Blind back-propagation method for fiber nonlinearity compensation with low computational complexity and high performance

Junhe Zhou; Junhe Zhou; Yuheng Wang; Yunwang Zhang

doi:10.1364/OE.387572

1. Introduction

With the requirement to further boost the transmission capacity of the fiber communication system, fiber nonlinearity mitigation has been a hot research topic recently [1]. While the all- optical fiber nonlinearity mitigation approach remains attractive [2–3], the digital signal processing (DSP) based nonlinearity compensation methods receive great attention. Back propagation (BP) method is one of the popular DSP approaches to combat the fiber nonlinearity [4–14].

The BP method intends to solve the nonlinear Schrödinger equation (NLSE) inversely and the input signal is expected to be recovered from the distortion caused by the fiber dispersion and the nonlinearity. Such numerical inverse solution of the NLSE is based on the split-step Fourier method (SSFM) [4–10], which contiguously evaluate the propagating signal in the frequency domain for dispersion compensation and in the time domain for nonlinearity compensation.

The main burden to implement the BP method is the computational cost. The iterative conversion of the signal in the frequency domain and the time domain requires numerous rounds of fast Fourier transforms (FFTs) and inverse fast Fourier transforms (IFFTs). The number of FFTs/IFFTs is directly proportional to the SSFM numerical step number in the BP method, which could be quite large in long haul transmission systems.

Several techniques have been proposed to improve the BP method performance and reduce its computational complexity. F. P. Guiomar et. al. analyzed the time domain Volterra series [8] and proposed a parallel BP method [9], which introduced the cross-polarization interaction term in the time domain nonlinear compensation step. X. Liang et. al. [10] proposed the perturbation theory based method to include the self phase modulation (SPM), the cross phase modulation (XPM), and the four wave mixing (FWM) terms in nonlinearity compensation step. C. Lin and A. Napoli et al. proposed an adaptive BP method which could tune the nonlinear power scaling factor [11]. A more advanced BP method with chromatic dispersion compensation and nonlinear compensation tuning was proposed in [12]. A low pass filter assisted BP method was elaborated in [13]. Combining [11–13], F. Zhang et. al [14] have proposed a blind BP method which uses the gradient search method to optimize the nonlinear scaling factor, the linear filter bandwidth and other numerical parameters. All these techniques have increased the accuracy of the BP method while keeping the numerical step size at a relatively large value. Therefore, it is expected to reduce the computational complexity by implementing these techniques.

Despite these inspiring achievements in the BP method [4–14], there is still further room to improve the algorithm so that it could be implemented more efficiently and more compactly. For example, the parallel BP method [9] and the perturbation method based BP method [10] require precise information of the transmission link so that the signal evolution can be accurately predicted. Such information might not be available in a dynamic optical network. Although a blind BP method [14] has already been developed, it does not offer a complete model to include the inter-symbol nonlinear effects within one BP step. As demonstrated in the perturbation theory based BP method, inclusion of these effects will significantly improve the nonlinear compensation performance. In addition, the various existing BP methods (except the blind BP method) have their nonlinear compensation coefficients evaluated analytically without considering the amplified spontaneous emission (ASE) noise. This indicates that these nonlinear tap coefficients might not be with the optimal values due to the existence of the Gordon Mollenauer noise [15–17] (the nonlinear noise arises from the interaction between the ASE noise and the nonlinear process). Hence, it is possible to further improve the nonlinearity compensation performance for the BP method. Finally, the main computational burden of the BP method arises from the total number of steps in the backward propagation. Therefore, if it is possible to reduce the step number, significant computational cost can be saved.

The perturbation theory based nonlinear compensator considers various nonlinear effects [10,17–20], and it can be adaptive [20]. If it is possible to make the perturbation theory based BP [10] to be purely blind, one may further improve the BP method performance. In this paper, a novel blind BP method with reduced complexity and optimized performance is proposed. Based on the perturbation theory, the time domain term in the proposed BP method takes into account the SPM, the intra-channel XPM and the intra-channel FWM effects [17–21] but with the coefficients optimized by the nonlinear least square method (NLSM). Such arrangement enables the BP method to be implemented blindly with a step size totally irrelevant to the link span length, which could be quite large to ease the computation. In comparison to the conventional BP method, the proposed method improves the performance and reduces the computational complexity.

2. Theory

The signal propagation in optical fibers is characterized by the full vector nonlinear Schrodinger equation [21].

(1)$$\begin{array}{l} \frac{{\partial {A_x}}}{{\partial z}} ={-} \frac{\alpha }{2}{A_x} - \frac{{i{\beta _2}}}{2}\frac{{{\partial ^2}{A_x}}}{{\partial {t^2}}} + i\gamma \left( {{{|{{A_x}} |}^2} + \frac{2}{3}{{|{{A_y}} |}^2}} \right){A_x} + \frac{{i\gamma }}{3}{A_x}^\ast {A_y}^2\\ \frac{{\partial {A_y}}}{{\partial z}} ={-} \frac{\alpha }{2}{A_y} - \frac{{i{\beta _2}}}{2}\frac{{{\partial ^2}{A_y}}}{{\partial {t^2}}} + i\gamma \left( {{{|{{A_y}} |}^2} + \frac{2}{3}{{|{{A_x}} |}^2}} \right){A_y} + \frac{{i\gamma }}{3}{A_y}^\ast {A_x}^2 \end{array}$$

where A_x and A_y stand for the amplitudes of the two signal polarizations, α the fiber attenuation, β₂ the chromatic dispersion coefficient, γ the fiber nonlinear coefficient. It is worth noting that random polarization rotation and the differential group delay should be included in the numerical integration of Eq. (1) so that the polarization mode dispersion (PMD) impact could be fully characterized. This can be done by dividing the fiber into multiple sections and multiplying the random matrices with differential group delay (DGD) on the Jones polarization vector (A_x A_y)^T. Due to the random polarization rotation in the optical fiber, the above Eq. (1) can be replaced by the well-known Manakov equation [7]

(2)$$\begin{array}{l} \frac{{\partial {A_x}}}{{\partial z}} ={-} \frac{\alpha }{2}{A_x} - \frac{{i{\beta _2}}}{2}\frac{{{\partial ^2}{A_x}}}{{\partial {t^2}}} + i\frac{8}{9}\gamma ({{{|{{A_x}} |}^2} + {{|{{A_y}} |}^2}} ){A_x}\\ \frac{{\partial {A_y}}}{{\partial z}} ={-} \frac{\alpha }{2}{A_y} - \frac{{i{\beta _2}}}{2}\frac{{{\partial ^2}{A_y}}}{{\partial {t^2}}} + i\frac{8}{9}\gamma ({{{|{{A_y}} |}^2} + {{|{{A_x}} |}^2}} ){A_y} \end{array}$$

The conventional BP method is accomplished by splitting Eq. (2) into the linear and the nonlinear steps. During the linear step, one conducts the Fourier transform and compensates the dispersion, and during the nonlinear step, one compensates the fiber nonlinearity. The proposed BP method also splits into the linear and the nonlinear steps which is demonstrated in Fig. 1.

Fig. 1. The procedure of the proposed BP method in the (a) initial optimization mode (b) normal processing mode.

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As shown in Fig. 1, the proposed BP method runs in two modes, i.e. the initial optimization mode and the normal processing mode. During the initial optimization mode, the algorithm tends to optimize the nonlinear step tap values, which are determined by the NLSM with a damping factor, i.e. the Levenberg-Marquardt method [22]. After the optimization process converges, the algorithm switches to the normal process mode and the nonlinear taps are fixed regardless the input power and the random fiber rotations. The change of the fiber link requires an update of the nonlinear compensation taps by the initial optimization mode. Minor link parameter changes, like temperature induced or bend induced loss/dispersion variations could also trigger the initialization process if the BER exceeds certain threshold. It is worth noting that the constant modulus algorithm (CMA) may be placed after the back propagation process so that the dispersion compensation step and the inverse dispersion compensation step may be omitted. However, it is found that the nonlinear compensation performance will be better if it is placed before the BP process because the compensation of the PMD in the link by CMA will make the channel response shorter and this will reduce the required nonlinear tap length in our proposed method.

As discussed above and shown in Fig. 1, the proposed BP method differs from the conventional BP method during the nonlinear step by employing the following equations according to the perturbation theory [10,17–20]

(3)$$\begin{array}{l} {E_{x,NL}} = \sum\limits_{m ={-} {N_{NL}}}^{{N_{NL}}} {\sum\limits_{l ={-} {N_{NL}}}^{{N_{NL}}} {w_{xx}^{ml}{x_{({n + l} )}}{x^\ast }_{({n + l + m} )}{x_{({n + m} )}}} } + \sum\limits_{m ={-} {N_{NL}}}^{{N_{NL}}} {\sum\limits_{l ={-} {N_{NL}}}^{{N_{NL}}} {w_{xy}^{ml}{y_{({n + l} )}}{y^\ast }_{({n + l + m} )}{x_{({n + m} )}}} } \\ {E_{y,NL}} = \sum\limits_{m ={-} {N_{NL}}}^{{N_{NL}}} {\sum\limits_{l ={-} {N_{NL}}}^{{N_{NL}}} {w_{yx}^{ml}{x_{({n + l} )}}{x^\ast }_{({n + l + m} )}{y_{({n + m} )}}} } + \sum\limits_{m ={-} {N_{NL}}}^{{N_{NL}}} {\sum\limits_{l ={-} {N_{NL}}}^{{N_{NL}}} {w_{yy}^{ml}{y_{({n + l} )}}{y^\ast }_{({n + l + m} )}{y_{({n + m} )}}} } \end{array}$$

where E_xNL and E_yNL stand for the nonlinear compensation term, N_NL the nonlinear tap length, w the nonlinear coefficient with its superscripts indicating the tap order and the subscripts indicating the nonlinear interactions, x and y the received signals for x and y polarizations. To reduce the complexity, it is usually assumed that w_xx=w_xy=w_yx=w_yy. Also, due to the symmetrical nature [10], we have

{w^{ml}} = {w^{lm}}

The nonlinear optimization of the coefficient w is done as follows. We equalize the signals with respect to the nonlinear tap vector w and the received symbols. One may expand them by the Taylor expansion as the following Eq. (4)

(4)$$\begin{array}{l} \hat{x} = {f_x}({x,{\mathbf w}} )\approx {f_x}({x,{{\mathbf w}_o}} )+ \frac{{\partial \hat{x}}}{{\partial {{\mathbf w}^T}}}\Delta {\mathbf w}\\ \hat{y} = {f_y}({y,{\mathbf w}} )\approx {f_y}({y,{{\mathbf w}_o}} )+ \frac{{\partial \hat{y}}}{{\partial {{\mathbf w}^T}}}\Delta {\mathbf w}\\ {\mathbf w} = {{\mathbf w}_o} + \Delta {\mathbf w} \end{array}$$

where $\hat{x}$ and $\hat{y}$ are the signals after nonlinear equalization, ${{\mathbf w}_o}$ the old nonlinear taps for the x and y polarizations (the two polarizations have been assumed with the same nonlinear taps), Δw the adjustment of the nonlinear taps. The optimal tap adjustment Δw_opt will tend to minimize the following target function J

(5)$$\begin{array}{l} \Delta {{\mathbf w}_{opt}} = \mathop {\arg \min (J )}\limits_{\Delta {\mathbf w}} \\ J = \left( {\sum\limits_{n = 0}^{{N_t}} {{{({\hat{x}(n )- {x_{DFE}}(n )} )}^2} + } \sum\limits_{n = 0}^{{N_t}} {{{({\hat{y}(n )- {y_{DFE}}(n )} )}^2}} } \right) \end{array}$$

where x_DFE and y_DFE are the estimated x and y signals after decision feedback, N_t the total length of the data block. According to the Levenberg-Marquardt method, the optimal tap values should be updated as

(6)$$\begin{array}{l} {{\mathbf w}_{opt}}({k + 1} )= {{\mathbf w}_{opt}}(k )+ {({{{({\nabla \hat{{\mathbf p}}} )}^H}\nabla \hat{{\mathbf p}} + \lambda {\mathbf I}} )^{ - 1}}{({\nabla \hat{{\mathbf p}}} )^H}({{{\mathbf p}_{DFE}} - \hat{{\mathbf p}}({{\mathbf p},{{\mathbf w}_{opt}}(k )} )} )\\ {\mathbf p} = \left( {\begin{array}{{c}} {\mathbf x}\\ {\mathbf y} \end{array}} \right)\\ \hat{{\mathbf p}} = \left( {\begin{array}{{c}} {\hat{{\mathbf x}}}\\ {\hat{{\mathbf y}}} \end{array}} \right)\\ {{\mathbf p}_{DFE}} = \left( {\begin{array}{{c}} {{{\mathbf x}_{DFE}}}\\ {{{\mathbf y}_{DFE}}} \end{array}} \right)\\ \nabla \hat{{\mathbf p}} = \left( {\begin{array}{{c}} {\nabla \hat{{\mathbf x}}}\\ {\nabla \hat{{\mathbf y}}} \end{array}} \right) = \left( {\begin{array}{{c}} {{{\left. {\frac{{\partial \hat{{\mathbf x}}}}{{\partial {{\mathbf w}^T}}}} \right|}_{{\mathbf w} = {{\mathbf w}_{opt}}(k )}}}\\ {{{\left. {\frac{{\partial \hat{{\mathbf y}}}}{{\partial {{\mathbf w}^T}}}} \right|}_{{\mathbf w} = {{\mathbf w}_{opt}}(k )}}} \end{array}} \right) \end{array}$$

where I is the identity matrix, λ the damping factor, which is variable during each step. The graphical illustration of the Levenberg-Marquardt optimization process is shown in Fig. 2. Initially, λ should be assigned with a small value, e.g. 0.001, and it will increase if the update of the nonlinear tap value does not bring the reduction of the target function J. Each unsuccessful update will enlarge λ by a factor, e.g. 5. The larger λ is, the closer the algorithm to the gradient search method will be. It is found that the tap coefficients usually converge within three iterations.

Fig. 2. The Nonlinear tap optimization by the Levenberg-Marquardt method.

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It is worth noting that the optimization of the nonlinear taps is done in one block of the received symbols, i.e. one may repeat the optimization process on the same data set.

3. Results and discussions

The proposed algorithm has been tested in the numerical simulations. The simulation parameters are the same as those in [20]. For the readers’ convenience, it is elaborated here.

The simulation is based on the vector NLSE, i.e. Eq. (1) with the random polarization rotation introduced between fiber sections, which have the lengths of several hundreds of meters. The signal wavelength is 1550 nm and the two polarizations are multiplexed with 28 Gbaud DQPSK signals or 16-QAM signal. Each symbol of the signal is simulated with 64 samples and only two samples per symbol are taken for DSP at the receiver side.

Two fiber links are considered in the simulations, which are the same as the ones in [20]. For the readers’ convenience, they are elaborated as follows. The first fiber link has the transmission distance of 4000 km, which is divided into 50 spans. Each span has the fiber length of 80 km and an EDFA with the gain of around 16 dB and noise figure (NF) of 5.5 dB. The second fiber link is a hybrid link, which is 6400 km long and is composed of two parts. The first part is 4000 km long and has 50 spans with the span length of 80 km. The EDFA after each span has the gain of 16 dB and the NF of 5.5 dB. The second part is 2400 km long and has 40 spans with the span length of 60 km. The EDFA gain and NF are 12 dB and 6 dB respectively. Both of the links have the standard single mode fiber (SSMF), with the attenuation as 0.2 dB/km, the chromatic dispersion as 16 ps/nm/km, the nonlinear coefficient as 1.3/W/km and the differential group delay (DGD) as 0.1 ps/sqrt(km). In order to illustrate the applicability of the proposed algorithm in the heterogeneous links with different fiber types, the third fiber link is introduced with 15 spans of 80 km SSMF and 15 spans of 80 km large effective area fiber (LEAF). The LEAF has the chromatic dispersion as 4.3 ps/nm/km and the effective fiber core area as 71.5 µm² (80 µm² for the SSMF). The EDFAs between the spans have the gains and NFs as 16 dB and 5.5 dB. The fourth link has 20 spans of 80 km LEAF. The EDFAs have the noise figures as 6.5 dB. The four fiber links are shown in Fig. 3, where PBC and PBS stand for the polarization beam combiner and polarization beam splitter. For the first three links, the DQPSK signals are implemented and the last link implements the 16-QAM signal.

Fig. 3. The four fiber links in the simulation (a) the 4000 km link (b) the 6400 km hybrid link. (c) the 2400 km hybrid heterogeneous link (d) the 1600 km LEAF fiber link

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For the first fiber link, the proposed method is used to compensate the fiber nonlinearity blindly and is compared with the conventional BP method. Six emulated BP steps are assumed in the method and the nonlinear tap length N_NL is assumed to be 2, 3, 4 and 5. The total nonlinear taps can be calculated as (2N_NL +1)² and they are 25, 49, 81 and 121 per step for one polarization respectively. Since the step number has been reduced from 50 to 6, the overall computational complexity of the proposed method is less than the conventional BP method with one span per step, which will be discussed thoroughly in the later sections. The initial nonlinear taps are assumed to be 0. The preliminary nonlinearity compensation performance can be seen from Fig. 4, which demonstrates the constellation of the received signal with only dispersion compensation (Fig. 4(a)) and with the nonlinearity compensation by the proposed method (Fig. 4(b)). The input signal power is 2 dBm.

Fig. 4. The constellation of the received signal after (a) chromatic dispersion compensation with the input power as 2 dBm (b) equalization by the proposed BP method with the number of steps as 6 and N_NL as 5 with the input power as 2 dBm.

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The convergence performance of the proposed algorithm in its initial optimization mode is shown in Fig. 5, where iteration 0 indicates the case before implementing the proposed nonlinearity compensation algorithm. It can be found that the convergence of the tap coefficients is quite fast and is accomplished within three iterations using the NLSM algorithm described in section 2 with totally 65532 symbols (32766 symbols on each polarization).

Fig. 5. The convergence performance of the proposed method in its initial optimization mode. The input signal power is 2 dBm and the number of steps is 6.

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For the first fiber link, the proposed method is used to compensate the fiber nonlinearity blindly and is compared with the conventional BP method. Six emulated BP steps are assumed in the method and the nonlinear tap length N_NL is assumed to be 2, 3, 4 and 5. The total nonlinear taps can be calculated as (2N_NL +1)² and they are 25, 49, 81 and 121 per step for one polarization respectively. Since the step number has been reduced from 50 to 6, the overall computational complexity of the proposed method is less than the conventional BP method with one span per step, which will be discussed thoroughly in the later sections. The initial nonlinear taps are assumed to be 0. The preliminary nonlinearity compensation performance can be seen from Fig. 4, which demonstrates the constellation of the received signal with only dispersion compensation (Fig. 4(a)) and with the nonlinearity compensation by the proposed method (Fig. 4(b)). The input signal power is 2 dBm.

The convergence performance of the proposed algorithm in its initial optimization mode is shown in Fig. 5, where iteration 0 indicates the case before implementing the proposed nonlinearity compensation algorithm. It can be found that the convergence of the tap coefficients is quite fast and is accomplished within three iterations using the NLSM algorithm described in section 2 with totally 65532 symbols (32766 symbols on each polarization).

The detailed BER performance of the proposed method is shown in Fig. 6. It is worth noting that the tap coefficients for the proposed method have been obtained when the input power is 2dBm and they are used for other cases without altering. It is found that the BER performance is the best with such signal input power and the calculated coefficients work well despite different input signal powers and different random rotations in the fiber. It can be seen from the curves that the proposed method outperforms the conventional BP method. In the regime where the signal power is less than 1 dBm, the proposed method even shows lower BER than the BP method with 1 km per step (the BP method performance limit). This is due to two facts. Firstly, the conventional BP method maintains a constant power scaling factor and this makes it difficult to compensate the Gordon Mollenauer noise [15–16]. The proposed method can adapt to the power scaling factor variations along the span and a better BER performance has been achieved. Secondly, the conventional BP method does not consider the SPM, the XPM, and the FWM terms during the nonlinear compensation step, while the proposed method finds the SPM, the XPM, and the FWM compensation taps through NLSM optimization. In order to further evaluate the performance of the proposed method, two modified BP methods, i.e. the parallel BP method [9] and the blind BP method [14] are used for comparison. The blind methods (including the proposed method and the blind BP method) have similar performance while the taps are optimized in the moderate input power regime. In order to have a fair comparison, the blind BP has its power scaling factor and the assisting low pass filter bandwidth optimized at the input power of 2 dBm, which is a moderate input power. Since the parallel BP method has its nonlinear scaling factor and averaging window optimized, it has similar performance as the blind BP in such a homogenous link. It can be seen from Fig. 6 that the proposed method still outperforms the modified BP methods in terms of minimum achievable BER. In most cases, the proposed method achieves better performance even with N_NL=3. The proposed method operates in the moderately nonlinear regime and achieves better performance. This might be attributed to the equalization of the Gordon Mollenauer noise, which becomes less dominant in the highly nonlinear regime.

Fig. 6. The BER curves for the proposed method with 6 assumed BP steps and different nonlinear tap lengths. The conventional BP methods with 80 km steps and 1 km step and the parallel BP method, the blind BP method with 80 km steps are used for comparison. The fiber link length is 4000 km.

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Thereafter, the proposed method is implemented in a hybrid fiber link (Fig. 3(b)) blindly along with the blind BP method. The conventional BP and the parallel BP were implemented with the known link parameters. The proposed method employs the similar parameters, i.e. N_NL=2, 3, 4 and 5, as the previous case. Since the link length is much longer, 9 steps are assumed for the BP process. The coefficients are evaluated when the input power is 1 dBm and they are used for other input powers without altering. The detailed nonlinear compensation performance is shown in Fig. 7. From the figure, it can be seen that the proposed method achieves satisfactory performance in such a hybrid link without any prior information. In most cases, the proposed method outperforms the conventional BP method with one span (80 km or 60 km) per step. Also, a superior performance is observed compared with other BP methods, such as the parallel BP method [9] and the blind BP method [14]. The blind BP uses the same parameters optimized at the input power of 1 dBm so that a fair comparison could be made. The minimum achievable BER is much lower for the proposed method.

Fig. 7. The BER curves for the proposed method with 9 assumed BP steps and different nonlinear tap lengths. The conventional BP methods with 80 km step and 1 km step and the parallel BP method, the blind BP method with 80 km steps are used for comparison. The fiber link length is 6400 km. The first 4000 km is 80 km per span and the latter 2400 km is 60 km per span.

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The heterogeneous link in Fig. 3(c) is studied with the results shown in Fig. 8. Due to the introduction of the LEAF, higher nonlinear interaction noise is observed. This is due to the fact that the LEAF has lower chromatic dispersion and relatively higher nonlinear coefficient. As expected, the blind BP method has much better performance in comparison with the parallel BP method and the conventional BP method in such a heterogeneous link with different fibers. The proposed method employs 6 steps along with a shorter tap length, i.e. N_NL=2 and 4, in comparison to the previous cases due to the lower chromatic dispersion. The coefficients are evaluated when the input power is 2 dBm. The proposed method demonstrated superior performance in comparison with other algorithms which validates its applicability in the heterogeneous links especially in the moderately nonlinear regime. The reason for the back propagation method with 1 km per step to behave poorly in this case is that the link is heterogeneous (with different fibers) and therefore different power scaling factors are required for different spans. However, the conventional BP can only use one power scaling factor for all spans of the links.

Fig. 8. The BER curves for the proposed method with 6 assumed BP steps and different nonlinear tap lengths. The conventional BP methods with 80 km step and 1 km step and the parallel BP method, the blind BP method with 80 km steps are used for comparison. The fiber link length is 2400 km. The first 1200 km is 80 km per span SSMF and the latter 1200 km is 80 km per span LEAF.

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On this link, the performance of the proposed method with respect to the step number is also investigated, which is shown in Fig. 9. The proposed method employs the tap length N_NL=2 with different step numbers. The BER is evaluated when the input power is 2 dBm. As demonstrated in the figure, when the step number increases from 2 to 8, the BER decreases accordingly. However, it will be shown in the following analysis that the increase of steps will bring computation complexity increase. Therefore, a balance of the performance and the computational cost should be considered while selecting the proper step number for the proposed method.

Fig. 9. The BER Vs step number on the 2400 km heterogeneous link.

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The link composed by the LEAF in Fig. 3(d) is studied with the results shown in Fig. 10 and Table 1. In order to demonstrate the applicability of the method on higher order modulation formats, 16-QAM signal is used. The signal baud rate is 28 Gbaud and it is multiplexed on two polarizations. The proposed method employs smaller number of steps, i.e. 2, and the tap length N_NL=6, because much less overall chromatic dispersion exists in this link. The signal input power is 1 dBm. Again, the proposed method has quite promising performance while dealing with the higher order modulation format, which can be seen in Fig. 10 and is demonstrated quantitatively in Table 1. In order to demonstrate the performance improvement with more metrics, Q factor is evaluated based on the BER. The proposed method has improved the Q factor by about 0.5 dB compared with the blind BP method.

Fig. 10. The constellation for the 16-QAM signal (a) with CD compensation only (b) compensated by the parallel BP method with 80 km step (c) compensated by the blind BP method with 80 km steps (d) compensated by the proposed method with 2 assumed BP steps along with the tap length as 6. The fiber link length is 1600 km, which is divided into 20 spans of 80 km per span LEAF. The noise figure of each EDFA is 6.5dB. The signal input power is 1 dBm.

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Table 1. The BER and Q factor for the 16-QAM signal equalized by different methods. The configuration is the same as Fig. 10.

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While the proposed method has reduced its step number due to less accumulated dispersion, the results in Table 1 indicates that the performance of the conventional BP with less steps degrades by 0.25 dB (two spans per step) and 0.88 dB (4 spans per step) for the Q factor compared with the one span per step case. The conventional BP's performance is sensitive to the step number reduction.

Complexity analysis of the algorithm is presented afterwards. Since the nonlinearity in the fiber link is quite stable and varies very slowly [14], the time for initial optimization mode for the proposed BP method will be negligible and the main computational cost for the proposed BP will be the one in the normal process mode. We may use the number of complex multiplications (CMs) as a measure for computational complexity. The number of CMs for FFTs/IFFTs in the conventional BP method and the proposed method (the normal process mode) equals the following expression [10,23]

(7)$$\begin{array}{l} \frac{{2{N_{step}}{N_{FFT}}{{\log }_2}{N_{FFT}}}}{{{N_{sym}}}}\\ {N_{FFT}} = K{N_{sym}} \end{array}$$

where N_FFT is the length of samples used for FFT, N_sym the length of the symbols, K the over-sampling rate (2 samples/symbol in our case, i.e. T/2 sampling), the factor of 2 relates to the FFT/IFFT pair for each step, N_step the total number of steps. The proposed method has the CM number in one nonlinear step per symbol as

(8)$$\begin{array}{l} K\left( {\frac{{N({N + 1} )}}{2} + N + N} \right)\\ N = 2{N_{NL}} + 1 \end{array}$$

The CM number in Eq. (8) has considered the symmetrical nature of the nonlinear terms and the recursive nature of the product terms in Eq. (3). Since the proposed method has longer step size and hence, the FFT block length should be longer. In our 4000 km link example, while the conventional BP adopts the FFT block size as 128 ✕ 2 with one span per step, the proposed method should use longer FFT block size, e.g. 1024 ✕ 2 (here 2 stands for the over-sampling rate). The dispersion compensation and inverse dispersion compensation process implement the FFT block size of 8192✕2. The FFT processing strategy is overlap-add. The overlap percentage between the blocks is determined by the chromatic dispersion channel response length per step. For example, it is about 20% for the BP method with one step per span. Since the percentage is relatively small, the impact of overlap percentage on the computational complexity is ignored for simplicity. According to the parameters of the link and Eq. (7-8), one may calculate the total CMs required by the conventional BP and the proposed method as shown in Table 2. The Table lookup row indicates the table lookup requirement to evaluate the exponential term in the conventional BP and its complexity is considered as one CM. It can be seen from the table that the proposed method does have lower computational complexity in comparison to the conventional BP method while maintaining a superior nonlinearity compensation performance. Only when N_NL=5, the complexity becomes higher in comparison to the conventional BP method.

Table 2. Number of complex multiplications per symbol per polarization for the conventional BP and the proposed method while compensating the nonlinearity of the 4000 km fiber link.

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4. Summary

In summary, we propose a novel blind BP method with lower computational complexity and higher nonlinearity compensation performance. The proposed method considers the SPM, the XPM, and the FWM terms in the nonlinear compensation step and uses the Levenberg-Marquardt method to optimize the corresponding coefficients. With the given total accumulated chromatic dispersion, the equalizer is free of the prior information of the transmission link and achieves performance improvement compared with the conventional BP method.

Funding

National Natural Science Foundation of China (61775168).

Disclosures

The authors declare no conflicts of interest.

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Methods	BER	Q factor
CD compensation only	2.60e-02	5.77
Conventional BP method with 320 km per step	1.29e-02	6.95
Conventional BP method with 160 km per step	8.31e-03	7.58
Conventional BP method with 80 km per step	6.88e-03	7.83
Parallel BP method	5.95e-03	8.01
blind BP method	5.60e-03	8.08
Proposed method	3.60e-03	8.59

	BP	proposed method
	BP	N_NL=2	N_NL = 3	N_NL = 4	N_NL = 5
N_FFT	256	2048	2048	2048	2048
Nstep	50	6	6	6	6
CM_FFT	1600	376	376	376	376
CM_linear	2	2	2	2	2
CM_nonlinear	2	100	168	252	352
Table lookup	2	0	0	0	0
Total CMs per symbol	1900	988	1396	1900	2500

Methods	BER	Q factor
CD compensation only	2.60e-02	5.77
Conventional BP method with 320 km per step	1.29e-02	6.95
Conventional BP method with 160 km per step	8.31e-03	7.58
Conventional BP method with 80 km per step	6.88e-03	7.83
Parallel BP method	5.95e-03	8.01
blind BP method	5.60e-03	8.08
Proposed method	3.60e-03	8.59

	BP	proposed method
	BP	N_NL=2	N_NL = 3	N_NL = 4	N_NL = 5
N_FFT	256	2048	2048	2048	2048
Nstep	50	6	6	6	6
CM_FFT	1600	376	376	376	376
CM_linear	2	2	2	2	2
CM_nonlinear	2	100	168	252	352
Table lookup	2	0	0	0	0
Total CMs per symbol	1900	988	1396	1900	2500

Blind back-propagation method for fiber nonlinearity compensation with low computational complexity and high performance

Abstract

1. Introduction

2. Theory

3. Results and discussions

4. Summary

Funding

Disclosures

References

Cited By

Figures (10)

Tables (2)

Equations (9)

Optics Express