Nonlinearities compensator based on SORBFNN in ultralonghaul highcapacity COOFDM systems
Abstract
Radial basis function neural network is recognized as being the most useful type of neural network for a nonlinear equalizer in the COOFDM system. In the history of nonlinear equalizers, the focus has always been on mitigation of nonlinearity. Despite this interest, no one to the best of our knowledge has studied ultralonghaul transmission length. Previous work has been limited to normal transmission length, i.e., up to 1400 km. In ultralonghaul optical fiber communication systems, the optical signal becomes very much complex at longer distance, i.e., around over 8000 km. It has not yet been established whether radial basis function neural network can do the same improvement in performance at ultralonghaul transmission length. A basic issue of a customary radial basis function neural system is its low performance at such a long distance. This paper presents a new approach to compensating nonlinearities in ultralonghaul optical fiber communication COOFDM systems. In this new approach, a radial basis function with nonlinear equalizer (first hidden layer with automatically adjusted structure, i.e., number of neuron nodes in the hidden layer and the different parameters of the structure through learning, and second hidden layer with fixed structure) based on two hidden layers has been recommended. The aim of this study is to examine the performance of a selforganizing radial basis function neural networkbased nonlinear equalizer (SORBFNNNLE) for the mitigation of nonlinearities in ultralonghaul COOFDM systems. Better performance of the SORBFNN is attributed due to selforganizing structure and training of network. From this study, it has been observed that the proposed SORBFNNNLE performs well regarding Qfactor and OSNR at an ultralong distance.
Keywords
Artificial neural network based nonlinear equalizer (ANNNLE) Coherent optical orthogonal frequency division multiplexing (COOFDM) Selforganizing radial basis function neural network based nonlinear equalizer (SORBFNNNLE)1 Introduction
The COOFDM system is attracting considerable interest due to its high spectral efficiency [1]. A challenging area in the field of COOFDM systems is high peaktoaverage power ratio [2], due to which nonlinear effects such as SPM and XPM have become a critical issue in the COOFDM system. Various approaches have been proposed to solve the issue of optical fiber nonlinearity compensation. Nonlinear equalizers are among the most widely used types of equalizers because linear equalizer has the main problem of linear decision limits [3]. It is well known that artificial neural networks can be used as nonlinear equalizers due to its complex input–output mapping [4]. Several studies, for instance [5, 6, 7, 8], have been carried out on artificial neural networkbased nonlinear equalizers. In [5], the authors studied a multilayer perceptron (MLP)based artificial neural networkbased nonlinear equalizer (ANNNLE) to compensate the nonlinearity in the COOFDM system with Reidmiller’s resilient backpropagation learning algorithm. Jarajreh and coworkers estimated ~ 0.7 dB improvement in Qfactor over a fiber length of 1000 km, at launch power of − 6 dBm, and payload of 40 Gbps [5]. The main pitfall of MLPbased ANNNLE is the slow rate of convergence and high reliance on starting values of weight parameters. It has now been suggested that functional link artificial neural networkbased nonlinear equalizer (FLANNNLE) is an alternative artificial neural network (ANN) technique to compensate the nonlinearity using a particle swarm optimizationbased backpropagation (PSOBP) learning algorithm with less computational complexity [6]. Unfortunately, it does not provide much amount of improvement in Qfactor as compared to MLPbased ANNNLE. More details can be found in our previous paper [6].
A recent review of the literature on this subject found that radial basis function neural network (RBFNN) with Kmeans clustering training algorithm is a new method for nonlinearities mitigation in COOFDM systems and concluded that it provide much improvement in Qfactor as it is able to produce proper decision boundaries [7, 8]. In [8], the authors investigated ~ 1.2 dB improvement in Qfactor at payload bit rate 40 Gbps. Until now, these methodologies have only been applied to normal fiber transmission distance, i.e., around 1000 km. However, there is still a need for studying the performance of radial basis function neural networkbased nonlinear equalizer (RBFNNNLE) for ultralonghaul COOFDM systems. It is common knowledge that optical signal becomes very much complex over ultralong distance. Unfortunately, RBFNN with the fixed structure fails to eliminate such complex nonlinearity effects. One of the major drawbacks to adopting this system is that at an ultralong distance the received 16point quadrature amplitude modulation (16QAM) complex signal is much more overlapped, and for the ordinary RBFNN, the most difficult task is to make proper decision boundaries and clusters. More details on this inability of RBFNN will be given in results. Selforganizing radial basis function neural networks (SORBFNNs) are attracting interest due to its flexible structure and better performance than other existing algorithms [9].
This paper focuses on how to compensate the complex nonlinearities generating at ultralong distances, i.e., around 8000 km. Selforganizing radial basis function neural networkbased nonlinear equalizer (SORBFNNNLE) is introduced in order to achieve better results than traditional RBFNNNLE. In this new methodology, two hidden layers are introduced in radial basis function neural network. A selforganizing algorithm is used for updating the positions and widths of the first hidden layer nodes. In second hidden layer, total number of nodes is fixed depending on the type of QAM modulation used in the COOFDM system. In this paper, least mean square algorithm is used for updating all the weights. Tracking the dynamic nonlinear impairments with any algorithm is directly related to the block length of the system and equalizer mapping ability. The tracking performance of the proposed trainingbased NLE for dynamic nonlinear impairments is approximately 71% improved as compared to that of RBFNNNLE. With 13% training overhead, reduction in nonlinear penalty is 3.5 dB at 100 Gbps payload in SORBFNNNLE. This solution improves on previous methods in terms of Qfactor due to the selforganizing training of network. This paper is divided into five sections. The first section gives an overview of introduction to this study. The second section presents simulation arrangement for a COOFDM system with the proposed nonlinear equalizer. A new methodology, i.e., SORBFNN, is outlined and the used learning algorithm through flowchart has been delineated in the third section. Obtained simulation results with proposed technique have been studied in the fourth section. Some conclusions are drawn in the final section.
2 SORBFNNNLEbased simulation model for COOFDM system
COOFDM systems are undergoing a revolution in terms of high capacity and ultralong transmission length. Most studies have only focused on high capacity. Ultralong transmission length has become a central issue in our study. The recent approach to performing the task of nonlinearities compensation is RBFNN as it is an efficient method in many applications [10, 11, 12, 13, 14]. The current solution to nonlinearities compensation is inefficient at ultralong transmission length as it is not able to make proper decision boundaries due to more spread in complex 16QAM received signal. For more complex signal, the very difficult and important task was the initial selection of a number of the RBF nodes in the hidden layer. An unsuitable initial selection of RBF nodes in the hidden layer results in extensive training time of the equalizer; therefore, stable error was tough to achieve to compensate more complex nonlinearities. Another important task in RBFNNNLE was the initial selection of RBF nodes widths and positions in the hidden layer of the network. An inappropriate selection of these nodes may result in the possibility of dropping into a wrong local minimum.
It is well known that for 16QAM modulation format number of nodes in the hidden layer should be equal to 16 [8]. Therefore, in this study, two hidden layers are used so that the requirement of total 16 clusters for 16QAM modulation and removal of abovementioned problems are accomplished simultaneously. Out of these two hidden layers, the first hidden layer is automatically adjusted with a selforganizing algorithm and the second hidden layer is with a fixed number of nodes. Another reason of this study is to maintain the balance between network complexity and its performance. Our simulation setup bears a close resemblance to the one proposed by Ahmad and Kumar [8].
3 Structure and working of SORBFNNNLE

Step 1 First hidden layer of l RBF nodes (initially set 20–30 RBF nodes) with radius r of the receptive field has been initialized. Moreover, this RBF node with width δ and position v has been randomly selected.

Step 2 Second fixed hidden layer of M RBF nodes (M is 16 for 16QAM modulation format) with fixed position (u) and width (σ) has been initialized.

Step 3 Weights \( (w_{tj} ) \) connecting the first hidden layer with the second hidden layer, and output layer weights of network w_{j} have been randomly selected from 0 to 1.

Step 4 Input pattern P(k) with their corresponding expected error (E_{d}) has been applied to the SORBFNNNLE structure.

Step 5 Winner node from the first hidden layer RBF nodes has been found by the formula presented in flowchart.

Step 6 After that, v_{t}(k) − P(k) > βr_{t} condition has been checked. Then, e(k) > αE_{d} and e(k) ≤ αE_{d} have been checked for v_{t}(k) − P(k) greater than βr_{t} and v_{t}(k) − P(k) less than or equal to βr_{t}, respectively, where α and β vary from 1 to 2.

Step 7 The control has been moved to step 8 or step 9 if e(k) > αE_{d} and e(k) ≤ αE_{d} have been satisfied or not satisfied, respectively.

Step 8 A RBF node in the first hidden layer has been inserted or deleted as explained in flowchart.

Step 9 First hidden layer widths and positions have been updated using formulas given in Eqs. 10 and 11.

Step 10 All the weights have been updated according to the LMS algorithm presented in Eq. 3.

Step 11 The output of the SORBFNNNLE has been estimated using Eqs. 4–7.

Step 12 Then, the value of cost function, i.e., MSE, using Eq. 2 has been calculated.

Step 13 At the end, achieved MSE has been compared with \( E_{d} \). The same procedure from step 4 has been repeated until the required cost function is achieved.
In this proposed method, the computational complexity and memory requirement are of the order of O (T_{s}NS^{2}) and O (T_{s}S), respectively, which are O (T_{s}R^{2}) and O (T_{s}R) in RBFNN method, where \( T_{\text{s}} \) is the training states, N is the number of times the proposed structure needs to be restructured, S = (M * l)(n + 1), and \( R = \left( M \right)\left( {n + 1} \right) \), in which M is the number of nodes in fixed hidden layer, l is the number of neurons required in first hidden layer, and n is the number of input variables.
4 Results and discussion
Figure 4 presents the constellation diagrams of received 16QAM data for the ultralonghaul COOFDM system. It is apparent from Fig. 4 that at an ultralong distance the received 16QAM data become much complex with more spreading and it cannot be efficiently compensated by conventional RBFNN. From Fig. 4, we can note that the proposed SORBFNNNLE technique (represented by green color) focuses on the constellation points more adequately at required positions in comparison with traditional RBFNNNLE (represented by blue color). As predicted, our experiments demonstrate that the proposed technique has the more ability to reduce fiber nonlinearities. In this paper, Qfactor is used as judgment factor to plot various graphs so that proper comparison of proposed SORBFNNNLE technique with the previously used RBFNNNLE technique can be presented. The formula for calculating the Qfactor from obtained BER is given as: \( Q = 20\log_{10} \left( {\sqrt 2 {\text{erfc}}^{  1} \left( {2 {\text{BER}}} \right)} \right) \). In this study, the threshold value of Qfactor is 6.4 dB obtained by assuming 20% overhead over commercially available FEC limit of soft decision (SD) on a typical value of bit error rate (BER). The threshold value of OSNR is 10.1 dB with a same SDFEC limit on the standard value of BER.
Figure 5 shows the input launch power (dBm) v/s Qfactor (dB) for the COOFDM system without NLE, with RBFNNNLE, and with proposed SORBFNNNLE over fiber length 8000 km, at payload bit rate 100 Gbps. As illustrated in Fig. 5, the value of Qfactor is improved in the whole range of discussed input launch powers, i.e., from − 5 to 5 dBm. This plot shows that maximum improvement is achieved at 0 dBm. As highlighted in Fig. 5, the improvement in Qfactor is ~ 1.1 and~ 3. 5 dB at 100 Gbps payload bit rate with RBFNNNLE and SORBFNNNLE, respectively. One downside regarding our methodology is that complexity of the system increases as compared to traditional RBFNNNLE technique. Despite this limitation of this method, our findings, however, suggest that in ultralonghaul systems nonlinearities are significantly compensated by proposed SORBFNNNLE technique due to its flexible structure and it is not possible to traditional RBFNNNLE technique. In fact, reduction in system complexity was not a primary goal in this research. The main aim of this study was to improve the performance of ultralonghaul COOFDM systems.
Figure 6 presents the payload bit rate versus Qfactor at 0 dBm input launch power, over 8000 km transmission length. Figure 6 shows that the improvement in Qfactor is ~ 1.1 and ~ 0.5 dB with conventional RBFNNNLE at a bit rate of 100 and 115 Gbps, respectively. Moreover, Fig. 6 shows that proposed SORBFNNNLE provides additional ~ 2.4 and ~ 1.7 dB improvement than conventional RBFNNbased NLE at bit rate of 100 and 115 Gbps, respectively. Figure 6 shows that the improvement in Qfactor with RBFNNNLE and SORBFNNNLE without NLE is more at lower signal bit rate, i.e., at 80 Gbps as compared to higher values of signal bit rates, i.e., 120 Gbps.
Figure 7 shows the plot of the Qfactor (dB) versus the optical fiber transmission length (km) at a bit rate of 100 Gbps. Figure 7 shows that the improvement in Qfactor is ~ 2.4 and ~ 2 dB with proposed SORBFNNNLE w.r.t. conventional RBFNNNLE at fiber length 8000 and 10000 km, respectively. Figure 7 shows that with the increase in the fiber length, the value of Qfactor decreases. It is concluded from this plot that for both the fiber transmission lengths (8000 and 10,000 km), almost comparable improvement in Qfactor is achieved.
Figure 8 shows OSNR (dB) versus the optical fiber transmission length (km) plot. Figure 8 indicates that the improvement in Qfactor is ~ 2.3 and ~ 7.6 dB at a bit rate of 100 Gbps over transmission length of 8000 km with RBFNNNLE and SORBFNNNLE, respectively. Figure 9 presents the plot of OSNR (dB) versus BER performance. Figure 9 shows that to achieve the standard value of BER, i.e., 4.7 × 10^{−3}, the required value OSNR is reduced in SORBFNNNLE than in conventional RBFNNNLE. The single most striking observation to emerge from the data comparison is that SORBFNNNLE performs significantly better than traditional RBFNNNLE technique.
5 Conclusion
This paper has highlighted the importance of selforganizing algorithm used in RBFNN for nonlinearity compensation. The indication from this study intimates that selforganizing algorithm reduces the burden of the selection of a number of RBF nodes in the first hidden layer and their position in the network structure. The numerical results of this work have proved that SORBFNNNLE performs better than the conventional RBFNNNLE. It has been demonstrated that network complexity with proposed technique is considerably more as compared to traditional RBFNNNLE. The current study is unable to reduce the complexity of the SORBFNNNLE. In conclusion, this work performs significantly in terms of Qfactor and OSNR due to flexible structure of SORBFNNNLE (i.e., the hidden neurons in the SORBFNNNLE can be added or removed) at ultralonghaul COOFDM systems.
References
 1.Shieh, W., Bao, H., Tang, Y.: Coherent optical OFDM: theory and design. Opt. Express 16(2), 842–859 (2008)CrossRefGoogle Scholar
 2.van Nee, R.: OFDM codes for peaktoaverage power reduction and error correction. In: Proceedings of IEEE Global Telecommunication Conference, pp. 740–744 (1996)Google Scholar
 3.Savazzi, P., Favalli, L., Costamagna, E., Mecocci, A.: A suboptimal approach to channel equalization based on the nearest neighbor rule. IEEE J. Sel. Areas Commun. 16(9), 1640–1648 (1998)CrossRefGoogle Scholar
 4.Charalabopoulos, G., Stavroulakis, P., Aghvami, A.H.: A frequency domain neural network equalizer for OFDM. In: Proceedings of IEEE Global Telecommunications Conference, vol. 2, pp. 571–575 (2003)Google Scholar
 5.Jarajreh, M.A., Giacoumidis, E., Aldaya, I., Le, S.T., Tsokanos, A., Ghassemlooy, Z., v, N.J.: artificial neural network nonlinear equalizer for coherent optical OFDM. IEEE Photonics Technol. Lett. 27(4), 387–390 (2015)CrossRefGoogle Scholar
 6.Kaur, G., Kaur, G.: Application of Functional Link Artificial Neural Network for Mitigating Nonlinear Effects in Coherent Optical OFDM. Springer, Optical and Quantum Electronics (2017)Google Scholar
 7.Gan, Q., Subramanian, R., Sundararajan, N., Saratchandran, P.: Design for centers of RBF neural networks for fast timevarying channel equalization. Electron. Lett. 32(25), 2333–2334 (1996)CrossRefGoogle Scholar
 8.Ahmad, S.T., Kumar, K.P.: Radial basis function neural network nonlinear equalizer for 16QAM coherent optical OFDM. IEEE Photonics Technol. Lett. 28(22), 2507–2510 (2016)CrossRefGoogle Scholar
 9.Han, H., Chen, Q., Qiao, J.: Research on an online selforganizing radial basis function neural network. Neural Comput. Appl. 19, 667–676 (2010). https://doi.org/10.1007/s0052100903236 CrossRefGoogle Scholar
 10.Nam, M.D., Thanh, T.C.: Approximation of function and its derivatives using radial basis function networks. Appl. Math. Model. 27(3), 197–220 (2003)CrossRefzbMATHGoogle Scholar
 11.Singh, J.K., Basu, D.K., Nasipuri, M., Kundu, M.: Face recognition using point symmetry distancebased RBF network. Appl. Soft Comput. 7(1), 58–70 (2007)CrossRefGoogle Scholar
 12.Zhao, T.: RBFNbased decentralized adaptive control of a class of largescale nonaffine nonlinear systems. Neural Comput. Appl. 1(4), 357–364 (2008)CrossRefGoogle Scholar
 13.Ram, D., Srivastava, L., Pandit, M., Sharma, J.: Corrective action planning using RBF neural network. Appl. Soft Comput. 7(3), 1055–1063 (2007)CrossRefGoogle Scholar
 14.Ibnkahla, M.: Applications of neural networks to digital communications—a survey. Sig. Process. 80(7), 1185–1215 (2000)CrossRefzbMATHGoogle Scholar