- Email: [email protected]
QAM classification methods by SVM machine learning for improved optical interconnection
Optics Communications 444 (2019) 1–8
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
QAM classification methods by SVM machine learning for improved optical interconnection Chang Wang a , Jiangbing Du a ,∗, Guoyao Chen a , Haoyang Wang a , Lin Sun a , Ke Xu b , Bo Liu c , Zuyuan He a a
State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China Department of Electronic and Information Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China c School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing, China b
ARTICLE
INFO
Keywords: Support vector machine Quadrature amplitude modulation Optical interconnection
ABSTRACT High-order quadrature amplitude modulation (QAM) formats are very effective for increasing the transmission capacity due to the highly increased spectral efficiency. However, the signal-to-noise-ratio (SNR) hungry and dense constellation of QAM make it very sensitive to nonlinear distortion. The nonlinear decision boundary adaptively generated by machine learning method of support vector machine (SVM) can be effectively used for the classification of the symbols. The different classification methods have different performance in terms of classification complexity. We experimentally investigated five SVM multi-classification methods for machine learning assisted adaptive nonlinear mitigation, including the one versus rest (OvR), the symbol encoding (SE), the binary encoding (BE), the constellation rows and columns (RC), and the in-phase and quadrature components (IQC). The comprehensive results with comparisons are demonstrated, indicating significant nonlinear mitigation with BER reductions. The SVM multi-classifier based on the in-phase and quadrature components is relatively optimal, considering the calculation and storage.
1. Introduction To meet the increasing demands of high capacity data communication, high-order QAM formats with significantly improved signaling efficiency (spectral efficiency) have been widely used for both intensitymodulation-direct-detection (IMDD) and coherent detection solutions. High-order QAM signal makes full use of amplitude and phase for carrying the information, which leads to great improvement of the transmission capacity [1]. However, the very dense constellation of high-order QAM leads to higher demand of SNR, which makes it very sensitive to the nonlinear distortion. The nonlinear distortion includes the modulation nonlinearity in short reach and Kerr nonlinear distortion in long haul. Thus, the correct detection of high-order QAM at the receiver side is usually very difficult due to the nonlinear distortion, particularly when the nonlinear distortion is varying. Transmission performance is also limited by amplified spontaneous emission (ASE) noise from in-line optical amplifiers which makes the tolerance to nonlinear distortion even smaller [2–7]. Therefore, adaptive nonlinear distortion mitigation methods are of key importance for high speed optical interconnection. Recently, digital signal processing based on machine learning has been studied for equalization and demodulation of optical interconnection systems [8–18]. In general, SVM-based signal processing is ∗
powerful with less complexity than deep learning of neural networks, which makes SVM very suitable for cost-sensitive IMDD optical interconnects. However, SVM has different classification methods from different demands. The different classification methods lead to different performance and complexity, which is important consideration when applying it to specific application scenarios, such as meter-scale multimode interconnection and kilometer-scale single-mode interconnection. T. Nguyen et al. demonstrated fiber nonlinearity equalizer based SVM classification for coherent optical orthogonal frequency division multiplexing (OFDM) [8]. In Ref. [12], one versus rest rule was employed for 16-QAM. The rule is that a received constellation point would belong to a certain cluster if that cluster accepted the received symbol and other clusters rejected it. This method would lead to uneven distribution of bit error ratio (BER) for each category of the constellation, which is caused by the number imbalance of different label points. D. Wang et al. applied the SVM decision method to the M-ary phase-shift keying (MPSK) based on the coherent optical system to mitigate the nonlinear phase noise. The authors designed the classification strategy for 16QAM with four SVMs and 64-QAM with six SVMs based on Gray code, respectively [13]. W. Chen et al. also used SVM for 64-QAM signal boundary decision based on the Gray code [14]. The idea that each classification is realized according to each bit of the symbol encoding
Corresponding author. E-mail address: [email protected] (J. Du).
https://doi.org/10.1016/j.optcom.2019.03.058 Received 15 January 2019; Received in revised form 13 March 2019; Accepted 23 March 2019 Available online 27 March 2019 0030-4018/© 2019 Published by Elsevier B.V.
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Optics Communications 444 (2019) 1–8
in Fig. 2(a), the hyperplane (straight line) in two-dimensional space divides data points into two categories. For the linearly inseparable case, mapping data to a high-dimensional space by kernel function 𝑘 is often used. In Fig. 2(b), linearly inseparable points in the original space can be transformed into linear separable problem in high-dimensional space by selecting specific mapping relationships 𝛷. This is equivalent to nonlinear classification in the original space. 𝑘 (𝑥, 𝑦) ∶= (𝛷 (𝑥) , 𝛷 (𝑦)) , 𝛷 ∶ RN → F
(1)
By learning from the training data, the feature space can be divided into two parts. The nonlinear classifier function can be described by the formula: ( 𝑙 ) ∑ ) ( 𝑓 (𝑥) = sign (2) 𝑎𝑖 ⋅ 𝑘 𝑥, 𝑥𝑖 + 𝑏
Fig. 1. (a) Ideal constellation and decision boundary. (b) Actual constellation and decision boundary with noise and nonlinearity.
𝑖=1
The parameters 𝑎𝑖 are computed as the solution of a quadratic programming problem and the sequential minimal optimization (SMO) algorithm is used to solve this problem. The algorithmic complexity of a trained model is determined by the number of support vectors, not by the dimensions of the data. The model trained by SVM relies entirely on support vectors. Even if all the non-supported vectors in the training set are removed, the training process will be repeated, and the result will still be exactly the same. If the number of support vectors obtained by training is small in an SVM, the model trained by SVM is easier to generalize.
is very reasonable. However, it would map the data to higher dimensions and require more support vectors to make the complex decision curves. G. Chen et al. demonstrated SVM decision method to mitigate modulation nonlinearity distortion for PAM-4 and PAM-8 VCSEL-MMF optical link [15]. Adaptive detection of PAM-4 signals modulated by silicon micro-ring modulator (Si-MRM) has also been demonstrated by SVM machine learning detection [16]. In Refs. [15,16], the authors proposed a new classification strategy based complete binary tree SVMs (CBT-SVMs) and applied for both PAM-4 and PAM-8 signals with lower complexity. Our objective of this study is to comprehensively investigate the classification methods of SVM machine learning detection with performance comparisons, so that we can realize the optimized QAM classification with balanced performance and complexity for a specified optical interconnection link. In this work, the SVM detection was implemented in a QAM-DMT optical transmission link based on the Mach– Zehnder modulator (M-ZM) and 10-km standard single mode fiber (SSMF), so that different QAM formats can be loaded onto different carriers. The experimental results in the case of back-to-back and 10-km transmission indicate that the SVM methods can well detect the nonlinear damage of signal and reduce the BER. Five multi-classification detection methods based on SVM have been investigated for QAM modulation formats. Both BER and complexity performance based on experimental results were investigated. The comparison indicates that SVM multi-classification based on the in-phase and quadrature component has the lowest complexity.
2.2. SVM multi-classification based on one versus one SVM multi-classification based on one versus one (OvO SVM) generates an optimal decision function between every two different categories of training sets. The advantage of OvO SVM is that the number of samples per training is relatively small, so the training speed of a single decision surface is faster and the precision is higher. However, since the N classification problem needs to train 𝑁 ∗ (𝑁 − 1)∕2 decision functions (𝑁 > 2), the total number of decision functions will be too large when N is large, and thus the prediction speed will be affected. 2.3. SVM multi-classification based on one versus rest The OvR SVM is to generate a hyperplane between a class of samples and the remaining multi-class samples, so as to achieve multiclass recognition. This method only requires an optimal hyperplane between a class of samples and the corresponding remaining samples, rather than classifying between the two samples. Therefore, if it is a N classification problem, then you need N SVMs (𝑁 > 2). Fig. 3(a) shows the training processing schematic diagrams of 16-QAM. When we use an SVM to make decision, the data is divided into two categories: the specific category to be decided with red color and the other category with green color. A total of 16 decision functions are obtained. The testing processing is shown in Fig. 3(b). Given a test data, substituting it into 16 decision functions and calculating the values, respectively. The value means that the score of the corresponding category is determined according to the distance between the hyperplane and constellation point. The category with the highest score is the final decision result. Compared with the OvO SVM, the number of generated hyperplanes is greatly reduced, so when the number of categories N is large, the prediction speed will be faster than the OvO SVM method. However, since it needs to use all the training sets every time it generates a hyperplane, it does not take less time to train than the OvO SVM. At the same time, since the remaining multi-classes are always regarded as one class during training, the positive and negative classes are extremely uneven in the number of training samples, which may affect the accuracy of prediction. In Refs. [13,14], the authors proposed the SVM classification based on the symbol encoding (SE) to improve the unbalance label.
2. SVM principle and classification methods Fig. 1 shows the ideal decision boundary for the original ideal constellation of 16-QAM modulation (as an example) and the actual decision boundary for a transmitted practical constellation using the SVM decision. Compared with the original ideal constellation, the transmitted practical constellation would be affected by both noise and nonlinearity. Thus, the decision boundary can no longer be a simple straight line for QAM. Therefore, SVM multi-classification to obtain adaptive nonlinear decision boundary is a good solution for optimizing the classification boundary which is particularly suitable for time-varying channels. 2.1. SVM principle The QAM constellation decision is actually a process of data classification and SVM is very suitable for dealing classification problems. An SVM can be understood as a binary classifier. In this work, with m bits for loading, the transmitted signal would be 2m -QAM and the demanded classification number is 𝑁 = 2m . To deal with 2m QAM classification, multi-SVMs are needed in order to implement multi-classification. For the linearly separable case, our goal is to find a hyperplane that separates points with different labels into the different side. As shown 2
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Optics Communications 444 (2019) 1–8
Fig. 2. (a) Data classification in two-dimensional space. (b) Nonlinear classification by high dimensional mapping.
Fig. 3. (a) A schematic diagram of the 16-QAM classification based on OvR. (b) OvR SVM classification flow chart.
Table 1 The comparison of SVM number for training model.
The SVM number for training
OvO-SVM
OvR-SVM
BT-SVM
𝑁 ∗ (𝑁 − 1)∕2
𝑁
𝑁 −1
2.4.1. BT-SVM based on constellation rows and columns Most of the constellations are rectangle, except for 32-QAM, 128QAM.etc. The label feature according to rows and columns is reasonable. Fig. 5(a) shows the hyperplane for the training processing of one category. The red and green colors respectively indicate the two categories to be distinguished, and the gray indicates the training sample set that is not required to be used. The training model of the modulation format such as 4-QAM, 8-QAM, 16-QAM, and 64-QAM is a complete binary tree. When testing, the decision requires log2 𝑁 SVMs. The training model of modulation format such as 32-QAM and 128QAM is a non-complete binary tree. When testing, the decision requires more than log2 𝑁 SVMs. Fig. 5(b) shows the number of SVMs required to distinguish each category of 32-QAM when testing. In the 32-QAM testing, an average of 5 SVMs is used at one point.
Table 2 The comparison of SVM number for testing.
The SVM number for testing
CBT-SVM
PBT-SVM
log2 𝑁
𝑁+1 2
−
1 𝑁
OBT-SVM log2 𝑁 ∼
𝑁+1 2
−
1 𝑁
2.4. SVM multi-classification based on binary tree Starting from the root node, the category contained in the node is divided into two subclasses, and then the two subclasses are further divided, and so on, until the subclass contains only one category, thus obtaining a binary tree. Finally, the SVM classifier is trained in each decision node of the binary tree to realize classification [19–22]. For a N classification problem, only N-1 SVM classifiers need to be generated and do not need to calculate all classifier decision functions when testing. There are fewer classifiers required than OvO or OvR method as shown in Table 1. There are many kinds of decision tree SVM for multi-classification methods. If 𝑁 = 2𝑚 , 𝑚 ∈ 𝑍 + , the tree is a complete binary tree (CBT) as shown in Fig. 4(a). Otherwise it is a partial binary tree (PBT) as shown in Fig. 4(b). The average number of SVM classifiers used in the CBT is log2 𝑁 when testing, the average SVM classifier used = 𝑁+1 − 𝑁1 . The binary tree with other in the PBT is (1+2+⋯+𝑁−1)+(𝑁−1) 𝑁 2 hierarchies (OBT) uses the average SVM classifiers between them. The complete tree requires the least SVM numbers, so the classifier speed of a complete binary tree with fewer support vectors is also faster. QAM constellation satisfies the complete binary tree structure, and the accuracy and efficiency of classification will be improved (see Table 2).
2.4.2. CBT-SVM based on binary encoding Since the QAM constellation symbol is based on binary encoding, the label feature based on binary encoding is also very obvious. The multi-classification is based on whether each bit is 0 or 1. Fig. 6(a) shows the training model for 16-QAM, 15 SVMs are needed, and the hyperplane generated by each SVM does not need all the training set data, which can effectively reduce the training time and improve the precision. When testing, you only need to substitute 4 SVM decision functions to get the result. Given the test set, after SVM 1–4, the test point is judged to be the category 1. Fig. 6(b) shows a schematic diagram of the hyperplane generated by node SVM for the training processing of category 1. The red and green colors respectively indicate the two categories to be distinguished, and the gray indicates the training sample set that is not required to be used. 2.4.3. BT-SVM based on in-phase and quadrature components According to the in-phase and quadrature component (IQC), the QAM signal is regarded as two PAM signals, which means splitting a binary tree training model into two binary trees to reduce the SVM 3
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Optics Communications 444 (2019) 1–8
Fig. 4. (a) The complete binary tree structure. (b) The partial binary tree structure.
Fig. 5. (a) A schematic diagram of hyperplane for training processing of one category. (b) The SVM numbers for each category of 32-QAM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. (a) The training model for QAM-16. (b) A schematic diagram of the hyperplane generated by node SVM for the training processing of category 1.
numbers. Fig. 7 shows the 16-QAM training model which used 15 SVMs spitted to the two simple tree with 3 SVMs, respectively. Fig. 8(a) shows the hyperplane for I-Path and Q-Path training processing of category 1, respectively. We can measure the point coded 0000 according to (2) and (4). However, not all QAM training models are completely binary trees. Fig. 8(b) shows the single path training model for QAM-32. Some categories need 2 SVMs and some categories need 3 SVMs.
at 1550 nm followed with external modulation by a Mach–ZehnderModulator (MZ-M). The optical signal is coupled into 10-km standard single-mode fiber (SSMF) for transmission. A photodetector (PD) with 22 GHz bandwidth is used for detecting the transmitted optical signal from the SSMF. A digital storage oscilloscope (DSO) (Keysight Z592A) with 59-GHz bandwidth is used to sample the signal with a sampling rate of 160 GSa/s for the off-line digital signal processing (DSP). DMT modulation is used for evaluating the SVM classification since different QAM can be used for different carriers so that comprehensive performance can be investigated. At transmitter (Tx) side, modulation parameters are adapted to the carrier SNR. For back-to-back (B2B) case, carrier 2 is modulated with 64-QAM, carrier 28 with 32-QAM, carrier 72 with 16-QAM, carrier 12 with 8-QAM, carrier 3 with QPSK and carrier 10 with OOK. For 10km-SSMF case, carrier 20 is modulated with 64-QAM, carrier 39 with 32-QAM, carrier 26 with 16-QAM, carrier 8 with 8-QAM, carrier 10 with QPSK and carrier 24 with OOK. It is 492 bits in total. Each DMT symbol contains 256 points, corresponding
3. Experimental results and discussions In the experiment, DMT modulation is utilized with different QAM for different carriers under different SNR and nonlinear distortions [23– 25]. Then, the SVM classification methods are investigated for different QAM. The experimental setup of QAM-DMT optical interconnection system is illustrated in Fig. 9. An arbitrary waveform generator (AWG) is used to generate the DMT signal with 54 GSa/s. The laser operates 4
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Optics Communications 444 (2019) 1–8
Fig. 7. The schematic diagram of 16-QAM training model.
Fig. 8. (a) A schematic diagram of the hyperplane for I-Path and Q-Path training processing of category 1. (b) The single path training model for 32-QAM.
Fig. 9. Experimental setup of transmission system. MZ-M: Mach–Zehnder Modulator; PD: Photodiode; AWG: arbitrary waveform generator; EA: electrical amplifier; DSO: digital storage oscilloscope; SSMF: standard single mode fiber; PC: polarization controller.
to 4.74 ns per DMT symbol. Thus, the aggregated transmission speed is 492 bits/4.74 ns=103.7 Gb/s. Given 3.19% cyclic prefix, the net transmission bit-rate is 100.3 Gb/s. At receiver (Rx) side, the signals in constellations are detected by SVM method. A total of 40,320 DMT symbols were sent in the experiment and 20% of the data has been used for training. The decoded signal in binary sequence after de-mapping is then off-line processed for the bit error rate (BER) measurement. Thus, SVM decision here for DMT is equivalent to that for QAM, only that the result is averaged (more reasonable) which covers various of loaded bit numbers. The received optical power is set at 4 dBm. The example constellation diagram in Fig. 9 is the 17th carrier for optical B2B case. The demodulation and decision for the DMT signal have been measured off-line. We use the different SVM classification methods for
decision and BER calculation. Figs. 10 to 13 illustrate the classification and multiple SVMs decision boundaries of the 17th carrier 64QAM in B2B experimental results for a single decision, respectively. From Fig. 10, we can see that the OvR takes 64 SVMs to complete the decision, and the decision boundary is an approximately circular closed curve that requires more support vectors. From Fig. 11, we can see that the BE takes 6 SVMs to complete the decision, and 3 decision boundaries are complex curves. From Fig. 12, we also face similar problems, but the complexity of the boundary line has been reduced. From Fig. 13, we can see that the IQC only need 6 slightly curved lines to complete the decision. The comparison of complexity and BER for the different classification methods are shown in Tables 3 and 4. The nonlinear distortion 5
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Optics Communications 444 (2019) 1–8
Fig. 10. SVM multi-classification based on OvR for 64-QAM.
Fig. 11. SVM multi-classification based on BE for 64-QAM.
Fig. 12. SVM multi-classification based on RC for 64-QAM.
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Optics Communications 444 (2019) 1–8
Fig. 13. SVM multi-classification based on IQC for 64-QAM. Table 3 BER Comparison.
the optimized QAM classification for a specified optical interconnection link.
BER
HD
OvR
SE
BE
RC
IQC
B2B (×10−3 ) 10-km SMF (×10−2 )
161.8 21.84
8.808 2.199
8.783 2.200
8.875 2.206
8.868 2.206
8.819 2.209
Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) (61875124, 61875049, 61675128).
Table 4 Complexity comparison for B2B. Complexity
OvR
SE
BE
RC
IQC
SVM number for training Support vector number (×105 ) Average SVM number for testing
2294 11.61 2294
492 3.154 492
2177 1.367 492
2177 1.376 492
804 1.997 492
References [1] S. Makovejs, et al., Characterization of long-haul 112 GB/s PDM-QAM-16 transmission with and without digital nonlinearity compensation, Opt. Express 18 (12) (2010) 12939–12947. [2] S. Kumar, et al., Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying, J. Lightw. Technol. 27 (21) (2009) 4722–4733. [3] N. Ekanayake, et al., Effect of nonlinear phase noise on the performance of M-ary PSK signals in optical fiber links, J. Lightw. Technol. 31 (3) (2013) 447–454. [4] J.P. Gordon, et al., Phase noise in photonic communications systems using linear amplifiers, Opt. Lett. 15 (23) (1990) 1351–1353. [5] L. Sun, J.B. Du, Z.Y. He, Multiband three-dimensional carrierless amplitude phase modulation for short reach optical communications, J. Lightw. Technol. 34 (13) (2016) 3103–3109. [6] L. Sun, J. Du, Z.Y. He, Experimental demonstration of 30-Gbit/s 3D-CAP modulation for short reach optical interconnection, in: Proc. OIC, San Diego, CA, USA, 2016, pp. WD1. [7] L. Liu, et al., Intrachannel nonlinearity compensation by inverse Volterra series transfer function, J. Lightw. Technol. 30 (3) (2012) 310–316. [8] T. Nguyen, et al., Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM, IEEE Photon. J. 8 (12) (2016) 1–9. [9] E. Giacoumidis, et al., Nonlinear inter-subcarrier intermixing reduction in coherent optical OFDM using fast machine learning equalization, in: Optical Fiber Communications Conference 2017, 16929919. [10] E. Giacoumidis, et al., Nonlinear blind equalization for 16-QAM coherent optical OFDM using support vector machines, in: European Conference on Optical Communication 2016, 978-3-8007-4274-5. [11] E. Giacoumidis, et al., Comparison of DSP-based nonlinear equalizers for intrachannel nonlinearity compensation in coherent optical OFDM, Opt. Lett. 41 (11) (2016) 2509–2512. [12] C.C. Chang, et al., LIBSVM: A Library for support vector machines, ACM Trans. Intell. Syst. Technol. 2 (3) (2011) 27:1–27:27. [13] D. Wang, et al., Nonlinear decision boundary created by a machine learningbased classifier to mitigate nonlinear phase noise, in: European Conference on Optical Communication 2015, paper 0720. [14] W. Chen, et al., Performance improvement of 64-QAM coherent optical communication system by optimizing symbol decision boundary based in support vector machine, Opt. Commun. 410 (2018) 1–7.
mitigation performances in terms of BER achieved by the five SVM classification methods are close between each of them, for both B2B case and 10-km SSMF transmission case, at received optical power of 4 dBm. The training and decision complexity of the SVM multi-classifier based on one-versus-rest are higher, and the number of support vectors is about 3 to 8 times higher than others. The smallest of SVM number for training is SE method, but the number of support vectors is larger. The IQ classification method also have a smaller SVM number for training and only requires about one-third of the support vectors compared with the SE methods, which is much simpler for implementation. One should carefully evaluate the requirement regarding different application scenarios (particularly different modulation formats and different nonlinear distortions) when choosing a specific classification method for SVM machine learning detection. 4. Conclusion In this paper, we experimentally investigated and compared five SVM multi-classification methods for machine learning assisted adaptive nonlinear mitigation, including OvR, SE, BE, RC, and IQC. The SVM detection was implemented in a QAM-DMT optical transmission link based on the M-ZM and 10-km SSMF. The results indicate significant nonlinear mitigation with BER reductions and the SVM multi-classifier based on the IQC is relatively optimal, considering the SVM number for training and the support vector number. Our objective of this study is to comprehensively investigate the classification methods of SVM machine learning detection with comparisons, so that we can realize 7
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Optics Communications 444 (2019) 1–8 [20] S. Cheong, Support vector machines with binary tree architecture for multi-class classification, Neural Inf. Process. 2 (3) (2004). [21] U. Sabanadesan, et al., Activity recognition using binary tree SVM, IEEE Stat. Signal Process. (2014) 248–251. [22] L. Zhang, et al., Decision tree support vector machine, Int. J. Artif. Intell. Tools 16 (01) (2007) 1–15. [23] S. Lee, et.al, 24-Gb/s Transmission over 730m of multimode fiber bydirect modulation of an 850-nm VCSEL using discrete multi-tone Modulation, in: Proc. NFOFC 2007, PDP 6. [24] R. Fischer, et al., A new loading algorithm for discrete multitone transmission, in: Proc. Global Telecommunications Conference, 1996, pp. 724–728. [25] K. Xu, C.Y. Wong, L. Zhang, L. Liu, G. Liu, C.W. Chow, H.K. Tsang, 56 Gbit/s DMT signal generated by an integrated silicon ring modulator, in: Proc. CLEO, San Jose, CA, USA, 2016, 1-2.
[15] G. Chen, J.B. Du, L. Sun, W.J. Zhang, K. Xu, X. Chen, G.T. Reed, Z.Y. He, Nonlinear distortion mitigation by machine learning of SVM classification for PAM-4 and PAM-8 modulated optical interconnection, J. Lightw. Technol. 36 (3) (2018) 650–657. [16] G. Chen, J.B. Du, L. Sun, L.F. Zheng, K. Xu, H.K. Tsang, X. Chen, G.T. Reed, Z.Y. He, Adaptive receiver for PAM-4 modulated optical interconnection based on silicon microring modulator, J. Lightw. Technol. 36 (18) (2018) 4106–4113. [17] Y. Han, et al., An SVM-based detection for coherent optical APSK systems with nonlinear phase noise, IEEE Photon. J. 6 (5) (2014) 1–10. [18] M. Jarajreh, et al., Artificial neural network nonlinear equalizer for coherent optical OFDM, IEEE Photon. Technol. Lett. 27 (4) (2015) 387–390. [19] B. Fei, et al., Binary tree of SVM: A new fast multiclass training and classification algorithm, IEEE Trans. Neural Netw. 17 (3) (2006).
8
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
QAM classification methods by SVM machine learning for improved optical interconnection Chang Wang a , Jiangbing Du a ,∗, Guoyao Chen a , Haoyang Wang a , Lin Sun a , Ke Xu b , Bo Liu c , Zuyuan He a a
State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China Department of Electronic and Information Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China c School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing, China b
ARTICLE
INFO
Keywords: Support vector machine Quadrature amplitude modulation Optical interconnection
ABSTRACT High-order quadrature amplitude modulation (QAM) formats are very effective for increasing the transmission capacity due to the highly increased spectral efficiency. However, the signal-to-noise-ratio (SNR) hungry and dense constellation of QAM make it very sensitive to nonlinear distortion. The nonlinear decision boundary adaptively generated by machine learning method of support vector machine (SVM) can be effectively used for the classification of the symbols. The different classification methods have different performance in terms of classification complexity. We experimentally investigated five SVM multi-classification methods for machine learning assisted adaptive nonlinear mitigation, including the one versus rest (OvR), the symbol encoding (SE), the binary encoding (BE), the constellation rows and columns (RC), and the in-phase and quadrature components (IQC). The comprehensive results with comparisons are demonstrated, indicating significant nonlinear mitigation with BER reductions. The SVM multi-classifier based on the in-phase and quadrature components is relatively optimal, considering the calculation and storage.
1. Introduction To meet the increasing demands of high capacity data communication, high-order QAM formats with significantly improved signaling efficiency (spectral efficiency) have been widely used for both intensitymodulation-direct-detection (IMDD) and coherent detection solutions. High-order QAM signal makes full use of amplitude and phase for carrying the information, which leads to great improvement of the transmission capacity [1]. However, the very dense constellation of high-order QAM leads to higher demand of SNR, which makes it very sensitive to the nonlinear distortion. The nonlinear distortion includes the modulation nonlinearity in short reach and Kerr nonlinear distortion in long haul. Thus, the correct detection of high-order QAM at the receiver side is usually very difficult due to the nonlinear distortion, particularly when the nonlinear distortion is varying. Transmission performance is also limited by amplified spontaneous emission (ASE) noise from in-line optical amplifiers which makes the tolerance to nonlinear distortion even smaller [2–7]. Therefore, adaptive nonlinear distortion mitigation methods are of key importance for high speed optical interconnection. Recently, digital signal processing based on machine learning has been studied for equalization and demodulation of optical interconnection systems [8–18]. In general, SVM-based signal processing is ∗
powerful with less complexity than deep learning of neural networks, which makes SVM very suitable for cost-sensitive IMDD optical interconnects. However, SVM has different classification methods from different demands. The different classification methods lead to different performance and complexity, which is important consideration when applying it to specific application scenarios, such as meter-scale multimode interconnection and kilometer-scale single-mode interconnection. T. Nguyen et al. demonstrated fiber nonlinearity equalizer based SVM classification for coherent optical orthogonal frequency division multiplexing (OFDM) [8]. In Ref. [12], one versus rest rule was employed for 16-QAM. The rule is that a received constellation point would belong to a certain cluster if that cluster accepted the received symbol and other clusters rejected it. This method would lead to uneven distribution of bit error ratio (BER) for each category of the constellation, which is caused by the number imbalance of different label points. D. Wang et al. applied the SVM decision method to the M-ary phase-shift keying (MPSK) based on the coherent optical system to mitigate the nonlinear phase noise. The authors designed the classification strategy for 16QAM with four SVMs and 64-QAM with six SVMs based on Gray code, respectively [13]. W. Chen et al. also used SVM for 64-QAM signal boundary decision based on the Gray code [14]. The idea that each classification is realized according to each bit of the symbol encoding
Corresponding author. E-mail address: [email protected] (J. Du).
https://doi.org/10.1016/j.optcom.2019.03.058 Received 15 January 2019; Received in revised form 13 March 2019; Accepted 23 March 2019 Available online 27 March 2019 0030-4018/© 2019 Published by Elsevier B.V.
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Optics Communications 444 (2019) 1–8
in Fig. 2(a), the hyperplane (straight line) in two-dimensional space divides data points into two categories. For the linearly inseparable case, mapping data to a high-dimensional space by kernel function 𝑘 is often used. In Fig. 2(b), linearly inseparable points in the original space can be transformed into linear separable problem in high-dimensional space by selecting specific mapping relationships 𝛷. This is equivalent to nonlinear classification in the original space. 𝑘 (𝑥, 𝑦) ∶= (𝛷 (𝑥) , 𝛷 (𝑦)) , 𝛷 ∶ RN → F
(1)
By learning from the training data, the feature space can be divided into two parts. The nonlinear classifier function can be described by the formula: ( 𝑙 ) ∑ ) ( 𝑓 (𝑥) = sign (2) 𝑎𝑖 ⋅ 𝑘 𝑥, 𝑥𝑖 + 𝑏
Fig. 1. (a) Ideal constellation and decision boundary. (b) Actual constellation and decision boundary with noise and nonlinearity.
𝑖=1
The parameters 𝑎𝑖 are computed as the solution of a quadratic programming problem and the sequential minimal optimization (SMO) algorithm is used to solve this problem. The algorithmic complexity of a trained model is determined by the number of support vectors, not by the dimensions of the data. The model trained by SVM relies entirely on support vectors. Even if all the non-supported vectors in the training set are removed, the training process will be repeated, and the result will still be exactly the same. If the number of support vectors obtained by training is small in an SVM, the model trained by SVM is easier to generalize.
is very reasonable. However, it would map the data to higher dimensions and require more support vectors to make the complex decision curves. G. Chen et al. demonstrated SVM decision method to mitigate modulation nonlinearity distortion for PAM-4 and PAM-8 VCSEL-MMF optical link [15]. Adaptive detection of PAM-4 signals modulated by silicon micro-ring modulator (Si-MRM) has also been demonstrated by SVM machine learning detection [16]. In Refs. [15,16], the authors proposed a new classification strategy based complete binary tree SVMs (CBT-SVMs) and applied for both PAM-4 and PAM-8 signals with lower complexity. Our objective of this study is to comprehensively investigate the classification methods of SVM machine learning detection with performance comparisons, so that we can realize the optimized QAM classification with balanced performance and complexity for a specified optical interconnection link. In this work, the SVM detection was implemented in a QAM-DMT optical transmission link based on the Mach– Zehnder modulator (M-ZM) and 10-km standard single mode fiber (SSMF), so that different QAM formats can be loaded onto different carriers. The experimental results in the case of back-to-back and 10-km transmission indicate that the SVM methods can well detect the nonlinear damage of signal and reduce the BER. Five multi-classification detection methods based on SVM have been investigated for QAM modulation formats. Both BER and complexity performance based on experimental results were investigated. The comparison indicates that SVM multi-classification based on the in-phase and quadrature component has the lowest complexity.
2.2. SVM multi-classification based on one versus one SVM multi-classification based on one versus one (OvO SVM) generates an optimal decision function between every two different categories of training sets. The advantage of OvO SVM is that the number of samples per training is relatively small, so the training speed of a single decision surface is faster and the precision is higher. However, since the N classification problem needs to train 𝑁 ∗ (𝑁 − 1)∕2 decision functions (𝑁 > 2), the total number of decision functions will be too large when N is large, and thus the prediction speed will be affected. 2.3. SVM multi-classification based on one versus rest The OvR SVM is to generate a hyperplane between a class of samples and the remaining multi-class samples, so as to achieve multiclass recognition. This method only requires an optimal hyperplane between a class of samples and the corresponding remaining samples, rather than classifying between the two samples. Therefore, if it is a N classification problem, then you need N SVMs (𝑁 > 2). Fig. 3(a) shows the training processing schematic diagrams of 16-QAM. When we use an SVM to make decision, the data is divided into two categories: the specific category to be decided with red color and the other category with green color. A total of 16 decision functions are obtained. The testing processing is shown in Fig. 3(b). Given a test data, substituting it into 16 decision functions and calculating the values, respectively. The value means that the score of the corresponding category is determined according to the distance between the hyperplane and constellation point. The category with the highest score is the final decision result. Compared with the OvO SVM, the number of generated hyperplanes is greatly reduced, so when the number of categories N is large, the prediction speed will be faster than the OvO SVM method. However, since it needs to use all the training sets every time it generates a hyperplane, it does not take less time to train than the OvO SVM. At the same time, since the remaining multi-classes are always regarded as one class during training, the positive and negative classes are extremely uneven in the number of training samples, which may affect the accuracy of prediction. In Refs. [13,14], the authors proposed the SVM classification based on the symbol encoding (SE) to improve the unbalance label.
2. SVM principle and classification methods Fig. 1 shows the ideal decision boundary for the original ideal constellation of 16-QAM modulation (as an example) and the actual decision boundary for a transmitted practical constellation using the SVM decision. Compared with the original ideal constellation, the transmitted practical constellation would be affected by both noise and nonlinearity. Thus, the decision boundary can no longer be a simple straight line for QAM. Therefore, SVM multi-classification to obtain adaptive nonlinear decision boundary is a good solution for optimizing the classification boundary which is particularly suitable for time-varying channels. 2.1. SVM principle The QAM constellation decision is actually a process of data classification and SVM is very suitable for dealing classification problems. An SVM can be understood as a binary classifier. In this work, with m bits for loading, the transmitted signal would be 2m -QAM and the demanded classification number is 𝑁 = 2m . To deal with 2m QAM classification, multi-SVMs are needed in order to implement multi-classification. For the linearly separable case, our goal is to find a hyperplane that separates points with different labels into the different side. As shown 2
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Fig. 2. (a) Data classification in two-dimensional space. (b) Nonlinear classification by high dimensional mapping.
Fig. 3. (a) A schematic diagram of the 16-QAM classification based on OvR. (b) OvR SVM classification flow chart.
Table 1 The comparison of SVM number for training model.
The SVM number for training
OvO-SVM
OvR-SVM
BT-SVM
𝑁 ∗ (𝑁 − 1)∕2
𝑁
𝑁 −1
2.4.1. BT-SVM based on constellation rows and columns Most of the constellations are rectangle, except for 32-QAM, 128QAM.etc. The label feature according to rows and columns is reasonable. Fig. 5(a) shows the hyperplane for the training processing of one category. The red and green colors respectively indicate the two categories to be distinguished, and the gray indicates the training sample set that is not required to be used. The training model of the modulation format such as 4-QAM, 8-QAM, 16-QAM, and 64-QAM is a complete binary tree. When testing, the decision requires log2 𝑁 SVMs. The training model of modulation format such as 32-QAM and 128QAM is a non-complete binary tree. When testing, the decision requires more than log2 𝑁 SVMs. Fig. 5(b) shows the number of SVMs required to distinguish each category of 32-QAM when testing. In the 32-QAM testing, an average of 5 SVMs is used at one point.
Table 2 The comparison of SVM number for testing.
The SVM number for testing
CBT-SVM
PBT-SVM
log2 𝑁
𝑁+1 2
−
1 𝑁
OBT-SVM log2 𝑁 ∼
𝑁+1 2
−
1 𝑁
2.4. SVM multi-classification based on binary tree Starting from the root node, the category contained in the node is divided into two subclasses, and then the two subclasses are further divided, and so on, until the subclass contains only one category, thus obtaining a binary tree. Finally, the SVM classifier is trained in each decision node of the binary tree to realize classification [19–22]. For a N classification problem, only N-1 SVM classifiers need to be generated and do not need to calculate all classifier decision functions when testing. There are fewer classifiers required than OvO or OvR method as shown in Table 1. There are many kinds of decision tree SVM for multi-classification methods. If 𝑁 = 2𝑚 , 𝑚 ∈ 𝑍 + , the tree is a complete binary tree (CBT) as shown in Fig. 4(a). Otherwise it is a partial binary tree (PBT) as shown in Fig. 4(b). The average number of SVM classifiers used in the CBT is log2 𝑁 when testing, the average SVM classifier used = 𝑁+1 − 𝑁1 . The binary tree with other in the PBT is (1+2+⋯+𝑁−1)+(𝑁−1) 𝑁 2 hierarchies (OBT) uses the average SVM classifiers between them. The complete tree requires the least SVM numbers, so the classifier speed of a complete binary tree with fewer support vectors is also faster. QAM constellation satisfies the complete binary tree structure, and the accuracy and efficiency of classification will be improved (see Table 2).
2.4.2. CBT-SVM based on binary encoding Since the QAM constellation symbol is based on binary encoding, the label feature based on binary encoding is also very obvious. The multi-classification is based on whether each bit is 0 or 1. Fig. 6(a) shows the training model for 16-QAM, 15 SVMs are needed, and the hyperplane generated by each SVM does not need all the training set data, which can effectively reduce the training time and improve the precision. When testing, you only need to substitute 4 SVM decision functions to get the result. Given the test set, after SVM 1–4, the test point is judged to be the category 1. Fig. 6(b) shows a schematic diagram of the hyperplane generated by node SVM for the training processing of category 1. The red and green colors respectively indicate the two categories to be distinguished, and the gray indicates the training sample set that is not required to be used. 2.4.3. BT-SVM based on in-phase and quadrature components According to the in-phase and quadrature component (IQC), the QAM signal is regarded as two PAM signals, which means splitting a binary tree training model into two binary trees to reduce the SVM 3
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Fig. 4. (a) The complete binary tree structure. (b) The partial binary tree structure.
Fig. 5. (a) A schematic diagram of hyperplane for training processing of one category. (b) The SVM numbers for each category of 32-QAM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. (a) The training model for QAM-16. (b) A schematic diagram of the hyperplane generated by node SVM for the training processing of category 1.
numbers. Fig. 7 shows the 16-QAM training model which used 15 SVMs spitted to the two simple tree with 3 SVMs, respectively. Fig. 8(a) shows the hyperplane for I-Path and Q-Path training processing of category 1, respectively. We can measure the point coded 0000 according to (2) and (4). However, not all QAM training models are completely binary trees. Fig. 8(b) shows the single path training model for QAM-32. Some categories need 2 SVMs and some categories need 3 SVMs.
at 1550 nm followed with external modulation by a Mach–ZehnderModulator (MZ-M). The optical signal is coupled into 10-km standard single-mode fiber (SSMF) for transmission. A photodetector (PD) with 22 GHz bandwidth is used for detecting the transmitted optical signal from the SSMF. A digital storage oscilloscope (DSO) (Keysight Z592A) with 59-GHz bandwidth is used to sample the signal with a sampling rate of 160 GSa/s for the off-line digital signal processing (DSP). DMT modulation is used for evaluating the SVM classification since different QAM can be used for different carriers so that comprehensive performance can be investigated. At transmitter (Tx) side, modulation parameters are adapted to the carrier SNR. For back-to-back (B2B) case, carrier 2 is modulated with 64-QAM, carrier 28 with 32-QAM, carrier 72 with 16-QAM, carrier 12 with 8-QAM, carrier 3 with QPSK and carrier 10 with OOK. For 10km-SSMF case, carrier 20 is modulated with 64-QAM, carrier 39 with 32-QAM, carrier 26 with 16-QAM, carrier 8 with 8-QAM, carrier 10 with QPSK and carrier 24 with OOK. It is 492 bits in total. Each DMT symbol contains 256 points, corresponding
3. Experimental results and discussions In the experiment, DMT modulation is utilized with different QAM for different carriers under different SNR and nonlinear distortions [23– 25]. Then, the SVM classification methods are investigated for different QAM. The experimental setup of QAM-DMT optical interconnection system is illustrated in Fig. 9. An arbitrary waveform generator (AWG) is used to generate the DMT signal with 54 GSa/s. The laser operates 4
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Fig. 7. The schematic diagram of 16-QAM training model.
Fig. 8. (a) A schematic diagram of the hyperplane for I-Path and Q-Path training processing of category 1. (b) The single path training model for 32-QAM.
Fig. 9. Experimental setup of transmission system. MZ-M: Mach–Zehnder Modulator; PD: Photodiode; AWG: arbitrary waveform generator; EA: electrical amplifier; DSO: digital storage oscilloscope; SSMF: standard single mode fiber; PC: polarization controller.
to 4.74 ns per DMT symbol. Thus, the aggregated transmission speed is 492 bits/4.74 ns=103.7 Gb/s. Given 3.19% cyclic prefix, the net transmission bit-rate is 100.3 Gb/s. At receiver (Rx) side, the signals in constellations are detected by SVM method. A total of 40,320 DMT symbols were sent in the experiment and 20% of the data has been used for training. The decoded signal in binary sequence after de-mapping is then off-line processed for the bit error rate (BER) measurement. Thus, SVM decision here for DMT is equivalent to that for QAM, only that the result is averaged (more reasonable) which covers various of loaded bit numbers. The received optical power is set at 4 dBm. The example constellation diagram in Fig. 9 is the 17th carrier for optical B2B case. The demodulation and decision for the DMT signal have been measured off-line. We use the different SVM classification methods for
decision and BER calculation. Figs. 10 to 13 illustrate the classification and multiple SVMs decision boundaries of the 17th carrier 64QAM in B2B experimental results for a single decision, respectively. From Fig. 10, we can see that the OvR takes 64 SVMs to complete the decision, and the decision boundary is an approximately circular closed curve that requires more support vectors. From Fig. 11, we can see that the BE takes 6 SVMs to complete the decision, and 3 decision boundaries are complex curves. From Fig. 12, we also face similar problems, but the complexity of the boundary line has been reduced. From Fig. 13, we can see that the IQC only need 6 slightly curved lines to complete the decision. The comparison of complexity and BER for the different classification methods are shown in Tables 3 and 4. The nonlinear distortion 5
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Fig. 10. SVM multi-classification based on OvR for 64-QAM.
Fig. 11. SVM multi-classification based on BE for 64-QAM.
Fig. 12. SVM multi-classification based on RC for 64-QAM.
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Fig. 13. SVM multi-classification based on IQC for 64-QAM. Table 3 BER Comparison.
the optimized QAM classification for a specified optical interconnection link.
BER
HD
OvR
SE
BE
RC
IQC
B2B (×10−3 ) 10-km SMF (×10−2 )
161.8 21.84
8.808 2.199
8.783 2.200
8.875 2.206
8.868 2.206
8.819 2.209
Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) (61875124, 61875049, 61675128).
Table 4 Complexity comparison for B2B. Complexity
OvR
SE
BE
RC
IQC
SVM number for training Support vector number (×105 ) Average SVM number for testing
2294 11.61 2294
492 3.154 492
2177 1.367 492
2177 1.376 492
804 1.997 492
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mitigation performances in terms of BER achieved by the five SVM classification methods are close between each of them, for both B2B case and 10-km SSMF transmission case, at received optical power of 4 dBm. The training and decision complexity of the SVM multi-classifier based on one-versus-rest are higher, and the number of support vectors is about 3 to 8 times higher than others. The smallest of SVM number for training is SE method, but the number of support vectors is larger. The IQ classification method also have a smaller SVM number for training and only requires about one-third of the support vectors compared with the SE methods, which is much simpler for implementation. One should carefully evaluate the requirement regarding different application scenarios (particularly different modulation formats and different nonlinear distortions) when choosing a specific classification method for SVM machine learning detection. 4. Conclusion In this paper, we experimentally investigated and compared five SVM multi-classification methods for machine learning assisted adaptive nonlinear mitigation, including OvR, SE, BE, RC, and IQC. The SVM detection was implemented in a QAM-DMT optical transmission link based on the M-ZM and 10-km SSMF. The results indicate significant nonlinear mitigation with BER reductions and the SVM multi-classifier based on the IQC is relatively optimal, considering the SVM number for training and the support vector number. Our objective of this study is to comprehensively investigate the classification methods of SVM machine learning detection with comparisons, so that we can realize 7
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