## Abstract

A recursive perturbation theory to model the fiber-optic system is developed. Using this perturbation theory, a multi-stage compensation technique to mitigate the intra-channel nonlinear impairments is investigated. The technique is validated by numerical simulations of a single-polarization single-channel fiber-optic system operating at 28 Gbaud, 32-quadrature amplitude modulation (32-QAM), and 40 × 80 km transmission distance. It is found that, with 2 samples per symbol, the multi-stage scheme with eight compensation stages increases the Q-factor as compared with linear compensation by 4.5 dB; as compared with single-stage compensation, the computational complexity is reduced by a factor of 1.3 and the required memory for storing perturbation coefficients is decreased by a factor of 13.

© 2014 Optical Society of America

## 1. Introduction

One of the dominant impairments in fiber-optic systems is caused by fiber intra-channel nonlinear effects. A number of techniques have been proposed to mitigate intra-channel nonlinear impairments such as digital back propagation (DBP) [1–8] and perturbation techniques [9–15]. In DBP, typically split-step Fourier scheme (SSFS) is used to solve the nonlinear Schrödinger equation (NLSE) in digital domain and it provides significant performance improvement if the step size is sufficiently small. In contrast, the existing perturbation-based compensation schemes use single-stage compensation, in which the dispersive and nonlinear distortions of the entire link are compensated all together at one step. Perturbation techniques have drawn significant attention for modeling the fiber-optic systems [16–26] as well as for compensation [9–15,27]. In a first order perturbation theory, the signal field propagating in a fiber is divided into two parts: (i) signal field due to fiber dispersion and loss in the absence of nonlinearity, which is the unperturbed solution; (ii) the first order field due to nonlinear distortions, which is the perturbation correction to the unperturbed solution. The first order correction of a given symbol is calculated by considering its nonlinear interaction with neighboring symbols, where single and double summations involving the perturbation coefficient matrix *X _{mn}* have to be carried out. For a symbol at 0th time slot,

*X*indicates the coefficient for the nonlinear interaction between the

_{mn}*m*th,

*n*th, and (

*m*+

*n*)th symbols. Generally, triple summation is required to calculate the first order field. The phase matching condition is usually applied to reduce the triple summation to double summation [16,17,21]. The single-stage compensation technique based on the first order perturbation theory has the following disadvantages: (i) the computational complexity is large in dispersion noncompensated links since the size of perturbation matrix

*X*is huge due to large accumulated dispersion; (ii) first order perturbation theory becomes inaccurate as the product of the transmission distance and launch power increases [20] and hence, the compensation performance decreases. In [28], a multi-stage nonlinear compensation technique based on a logarithmic perturbation analysis was shown to allow significant complexity reduction without sacrificing performance.

_{mn}In this paper, we develop a novel recursive perturbation theory to model a fiber-optic link. Suppose the fiber-optic link consisting of *N _{tot}* spans is divided into

*N*stages so that

_{stg}*N*=

_{tot}*N*×

_{stg}*N*, where

_{spn}*N*is the number of spans per stage. The first order field due to the

_{spn}*k*th stage is calculated using the conventional perturbation theory and it is added to the linear field at the end of the

*k*th stage. The combined field is used as the input to the (

*k*+ 1)th stage. However, dispersive effect as well as the addition of the first order field alters the pulse shape of the signal field and hence the perturbation coefficients of the

*k*th stage cannot be used for (

*k*+ 1)th stage without adjusting the pulse shape. For this purpose, we project the signal field on the basis of suitably chosen sampling functions. At the end of each stage, weights of the combined field (i.e., linear field + nonlinear distortion) on the basis of the sampling functions are computed. The distorted output signal of the

*k*th stage can be expressed in the same form as the fiber input, except that the input data {$\text{\hspace{0.05em}}{a}_{n}^{in}$} is modified as {$\text{\hspace{0.05em}}{a}_{n}^{(k)}$}, so that perturbation coefficient matrix

*X*is the same for all the stages. The proposed technique has the following advantages: (i) the size of the matrix

_{mn}*X*and hence the implementation complexity is significantly reduced since the size of

_{mn}*X*is determined by the accumulated dispersion in each stage rather than that of the entire fiber-optic link; (ii) the accuracy of the perturbation technique is enhanced, resulting in better performance. This is because, in each stage the first order theory is used to calculate the nonlinear distortions occurring in a shorter distance, which improves the accuracy and the summation of the signal field and the first order field of the previous stage is used as the unperturbed solution for the next stage, which further improves the accuracy. We investigated a 28 Gbaud single channel system with 32-quadrature amplitude modulation (32-QAM) format and 3,200 km transmission distance. With 2 samples per symbol, the multi-stage scheme with

_{mn}*N*= 8 increases the Q-factor as compared with linear compensation by 4.5 dB; as compared with single-stage compensation, the computational complexity is reduced by a factor of 1.3 and the required memory for storing matrix

_{stg}*X*is decreased by a factor of 13. For systems with quadrature phase-shift keying (QPSK) or 16-QAM, we expect similar performance improvements using the multi-stage perturbation technique. In this paper, the multi-stage perturbation technique is implemented at the receiver; however, this technique can also be deployed at the transmitter and similar performance improvements are expected. The simulations in this paper are based on single-polarization systems; however, the multi-stage perturbation technique can be properly modified to apply to dual-polarization systems and similar performance improvement is expected.

_{mn}For DBP, increasing the number of stages (or steps) increases the computational complexity (and improves the performance, too). In contrast, for multi-stage perturbation technique, it is the opposite unless the number of stages is too large. This is because a given pulse interacts nonlinearly with *K* neighboring pulses over a distance *L* and *K* is roughly proportional to *L*. Total number of nonlinear interactions (and hence the number of complex multiplications) scales as ~*K*^{2}*N _{p}* (or ~

*L*

^{2}

*N*), where

_{p}*N*is the number of signal pulses. If

_{p}*L*is divided into

*N*equal segments, total number of nonlinear interactions scales as ~

_{stg}*N*(

_{stg}*L/N*)

_{stg}^{2}

*N*.

_{p}## 2. Recursive perturbation theory

In this paper, we develop a recursive perturbation theory to model the evolution of optical signals propagating in fiber-optic links. A fiber-optic link is divided into multiple stages and the dispersive and nonlinear effects of each stage are calculated using a first order perturbation technique. The input and output signals of different stages are expressed using the same basis functions such that the perturbation calculations of different stages can be done in a similar way using an identical perturbation coefficient matrix *X _{mn}*. Perturbation calculations are implemented recursively, that is, the output signal of the

*k*th perturbation stage is used as the input signal of the (

*k*+ 1)th perturbation stage. The optical signal propagation is described by the nonlinear Schrödinger equation (NLSE):

*q*is the optical field envelope;

*α*,

*β*, and

_{2}*γ*are the loss, dispersion and nonlinear coefficients of the optical fiber, respectively. Using the transformationwhere $w(z)={\displaystyle {\int}_{0}^{z}\alpha (s)ds}$, we obtain the NLSE in the lossless form as

_{0}*P*is the signal power,

*d*is the data,

_{n}*p*(0,

*T*) is the pulse shape function,

*T*

_{0}is the symbol period, and

*N*( = 2

_{s}*N*+ 1) is the number of symbols. Using the Nyquist sampling theorem, we express the input signal aswhere

_{sym}*N*is the number of samples,

*a*is the data sample,

_{n}*g*(0,

*T*) is a sampling function, and

*T*is the sampling period. In this paper, we choose an inter-symbol interference (ISI)-free sampling pulse

_{s}*g*(0,

*T*) = sinc(

*T*/

*T*), so that

_{s}*a*is simply the data sample at

_{n}*T*=

*nT*. Note that

_{s}*g*(0,

*T*) is the sampling function, not necessarily the same as the symbol pulse shape

*p*(0,

*T*). The symbol pulse shape can be arbitrary. Also, the mathematical derivation can be applied to the cases of multiple samples per symbol by properly choosing

*T*. For example, if

_{s}*T*equals to half the symbol period, the perturbation calculations will be carried out with two samples per symbol. Using the perturbation technique, we assume that the leading order solution of Eq. (3) is linear and treat the nonlinear terms on the right-hand side as perturbations. We expand the optical field into a series [19,20]where

_{s}*u*denotes the

^{(m)}*m*th-order solution. The linear solution is found as

*F*is the Fourier transform operator. The solution to Eq. (10) with the initial condition ${\tilde{u}}^{(1)}(0,\omega )=0$ is

*g*(

*z*,

*T*−

*nT*) [24], we rewrite the nonlinear distortion aswhere the modification to the input data due to nonlinear effects is found as [17,23,24]

_{s}*j*. Substituting Eq. (7) into Eq. (14) and using the phase matching condition

*m*+

*n*−

*l*=

*j*, where

*m*,

*n*, and

*l*are sample indices, we obtain [11,17,23,24]

*K*is the number of neighboring samples that interacts nonlinearly with the

*j*th sample,

*X*is the perturbation coefficient matrix. The integration in Eq. (16) is evaluated numerically using Simpson’s 1/3 rule with step sizes

_{mn}*dT*=

*T*/ 8 and

_{s}*ds*= 0.1 km. Including the nonlinear distortions, the output signal of the first perturbation stage can be written as

*L*is the fiber length of the first perturbation stage.

To make the perturbation coefficient matrix *X _{mn}* independent of the stages, we express the output signal of the first stage on the basis

*g*(

*0*,

*T*−

*nT*) rather than

_{s}*g*(

*L*,

*T*−

*nT*). We rewrite the output signal of the first stage as

_{s}*b*is the weight under the basis

_{m}*g*(0,

*T*−

*mT*). Since

_{s}*g*(0,

*T*−

*mT*) is a sinc function centered at

_{s}*mT*, ${b}_{m}\sqrt{P}$ is simply the sample of

_{s}*u*(

*L*,

*T*) at

*mT*. Using Eq. (17), we findLetUsing Eq. (8),

_{s}*c*is calculated by

_{n}*DFT*and

*IDFT*indicate the discrete Fourier transform and the inverse discrete Fourier transform, respectively. Now, the output signal [Eq. (18)] of the first perturbation stage is expressed in the same form as the input signal [Eq. (5)], so the output signal of the second perturbation stage can be calculated recursively by using the output data of the first stage as its input data.

So far, we considered an additive perturbation model, where the nonlinear distortion is added to the linear field (i.e., ${{a}^{\prime}}_{n}={a}_{n}+{\gamma}_{0}{a}_{n}^{(1)}$). Equation (15) shows that the nonlinear distortion ${a}_{n}^{(1)}$ includes self-phase modulation (SPM), intra-channel cross-phase modulation (IXPM) and intra-channel four wave mixing (IFWM) effects. In order to accurately model the phase noise characteristics of SPM and IXPM effects, it is useful to introduce a multiplicative model for the SPM and IXPM effects while keeping the additive model for IFWM effect [29]. At symbol slot 0, we expand Eq. (15) as

*j*can be calculated similarly using Eqs. (28) and (29) with the time window shifted by

*j*T

_{s}.

## 3. Multi-stage compensation scheme based on recursive perturbation theory

A multi-stage compensation scheme is developed based on the recursive perturbation theory. Consider a fiber-optic link consisting of *N _{tot}* spans. We divide the compensation process into

*N*stages, and the number of fiber spans of each perturbation stage is

_{stg}*N*=

_{spn}*N*/

_{tot}*N*. At the receiver, the data samples

_{stg}*r*of the distorted signal are used as the input to the first compensation stage. The signal impairments due to propagating in the last

_{n}*N*spans of transmission fibers are removed in the first compensation stage based on the A-M model, using the following steps:

_{spn}*a*replaced by

_{n}*h*. In the next compensation stage, the output data of the previous stage is used as the input data, and the same compensation procedures are implemented using an identical perturbation coefficient matrix

_{n}*X*as the first stage. In the multi-stage perturbation technique, the propagation of signal field is reversed in a multi-stage flow where dispersion and nonlinearity are removed in each stage, similar to the DBP scheme. The difference between the two algorithms is that the multi-stage perturbation technique provides a better approximation of the nonlinear operation in each stage. The multi-stage perturbation technique employs a perturbation analysis and calculates the nonlinear distortions by summing over multiple triplets (using the perturbation coefficient matrix

_{mn}*X*, as shown in Eqs. (25) and (26)), while DBP involves only a phase rotation that is proportional to the energy of each sample.

_{mn}## 4. Results and discussions

#### 4.1 Comparisons of the additive model and the additive-multiplicative model

First we compare the additive perturbation model with the additive-multiplicative (A-M) perturbation model for optical signals propagating in a dispersion noncompensated fiber-optic link. The input signal is 28 Gbaud, modulated with 32-QAM format using a raised cosine pulse with a roll-off factor of 0.1. We note that better performance at the linear regime can be achieved using a square-root raised cosine pulse at the transmitter and a corresponding matched filter at the receiver. The launch power is 2 dBm. The fiber-optic link consists of 40 spans of standard single mode fiber (SSMF) with 80 km amplifier spacing. The loss, dispersion, and nonlinear coefficients of the SSMF are 0.2 dB/km, −21 ps^{2}/km, and 1.1 W^{−1}km^{−1}, respectively. An in-line amplifier is used after each fiber span to fully compensate the fiber loss. The output signals calculated using the additive model and the A-M model are compared with the numerical simulation result based on SSFS. The amplifier noise is turned off to obtain the results in Fig. 1. Figure 1(a) shows that the result of the A-M model agrees well with the numerical result, while the result of the additive model shows the overestimated signal amplitude which has been observed in [27]. Figure 1(b) shows the signal square errors of the two models. The signal square error is calculated using ${\left|{q}_{\text{model}}\right|}^{2}-{\left|{q}_{SSFS}\right|}^{2}$, where *q _{model}* and

*q*are the output signal fields obtained using the additive or A-M models and the SSFS simulation, respectively. The mean square errors are 0.72 mW and −0.06 mW for the additive model and the A-M model, respectively. Figure 1(c) shows the signal power as a function of the number of perturbation stages, where each stage includes one fiber span. It shows that the over-estimation of signal power in the additive model accumulates over perturbation stages. This power over-estimation is due to the neglecting of phase noise characteristics of SPM and IXPM effects. Comparing Eqs. (28) and (29), we note that the power is conserved in the A-M model since ${\left|{a}_{0}\mathrm{exp}\left(i{\varphi}_{nl}\right)\right|}^{2}={\left|{a}_{0}\right|}^{2}$, while for the additive model, ${\left|{a}_{0}\left(1+i{\varphi}_{nl}\right)\right|}^{2}>{\left|{a}_{0}\right|}^{2}$, which is the reason of power over-estimation.

_{SSFS}Enhanced regular perturbation (ERP) in [19] is an interesting technique to remove the average nonlinear phase before applying the perturbative analysis. ERP is an alternative technique that could help to partially cope with the energy divergence problem. Both A-M and ERP techniques are expected to provide similar results.

#### 4.2 Compensation performance and computational complexity

We investigated a single-channel fiber-optic system to study the performance of the multi-stage compensation scheme based on the recursive perturbation theory, as shown in Fig. 2. The system configuration is as follows: symbol rate = 28 Gbaud, modulation format = 32-QAM, amplifier spacing = 80 km, total number of fiber spans *N _{tot}* = 40, transmission fiber = SSMF, linewidth of transmitter and local oscillator lasers = 100 kHz. In each Monte-Carlo simulation, 65,536 symbols are transmitted all together. The optical signal is modulated using a raised cosine pulse with a roll-off factor of 0.1. A second order Gaussian band pass filter (BPF) with 3-dB bandwidth of 40 GHz is used before the coherent receiver. After multi-stage perturbation-based compensation, a second order Gaussian low pass filter (LPF) with 3-dB bandwidth of 15 GHz is used to limit out-of-band noise. Carrier phase recovery (CPR) is then implemented using the feedforward method [30]. Bit error rate (BER) is calculated by counting the number of error bits, and Q-factor is converted from BER using $Q=\sqrt{2}erf{c}^{-1}(2\times BER)$.

We studied compensation schemes with different number of stages. The signal impairments of *N _{spn}* ( =

*N*/

_{tot}*N*) fiber spans is compensated in one stage. The computational complexity of the perturbation-based compensation scheme is proportional to the size of the coefficient matrix

_{stg}*X*(as shown in Eqs. (25) and (26)), which is dependent on

_{mn}*N*. In practical implementations, the insignificant components in the matrix

_{stg}*X*are truncated to reduce the computational complexity. We use the following truncation criterion: $20{\mathrm{log}}_{10}\left|{X}_{mn}/{X}_{m=0,n=0}\right|>x\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}dB$, where

_{mn}*x*is a truncation threshold [9]. Figure 3 shows the perturbation coefficient matrices for different values of

*N*, where 2 samples/symbol are used (i.e.,

_{stg}*T*equals half the symbol period). At −40 dB truncation threshold, the numbers of significant terms in

_{s}*X*are 173, 6193, and 23289 for the cases when

_{mn}*N*is 40, 8, and 1, respectively. Due to the reduced dispersive effect within one stage, the matrix size and hence the required memory in DSP for storing

_{stg}*X*are significantly reduced in multi-stage schemes. Also, the number of neighboring samples required for perturbation calculation is reduced. Figure 3 shows that the required numbers of neighboring samples are 28, 194, and 1544 for the cases when

_{mn}*N*is 40, 8, and 1, respectively.

_{stg}Figure 4 shows the compensated signal constellations, in comparison with the input signal constellation. The launch power is 2 dBm and *N _{stg}* = 8. The constellation of the compensated signal obtained using the additive model is expanded, indicating over-estimation of signal power, while the power of the compensated signal using the A-M model is conserved, which is consistent with the discussion in section 4.1.

Figures (5) and (6) show the compensation performance of the A-M model, where 1 sample/symbol and 2 samples/symbol are used, respectively. In the case of 1 sample/symbol, Fig. 5(a) shows that Q-factor improvements over the linear compensation scheme are 1.9, 2.0, 2.3, 2.5, and 2.6 dB, when *N _{stg}* is 1, 2, 4, 8 and 40, respectively. In the case of 2 samples/symbol, Fig. 6(a) shows that the Q-factor improvements are 2.9, 3.6, 4.0, 4.5, and 5.3 dB, when

*N*is 1, 2, 4, 8 and 40, respectively. In this paper, we have simulated a single-channel transmission system. In the case of wavelength division multiplexing (WDM) systems, the performance improvements are expected to be significantly lower unless inter-channel impairments due to cross-phase modulation (XPM) are mitigated [27]. The compensation schemes using 2 samples/symbol bring 1.0~2.7 dB improvements in Q-factor as compared with 1 sample/symbol cases, due to the fact that the nonlinear effects broaden the signal spectrum and hence the sampling rate should exceed the symbol rate which roughly equals the bandwidth of the linear signal. As compared with single-stage compensation schemes, multi-stage schemes with

_{stg}*N*= 8 bring 0.6 and 1.6 dB improvements in Q-factor for the cases of 1 sample/symbol and 2 samples/symbol, respectively; multi-stage schemes with

_{stg}*N*= 40 bring 0.7 and 2.4 dB improvements in Q-factor for the cases of 1 sample/symbol and 2 samples/symbol, respectively. Figures 5(b) and 6(b) show that compensation performance degrades as the truncation threshold increases. The optimal launch powers that maximize Q-factors, obtained from Figs. 5(a) and 6(a) are chosen as the launch powers for Figs. 5(b) and 6(b), respectively. When the truncation threshold is too high (> −30 dB), the number of coefficients in

_{stg}*X*is not adequate to provide an accurate estimation of nonlinear distortions. This inaccuracy will accumulate with the number of perturbation stages. The build-up of uncompensated nonlinear distortions in a multi-stage scheme could be larger than that in a single-stage in some cases, as shown in Fig. 6(b) at high truncation thresholds.

_{mn}In order to reduce the implementation complexity, we select the truncation threshold based on a criterion that the reduction of Q-factor from its value at −50 dB truncation threshold is within 0.1 dB. To estimate the computational complexity, the numbers of complex multiplications per symbol is found as *M _{c}* =

*N*[3

_{stg}*M*+ (

*N*log

_{2}

*N*)

*/ N*], where

_{s}*M*is the number of significant perturbation coefficient in

*X*for one compensation stage,

_{mn}*N*is the number of samples,

*N*= (2

_{s}*N*+ 1) is the number of symbols, the factor 3 accounts for the three multiplications for each coefficient in

_{sym}*X*, as shown in Eq. (26), and the logarithm terms correspond to the fast Fourier transforms (FFTs) in Eq. (30). Table 1. compares the Q-factor improvement over the linear compensation scheme, computational complexity, and required memory for multi-stage compensation schemes with different

_{mn}*N*. As compared with single-stage schemes, in case of 1 sample/symbol and

_{stg}*N*= 8, the computational complexity and the required memory of storing

_{stg}*X*are reduced by factors of 4.2 and 34, respectively; in case of 2 samples/symbol and

_{mn}*N*= 8, the computational complexity and the required memory are reduced by factors of 1.3 and 13, respectively. The reference Q-factor correspondes to the value obtained by using only the linear compensation at the receiver. Table 2. and Table 3. compare the implementation complexities of schemes with different values of

_{stg}*N*to achieve Q-factor improvements of 2.0 dB and 3.0 dB over the scheme consisting of linear compensation only, respectively. The “—” sign indicate that the given Q-factor is not achievable by the scheme. The required memory is related with the number of samples per symbol, the accumulated dispersion within one perturbation stage, and the truncation threshold. When we increases from 1 sample/symbol to 2 samples/symbol, the pulse width of the sampling function (

_{stg}*g*(0,

*T*) in Eq. (5)) decreases by half, and as a result, the broadening in the pulse width of

*g*(

*z*,

*T*) is increased for a given transmission distance, which increases the required memory.

Figures 7(a) and 7(b) show that as the number of perturbation stage increases, the compensation performance increases and also the computational complexity decreases. Compared with the conventional single-stage compensation scheme with 1 sample/symbol, the multi-stage scheme with *N _{stg}* = 8 and 2 samples/symbol enhances the Q-factor improvement by 2.6 dB. To determine the optimum number of stages in terms of computational complexity for the given Q-factor gain, we first calculated the truncation threshold for the fixed Q-factor gain for various number of perturbation stages, as shown in Fig. 8(a). Since the computational complexity depends on both the truncation threshold and the number of perturbation stages, we calculate the number of complex multiplications per symbol as a function of number of perturbation stages for the fixed Q-factor gains of 3.0 dB and 4.0 dB, which is shown in Fig. 8(b). As can be seen, the number of multiplications per symbol decreases initially and then it increases when the number of stages is large. In this example, the optimum number of stages is 40. However, this optimal value may depend on the system configuration.

## 5. Conclusions

We have investigated a multi-stage scheme based on a recursive perturbation theory to compensate for intra-channel nonlinear impairments. The input signals of different compensation stages are expressed using the same basis functions so that the perturbation coefficient matrix *X _{mn}* is the same for all stages. The multi-stage compensation is implemented recursively. In each stage the first order theory is used to calculate the nonlinear distortions occurring in a distance much shorter than the entire fiber-optic link, which improves the accuracy and the summation of the signal field and the first order field of the previous stage is used as the unperturbed solution for the next stage, which further improves the accuracy. Moreover, the accumulated dispersion in one compensation stage is much smaller than that of the entire link which significantly reduces the size of the matrix

*X*, leading to reductions both in computational complexity and the required memory for storing

_{mn}*X*. Numerical simulations of a 28 Gbaud single-polarization single-channel fiber-optic system with 32-QAM and 40 × 80 km transmission distance show that, with 2 samples per symbol, the multi-stage scheme with eight compensation stages increases the Q-factor as compared with the linear compensation scheme by 4.5 dB; as compared with single-stage compensation, the computational complexity is reduced by a factor of 1.3 and the required memory for storing perturbation coefficients is decreased by a factor of 13.

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