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Abstract
The optical signal-to-noise ratio (OSNR) and fiber nonlinearity are critical factors in evaluating the performance of high-speed optical fiber communication systems. Recently, several deep learning based methods have been put forward to monitor OSNR of a fiber communication system. In this work, we propose a long short-term memory (LSTM) network based method to simultaneously estimate OSNR and nonlinear noise power caused by fiber nonlinearity. In the training step, LSTM network extracts the essential features in frequency domain of the input signal. Then, with the built model in the training step, the LSTM output the OSNR and nonlinear noise power of the signal under test. The simulation by VPI software is carried on a 5-channel long haul optical transmission system with the launched optical power of −3.0~ + 3.0dBm per channel. The results show that the test error of OSNR is less than 1.0dB with the reference OSNR from 15 to 30dB for QPSK, 16QAM and 64QAM signal. The test error of nonlinear noise power is less than 1.0dB for QPSK and 16QAM signal when the Laser linewidth is 6 KHz and 100 KHz respectively. The proposed method is a promising candidate for nonlinearity-insensitive OSNR and accurate nonlinear noise power estimation in multi-channel long haul optical fiber communication systems.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Monitoring the quality of optical signals and transmission links precisely is an important task for the long haul coherent optical fiber communication systems. The optical signal-to-noise ratio (OSNR) and nonlinearity are critical factors to measure the performance of the systems and have become hot research topics. With the help of coherent receiver and digital signal processing (DSP), error vector magnitude (EVM) is proved to be an appropriate metric to measure OSNR [1]. The noise power in OSNR refers to the amplified spontaneous emission (ASE) noise specifically. However, with the increase of transmission distance and launched power, the noise or distortion caused by fiber nonlinearity cannot be ignored. The method based on EVM will underestimate the OSNR since the contribution of fiber nonlinearity is not excluded. Hence the effective nonlinearity-insensitive OSNR monitoring technique is the focus of research [2–7]. Some statistical OSNR estimation methods have been proposed which are robust against the impact of nonlinearity, such as the differential pilot aided technique [2], a correlation function with calibration factor method [3,6], and time domain pilot aided with fractional Fourier transformation technique [4]. However, all of these methods need manual operation that takes a lot of time and requires domain expertise, and it is difficult to get accurate results due to personal error.
Recently, deep learning (DL) is a hot research area in the field of machine learning, which is considered as deep neural networks with multiple nonlinear layers. DL algorithms can extract more abstract features from high-dimensional data with the ability of self-learning and need less human operation, which mainly include deep neural network (DNN), convolutional neural network (CNN), recurrent neural network (RNN), long short-term memory (LSTM) network, etc. These methods have dramatically improved the state-of-the-art in various fields, such as speech recognition, target detection, natural language processing, and medical analysis [8,9]. Since the constellation or eye diagrams of coherently received optical signal vary with different OSNR, several OSNR estimation methods based on DL have been put forward [10–12]. The estimation of OSNR was realized by training CNN to extract features from the eye or constellation diagrams [10,11]. However, with the increase of OSNR, the change of constellation or eye diagram gets less significant, and the estimation accuracy of CNN recognition based method will decrease. The method for OSNR monitoring by combining DNN and amplitude histogram of signal is also been put forward [12]. However, all these methods based on DL do not take into account the impact of fiber nonlinearity. In addition, some of these techniques only consider a single channel and back-to-back communication system.
In a multi-channel long haul optical fiber communication system, the phase noise or distortion caused by fiber nonlinearity is the most important factor limiting its performance. The nonlinearity-insensitive OSNR estimation methods also require the accurate estimation of fiber nonlinearity. Nonlinear noise power is commonly used to characterize the impact of fiber nonlinearity [4,13–18]. Several artificial neural network (ANN) based methods for nonlinearity monitoring and equalization have been reported, such as the nonlinear SNR monitoring [18,19] and the nonlinear equalizer [20]. The capability of ANN to capture and learn the characteristics of data is limited because it contains a single hidden layer. As a result, the time-consuming and complicated preprocessing, like the calculation of phase noise and the second moment statistics of the received signal data, are needed in the ANN based method [18]. Besides that, the statistical preprocessing will cause additional personal error. To the best of our knowledge, few deep learning-based methods for the estimation of nonlinear noise power have been reported.
Long haul fiber transmission leads to the accumulation of nonlinearity-induced distortion, which is mainly reflected by the crosstalk between symbols in time domain, and the spectrum broadening in frequency domain. The LSTM network is a special RNN architecture that has powerful modeling capabilities for long-term dependencies and can extract the correlation between the past and current data [21,22]. In this paper, we propose a LSTM network based method to simultaneously estimate the OSNR and nonlinear noise power. The simulation by VPI software is carried on a 5-channel long haul optical fiber transmission system with the launched optical power of −3.0~ + 3.0dBm per channel and the transmission distance is up to 1000 km. In our method, the coherently received signal is first transformed into frequency domain by Fast Fourier Transformation (FFT) with the length of 1024. Then, we use a LSTM network of two hidden layers to estimate nonlinear noise power and OSNR simultaneously. For comparison, the EVM based OSNR estimation method is also performed to verify that our method is tolerant to fiber nonlinearity. The proposed method can estimate OSNR with the mean absolute error (MAE) of 0.04dB, 0.04dB, 0.06dB for QPSK,16QAM, and 64QAM signal respectively in the OSNR range of 15.0~30.0 dB. The maximum estimation error of nonlinear noise power is less than 1.0dB for QPSK and 16QAM signal when the Laser linewidth is 6 KHz and 100 KHz respectively, which outperforms the existing nonlinear noise power estimation techniques [4].
2. Operating principle
In multichannel optical fiber communication systems, the inter-channel cross-phase-modulation (XPM) constitutes the predominant contribution to nonlinear noise. XPM refers to the nonlinear phase shift of an optical field induced by a co-propagating field at a different wavelength, which broadens the spectrum of optical signal propagating through the fiber [23]. Long haul transmission and high launched power will lead to the accumulation of nonlinearity-induced distortion and the spectrum broadening. In our work, we use the LSTM network to train the data of broadened spectrum and test the OSNR and nonlinear noise power. In the training stage, the frequency domain data of coherently received signal under different conditions, like launched power, OSNR, number of channels and transmission spans, and combinations of different fibers, are input to the LSTM network. With the internal self-learning of the LSTM network, the frequency domain features of optical signal imposed by ASE noise and fiber nonlinearity, like the statistics of amplitude and phase shift can be learned and extracted from the real and imaginary data. In order to confirm the test accuracy of nonlinear noise power by our method, the reference nonlinear noise power is obtained with a de-correlation algorithm in [4], in which de-correlation and Wiener filter is utilized to subtract the signal without nonlinearity from the signal with nonlinearity [24].
LSTM is a special RNN architecture that has powerful modeling capabilities for long-term dependencies. In the LSTM network, a recurrent LSTM memory cell replaces traditional node (a simple activation function) in the hidden layer of a standard RNN, as shown in Fig. 1(a). The LSTM network can be unfolded to several steps like the right graph in Fig. 1(a) for the purpose of understanding [22,25]. In each step, the LSTM cell will output the current hidden memoryand the current cell stateto the next step.
Fig. 1 (a) Schematic diagram of LSTM based OSNR and nonlinear noise power estimation. (b) The internal structure of a LSTM cell. The activation function is sigmoid, is tanh.
The internal structure of a LSTM cell is shown in Fig. 1(b), which contains one cell state and three “gate” structures. A gate is so-called because its value is used to multiply the value of another node to let information through. The cell state is the heart of each LSTM cell that stores historical information. Combined with the previous cell state, the previous hidden memoryand the current input, the three “gates”, i.e. input gate, forget gate and output gateare utilized to remove or add information to the current cell state by using activation functions and some linear operations [22]. The red dotted line in Fig. 1(b) demonstrates the activation functions, also called neurons, in a LSTM cell. The computation in the LSTM cell proceeds according to the following formulas at each step, i.e. the forward propagation algorithm [21,22]:
In these formulas, all W variables are weights and b variables are biases, which are learned and optimized during the training stage by the back propagation algorithm. In the back propagation algorithm, the partial derivatives of the loss function to all W and b variables are calculated, whereby all the W and b variables are iteratively updated through gradient descent method. Normally, the mathematical relation between the LSTM output and the actual output, like cross entropy function or mean square deviation, constitutes the loss function of the network. In our work, the loss function of LSTM is mean square deviation. Due to the special structure of LSTM network, the correlation between the past and current data can be extracted well.In our proposed method, the LSTM network has two hidden layers, each hidden layer has two cells, and each cell has 15 neurons. To take into account the dependencies of frequency components in a nonlinearity distorted signal, the coherently received signal is transformed to frequency domain by 1024-point FFT and then input the LSTM network. The input layer of LSTM network has one neuron, and a frequency-domain sequence of real and imaginary parts is input this neuron. The output layer of LSTM network also has one neuron, in which the linear function is used as activation function to obtain the estimated OSNR and nonlinear noise power. The activation functions in the LSTM cells are sigmoid and tanh [26]. The loss function is the mean square deviation of LSTM output referring to the actual output.
The convergence of LSTM network should be studied with different network parameters. To deal with this issue, the choice of optimizers with different gradient descent algorithms and learning rate are mainly considered in the training stage [27]. It is empirical that the loss function can be used to evaluate the convergence of LSTM network. Stochastic Gradient Descent (SGD) and Adam optimizers are commonly employed to optimize the DL network. Figure 2(a) compares the loss function of LSTM network with SGD and Adam optimizers when the learning rate is 0.01. When the iterations grow to 200, the loss function of using SGD algorithm remains constant around 20 and no longer declines, indicating that it is difficult to achieve convergence. On the contrary, the Adam optimizer has good convergence when the iterations grow to 1500. Figure 2(b) shows that the Adam optimizer can converge when the learning rate is 0.01 or 0.1.
Fig. 2 The loss function evolves with the iterations. (a) SGD and Adam optimizer. (b) Adam optimizer with different learning rate: 0.001, 0.01, 0.1, 1.0.
The LSTM network in our method is iteratively trained with a batch size of 100, and the loss function is used to evaluate its convergence under different instances. When the value of loss function drops to 0.005, the training is early terminated to prevent overfitting [28]. It is worth pointing here that, the network parameters are randomly initialized in each training, and the optimization trajectories may be different, even with the same input data and structure of network. However, the performance of trained network and its test results are consistent and similar.
3. System setup and results
3.1 OSNR estimation
We set up the simulation system with the commercial software Virtual Photonics Inc. (VPI) Transmission Maker. The schematic diagram of OSNR estimation method based on frequency domain LSTM network for a 5-channel optical fiber transmission system is shown in Fig. 3.
Fig. 3 Schematic diagram of LSTM based OSNR estimation method for a 5-channel optical fiber transmission system.
The symbol rate is 28 GB and the launched power per channel is in the range of −3.0~ + 3.0dBm. The channel spacing is 50 GHz and the center channel is under the test. The modulated optical signal is amplified using an erbium-doped fiber amplifier (EDFA) and sent to a fiber link, each span consists of a 100 km long standard single-mode fiber (SSMF) and an EDFA. The chromatic dispersion (CD) and nonlinear refractive index of SSMF are andrespectively. There is no ASE noise added in each EDFA and the OSNR is controlled by adding ASE at the end of the fiber link. The reference OSNR for QPSK, 16QAM and 64QAM is varied in the range of 15~30 dB with step of 1.0dB. The total transmission fiber length is varied from 100~1000km. At the receiver end, the optical signal is filtered using a 50 GHz bandwidth optical band-pass filter (OBPF) and then detected by a coherent receiver. The simulation parameters are summarized in Table 1. After the CD equalization by DSP, the signal is transformed to frequency domain by 1024-point FFT, and input to LSTM network. In our method, the LSTM is built based on the Spyder platform and the TensorFlow library.
Table 1. The simulation parameters of the 5-channel optical fiber communication system
Based on the above system, we collect 1120 sets of frequency domain data for each modulation format, corresponding to different OSNR, launched power and spans. The whole collection set is randomly divided into training data (70%) and testing data (30%). During training process, the essential features of input signal are extracted by LSTM network.
The OSNR estimation results for QPSK, 16QAM and 64QAM are shown in Fig. 4. The maximum, minimum and average estimated values for each reference OSNR are marked in these figures. It can be seen that OSNR estimation is quite accurate even in the impact of fiber nonlinearity. The maximum test error is 0.30 dB, 0.48 dB, 0.95 dB for QPSK, 16QAM and 64QAM signal respectively, and the mean absolute error (MAE) of all tests are 0.04dB, 0.04 dB, 0.06 dB respectively. The maximum MAE for three modulation signals is less than 0.15dB and the maximum normalized error is less than 1.0%, as shown in Fig. 4(d). It is noted that the OSNR estimation error bars increase with the greater reference OSNR. This can be explained that, with the increase of OSNR, the power of ASE noise is too low and the distortion caused by nonlinearity becomes the main noise of the signal. In this case, OSNR estimation will have more deviation. The error bars are greater when the modulation format is from QPSK to 64QAM. This is because that the distortion caused by fiber nonlinearity is mainly reflected in the crosstalk between transmission symbols. High order modulation signals are more susceptible to fiber nonlinearity. Therefore, the estimation error for mQAM signal is greater. For comparison, the EVM based method is also performed to approve that our method is tolerant to fiber nonlinearity for QPSK signal. The EVM based method mainly relies on the distribution of points in the constellation diagram of signal [1]. Long haul transmission leads to the accumulation of nonlinearity-induced distortion, and causes the crosstalk between symbols in time domain. As a result, the distribution of points in the constellation diagram is expanded and the corresponding EVM is greater. Therefore, EVM based method will underestimate the OSNR since the contribution of fiber nonlinearity is not excluded [6].
Fig. 4 Test results of OSNR by LSTM based method: (a) QPSK, (b) 16QAM, (c) 64QAM, (d) Mean absolute error and normalized error (Norm-error) of OSNR estimation for QPSK, 16QAM, 64QAM signal.
In Fig. 5, the MAE of the proposed method is much less than that of EVM based method in the presence of fiber nonlinearity. Compared with other existing methods based on DL [10–12], the proposed method has superior advantages in that our method takes full account of the frequency domain characteristics of data and achieves fairly high accuracy. The above results on a 5-channel long haul optical transmission system with the launched optical power of −3.0~ + 3.0dBm per channel and the transmission distance up to 1000 km demonstrate that our method can be implemented for real optical fiber communication systems even with the existence of fiber nonlinearity.
Fig. 5 OSNR test error of QPSK signal with EVM or LSTM based method. The transmission fiber length is from 100~1000km.
3.2 Nonlinear noise power estimation
Figure 6 is the system setup for nonlinear noise power estimation with our method. The polarization multiplexing QPSK and 16QAM signal are generated by using polarization beam splitter. Two stages are included in our method. In the first stage, the reference nonlinear noise power is calculated by making simulation on a 5-channel transmission system with and without nonlinearity (NL), in which the ASE noise and Laser linewidth are not considered. The nonlinear noise power of the channel under test is obtained by Wiener filter de-correlation [4,24]. In the second stage, a set of simulations are conducted on the 5-channel transmission system with the consideration of nonlinearity, OSNR and different Laser linewidth, and the results serve as the input of training LSTM. When the training is completed, the test signal is input to the trained LSTM, and the output is nonlinear noise power contained in the test signal.
Fig. 6 Schematic diagram of LSTM based nonlinear noise power estimation method for a 5-channel optical fiber transmission system.
In the second stage, the coherently received signal is transformed into frequency domain by 1024-point FFT after the CD equalization. To train the LSTM, we collect 2240 sets of frequency domain data for each modulation format corresponding to different OSNR, linewidth, launched power and spans. The whole collection set is randomly divided into training data set (70%) and testing data set (30%). The OSNR ranges from 15 to 30dB. The Laser linewidth is set as 6 KHz and 100 KHz respectively.
Figure 7 shows the estimated nonlinear noise power and the error for QPSK and 16QAM optical fiber communication system with 8 spans transmission distance. The Laser linewidth is 100 KHz and launched power is from −2.0 to 2.0dBm. Figures 7(a) and 7(b) demonstrate that the absolute error of test is less than 1.0dB and 0.6dB for a QPSK and 16QAM system respectively when OSNR is from 15 to 30dB. It can be conclude that the estimated nonlinear noise power is accurate when comparing it to the reference nonlinear noise power, and the estimation error barely changes for different OSNR. This indicates that our method is not significantly affected by ASE noise. Therefore, the estimated nonlinear noise power can be represented by averaging its value under different OSNR. Then we estimate the nonlinear noise power with different launched power. It can be seen from Figs. 8(a) and 8(b) that the increase of launched power by 1.0dB will lead to about 3.0dB increase of nonlinearity power irrespective of different Laser linewidth and transmission distance, which matches well the conclusion that the nonlinearity power is proportional to the third power of launched power [29,30]. The estimation error is shown in Figs. 8(c) and 8(d). The maximum error is less than 1.0dB and maximum normalized error is less than 10% for QPSK and 16QAM transmission system. The MAE of test is 0.48dB, 0.28dB, 0.34dB, 0.33dB respectively for a QPSK system with the transmission length and Laser linewidth of 3 spans and 6 KHz, 5 spans and 6 KHz, 8 spans and 100 KHz, 10 spans and 100 KHz. For a 16QAM transmission system, the MAE of test is 0.20dB, 0.22dB, 0.39B, 0.24dB respectively. There is no similar conclusion and trend that the low or large launched power corresponding to large error, as in other statistical methods [2–7]. In the LSTM training analysis module, we expound that the achieved performance is consistent for all the trainings when the value of loss function drops to 0.005. The MAE of each training is not exactly the same, but falls within the 1.0dB error with our method.
Fig. 7 The estimated nonlinear noise power of an 8-span optical fiber transmission system. (a) QPSK, (b) 16QAM. The launched power is from −2.0 to 2.0dBm.
Fig. 8 Estimated nonlinear noise power and its error with our method. (a) Nonlinear noise power estimation for QPSK. (b) Nonlinear noise power estimation for 16QAM. (c) Mean absolute error and normalized error (Norm-err) for QPSK. (d) Mean absolute error and normalized error (Norm-err) for 16QAM. Each data represents the average value under OSNR from 15 to 30dB.
By transforming the coherently received signal into frequency domain by FFT and training the LSTM network for self-learning and feature extraction, the proposed method for simultaneously estimating OSNR and nonlinear noise power requires little knowledge of the optical fiber transmission system or complicated statistical operations as in other statistical methods [2–7]. In addition, our method based on intelligent operation needs less manual intervention and avoids the personal error. Once the training process of the neural network is completed offline, the test data can directly enter into the network to output the estimated results. In this sense, the proposed method is more suitable for real-time optical performance monitoring for the fiber transmission link.
4. Conclusions
In this paper, we propose a LSTM network based method to simultaneously estimate OSNR and nonlinear noise power. For OSNR estimation, three widely-used modulation formats, QPSK, 16QAM and 64QAM are investigated and the test error is less than 1.0dB when the reference OSNR is from 15 to 30dB. Compared with the EVM based method, our method for estimating OSNR is tolerant to fiber nonlinearity. With our method, the estimation error of nonlinear noise power is less than 1.0dB for QPSK and 16QAM signal with different Laser linewidth, launched power and long haul transmission distance, which outperforms the existing techniques. The results demonstrate that our method is a promising candidate for nonlinearity-insensitive OSNR and accurate nonlinear noise power estimation in multi-channel long haul optical communication systems.
Funding
National Natural Science Foundation of China (NSFC) (61427813, 61331010).
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