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Nonlinearities compensator based on SORBFNN in ultra-long-haul high-capacity CO-OFDM systems

  • Gurpreet Kaur
  • Gurmeet Kaur
Original Paper

Abstract

Radial basis function neural network is recognized as being the most useful type of neural network for a nonlinear equalizer in the CO-OFDM 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 ultra-long-haul transmission length. Previous work has been limited to normal transmission length, i.e., up to 1400 km. In ultra-long-haul 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 ultra-long-haul 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 ultra-long-haul optical fiber communication CO-OFDM 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 self-organizing radial basis function neural network-based nonlinear equalizer (SORBFNN-NLE) for the mitigation of nonlinearities in ultra-long-haul CO-OFDM systems. Better performance of the SORBFNN is attributed due to self-organizing structure and training of network. From this study, it has been observed that the proposed SORBFNN-NLE performs well regarding Q-factor and OSNR at an ultra-long distance.

Keywords

Artificial neural network based nonlinear equalizer (ANN-NLE) Coherent optical orthogonal frequency division multiplexing (CO-OFDM) Self-organizing radial basis function neural network based nonlinear equalizer (SORBFNN-NLE) 

1 Introduction

The CO-OFDM system is attracting considerable interest due to its high spectral efficiency [1]. A challenging area in the field of CO-OFDM systems is high peak-to-average power ratio [2], due to which nonlinear effects such as SPM and XPM have become a critical issue in the CO-OFDM 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 network-based nonlinear equalizers. In [5], the authors studied a multilayer perceptron (MLP)-based artificial neural network-based nonlinear equalizer (ANN-NLE) to compensate the nonlinearity in the CO-OFDM system with Reidmiller’s resilient back-propagation learning algorithm. Jarajreh and co-workers estimated ~ 0.7 dB improvement in Q-factor over a fiber length of 1000 km, at launch power of − 6 dBm, and payload of 40 Gbps [5]. The main pitfall of MLP-based ANN-NLE 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 network-based nonlinear equalizer (FLANN-NLE) is an alternative artificial neural network (ANN) technique to compensate the nonlinearity using a particle swarm optimization-based back-propagation (PSO-BP) learning algorithm with less computational complexity [6]. Unfortunately, it does not provide much amount of improvement in Q-factor as compared to MLP-based ANN-NLE. 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 K-means clustering training algorithm is a new method for nonlinearities mitigation in CO-OFDM systems and concluded that it provide much improvement in Q-factor as it is able to produce proper decision boundaries [7, 8]. In [8], the authors investigated ~ 1.2 dB improvement in Q-factor 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 network-based nonlinear equalizer (RBFNN-NLE) for ultra-long-haul CO-OFDM systems. It is common knowledge that optical signal becomes very much complex over ultra-long 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 ultra-long distance the received 16-point quadrature amplitude modulation (16-QAM) 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. Self-organizing 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 ultra-long distances, i.e., around 8000 km. Self-organizing radial basis function neural network-based nonlinear equalizer (SORBFNN-NLE) is introduced in order to achieve better results than traditional RBFNN-NLE. In this new methodology, two hidden layers are introduced in radial basis function neural network. A self-organizing 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 CO-OFDM 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 training-based NLE for dynamic nonlinear impairments is approximately 71% improved as compared to that of RBFNN-NLE. With 13% training overhead, reduction in nonlinear penalty is 3.5 dB at 100 Gbps payload in SORBFNN-NLE. This solution improves on previous methods in terms of Q-factor due to the self-organizing 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 CO-OFDM 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 SORBFNN-NLE-based simulation model for CO-OFDM system

CO-OFDM systems are undergoing a revolution in terms of high capacity and ultra-long transmission length. Most studies have only focused on high capacity. Ultra-long 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 ultra-long transmission length as it is not able to make proper decision boundaries due to more spread in complex 16-QAM 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 RBFNN-NLE 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 16-QAM 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 16-QAM modulation and removal of above-mentioned problems are accomplished simultaneously. Out of these two hidden layers, the first hidden layer is automatically adjusted with a self-organizing 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].

The simulation setup is equipped with a proposed SORBFNN-NLE depicted in Fig. 1. As the equalization is performed at individual subcarrier of the CO-OFDM system, the SORBFNN-NLE block present in the simulation model contains k sub-networks, where each sub-network performs the task of equalization over each subcarrier signal using the proposed method. For the complex OFDM signal \( P\left( k \right), \) the equalization is performed by dividing this kth subcarrier signal into real and imaginary parts as Pr(k) and Pi(k), respectively. After applying SORBFNN-NLE to the real part and imaginary part of the received complex signal, the estimated output of SORBFNN-NLE-based sub-network is as follows: The real part is \( \hat{F}_{\text{real}} \left( k \right) \) and the imaginary part is \( \hat{F}_{\text{imag}} \left( k \right). \) Estimated output for the kth subcarrier with SORBFNN-NLE, i.e., the estimated real part and the estimated imaginary part, is recombined to generate the total estimated CO-OFDM signal and is calculated as:
$$ \hat{F}\left( k \right) = \hat{F}_{\text{real}} \left( k \right) + j\hat{F}_{\text{imag}} \left( k \right). $$
(1)
Fig. 1

Simulation arrangement of a CO-OFDM system using SORBFNN-based nonlinearity compensator

Suppose F(k) is used as the desired signal for the training of SORBFNN-based nonlinear equalizer. The cost function for the training of neural network parameters is mean squared error (MSE) for kth subcarrier and calculated as:
$$ e\left( k \right) = \frac{1}{{N_{\text{s}} }}\mathop \sum \limits_{t = 0}^{{N_{\text{s}} }} \left( {F_{t} \left( k \right) - \hat{F}_{t} \left( k \right)} \right)^{2} , $$
(2)
where \( N_{\text{s}} \) is the number of the samples in kth subcarrier, \( F_{t} \left( k \right) \) the expected output for tth sample, and \( \hat{F}_{t} \left( k \right) \) the estimated output for tth sample using proposed SORBFNN-NLE. Our aim of the training is to get e(k) < Ed by learning, where \( E_{d } \) is the expected steady error. This condition of desired tolerable error is achieved by updating the weights of the network and various other parameters. In this method, LMS algorithm is used for updating the weights of output layer and is given as:
$$ {\text{new}}\;{\text{weight}} = {\text{old}}\;{\text{weight}} - a_{1} .\frac{{\partial \left( {\text{MSE}} \right)}}{{\partial \left( {{\text{old}}\;{\text{weight}}} \right)}}, $$
(3)
where a1 is a positive constant and is always less than 1.

3 Structure and working of SORBFNN-NLE

In this section, the SORBFNN-NLE structure to mitigate the optical fiber nonlinearities in an ultra-long-haul CO-OFDM system is described and is presented in Fig. 2.
Fig. 2

Structure of SORBFNN-NLE

The output for the kth subcarrier of the SORBFNN-based NLE (for imaginary part) is calculated as:
$$ \hat{F}_{\text{imag}} \left( k \right) = \mathop \sum \limits_{j = 1}^{M} \varphi_{j} \left( {\varvec{x}\left( \varvec{k} \right) - u_{j} \left( k \right)} \right)w_{j} , $$
(4)
where M is the neurons or nodes present in the second fixed hidden layer of the network which is 16 in 16-QAM modulation; wj is output layer weights ranging from 0 to 1. In the above equation, \( \varvec{x}\left( \varvec{k} \right) \) is calculated as:
$$ \varvec{x}\left( \varvec{k} \right) = \varvec{s}\left( k \right)\varvec{w}_{j} , $$
(5)
where \( \varvec{s}\left( k \right) = \left\{ {s_{1} \left( k \right) \cdots s_{t} \left( k \right) \cdots s_{l} \left( k \right)} \right\} \) is the first hidden layer outputs and \( \varvec{w}_{j} = \left\{ {w_{1j} \cdots w_{tj} \cdots w_{lj} } \right\} \) is the connection weight matrix between the first hidden layer and second hidden layer.
The hidden layer nonlinear output is defined as φj(·) and radially symmetrical. The Gaussian function is chosen in this study, and φ(·) is calculated as:
$$ \varphi \left( x \right) = e^{{\left( { - \frac{{\left( {\varvec{x} - u} \right)^{2} }}{{\sigma^{2} }}} \right)}} , $$
(6)
where u and σ are the fixed position and width of the second hidden layer RBF nodes, respectively, and (‖·‖) is the Euclidean distance. And the output of first hidden layer with self-organizing algorithm is calculated as:
$$ s_{t} \left( k \right) = {\text{med}}\left( {\varphi_{t} \left( {P_{q} \left( k \right) - v_{t} \left( k \right)} \right)} \right), $$
(7)
where q varies from 1 to n, t varies from 1 to l, and the value of l is changed by using self-organizing algorithm depending on the complexity introduced into the received signal due to nonlinearities at an ultra-long distance. Generally, the value of l is greater than of M. In the first hidden layer of the network, v (position) and \( \delta \) (width) of the RBF nodes are automatically adjusted by the self-organizing algorithm.
Basically, in this section working of a self-organizing radial basis function-based nonlinear equalization for imaginary signal has been explained with the help of flowchart presented in Fig. 3. The procedure of equalization with the proposed algorithm is summarized by the following steps:
Fig. 3

Flowchart for working of self-organizing radial basis function

  • 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 16-QAM 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 wj have been randomly selected from 0 to 1.

  • Step 4 Input pattern P(k) with their corresponding expected error (Ed) has been applied to the SORBFNN-NLE 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, vt(k) − P(k) > βrt condition has been checked. Then, e(k) > αEd and e(k) ≤ αEd have been checked for vt(k) − P(k) greater than βrt and vt(k) − P(k) less than or equal to βrt, 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) > αEd and e(k) ≤ αEd 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 SORBFNN-NLE has been estimated using Eqs. 47.

  • 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.

Besides the learning process, the hidden layer nodes with receptive field radius are accustomed based on the winner times of a particular node x as rx = ɛxrx in which x is a winner node and rx = τxrx in which x is not a winner node. Here, ɛx, τx are the hidden layer nodes weights valued changing parameters. Actually, the value of ɛx is always more than 1 and τx is always less than 1. These two parameters are calculated as:
$$ \varepsilon_{x} = 1 + \frac{1}{{\sqrt {2W_{x} } }}, $$
(8)
$$ \tau_{x} = 1 - \frac{1}{{10\sqrt {W_{x} } }}, $$
(9)
where \( W_{x} \) is the winner times of the hidden layer RBF node \( v_{x} . \) Constant values of these parameters lead to slow changes in network structures, and then, barely optimal networks are obtained. Therefore, these values should be changing during network optimization. And the formulas used to update the positions and widths of the RBF nodes are given as follows:
$$ {\text{new}}\;{\text{width}} = {\text{old}}\;{\text{width}} - a_{2} .\frac{{\partial \left( {\text{MSE}} \right)}}{{\partial \left( {{\text{old}}\;{\text{width}}} \right)}}, $$
(10)
$$\begin{aligned} &{\text{new}}\;{\text{position}}\\ &= \left\{ {\begin{array}{*{20}c} {{\text{old}}\;{\text{position}} - a_{3} .\left( {P\left( k \right) - v_{t} \left( k \right)} \right)} & {{\text{for}}\;v_{t} \left( k \right){\text{winner}}} \\ {{\text{old}}\;{\text{position}} - a_{4} .\frac{{\partial \left( {\text{MSE}} \right)}}{{\partial \left( {{\text{old}}\;{\text{position}}} \right)}}} & {{\text{for}}\;{\text{others}}} \\ \end{array} } \right.,\end{aligned} $$
(11)
where a2, a3, and a4 are positive constants and are always less than 1.

In this proposed method, the computational complexity and memory requirement are of the order of O (TsNS2) and O (TsS), respectively, which are O (TsR2) and O (TsR) 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

In this section, Figs. 4, 5, 6, 7, 8, and 9 show the obtained results for the remuneration of nonlinearities with proposed SORBFNN–NLE in high-capacity ultra-long-haul distance CO-OFDM system.
Fig. 4

Constellation diagrams of received 16-QAM data at payload bit rate 100 Gbps, over transmission distance—8000 km (Color figure online)

Fig. 5

Plot of input launch power (dBm) versus Q-factor (dB) over 8000 km transmission length and 100 Gbps payload bit rate

Fig. 6

Plot of payload bit rate (Gbps) versus Q-factor (dB) over 8000 km transmission length

Fig. 7

Plot of transmission length (Km) versus Q-factor (dB) at 100 Gbps payload bit rate

Fig. 8

Plot of transmission length (Km) versus OSNR (dB) at 100 Gbps payload bit rate

Fig. 9

Plot of OSNR (dB) versus BER over 8000 km transmission length and 100 Gbps payload bit rate

Figure 4 presents the constellation diagrams of received 16-QAM data for the ultra-long-haul CO-OFDM system. It is apparent from Fig. 4 that at an ultra-long distance the received 16-QAM 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 SORBFNN-NLE technique (represented by green color) focuses on the constellation points more adequately at required positions in comparison with traditional RBFNN-NLE (represented by blue color). As predicted, our experiments demonstrate that the proposed technique has the more ability to reduce fiber nonlinearities. In this paper, Q-factor is used as judgment factor to plot various graphs so that proper comparison of proposed SORBFNN-NLE technique with the previously used RBFNN-NLE technique can be presented. The formula for calculating the Q-factor 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 Q-factor 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 SD-FEC limit on the standard value of BER.

Figure 5 shows the input launch power (dBm) v/s Q-factor (dB) for the CO-OFDM system without NLE, with RBFNN-NLE, and with proposed SORBFNN-NLE over fiber length 8000 km, at payload bit rate 100 Gbps. As illustrated in Fig. 5, the value of Q-factor 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 Q-factor is ~ 1.1 and~ 3. 5 dB at 100 Gbps payload bit rate with RBFNN-NLE and SORBFNN-NLE, respectively. One downside regarding our methodology is that complexity of the system increases as compared to traditional RBFNN-NLE technique. Despite this limitation of this method, our findings, however, suggest that in ultra-long-haul systems nonlinearities are significantly compensated by proposed SORBFNN-NLE technique due to its flexible structure and it is not possible to traditional RBFNN-NLE 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 ultra-long-haul CO-OFDM systems.

Figure 6 presents the payload bit rate versus Q-factor at 0 dBm input launch power, over 8000 km transmission length. Figure 6 shows that the improvement in Q-factor is ~ 1.1 and ~ 0.5 dB with conventional RBFNN-NLE at a bit rate of 100 and 115 Gbps, respectively. Moreover, Fig. 6 shows that proposed SORBFNN-NLE provides additional ~ 2.4 and ~ 1.7 dB improvement than conventional RBFNN-based NLE at bit rate of 100 and 115 Gbps, respectively. Figure 6 shows that the improvement in Q-factor with RBFNN-NLE and SORBFNN-NLE 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 Q-factor (dB) versus the optical fiber transmission length (km) at a bit rate of 100 Gbps. Figure 7 shows that the improvement in Q-factor is ~ 2.4 and ~ 2 dB with proposed SORBFNN-NLE w.r.t. conventional RBFNN-NLE at fiber length 8000 and 10000 km, respectively. Figure 7 shows that with the increase in the fiber length, the value of Q-factor decreases. It is concluded from this plot that for both the fiber transmission lengths (8000 and 10,000 km), almost comparable improvement in Q-factor is achieved.

Figure 8 shows OSNR (dB) versus the optical fiber transmission length (km) plot. Figure 8 indicates that the improvement in Q-factor is ~ 2.3 and ~ 7.6 dB at a bit rate of 100 Gbps over transmission length of 8000 km with RBFNN-NLE and SORBFNN-NLE, 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 SORBFNN-NLE than in conventional RBFNN-NLE. The single most striking observation to emerge from the data comparison is that SORBFNN-NLE performs significantly better than traditional RBFNN-NLE technique.

5 Conclusion

This paper has highlighted the importance of self-organizing algorithm used in RBFNN for nonlinearity compensation. The indication from this study intimates that self-organizing 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 SORBFNN-NLE performs better than the conventional RBFNN-NLE. It has been demonstrated that network complexity with proposed technique is considerably more as compared to traditional RBFNN-NLE. The current study is unable to reduce the complexity of the SORBFNN-NLE. In conclusion, this work performs significantly in terms of Q-factor and OSNR due to flexible structure of SORBFNN-NLE (i.e., the hidden neurons in the SORBFNN-NLE can be added or removed) at ultra-long-haul CO-OFDM systems.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of ECEPunjabi UniversityPatialaIndia

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