Correlated digital back propagation based on perturbation theory

Xiaojun Liang; Shiva Kumar

doi:10.1364/OE.23.014655

1. Introduction

Digital back propagation (DBP) is an effective technique to mitigate dispersion and nonlinear impairments in fiber-optic transmission systems [1–6]. In DBP, virtual fibers are implemented in the digital signal processing (DSP) unit of the coherent receiver whose dispersion, loss and nonlinear profiles are inverse of transmission fibers. The signal propagation in the virtual fibers is governed by the nonlinear Schrödinger equation (NLSE) and typically, split-step Fourier method (SSFM) is used to solve the NLSE in digital domain. In enhanced or correlated DBP [7–12], the nonlinear phase shift on one symbol is not only related to the power of that symbol but also related to the powers of the neighboring symbols due to pulse broadening.

Time domain perturbation techniques have drawn significant attention for modeling [13–20] of fiber-optic system as well as for digital compensation of fiber nonlinearities [21–27]. In multi-stage perturbation technique [27], the fiber-optic link is divided into several stages. In each stage, nonlinear distortion is calculated using a first order perturbation theory and it is added to the unperturbed signal field. The signal field is expressed as a superposition of suitably chosen orthonormal basis functions and the weighting factors of the basis functions are updated at each stage – first by calculating the first order perturbation correction which takes into account the interplay between dispersion and nonlinearity and then taking into account the dispersive effect to the unperturbed solution. Hence, this technique may be considered as a form of asymmetric split-step scheme. First order perturbation correction terms can be divided into two groups: (i) phase-independent terms (similar to self-phase modulation (SPM) and intra-channel cross-phase modulation (IXPM)) and (ii) phase-dependent terms (similar to intra-channel four-wave mixing (IFWM)). The computational cost per sample of calculating phase-dependent terms in each stage scales as ${(2 N_{n b})}^{2}$ where $N_{n b}$ is the number of interacting samples on one side, while that of phase-independent terms scales as $N_{n b} / 4$ . In this paper, we modify the technique of [27] and make connections with the correlated or enhanced DBP developed in [7–12]. We make the following modifications to the technique of [27]: (a) a symmetric split-step scheme is realized by introducing the nonlinear correction (based on the first order perturbation theory) in the middle of the step rather than at the end. (b) in [27], perturbation terms due to both phase-independent and phase-dependent terms are included. Although the inclusion of the phase-dependent terms improves the accuracy, due to the enormous computational cost of these calculations, we ignore those terms in this paper. The scheme investigated in this paper is similar to correlated or enhanced DBP [7–12], yet it is different in the following ways: (i) nonlinear phase shifts due to neighboring symbols are added with certain coefficients which are obtained based on measured signal correlation [7], perturbation analysis and numerical optimization [8,9], time-domain filters [10,11], or adaptive numerical optimization [12]. In our approach, we develop an analytic expression for these correlation coefficients using the perturbation theory. When the knowledge of the system characteristics is approximately known, one may have to resort to numerical optimization and analytic expressions developed here could guide the numerical optimization. (ii) in this paper, weighting factors of the signal field are calculated using sinc basis functions and periodically updated whereas in standard DBP [1,2] or correlated DBP [7–12], signal field is expressed as a superposition of sinusoids using discrete Fourier transform (DFT).

Our simulation results show that for a single-channel single-polarization 32-Quadrature amplitude modulation (QAM), 40 × 80 km fiber-optic system, correlated DBP with number of steps, N_stp = 4, provides roughly the same performance as that of standard DBP with N_stp = 40 with the reduction in computational cost by a factor of 6.7.

2. Digital back propagation and split-step Fourier method

The signal propagation in a fiber-optic link is described by the nonlinear Schrödinger equation (NLSE)

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

where q(z,t) is the optical field envelope; α(z), β₂(z), and γ₀ are the loss, dispersion and nonlinear coefficients of the optical fiber, respectively. The NLSE can be written in the lossless form as

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

using the transformation

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

where

γ (z) = γ_{0} e^{- w (z)}

and

w (z) = \int_{0}^{z} α (s) d s

. In multi-span fiber optic systems,

w (z) = \mod (z, L_{a}) α

, where L_a is the amplifier spacing. Signal is distorted due to the interplay of fiber dispersion and nonlinearity. Digital back propagation (DBP) can undo the distortions by reversing the signal propagation in a virtual fiber. The output optical signal of the real fiber-optic link is converted to the digital domain using a coherent receiver. Then the digital signal is down-sampled and used as the input to the virtual fiber. The virtual fiber has parameters whose signs are opposite of those of the transmission fiber: α_b = -α, β_2,b = -β₂, and γ_0,b = -γ₀. Signal propagation in the virtual fiber is solved in the digital domain and its output signal becomes the same as the input field to the real fiber-optic link, if amplifier noises are ignored.

SSFM is usually used to solve the NLSE. The fiber link is divided into N_stp steps and the linear and nonlinear operators are implemented separately within each step, which are defined as

\hat{D} = - i \frac{β_{2} (z)}{2} \frac{\partial^{2}}{\partial t^{2}},

\hat{N} = i γ_{0} e^{- α z} {| u (z, t) |}^{2} .

In symmetric SSFM, three operations are carried out in each step [28]:

u (Δ z, t) = \exp (\frac{Δ z}{2} \hat{D}) \exp [i γ_{0} \int_{0}^{Δ z} e^{- α z} {| u (z, t) |}^{2} d z] \exp (\frac{Δ z}{2} \hat{D}) u (0, t),

where

Δ z

is the step size. The linear operation,

\exp (\frac{Δ z}{2} \hat{D}) u

can be efficiently calculated in the frequency domain using fast Fourier transforms (FFTs). The phase term in the nonlinear operation is difficult to calculate due to the dependence of u on z inside the integral. To reduce computational cost, the pulse broadening is ignored in the evaluation of the integral while the power change is included using an effective step length. The nonlinear phase is calculated at the middle of a step as [29,30]

θ (t) = i γ_{0} {| u (Δ z / 2, t) |}^{2} L_{e f f},

where

L_{e f f} = \frac{1 - \exp (- α Δ z)}{α} .

The accuracy of SSFM is determined by the step size. Smaller step size leads to accurate results, but the computational cost increases and hence the step size is chosen based on the desired accuracy.

3. Correlated DBP based on perturbation theory

In DBP based on standard SSFM, the nonlinear phase shift is calculated using the instantaneous signal power and the signal evolution within the step is ignored, as shown in Eq. (7). The nonlinear phase shift of each signal sample is proportional to the instantaneous power of the same sample only, without taking into account the correlation between neighboring samples. The accuracy of SSFM can be substantially increased when the sample correlations are included [12]. The sample correlations can be included in the calculation of nonlinear phase shift by monitoring the signal evolution within the step size. In this paper, we derive analytical expressions for the correlation coefficients between neighboring weighting factors using a first order perturbation theory.

Consider an input signal to the fiber-optic link as

u (0, t) = \sqrt{P} \sum_{n = - N_{s y m} / 2 + 1}^{N_{s y m} / 2} 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_sym is the number of symbols. Using the Nyquist sampling theorem, we rewrite the input signal as [27]

u (0, t) = \sqrt{P} \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} a_{n} g (0, t - n T_{s}),

where N_smp 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. Multiplying Eqs. (9) and (10) by

g^{*} (0, t - n T_{s}) / \sqrt{T_{s}}

and integrating from -∞ to ∞, we obtain

a_{n} = \sum_{k = - N_{s y m} / 2 + 1}^{N_{s y m} / 2} d_{k} \int_{- \infty}^{\infty} \frac{1}{\sqrt{T_{s}}} p (0, t - k T_{0}) g^{*} (0, t - n T_{s}) d t .

From Eq. (11), we find that a_n depends on the pulse shape function

p (0, t)

because of the overlap integral. 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. This sampling function g(0,t) could be quite different from the symbol pulse shape function p(0,t). Using a perturbation technique, we assume that the leading order solution of Eq. (2) is linear and treat the nonlinear terms on the right-hand side as perturbations. We expand the field into a series

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

where u^(m)(z,t) denotes the mth-order solution. Substituting Eq. (12) into Eq. (2), we find the governing equation for the first order correction as

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

The linear solution is given by

u^{(0)} (z, t) = \sqrt{P} \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} a_{n} g (z, t - n T_{s}),

where

g (z, t) = \hat{P} (z) {g (0, t)},

and the propagation operator is defined as

\hat{P} (z) = \exp [- i \frac{S (z)}{2} \frac{\partial^{2}}{\partial t^{2}}],

and

S (z) = \int_{0}^{z} β_{2} (s) d s

. When

g (0, t) = \sin c (t / T_{s})

, Eq. (15) becomes

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 .

At the middle of a DBP step, the linear solution is

u^{(0)} (Δ z / 2, t) = \sqrt{P} \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} a_{n} g (Δ z / 2, t - n T_{s}) .

We can rewrite this solution using the basis functions g(0,t − nT_s) as

u^{(0)} (Δ z / 2, t) = \sqrt{P} \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} b_{n} g (0, t - n T_{s}) .

Since the basis functions

g (0, t - n T_{s})

are orthogonal, it follows that [27]

b_{k} = \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} a_{n} v_{k - n},

where

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

The convolution in Eq. (20) can be conveniently computed using DFT and inverse DFT (IDFT) as [27]

b_{m} = I D F T {D F T {a_{n}} \times D F T {v_{n}}} .

The nonlinear distortion accumulated at the end of a step is found by solving Eq. (13) as [17,27]

\begin{array}{l} Δ u (Δ z, t) = γ_{0} u^{(1)} (Δ z, t) \\ = γ_{0} \int_{0}^{Δ z} \hat{P} (Δ z - s) {F (s, t)} d s, \end{array}

where

F (s, t) = i e^{- w (s)} {| u^{(0)} (s, t) |}^{2} u^{(0)} (s, t) .

In order to facilitate a symmetric split-step technique, we linearly propagate the nonlinear distortion of Eq. (23) backward by a half step size to reach the middle point.

\begin{array}{l} Δ u^{'} (Δ z, t) = \hat{P} (- Δ z / 2) {Δ u (Δ z, t)} \\ = γ_{0} \int_{0}^{Δ z} \hat{P} (Δ z / 2 - s) {F (s, t)} d s . \end{array}

Δ u^{'}

can also be rewritten using the basis functions g(0,t − kT_s) as

Δ u^{'} (Δ z, t) = γ_{0} \sqrt{P} \sum_{k = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} b_{k}^{(1)} g (0, t - k T_{s}) .

Multiplying Eqs. (25) and (26) by

g^{*} (0, t - k T_{s}) / \sqrt{T_{s}}

and integrating from -∞ to ∞, we find the nonlinear distortion of the kth sample as

b_{k}^{(1)} = \frac{i}{T_{s} \sqrt{P}} \int_{0}^{Δ z} d s e^{- w (s)} \int_{- \infty}^{\infty} d t {| u^{(0)} (s, t) |}^{2} u^{(0)} (s, t) g^{*} (s - Δ z / 2, t - k T_{s}) .

γ_{0} b_{k}^{(1)}

may be interpreted as the first order change in the unperturbed weight b_k (see Eqs. (19) and (26)) due to fiber nonlinear effects. In order to calculate

u^{(0)} (s, t)

appearing in Eq. (27), we proceed as follows. From Eq. (19), we find that the unperturbed signal field at Δz/2 is the superposition of basis functions g(0,t-nT_s) with weighting factors b_n. The unperturbed field at

s \in [0, Δ z]

may be written as

u^{(0)} (s, t) = \sqrt{P} \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} b_{n} g (s - Δ z / 2, t - n T_{s}) .

Equation (28) indicates that the unperturbed field for s < Δz/2 is calculated by back-propagating

u^{(0)} (Δ z / 2, t)

(linearly) a distance of (Δz/2 - s). Substituting Eq. (28) into Eq. (27) and using the phase matching condition m + n − l = k, where m, n, and l are sample indices, we obtain

b_{k}^{(1)} = i P \sum_{m = - N_{n b}}^{N_{n b}} \sum_{n = - N_{n b}}^{N_{n b}} b_{m + k} b_{n + k} b_{m + n + k}^{*} X_{m n},

where N_nb is the number of neighboring weighting factors on one side, up to which the nonlinear interaction is significant and

X_{m n} = \frac{1}{T_{s}} \int_{0}^{Δ z} d s e^{- w (s)} \int_{- \infty}^{\infty} d t g^{*} (s - \frac{Δ z}{2}, t) g (s - \frac{Δ z}{2}, t - m T_{s}) g (s - \frac{Δ z}{2}, t - n T_{s}) g^{*} (s - \frac{Δ z}{2}, t - (m + n) T_{s}) .

The nonlinear distortion

b_{k}^{(1)}

includes phase-independent perturbation due to

{| b_{m} |}^{2} b_{n}

(similar to SPM / IXPM) and phase-dependent perturbation due to

b_{m} b_{n} b_{l}^{*}

(similar to IFWM). Ignoring the phase-dependent perturbation terms that require enormous computational resources, we obtain the modified data at the middle point using Eqs. (19) and (26)

{b^{'}}_{k} = b_{k} + γ_{0} b_{k}^{(1)} \approx b_{k} \exp (i ϕ_{n l, k}),

ϕ_{n l, k} = γ_{0} P ({| b_{k} |}^{2} X_{k} + 2 \sum_{n = 1}^{N_{n b}} [({| b_{n} |}^{2} + {| b_{- n} |}^{2}) X_{n}]),

X_{n} \equiv X_{m = 0, n} = \frac{1}{T_{s}} \int_{0}^{Δ z} d s e^{- w (s)} \int_{- \infty}^{\infty} d t {| g (s - \frac{Δ z}{2}, t) |}^{2} {| g (s - \frac{Δ z}{2}, t - n T_{s}) |}^{2},

where we have used the symmetry

X_{n} = X_{- n}

. The integrals in Eq. (33) are evaluated numerically using Simpsons 1/3 rule with step sizes dt = T_s / 8 and ds = 0.1 km. Equation (33) shows that X_n are real numbers. As a result, we need to do real multiplications as opposed to complex multiplications required in digital compensation schemes based on perturbation theory [21,24–27]. The signal field after taking nonlinear distortion into account is

u (Δ z / 2, t) = \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} {b^{'}}_{n} g (0, t - n T_{s}) .

The main difference between the conventional DBP and this approach is that in conventional DBP nonlinear phase introduced at

Δ z / 2

is proportional to the absolute square of the signal at

Δ z / 2

while in our approach the nonlinear distortion is calculated by integrating the distortion occurring within the interval

[0 Δ z]

as done through Eqs. (23-33). As a result,

Δ z

can be chosen sufficiently large without losing much accuracy. In the multi-stage perturbation theory of [27], phase-dependent terms of Eq. (29) are not ignored which helps to reduce the number of perturbation stages; however, overall computational cost increases since phase-dependent terms in Eq. (29) scale as

~ {(2 N_{n b})}^{2}

while phase-independent terms scale as

2 N_{n b}

. In standard DBP, only SPM term is included and hence, the step size has to be really small. The signal field at the end of the step is obtained by the linear propagation equations,

c_{n} = \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} {b^{'}}_{n} v_{n - m},

u (Δ z, t) = \sum_{n = - N_{s m p} / 2 + 1}^{N_{s m p} / 2} c_{n} g (0, t - n T_{s}) .

The convolution in Eq. (35) can be conveniently computed using DFT and IDFT as in Eq. (22).

The modified DBP based on the perturbation theory can be summarized as follows. Similar to standard DBP with symmetric SSFS, there are also three operations within each step in the modified DBP. Given the data a_n at the fiber input: (i) Calculate the data b_n at the middle of the step using Eq. (20), corresponding to the first $\exp (\frac{Δ z}{2} \hat{D})$ operation in Eq. (6); (ii) Calculate the nonlinear phase shift of b_n using Eqs. (31) and (32); (iii) Calculate the output weighting factor at the end of the step using Eq. (35), corresponding to the second $\exp (\frac{Δ z}{2} \hat{D})$ operation in Eq. (6). The convolutions appearing in Eqs. (20) and (35) can be calculated using a pair of FFTs, just like the dispersion operation in standard DBP. The difference lies in the nonlinear operation. In the standard DBP, the nonlinear phase shift of a single sample is only proportional to the instantaneous power of the same sample, as shown in Eq. (7). On the other hand, the contribution of the neighboring weighting factors to the nonlinear phase shift is included in the perturbation analysis, as shown in Eq. (32). As the modified DBP includes the correlation between neighboring weighting factors, this technique may be considered as a form of correlated DBP [7–12]. In practice, the only change that is needed for the implementation of correlated DBP is to add the phase contribution from the neighboring weighting factors according to Eq. (32). The required additional number of complex multiplications is $N_{n b} / 4$ , considering that one complex multiplication has four real multiplications. This additional computational cost is insignificant since the computational cost is dominated by the cost of FFTs in the dispersion operation. The weighting factors b_n appearing in Eq. (32) may not be the samples of the neighboring symbols, but the weighting factors of the neighboring samples when the signal field at the midpoint is expressed as the superposition of suitably chosen orthogonal basis functions.

4. Results and discussions

We studied the performance of the correlated DBP using a single-channel single-polarization fiber-optic system, shown in Fig. 1. The system operates at 28 Gbaud symbol rate with 32-quadrature amplitude modulation (32-QAM). 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. In each Monte-Carlo simulation, 65,536 symbols are transmitted. Standard SSFM is used to simulate the fiber-optic system with 8 samples per symbol. The linewidth of transmitter and local oscillator lasers = 100 kHz. 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 42 GHz is used before the coherent receiver. Different types of DBP with 2 samples/symbol are then implemented to compensate for dispersive and nonlinear distortions. After DBP, 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 [31]. 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. 1 Schematic of a single-channel fiber-optic system. Tx: transmitter, BPF: band-pass filter, DBP: digital back propagation, LPF: low-pass filter, CPR: carrier phase recovery.

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Figure 2 shows the compensation performance of DBP schemes. The dotted, dashed and solid lines represent results for dispersion compensation only, standard DBP compensation and correlated DBP compensation, respectively. The maximum Q-factor for the case of linear compensation only is 6.1 dB at a launch power of −3 dBm. Standard DBPs bring 1.3 dB and 0.5 dB improvements in Q-factor as compared to the case of dispersion compensation only when the step sizes are 80 km and 2 × 80 km, respectively. For correlated DBP, the Q-factor improvements are 2.0 dB, 2.1 dB, 2.3 dB, 1.5 dB and 1.0 dB, when the step sizes are 80 km, 2 × 80 km, 4 × 80 km, 10 × 80 km and 20 × 80 km, respectively. An interesting phenomenon is that the compensation performance of correlated DBP increases when the step size increases from 80 km to 4 × 80 km, but it decreases as the step size further increases. In this paper, we have ignored the phase-dependent contributions to nonlinear distortion and included only phase-independent contributions (see Eq. (32)). However, in the previous work [27], both types of contributions were included and results showed that as the step size decreases from 5 × 80 km to 1 × 80 km, the performance improves by 0.8 dB. So, the slight performance improvement with the increase in step size over a limited range (1 × 80 km – 4 × 80 km) in this paper is due to the fact that we have ignored the phase-dependent nonlinear distortions. The relative contribution of the phase-dependent nonlinear distortions decreases as the step size increases from 1 × 80 km to 4 × 80 km. However, as the step size become too large (> 4 × 80 km), the accuracy of the perturbation theory decreases, leading to performance degradation. The number of DBP steps is reduced by a factor of 10 using correlated DBP to achieve a same Q-factor improvement, as compared with standard DBP.

Fig. 2 Q-factor versus launch power. Transmission distance = 40 × 80 km.

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Figure 3 shows the X_n profiles for cases of different step sizes. It can be seen that the number of nonlinearly interacting neighbors increases with the DBP step size, due to increase in accumulated dispersion within the step size.

Fig. 3 Correlation coefficients calculated using perturbation theory for correlated DBP.

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Figure 4 shows the dependence of compensation performance on the number of neighbors on one side, N_nb in correlated DBP. For correlated DBP with step sizes less than 4 × 80 km, there exists optimal values of N_nb. When N_nb is small (<10) and the step size is larger than 4 × 80 km, from Fig. 3 it can be seen that X_n is substantially large for $| n | < N_{n b}$ . Hence, ignoring the terms $| n | > N_{n b}$ in Eq. (32) leads to performance degradation. When N_nb is large, the Q-factor saturates since X_n becomes negligible for large |n|. Although we do not have a good explanation as to why there is an optimal N_nb, we find an interesting correlation between the pulse broadening of sinc pulses over the step size and the optimal N_nb. For example, when the step size is 80 km, the effective step size, $Δ z_{e f f}$ (defined as $[1 - \exp (- α Δ z)] / α$ ) is 21 km. After propagating 21 km, the pulse width of the sinc pulse is 7 T_s, which corresponds to N_nb of 3.5 that is close to the optimal N_nb of 3 in Fig. 4.When the step size is 4 × 80 km, the effective step size is 84 km. After 84 km propagation, the pulse width of the sinc pulse is 30 T_s, which corresponds to N_nb of 15 that is close to the optimal N_nb of 16.

Fig. 4 Q-factor vs. the number of neighboring weighting factors.

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Figure 5 compares the compensation performance and computational cost of correlated DBP with standard DBP. Q-factor improvement is the difference between different types of DBP and the case of dispersion compensation. The computational cost is measured in terms of number of complex multiplications per symbol. For correlated DBP with 2 samples/symbol, the number of complex multiplications per symbol equals $2 N_{s t p} [\log_{2} (2 N_{s y m}) + N_{n b} / 4]$ , where N_stp is the total number of DBP steps for the entire fiber-optic link, N_sym is the number of symbols, and N_nb is the number of interacting neighbors on one side. For standard DBP with 2 samples/symbol, the same equation can be applied by setting N_nb = 0. Correlated DBP with N_stp = 20 provides a Q-factor improvement of 1.6 dB higher than that of standard DBP with N_stp = 20 at almost the same computational cost. As compared to standard DBP with N_stp = 40, correlated DBP with N_stp = 4 provides a slight better performance (gain of 0.2 dB) with the reduction in the computational cost by a factor of 6.7. Even a correlated DBP with 2 steps provides about one dB improvement in Q-factor.

Fig. 5 Q-factor improvement over dispersion compensation only versus computational cost. N_stp is the number of DBP steps for the 40 × 80km fiber-optic link.

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5. Conclusions

We have investigated a correlated DBP scheme based on a perturbation theory. In correlated DBP, the nonlinear phase shift is calculated by integrating the distortion occurring within the step length using a perturbation theory, while in conventional DBP only the instantaneous power is used for nonlinear phase calculation. The perturbation theory includes the correlation between neighboring weighting factors and an analytical expression is derived to calculate the correlation coefficients. Simulations of a 28 Gbaud system with 32-QAM format and 40 × 80 km distance show that, as compared to standard DBP with 40 steps, correlated DBP with 4 steps provides a slight better performance (gain of 0.2 dB) with the reduction in the computational cost by a factor of 6.7. Even a correlated DBP with 2 steps provides about one dB improvement in Q-factor over linear compensation.

References and links

1. 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]

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

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. 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]

5. 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]

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

7. L. Li, Z. Tao, L. Dou, W. Yan, S. Oda, T. Tanimura, T. Hoshida, and J. C. Rasmussen, “Implementation efficient nonlinear equalizer based on correlated digital backpropagation,” in Optical Fiber Communication Conference, (Optical Society of America, 2011), paper OWW3. [CrossRef]

8. W. Yan, Z. Tao, L. Dou, L. Li, S. Oda, T. Tanimura, T. Hoshida, and J. Rasmussen, “Low complexity digital perturbation back-propagation,” in European Conference and Exhibition on Optical Communication (ECOC 2011), Tu.3.A.2. [CrossRef]

9. Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

10. D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011). [CrossRef] [PubMed]

11. D. Rafique, M. Mussolin, J. Mårtensson, M. Forzati, J. K. Fischer, L. Molle, M. Nölle, C. Schubert, and A. D. Ellis, “Polarization multiplexed 16QAM transmission employing modified digital back-propagation,” Opt. Express 19(26), B805–B810 (2011). [CrossRef] [PubMed]

12. 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. [CrossRef]

13. 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]

14. 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]

15. A. Mecozzi and R.-J. Essiambre, “Nonlinear shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2011–2024 (2012). [CrossRef]

16. S. Kumar, S. N. Shahi, and D. Yang, “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20(25), 27740–27755 (2012). [CrossRef] [PubMed]

17. 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]

18. R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Opt. Lett. 39(2), 398–401 (2014). [CrossRef] [PubMed]

19. 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]

20. 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]

21. 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]

22. 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), 1209–1219. [CrossRef]

23. 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]

24. 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]

25. 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]

26. 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]

27. X. Liang and S. Kumar, “Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links,” Opt. Express 22(24), 29733–29745 (2014). [CrossRef] [PubMed]

28. G. P. Agrawal, Nonlinear Fiber Optics, 4th Ed. (Academic Press, 2007).

29. J. Shao, X. Liang, and S. Kumar, “Comparison of split-step Fourier schemes for simulating fiber optic communication systems,” IEEE Photonics J. 6(4), 7200515 (2014).

30. 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]

31. 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]

Correlated digital back propagation based on perturbation theory

Abstract

1. Introduction

2. Digital back propagation and split-step Fourier method

3. Correlated DBP based on perturbation theory

4. Results and discussions

5. Conclusions

References and links

Cited By

Figures (5)

Equations (36)

Optics Express