity pre-compensation at a FEC threshold of 2 × 10^{−2}.

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