Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links

Xiaojun Liang; Shiva Kumar

doi:10.1364/OE.22.029733

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_mn indicates the coefficient for the nonlinear interaction between the mth, nth, 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_mn 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.

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_stg stages so that N_tot = N_stg × N_spn, where N_spn is the number of spans per stage. The first order field due to the kth stage is calculated using the conventional perturbation theory and it is added to the linear field at the end of the kth 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 kth 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 kth stage can be expressed in the same form as the fiber input, except that the input data { $a_{n}^{i n}$ } is modified as { $a_{n}^{(k)}$ }, so that perturbation coefficient matrix X_mn is the same for all the stages. The proposed technique has the following advantages: (i) the size of the matrix X_mn and hence the implementation complexity is significantly reduced since the size of X_mn 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 N_stg = 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 X_mn 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.

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 ~K²N_p (or ~L²N_p), where N_p is the number of signal pulses. If L is divided into N_stg equal segments, total number of nonlinear interactions scales as ~N_stg(L/N_stg)²N_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 kth 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):

\frac{\partial q}{\partial z} + \frac{α}{2} q + i \frac{β_{2}}{2} \frac{\partial^{2} q}{\partial T^{2}} - i γ_{0} {| q |}^{2} q = 0,

where q is the optical field envelope; α, β₂, and γ₀ are the loss, dispersion and nonlinear coefficients of the optical fiber, respectively. Using the transformation

q (z, T) = e^{- w (z) / 2} u (z, T),

where

w (z) = \int_{0}^{z} α (s) d s

, we obtain the NLSE in the lossless form as

i \frac{\partial u}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} u}{\partial T^{2}} = - γ {| u |}^{2} u,

where

γ = γ_{0} e^{- w (z)}

. Let the input signal be

u (0, T) = \sqrt{P} \sum_{n = - N_{s y m}}^{N_{s y m}} d_{n} p (0, T - n T_{0}),

where P is the signal power, d_n is the data, p(0,T) is the pulse shape function, T₀ is the symbol period, and N_s ( = 2N_sym + 1) is the number of symbols. Using the Nyquist sampling theorem, we express the input signal as

u (0, T) = \sqrt{P} \sum_{n = - N / 2 + 1}^{N / 2} a_{n} g (0, T - n T_{s}),

where N is the number of samples, a_n is the data sample, g(0,T) is a sampling function, and T_s is the sampling period. In this paper, we choose an inter-symbol interference (ISI)-free sampling pulse g(0,T) = sinc(T / T_s), so that a_n is simply the data sample at T = nT_s. Note that 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_s. For example, if T_s 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]

u = u^{(0)} + γ_{0} u^{(1)} + γ_{0}^{2} u^{(2)} + ...,

where u^(m) denotes the mth-order solution. The linear solution is found as

u^{(0)} (z, T) = \sqrt{P} \sum_{n} a_{n} g (z, T - n T_{s}),

where

g (z, T) = \frac{T_{s}}{2 π} \int_{- π / T_{s}}^{π / T_{s}} \exp [i S (z) ω^{2} / 2 - i ω T] d ω,

and

S (z) = \int_{0}^{z} β_{2} (s) d s

. Substituting Eq. (6) into Eq. (3), we find the governing equation for the first order correction as

i \frac{\partial u^{(1)}}{\partial z} - \frac{β_{2}}{2} \frac{\partial^{2} u^{(1)}}{\partial T^{2}} = - e^{- w (z)} {| u^{(0)} |}^{2} u^{(0)} .

Taking the Fourier transform of Eq. (9), we find

\frac{\partial {\tilde{u}}^{(1)}}{\partial z} - i \frac{β_{2}}{2} ω^{2} {\tilde{u}}^{(1)} = \tilde{G} (z, ω),

where

{\tilde{u}}^{(1)} (z, ω) = F {u^{(1)} (z, T)}

,

\tilde{G} (z, ω) = F {G (z, T)}

,

G (z, T) = i e^{- w (z)} {| u^{(0)} |}^{2} u^{(0)}

, and F is the Fourier transform operator. The solution to Eq. (10) with the initial condition

{\tilde{u}}^{(1)} (0, ω) = 0

is

{\tilde{u}}^{(1)} (z, ω) = \int_{0}^{z} \tilde{G} (s, ω) \exp {i [S (z) - S (s)] ω^{2} / 2} d s .

The first order nonlinear distortion is found as

Δ u (z, T) = γ_{0} u^{(1)} (z, T) = γ_{0} F^{- 1} {{\tilde{u}}^{(1)} (z, ω)},

where

F^{- 1}

is the inverse Fourier transform operator. Using the orthogonality of the linear output waveforms g(z,T − nT_s) [24], we rewrite the nonlinear distortion as

Δ u (z, T) = γ_{0} \sqrt{P} \sum_{n} a_{n}^{(1)} g (z, T - n T_{s}),

where the modification to the input data due to nonlinear effects is found as [17,23,24]

\begin{array}{l} a_{n}^{(1)} = \int_{- \infty}^{\infty} \frac{Δ u (z, T)}{T_{s} γ_{0} \sqrt{P}} g^{*} (z, T - n T_{s}) d T, \\ = \frac{i}{T_{s} \sqrt{P}} \int_{0}^{z} d s e^{- w (s)} \int_{- \infty}^{\infty} d T g^{*} (s, T - n T_{s}) {| u^{(0)} (s, T) |}^{2} u^{(0)} (s, T) . \end{array}

Consider the pulse at symbol slot 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]

a_{j}^{(1)} = i P \sum_{m = - K / 2}^{K / 2} \sum_{n = - K / 2}^{K / 2} a_{m + j} a_{n + j} a_{m + n + j}^{*} X_{m n},

X_{m n} = \frac{1}{T_{s}} \int_{0}^{z} d s e^{- w (s)} \int_{- \infty}^{\infty} d T g^{*} (s, T) g (s, T - m T_{s}) g (s, T - n T_{s}) g^{*} (s, T - (m + n) T_{s}),

where K is the number of neighboring samples that interacts nonlinearly with the jth sample, X_mn is the perturbation coefficient matrix. The integration in Eq. (16) is evaluated numerically using Simpson’s 1/3 rule with step sizes dT = T_s / 8 and ds = 0.1 km. Including the nonlinear distortions, the output signal of the first perturbation stage can be written as

u (L, T) = \sqrt{P} \sum_{n} {a^{'}}_{n} g (L, T - n T_{s}),

where the modified data is given as

{a^{'}}_{n} = a_{n} + γ_{0} a_{n}^{(1)}

, and 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_s) rather than g(L,T − nT_s). We rewrite the output signal of the first stage as

u (L, T) = \sqrt{P} \sum_{m} b_{m} g (0, T - m T_{s}),

where b_m is the weight under the basis g(0,T − mT_s). Since g(0,T − mT_s) is a sinc function centered at mT_s,

b_{m} \sqrt{P}

is simply the sample of u(L,T) at mT_s. Using Eq. (17), we find

b_{m} = u (L, T = m T_{s}) / \sqrt{P} = \sum_{n} {a^{'}}_{n} g (L, (m - n) T_{s}) .

Let

c_{m - n} = g (L, (m - n) T_{s}) .

Using Eq. (8), c_n is calculated by

c_{n} = \frac{T_{s}}{2 π} \int_{- π / T_{s}}^{π / T_{s}} \exp [i S (L) ω^{2} / 2 - i ω n T_{s}] d ω .

Substituting Eq. (20) in Eq. (19), we obtain

b_{m} = \sum_{n} {a^{'}}_{n} c_{m - n} .

The convolution in Eq. (22) can be conveniently computed using discrete Fourier transforms as

b_{n} = D F T {I D F T {a'_{n}} \times I D F T {c_{n}}},

where 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., ${a^{'}}_{n} = a_{n} + γ_{0} a_{n}^{(1)}$ ). 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

a_{n = 0}^{(1)} = i P a_{0} [{| a_{0} |}^{2} X_{m = 0, n = 0} + 2 \sum_{n \neq 0} {| a_{n} |}^{2} X_{m = 0, n}] + i P \sum_{m \neq 0} \sum_{n \neq 0} a_{m} a_{n} a_{m + n}^{*} X_{m n},

where the first, second and third terms account for SPM, IXPM and IFWM effects, respectively. Using the following definitions,

ϕ_{n l} = γ_{0} P ({| a_{0} |}^{2} X_{m = 0, n = 0} + 2 \sum_{n \neq 0} {| a_{n} |}^{2} X_{m = 0, n}),

Δ a_{I F W M} = i γ_{0} P \sum_{m \neq 0} \sum_{n \neq 0} a_{m} a_{n} a_{m + n}^{*} X_{m n},

we find the modified data at symbol slot 0 as

{a^{'}}_{n = 0} = a_{0} + γ_{0} a_{n = 0}^{(1)}

= a_{0} (1 + i ϕ_{n l}) + Δ a_{I F W M}

\approx a_{0} \exp (i ϕ_{n l}) + Δ a_{I F W M} .

Equation (28) represents the additive perturbation model, while Eq. (29) corresponds to an additive-multiplicative (A-M) model where the phase noise characteristics of SPM and IXPM effects are included. The sample at time slot j can be calculated similarly using Eqs. (28) and (29) with the time window shifted by jT_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_stg stages, and the number of fiber spans of each perturbation stage is N_spn = N_tot / N_stg. At the receiver, the data samples r_n of the distorted signal are used as the input to the first compensation stage. The signal impairments due to propagating in the last N_spn spans of transmission fibers are removed in the first compensation stage based on the A-M model, using the following steps:

h_{n} = D F T {I D F T {r_{n}} \times I D F T {{c^{'}}_{n}}},

{c^{'}}_{n} = \frac{T_{s}}{2 π} \int_{- π / T_{s}}^{π / T_{s}} \exp [- i S (L) ω^{2} / 2 - i ω n T_{s}] d ω,

v_{n} = (h_{n} - Δ a_{I F W M}) \exp (- i ϕ_{n l}),

where

ϕ_{n l}

and

Δ a_{I F W M}

are respectively calculated using Eqs. (25) and (26), with the data a_n replaced by h_n. 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 X_mn 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 X_mn, as shown in Eqs. (25) and (26)), while DBP involves only a phase rotation that is proportional to the energy of each sample.

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 ps²/km, and 1.1 W⁻¹km⁻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 ${| q_{model} |}^{2} - {| q_{S S F S} |}^{2}$ , where q_model and q_SSFS 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 ${| a_{0} \exp (i ϕ_{n l}) |}^{2} = {| a_{0} |}^{2}$ , while for the additive model, ${| a_{0} (1 + i ϕ_{n l}) |}^{2} > {| a_{0} |}^{2}$ , which is the reason of power over-estimation.

Fig. 1 Comparisons of the additive perturbation model and the additive-multiplicative perturbation model (40-span SSMF_S, power = 2dBm): (a) output signals vs. time, (b) signal square error vs. time, and (c) signal power vs. number of perturbation stages.

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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 $Q = \sqrt{2} e r f c^{- 1} (2 \times B E R)$ .

Fig. 2 Schematic of a single-channel fiber-optic system with multi-stage perturbation-based compensation. Tx: transmitter, BPF: band-pass filter, LPF: low-pass filter, CPR: carrier phase recovery.

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We studied compensation schemes with different number of stages. The signal impairments of N_spn ( = N_tot / N_stg) 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 X_mn (as shown in Eqs. (25) and (26)), which is dependent on N_stg. In practical implementations, the insignificant components in the matrix X_mn are truncated to reduce the computational complexity. We use the following truncation criterion: $20 \log_{10} | X_{m n} / X_{m = 0, n = 0} | > x d B$ , where x is a truncation threshold [9]. Figure 3 shows the perturbation coefficient matrices for different values of N_stg, where 2 samples/symbol are used (i.e., T_s equals half the symbol period). At −40 dB truncation threshold, the numbers of significant terms in X_mn are 173, 6193, and 23289 for the cases when N_stg 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 X_mn 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 N_stg is 40, 8, and 1, respectively.

Fig. 3 Perturbation coefficient matrices, 20log₁₀|X_mn|: (a) N_stg = 40, (b) N_stg = 8, (c) N_stg = 1.

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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.

Fig. 4 Signal constellations: (a) input signal, (b) compensated signal using the additive model, (c) compensated signal using the additive-multiplicative model.

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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_stg 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 N_stg = 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 N_stg = 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 X_mn 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.

Fig. 5 Q-factor versus (a) launch power and (b) truncation threshold, 1 sample/symbol is used.

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Fig. 6 Q-factor versus (a) launch power and (b) truncation threshold, 2 samples/symbol are used.

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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_stg[3M + (Nlog₂N) / N_s], where M is the number of significant perturbation coefficient in X_mn for one compensation stage, N is the number of samples, N_s = (2N_sym + 1) is the number of symbols, the factor 3 accounts for the three multiplications for each coefficient in X_mn, 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 N_stg. As compared with single-stage schemes, in case of 1 sample/symbol and N_stg = 8, the computational complexity and the required memory of storing X_mn are reduced by factors of 4.2 and 34, respectively; in case of 2 samples/symbol and N_stg = 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 N_stg 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 (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.

Table 1. Q-factor improvement (ΔQ), number of complex multiplications per symbol (M_c), and required memory for storing X_mn. Reference Q-factor = 6.2 dB.

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Table 2. Comparison of scheme complexity to achieve ΔQ = 2.0 dB. Reference Q-factor = 6.2 dB.

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Table 3. Comparison of scheme complexity to achieve ΔQ = 3.0 dB. Reference Q-factor = 6.2 dB.

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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.

Fig. 7 Comparisons of multi-stage perturbation-based compensation scheme of different number of stages: (a) compensation performance, (b) computational complexity.

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Fig. 8 (a) Q-factor gain vs. truncation threshold, (b) number of complex multiplications per symbol vs. number of perturbation stages for given Q-factor gains. (2 samples per symbol are used. Reference Q-factor is 6.2 dB.)

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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_mn, leading to reductions both in computational complexity and the required memory for storing X_mn. 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.

References and links

1. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

2. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008). [CrossRef] [PubMed]

3. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010). [CrossRef] [PubMed]

4. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010). [CrossRef]

5. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011). [CrossRef] [PubMed]

6. Y. Gao, J. H. Ke, K. P. Zhong, J. C. Cartledge, and S. S.-H. Yam, “Assessment of intra-channel nonlinear compensation for 112 Gb/s dual polarization 16QAM systems,” J. Lightwave Technol. 30(24), 3902–3910 (2012). [CrossRef]

7. J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013). [CrossRef]

8. S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications, (Wiley, 2014), Chapters. 10 and 11.

9. Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011). [CrossRef]

10. Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Pd3.E.5. [CrossRef]

11. Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014). [CrossRef] [PubMed]

12. Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8. [CrossRef]

13. Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6. [CrossRef]

14. T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3. [CrossRef]

15. Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014). [CrossRef]

16. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000). [CrossRef]

17. A. Mecozzi and R.-J. Essiambre, “Nonlinear shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2010–2024 (2011). [CrossRef]

18. R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunication IVB, I. P. Kaminow and T. Li, ed. (Academic Press, 2002).

19. A. Vannucci, P. Serena, and A. Bononi, “The RP method: A new tool for the iterative solution of the nonlinear Schrödinger equation,” J. Lightwave Technol. 20(7), 1102–1112 (2002). [CrossRef]

20. S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23(6), 2073–2080 (2005). [CrossRef]

21. D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightwave Technol. 27(14), 2916–2923 (2009). [CrossRef]

22. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009). [CrossRef]

23. A. Mecozzi, “A unified theory of intrachannel nonlinearity in pseudolinear transmission,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (Springer, 2011).

24. 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] [PubMed]

25. S. N. Shahi, S. Kumar, and X. Liang, “Analytical modeling of cross-phase modulation in coherent fiber-optic system,” Opt. Express 22(2), 1426–1439 (2014). [CrossRef] [PubMed]

26. X. Liang and S. Kumar, “Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems,” Opt. Express 22(9), 10579–10592 (2014). [CrossRef] [PubMed]

27. X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014). [CrossRef] [PubMed]

28. M. Secondini, D. Marsella, and E. Forestieri, “Enhanced split-step Fourier method for digital backpropagation,” in 40th European Conference and Exhibition on Optical Communication (ECOC 2014), We.3.3.5.

29. 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 38th European Conference and Exhibition on Optical Communication (ECOC 2012), We.2.C.3. [CrossRef]

30. 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). [CrossRef]

N_stg		1	2	4	8	40
1 sample per symbol	ΔQ (dB)	1.8	1.9	2.2	2.4	2.5
	M_c	32,227	10,478	9,820	7,736	2,200
	memory	10,737	1,741	813	317	13
2 samples per symbol	ΔQ (dB)	2.8	3.5	3.9	4.4	5.2
	M_c	248,089	392,306	326,836	148,904	22,120
	memory	82,685	65,373	27,225	6,193	173

N_stg		1	2	4	8	40
1 sample per symbol	M_c	—	—	6,028	3,032	1,240
1 sample per symbol	memory	—	—	497	121	5
2 samples per symbol	M_c	55,477	45,986	39,220	33,320	8,680
2 samples per symbol	memory	18,481	7,653	3,257	1,377	61

N_stg		1	2	4	8	40
1 sample per symbol	ΔQ (dB)	1.8	1.9	2.2	2.4	2.5
	M_c	32,227	10,478	9,820	7,736	2,200
	memory	10,737	1,741	813	317	13
2 samples per symbol	ΔQ (dB)	2.8	3.5	3.9	4.4	5.2
	M_c	248,089	392,306	326,836	148,904	22,120
	memory	82,685	65,373	27,225	6,193	173

N_stg		1	2	4	8	40
1 sample per symbol	M_c	—	—	6,028	3,032	1,240
1 sample per symbol	memory	—	—	497	121	5
2 samples per symbol	M_c	55,477	45,986	39,220	33,320	8,680
2 samples per symbol	memory	18,481	7,653	3,257	1,377	61

Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links

Abstract

1. Introduction

2. Recursive perturbation theory

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

4. Results and discussions

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

4.2 Compensation performance and computational complexity

5. Conclusions

References and links

Cited By

Figures (8)

Tables (3)

Equations (32)

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