簡易檢索 / 詳目顯示

回結果列表
研究生： 高文
Govind sharan Yadav
論文名稱： 基於稀疏架構學習的低複雜度弗爾特拉與人工智慧非線性等化器應用於光纖通訊系統
Reducing Complexity of Volterra and Artificial Intelligence-based Nonlinear Equalizers using Structural Sparsity Learning Techniques for Optical Fiber Communication Systems
指導教授： 馮開明
Feng, Kai-Ming
口試委員: 賴暎杰
Lai, Yin-Chieh
鄒志偉‬
‪Chow, Chi-Wai
陳杰紅
Chen, Jyehong
黃勝廣
Hwang, Sheng-Kwang
學位類別： 博士
Doctor
系所名稱： 電機資訊學院 - 光電工程研究所
Institute of Photonics Technologies
論文出版年： 2021
畢業學年度： 109
語文別： 英文
論文頁數： 106
中文關鍵詞： 基於稀疏架構學習的低複雜度弗爾特拉與人工智慧非線性等化器應用於光纖通訊系統 、非線性均衡器 、光纖通訊 、人工智能 、結構稀疏學習技術 、數字信號處理
外文關鍵詞： Reducing Complexity of Volterra and Artificial Intelligence-based Nonlinear Equalizers using Structural Sparsity Learning Techniques for Optical Fiber Communication Systems, Nonlinear Equalizers, Optical Fiber Communication, Artificial Intelligence, Structural Sparsity Learning Techniques, digital signal processing
相關次數： 點閱：2  下載：0 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 現今的光通訊系統的通訊容量需要更升級，方可滿足數位通訊的需求。目前，因為嚴重的傳輸劣化與缺乏實用的數位訊號處理方法，光通訊系統受到諸多掣肘。時至今日，基於人工智慧的訊號處理技術成為相當適合的候選技術，足以應付未來更高容量與更複雜系統的挑戰。
    本論文的研究範疇包含三個光纖通訊系統的非線性補償機制。
    首先，我們採用16-APSK調變技術建立長距離分波多工(WDM)傳輸系統，並且提出採用提前停止機制的一項「動態人工智慧非線性補償」(AI-NLC)技術，在單通道與分波多工通道的長距離光通訊系統都能減緩非線性效應的影響。我們的電腦模擬結果顯示，在使用適當的光訊號輸出功率之時，這項動態人工智慧非線性補償技術比現行技術提供更好的訊號訊雜比(SNR)。在2,400公里的長距離傳輸系統，相較於線性頻域等化器(LFDE)、靜態人工智慧非線性補償(static AI-NLC)、分離步驟傅立葉方式的數位反向傳播算法(SSFM-DBP)等三項現有技術，動態人工智慧非線性補償技術分別擁有2.25 dB、1.65 dB、1 dB的訊雜比優勢。在4,800公里的長距離傳輸系統，這樣的訊雜比優勢則分別是2.11 dB、1.50 dB、1.01 dB。同時，我們的研究顯示，動態人工智慧非線性補償技術具有更好的電腦運算效率。
    其次，我們採用短距離傳輸系統。一般而言，弗爾特拉等化器(VE)在增強高速光通訊訊號的訊號能獲得相當好的成果。然而，弗爾特拉等化器的運算複雜度相當高，導致實際應用受到諸多限制。為此，我們提出一項「彈性正規化精簡弗爾特拉等化器」(EN-PVE)技術，並透過實驗驗證之。這項等化器技術不但降低運算複雜度，也與一般的弗爾特拉等化器擁有相似的訊號補償表現。藉由下述的三個階段設置，我們的架構得以精簡多餘的權重係數。第一、透過一組適當的彈性正規化機制預先訓練弗爾特拉等化器，辨識主要權重係數；第二、刪除這些主要權重係數；第三、透過其他剩下的權重係數重新訓練以微調等化器。這項實驗展示為40公里單模光纖傳輸系統，並且採用坐落於O Band的80 Gbps PAM-4光訊號。在接收光功率-4 dBm時，我們提出的彈性正規化精簡弗爾特拉等化器技術擁有良好的運算複雜度。相較於一般的弗爾特拉等化器和L1正規化弗爾特拉等化器(L1VE)，分別降低97.4%與20%。

    因此，我們提出一創新的「稀疏結構深度神經網絡非線性等化技術」(SSLDNN-NLM)。實驗系統採用振幅調變直接接收(IM-DD)，藉由外調雷射(EML)產生112 Gbps PAM-4的光訊號，並傳輸40公里標準單模態光纖(SSMF)。我們提出的等化技術不僅降低運算複雜度，也能有效改善訊號品質。這項創新技術採用L2補償正規法計算損失函數，並且藉由辨別每個權重的重要性而採用精簡深度神經網絡模型。因此，我們可以刪除重要性低的權重，並且依然保有良好的系統表現。實驗結果顯示，這項創新技術能有效克服光纖通訊系統造成的非線性效應的影響。更甚者，這項稀疏結構深度神經網絡非線性等化技術結合L2正規法，得以在與一般深度神經網絡學習相同複雜度的情形之下，得到更好的誤碼率(BER)表現。在相同的光纖傳輸系統（背對背、40公里單模態光纖），這項創新技術比L2正規弗爾特拉等化器(L2VE)降42%的系統複雜度。在背對背的實驗系統中，在接收光功率(ROP)為-3 dBm時，與L2正規弗爾特拉等化器、深度神經網絡等化器、稀疏L2正規弗爾特拉等化器、稀疏深度神經網絡等化器四種技術相比較，這項稀疏結構深度神經網絡非線性等化技術得以透過刪除低重要權重，分別降低86%、75%、70%、61%的運算複雜度。而在40公里單模態光纖傳輸的系統、接收光功率-5 dBm時，降低的複雜度則為79%、63%、58%、57%，卻仍然絲毫沒有減損系統的整體表現。
    總結而言，我們的研究成果提出有效的非線性訊號補償技術，也立下重要的里程碑。同時，對於非線性的光纖通訊系統，我們的技術實現降低運算複雜度的成效。

    The capabilities of existing optical communication systems need to be further upgraded to meet the future demands of digital information. At present, they are reaching their limits due to severe optical transmission link distortions and the lack of practical digital signal processing (DSP) solutions. Artificial intelligence-based methods have now emerged as a suitable candidate for establishing newer directions to meet the upcoming challenges posed by higher capacity requirements and more complex systems.
    The main research focus of this thesis includes three approaches for nonlinear compensation of optical fiber transmission systems. Firstly, we address long-haul wavelength-division multiplexing (WDM) transmission systems using 16-APSK modulation formats. An early-stopping technique solution for dynamic artificial intelligence-based nonlinear compensator (AI-NLC) which can mitigate nonlinear effects on both single and 7-WDM channels is proposed. Our simulation results show that at optimal launch power, dynamic AI-NLC outperforms conventional linear frequency domain equalizer (LFDE), static AI-NLC, and split-step Fourier method-based digital back-propagation (SSFM-DBP) in signal to noise ratio (SNR) by 2.25 dB, 1.65 dB, and 1 dB, respectively for a 2400 km transmission case. For a 4800 km transmission with 16-APSK signals, AI-NLC outperforms the above-mentioned nonlinear compensation methods in SNR by2.11 dB, 1.50 dB, and 1.01 dB in the same order. Moreover, our investigation shows that AI-NLC is computationally more effective than other nonlinear compensators. Secondly, we address short-reach transmission systems. Conventionally, the Volterra equalization (VE) method is a promising solution to obtain substantial performance enhancement in high-speed optical signals. However, it suffers from high computational complexity, which subsequently limits its physical implementation scopes. To address these limitations, we propose and experimentally demonstrate an elastic net regularization-based pruned Volterra equalization (EN-PVE) method which can not only reduce computational complexity but also maintain system performance. Our proposed scheme prunes redundant weight coefficients via a three-phase configuration. Firstly, the VE is pre-trained with an adaptive elastic net-regularizer to identify significant weights. Next, the insignificant weights are pruned away. Finally, the equalizer is re-trained by fine-tuning the remaining weight coefficients. When compared to a conventional VE and L1-regularization-based Volterra equalizer (L1VE) for an O-band 80-Gbps PAM-4 signal at a received optical power of −4 dBm after 40 km SMF transmission, the proposed EN-PVE method demonstrates complexity reduction of 97.4% and 20.2%, respectively. The final approach tackles the requirement for faster short-reach transceivers data center interconnects by proposing deep learning algorithms. Conventional fully-connected deep neural network-based nonlinear equalizers (DNN-NLEs) are a powerful mechanism in modern artificial intelligence applications. However, their inherent high computation complexity limits their feasibility and compatibility with off-the-shelf hardware deployments. Thus, we propose and experimentally demonstrate a novel structural sparsity learning deep neural network nonlinear equalization method (SLDNN-NLM), which reduces computation complexity by substantially improving system performance for 112-Gbps PAM-4 IM-DD with an externally modulated laser (EML) transmission links over 40 km standard single-mode fiber (SSMF). This novel method employs the L2-penalty regularization term into the loss function to adapt the compact DNN model by distinguishing the significance of each connection. We then prune away insignificant weight connections using the pruning technique, still maintaining excellent system performance. The experimental outcome illustrating the proposed method shows great potential to significantly overcome nonlinear distortions caused by the optical transmission systems. Furthermore, the proposed SLDNN-NLM with L2-regularization achieves better BER performance capability with the same complexity as conventional DNN, while it is reduced by 42% more complexity than the L2-regularized Volterra equalizer (L2VE) at each ROP for both BTB and 40-km transmission distance. Compared with L2VE, conventional DNN, sparse L2VE, and sparse conventional DNN, the proposed SLDNN-NLM after pruning attains 86%, 75%, 70%, and 61% complexity reduction for BTB at the ROP of -3dBm, respectively, while 79%, 63%, 58%, and 57% drops in computational complexity are accomplished when compared with L2VE, conventional DNN, sparse L2VE, and conventional sparse DNN for 40-km transmission at the ROP of −5 dBm, respectively, without degrading the system performance. In conclusion, these presented research solutions are key milestones towards promising solutions to effectively deal with nonlinear signal distortions, while also reducing computation complexity for data transmission over nonlinear fiber systems at the same time.

    Table of Contents Abstract i Acknowledgments v List of Figures I List of Table IV Chapter 1 1 Introduction 1 1.1. Motivation 1 1.1.1. Capacity demands in Long-Haul Transmission Systems 1 1.1.2. Capacity demands in Data Center Interconnects 2 1.2. The proposed works 3 1.2.1. The Proposed AI-NLC for Long-Haul WDM Systems 3 1.2.2. The Proposed ENPVE for Data Center Interconnects 5 1.2.3. The Proposed SLDNN-NLM for Data Center Interconnects 7 1.3. The Organization of thesis 8 1.4. List of publications 10 Chapter 2 12 Theory of optical communication systems 12 2.1. Introduction 12 2.1. The architecture of optical transmission systems 12 2.2. The Transmitter 13 2.2.1. Digital-to-analog converter (DAC) 13 2.2.2. Electro-optical conversion 14 2.3. Optical link and Impairments 15 2.3.1. Chromatic dispersion 16 2.3.2. Attenuation 17 2.3.3. ASE Noise 19 2.3.4. Nonlinear effects 19 2.4. The receiver 21 2.4.1. Coherent-detection scheme 21 2.4.2. Direct-detection scheme 21 2.4.3. Analog-to-digital conversion 22 2.4.4. Important Digital Signal Processing steps 23 2.4.5. Performance Evaluation Metrics 23 Chapter 3 25 Theory of Volterra filters, artificial neural networks, and deep neural network nonlinear equalization methods 25 3.1. Introduction 25 3.1. Volterra Filter 26 3.1.1. Standard Volterra Filter 26 3.1.2. Symmetrical Redundancy-Removed Volterra Filter 27 3.2. Learning Algorithms 29 3.2.1. Supervised Learning 30 3.2.2. Unsupervised Learning 30 3.1. Artificial neural networks 30 3.1.1. Artificial Neuron model 31 3.1.2. Types of ANN 34 3.2. Feedforward ANN framework 34 3.3. Radial basis function neural network 36 3.4. Deep Neural Networks 37 Chapter 4 39 Resolving overfitting and complexity challenges in artificial intelligence-based nonlinear compensator using an early-stopping approach in long-haul WDM 16-APSK transmission systems 39 4.1. Introduction 39 4.2. The proposed AI-NLC in long-haul WDM transmission systems 39 4.2.1. Training phase 40 4.2.2. Prediction phase 42 4.3. Fiber transmission system model 43 4.4. Evaluation of optimize dynamic AI-NLC model 45 4.5. Comparison of dynamic AI-NLC performance with LFDE, static AI-NLC and SSFM-DBP 49 4.6. Computation efforts 51 4.7. Conclusions 55 Chapter 5 56 Structured sparsity learning-based pruned retraining Volterra equalization for data-center interconnects 56 5.1. Introduction 56 5.2. Principles of ENPVE 56 5.2.1. Equalizer principle and complexity of the Volterra equalizer 56 5.2.2. Sparsity metric 58 5.2.3. Principles of Volterra pruning and retraining 58 5.3. Experimental setup 61 5.4. Results and discussion 62 5.5. Conclusion 70 Chapter 6 71 Structural sparsity learning deep neural network nonlinear equalization method for 112Gbps PAM4 short-reach transmission links 71 6.1. Introduction 71 6.2. Concept of L2-regularization and pruning methods 72 6.3. The proposed structural sparsity learning deep neural network nonlinear equalization method (SLDNN-NLM) 73 6.4. Experimental setup 77 6.5. Results and Discussions 78 6.5.1. Complexity and performance using conventional DNN equalizer 78 6.5.2. The proposed SLDNN-NLM after L2-regularization but before pruning 81 6.5.3. The proposed SLDNN-NLM after pruning 84 6.6. Conclusion 91 Chapter 7 92 Conclusions 92 References 94 Glossary of terms -----------------------------------------------------------------------------104

    [1] G. Forecast, "Cisco visual networking index: global mobile data traffic forecast update, 2017–2022," Update, vol. 2017, p. 2022, 2019.
    [2] C. Cisco, "Cisco Global Cloud Index: Forecast and Methodology, 2016–2021," San Jose: Cisco, 2018.
    [3] P. J. Winzer, D. T. Neilson, and A. R. Chraplyvy, "Fiber-optic transmission and networking: the previous 20 and the next 20 years," Opt. Express, vol. 26, no. 18, pp. 24190-24239, 2018.
    [4] R. W. Tkach, "Scaling optical communications for the next decade and beyond," Bell Labs Tech. J., vol. 14, no. 4, pp. 3-9, 2010.
    [5] A. D. Ellis, J. Zhao, and D. Cotter, "Approaching the non-linear Shannon limit," J. Lightw. Technol., vol. 28, no. 4, pp. 423-433, 2009.
    [6] M. Secondini, E. Forestieri, and G. Prati, "Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps," J. Lightw. Technol., vol. 31, no. 23, pp. 3839-3852, 2013.
    [7] J. Renaudier, G. Charlet, P. Tran, M. Salsi, and S. Bigo, "A performance comparison of differential and coherent detections over ultra long haul transmission of 10Gb/s BPSK," in Optical Fiber Communication and the National Fiber Optic Engineers Conference, Anaheim, CA, USA, p. OWM1.
    [8] P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature, vol. 411, no. 6841, p. 1027, 2001.
    [9] A. Napoli et al., "Reduced complexity digital back-propagation methods for optical communication systems," J. Lightw. Technol., vol. 32, no. 7, pp. 1351-1362, 2014.
    [10] A. J. Lowery, "Fiber nonlinearity pre-and post-compensation for long-haul optical links using OFDM," Opt. Express, vol. 15, no. 20, pp. 12965-12970, 2007.
    [11] A. Bakhshali et al., "Frequency-domain Volterra-based equalization structures for efficient mitigation of intrachannel Kerr nonlinearities," J. Lightw. Technol., vol. 34, no. 8, pp. 1770-1777, 2015.
    [12] T. Tanimura et al., "Deep learning based OSNR monitoring independent of modulation format, symbol rate and chromatic dispersion," in European Conference on Optical Communication, Dusseldorf, Germany pp. 1-3.
    [13] B. Karanov et al., "End-to-end deep learning of optical fiber communications," J. Lightw. Technol., vol. 36, no. 20, pp. 4843-4855, 2018.
    [14] T. Zhao, L. Deng, W. Wang, D. S. Elson, and L. Su, "Bayes’ theorem-based binary algorithm for fast reference-less calibration of a multimode fiber," Opt. Express, vol. 26, no. 16, pp. 20368-20378, 2018.
    [15] J. Zhang, W. Chen, M. Gao, and G. Shen, "K-means-clustering-based fiber nonlinearity equalization techniques for 64-QAM coherent optical communication system," Opt. Express, vol. 25, no. 22, pp. 27570-27580, 2017.
    [16] N. G. Gonzalez, D. Zibar, X. Yu, and I. T. Monroy, "Optical phase-modulated radio-over-fiber links with k-means algorithm for digital demodulation of 8PSK subcarrier multiplexed signals," in Optical Fiber Communication Conference, San Diego, CA , USA, p. OML3.
    [17] G. Chen et al., "Nonlinear distortion mitigation by machine learning of SVM classification for PAM-4 and PAM-8 modulated optical interconnection," J. Lightw. Technol., vol. 36, no. 3, pp. 650-657, 2018.
    [18] M. A. Jarajreh et al., "Artificial neural network nonlinear equalizer for coherent optical OFDM," IEEE Photon. Technol. Lett., vol. 27, no. 4, pp. 387-390, 2014.
    [19] E. Giacoumidis et al., "Fiber nonlinearity-induced penalty reduction in CO-OFDM by ANN-based nonlinear equalization," Opt. Lett., vol. 40, no. 21, pp. 5113-5116, 2015.
    [20] S. T. Ahmad and K. P. Kumar, "Radial basis function neural network nonlinear equalizer for 16-QAM coherent optical OFDM," IEEE Photon. Technol. Lett., vol. 28, no. 22, pp. 2507-2510, 2016.
    [21] V. Kamalov et al., "Evolution from 8QAM live traffic to PS 64-QAM with neural-network based nonlinearity compensation on 11000 km open subsea cable," pp. Th4D-5: Optical Society of America.
    [22] S. Zhang et al., "Field and lab experimental demonstration of nonlinear impairment compensation using neural networks," Nature communications, vol. 10, no. 1, pp. 1-8, 2019.
    [23] E. Giacoumidis, A. Matin, J. Wei, N. J. Doran, L. P. Barry, and X. Wang, "Blind nonlinearity equalization by machine-learning-based clustering for single-and multichannel coherent optical OFDM," J. Lightw. Technol., vol. 36, no. 3, pp. 721-727, 2018.
    [24] in Cisco Annual Internet Report ,[Online]:https://www.cisco.com/c/en/us/solutions/collateral/executive-perspectives/annual-internet-report/white-paper-c11-741490.html, ed.
    [25] N. Bitar, S. Gringeri, and T. J. Xia, "Technologies and protocols for data center and cloud networking," IEEE Commun. Mag., vol. 51, no. 9, pp. 24-31, 2013.
    [26] H. Mardoyan et al., "84-, 100-, and 107-GBd PAM-4 intensity-modulation direct-detection transceiver for datacenter interconnects," J. Lightw. Technol., vol. 35, no. 6, pp. 1253-1259, 2016.
    [27] K. Zhong et al., "Experimental study of PAM-4, CAP-16, and DMT for 100 Gb/s short reach optical transmission systems," Opt Express, vol. 23, no. 2, pp. 1176-89, Jan 26 2015.
    [28] M. S. Alfiad, D. Van den Borne, F. N. Hauske, A. Napoli, A. M. J. Koonen, and H. de Waardt, "Maximum-likelihood sequence estimation for optical phase-shift keyed modulation formats," J. Lightw. Technol., vol. 27, no. 20, pp. 4583-4594, 2009.
    [29] D. Maiti and M. Brandt-Pearce, "Modified nonlinear decision feedback equalizer for long-haul fiber-optic communications," J. Lightw. Technol., vol. 33, no. 18, pp. 3763-3772, 2015.
    [30] N. Stojanovic, F. Karinou, Z. Qiang, and C. Prodaniuc, "Volterra and Wiener equalizers for short-reach 100G PAM-4 applications," J. Lightw. Technol., vol. 35, no. 21, pp. 4583-4594, 2017.
    [31] R. D. Nowak and B. D. Van Veen, "Volterra filter equalization: A fixed point approach," IEEE Trans. Signal Process., vol. 45, no. 2, pp. 377-388, 1997.
    [32] J. Tsimbinos and K. V. Lever, "Computational complexity of Volterra based nonlinear compensators," Electron. Lett., vol. 32, no. 9, pp. 852-854, 1996.
    [33] Y. Yu, M. R. Choi, T. Bo, Z. He, Y. Che, and H. Kim, "Low-Complexity Second-Order Volterra Equalizer for DML-Based IM/DD Transmission System," J. Lightw. Technol., vol. 38, no. 7, pp. 1735-1746, 2020.
    [34] N.-P. Diamantopoulos, H. Nishi, W. Kobayashi, K. Takeda, T. Kakitsuka, and S. Matsuo, "On the complexity reduction of the second-order Volterra nonlinear equalizer for IM/DD systems," J. Lightw. Technol., vol. 37, no. 4, pp. 1214-1224, 2019.
    [35] Y. Yu, T. Bo, Y. Che, D. Kim, and H. Kim, "Low-complexity nonlinear equalizer based on absolute operation for C-band IM/DD systems," Opt. Express, vol. 28, no. 13, pp. 19617-19628, 2020.
    [36] W.-J. Huang et al., "93% complexity reduction of Volterra nonlinear equalizer by ℓ 1-regularization for 112-Gbps PAM-4 850-nm VCSEL optical interconnect," presented at the Optical Fiber Communications Conference and Exposition OSA Technical Digest (CD) (Optical Society of America, 2018) ,San Diego, CA, USA, Paper,pp. M2D.7,
    [37] S.-Y. Lu, C.-C. Wei, C.-Y. Chuang, Y.-K. Chen, and J. Chen, "81.7% complexity reduction of Volterra nonlinear equalizer by adopting L1 regularization penalty in an OFDM long-reach PON," presented at the European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 2017.
    [38] L. Shu et al., "Single-photodiode 112-Gbit/s 16-QAM transmission over 960-km SSMF enabled by Kramers-Kronig detection and sparse I/Q Volterra filter," Opt. Express, vol. 26, no. 19, pp. 24564-24576, 2018.
    [39] S. Zeng, J. Gou, and L. Deng, "An antinoise sparse representation method for robust face recognition via joint l1 and l2 regularization," Expert Systems with Applications, vol. 82, pp. 1-9, 2017.
    [40] N. Eiselt et al., "Experimental demonstration of 84 Gb/s PAM-4 over up to 1.6 km SSMF using a 20-GHz VCSEL at 1525 nm," J. Lightw. Technol., vol. 35, no. 8, pp. 1342-1349, 2017.
    [41] H. Mardoyan et al., "84-, 100-, and 107-GBd PAM-4 intensity-modulation direct-detection transceiver for datacenter interconnects," J. Lightw. Technol., vol. 35, no. 6, pp. 1253-1259, 2016.
    [42] J. Zhang, J. Yu, and H.-C. Chien, "EML-based IM/DD 400G (4× 112.5-Gbit/s) PAM-4 over 80 km SSMF based on linear pre-equalization and nonlinear LUT pre-distortion for inter-DCI applications," in Optical fiber communication Conference Los Angeles,CA,USA,Paper,W3A.4.
    [43] E. Alliance, "The 2018 Ethernet Roadmap," Ethernet Alliance, 2018.
    [44] E. T. Force, "IEEE Standard P802. 3bs 200 Gb/s and 400 Gb/s," 2017.
    [45] C. Chen, X. Tang, and Z. Zhang, "Transmission of 56-Gb/s PAM-4 over 26-km single mode fiber using maximum likelihood sequence estimation," in Optical Fiber Communication Conference, Los Angeles, California United States ,Paper,Th4A.5, pp. Th4A-5.
    [46] S. Zhou, X. Li, L. Yi, Q. Yang, and S. Fu, "Transmission of 2× 56 Gb/s PAM-4 signal over 100 km SSMF using 18 GHz DMLs," Optics letters, vol. 41, no. 8, pp. 1805-1808, 2016.
    [47] X. Han and C.-H. Cheng, "Nonlinear filter based decision feedback equalizer for optical communication systems," Opt. Express, vol. 22, no. 7, pp. 8712-8719, 2014.
    [48] J. Zhang, M. Gao, W. Chen, and G. Shen, "Non-data-aided k-nearest neighbors technique for optical fiber nonlinearity mitigation," J. Lightw. Technol., vol. 36, no. 17, pp. 3564-3572, 2018.
    [49] L. Sun et al., "Intelligent 2-dimensional soft decision enabled by K-means clustering for VCSEL-based 112-Gbps PAM-4 and PAM-8 optical interconnection," J. Lightw. Technol., vol. 37, no. 24, pp. 6133-6146, 2019.
    [50] Z. Xu, C. Sun, T. Ji, J. H. Manton, and W. Shieh, "Computational complexity comparison of feedforward/radial basis function/recurrent neural network-based equalizer for a 50-Gb/s PAM4 direct-detection optical link," Opt. Express, vol. 27, no. 25, pp. 36953-36964, 2019.
    [51] C.-Y. Chuang et al., "Employing deep neural network for high speed 4-PAM optical interconnect," in 2017 European Conference on Optical Communication (ECOC), 2017, pp. 1-3: IEEE.
    [52] P. J. Freire et al., "Performance versus complexity study of neural network equalizers in coherent optical systems," arXiv preprint arXiv:2103.08212, 2021.
    [53] O. Sidelnikov, A. Redyuk, and S. Sygletos, "Equalization performance and complexity analysis of dynamic deep neural networks in long haul transmission systems," Opt. Express, vol. 26, no. 25, pp. 32765-32776, 2018.
    [54] L. Ge, W. Zhang, C. Liang, and Z. He, "Compressed Neural Network Equalization Based on Iterative Pruning Algorithm for 112-Gbps VCSEL-Enabled Optical Interconnects," J. Lightw. Technol., vol. 38, no. 6, pp. 1323-1329, 2020.
    [55] Z. Wan et al., "Nonlinear equalization based on pruned artificial neural networks for 112-Gb/s SSB-PAM4 transmission over 80-km SSMF," Opt. Express, vol. 26, no. 8, pp. 10631-10642, 2018.
    [56] A. G. Reza and J.-K. K. Rhee, "Nonlinear equalizer based on neural networks for PAM-4 signal transmission using DML," IEEE Photon. Technol. Lett., vol. 30, no. 15, pp. 1416-1419, 2018.
    [57] C. Laperle and M. O’Sullivan, "Advances in high-speed DACs, ADCs, and DSP for optical coherent transceivers," J. Lightw. Technol., vol. 32, no. 4, pp. 629-643, 2014.
    [58] K. Gentile, "The effect of DAC resolution on spurious performance," A technical tutorial on digital signal synthesis (Analog Devices Inc.), pp. 16-38, 1999.
    [59] K. Ying, Z. Yu, R. J. Baxley, and G. T. Zhou, "Optimization of signal-to-noise-plus-distortion ratio for dynamic-range-limited nonlinearities," Digital Signal Processing, vol. 36, pp. 104-114, 2015.
    [60] G. P. Agrawal, Fiber-optic communication systems. John Wiley & Sons, 2012.
    [61] G. P. Agrawal, "Nonlinear fiber optics," in Nonlinear Science at the Dawn of the 21st Century: Springer, 2000, pp. 195-211.
    [62] G. Keiser, "Optical fiber communications," Wiley encyclopedia of telecommunications, 2003.
    [63] http://www.fowiki.com/b/understand-fiber-attenuation/, ed.
    [64] Y. Colombe, D. H. Slichter, A. C. Wilson, D. Leibfried, and D. J. Wineland, "Single-mode optical fiber for high-power, low-loss UV transmission," Opt. Express, vol. 22, no. 16, pp. 19783-19793, 2014.
    [65] S. Donati and G. Giuliani, "Noise in an optical amplifier: Formulation of a new semiclassical model," IEEE J. Quantum Electron., vol. 33, no. 9, pp. 1481-1488, 1997.
    [66] D. J. T. Elson, "Maximising Achievable Rates of Experimental Nonlinear Optical Fibre Transmission Systems," UCL (University College London), 2019.
    [67] G. P. Agrawal, "Modulation instability induced by cross-phase modulation," Phys. Rev. Lett., vol. 59, no. 8, p. 880, 1987.
    [68] E. Ip, A. P. T. Lau, D. J. Barros, and J. M. Kahn, "Coherent detection in optical fiber systems," Opt. Express, vol. 16, no. 2, pp. 753-791, 2008.
    [69] W. Waters and B. Jarrett, "Tests of direct-sampling coherent detection with a laboratory analog-to-digital converter," IEEE transactions on aerospace and electronic systems, no. 3, pp. 430-432, 1985.
    [70] M. Chagnon, "Optical communications for short reach," J. Lightw. Technol., vol. 37, no. 8, pp. 1779-1797, 2019.
    [71] K. Takemura, M. Kurihara, T. Uemura, A. Ukita, K. Yashiki, and K. Kurata, "Chip-scale packaging of hybrid-integrated Si photonic transceiver: Optical I/O core," in 2015 IEEE CPMT Symposium Japan (ICSJ), 2015, pp. 191-194: IEEE.
    [72] C. Elster, "Evaluation of measurement uncertainty in the presence of combined random and analogue-to-digital conversion errors," Measurement Science and Technology, vol. 11, no. 9, p. 1359, 2000.
    [73] G. Shulkind and M. Nazarathy, "Efficient nonlinear compensator for coherent OFDM based on volterra series transfer function (VSTF) factorization," in 2013 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), 2013, pp. 1-3: IEEE.
    [74] I. T. Lima, "Comparison of Digital Back-Propagation with Nonlinear Fourier Transform and Split-Step Fourier for Nonlinear Mitigation in Optical Fiber Systems," in 2018 IEEE Research and Applications of Photonics In Defense Conference (RAPID), 2018, pp. 1-4: IEEE.
    [75] W.-H. Sheen and G. L. Stuber, "MLSE equalization and decoding for multipath-fading channels," IEEE transactions on communications, vol. 39, no. 10, pp. 1455-1464, 1991.
    [76] J. Lee, H. Choi, S. Shin, and Y. C. Chung, "A review of the polarization-nulling technique for monitoring optical-signal-to-noise ratio in dynamic WDM networks," J. Lightw. Technol., vol. 24, no. 11, pp. 4162-4171, 2006.
    [77] V. J. Mathews and G. Sicuranza, Polynomial signal processing. John Wiley & Sons, Inc., 2000.
    [78] B. Farhang-Boroujeny, Adaptive filters: theory and applications. John Wiley & Sons, 2013.
    [79] K. Ozeki and T. Umeda, "An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties," Electron. and Commun. Jpn vol. 67, no. 5, pp. 19-27, 1984.
    [80] R. Caruana and A. Niculescu-Mizil, "An empirical comparison of supervised learning algorithms," in Proceedings of the 23rd international conference on Machine learning, 2006, pp. 161-168.
    [81] M. E. Celebi and K. Aydin, Unsupervised learning algorithms. Springer, 2016.
    [82] D. O. Hebb, The organization of behavior: A neuropsychological theory. Psychology Press, 2005.
    [83] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning representations by back-propagating errors," nature, vol. 323, no. 6088, pp. 533-536, 1986.
    [84] K. Hara, D. Saito, and H. Shouno, "Analysis of function of rectified linear unit used in deep learning," in 2015 international joint conference on neural networks (IJCNN), 2015, pp. 1-8: IEEE.
    [85] P. Chandra, "Sigmoidal function classes for feedforward artificial neural networks," Neural Processing Letters, vol. 18, no. 3, pp. 205-215, 2003.
    [86] A. H. Namin, K. Leboeuf, R. Muscedere, H. Wu, and M. Ahmadi, "Efficient hardware implementation of the hyperbolic tangent sigmoid function," in 2009 IEEE International Symposium on Circuits and Systems, 2009, pp. 2117-2120: IEEE.
    [87] K. Suzuki, Artificial neural networks: Architectures and applications. BoD–Books on Demand, 2013.
    [88] Y. Zhang and N. Tan, "Weights direct determination of feedforward neural networks without iterative BP-training," in Intelligent Soft Computation and Evolving Data Mining: Integrating Advanced Technologies: IGI Global, 2010, pp. 197-225.
    [89] S. Haykin, Neural networks: a comprehensive foundation. Prentice Hall PTR, 1994.
    [90] N. Benvenuto and F. Piazza, "On the complex backpropagation algorithm," IEEE Trans. Signal Process., vol. 40, no. 4, pp. 967-969, 1992.
    [91] S. E. Vt and Y. C. Shin, "Radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems," IEEE transactions on neural networks, vol. 5, no. 4, pp. 594-603, 1994.
    [92] N. Benvenuto and F. Piazza, "On the complex backpropagation algorithm," Ieee T Signal Proces, vol. 40, no. 4, pp. 967-969, 1992.
    [93] M. A. Nielsen, Neural networks and deep learning. Determination press San Francisco, CA, 2015.
    [94] T. T. Truong and H.-T. Nguyen, "Backtracking Gradient Descent Method and Some Applications in Large Scale Optimisation. Part 2: Algorithms and Experiments," Applied Mathematics & Optimization, pp. 1-30, 2020.
    [95] F. Seide, H. Fu, J. Droppo, G. Li, and D. Yu, "On parallelizability of stochastic gradient descent for speech DNNs," in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2014, pp. 235-239: IEEE.
    [96] S. Ioffe and C. Szegedy, "Batch normalization: Accelerating deep network training by reducing internal covariate shift," presented at the International Conference on Machine,
    Learning Lille, France, 2015.
    [97] W. Sung, S. Kang, P. Kim, D. I. Chang, and D. J. Shin, "Performance analysis of APSK modulation for DVB‐S2 transmission over nonlinear channels," International Journal of Satellite Communications and Networking, vol. 27, no. 6, pp. 295-311, 2009.
    [98] R. S. Michalski, J. G. Carbonell, and T. M. Mitchell, Machine learning: An artificial intelligence approach. Springer Science & Business Media, 2013.
    [99] J. Estaran et al., "Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IM/DD systems," pp. 1-3: VDE.
    [100] B. Spinnler, "Equalizer design and complexity for digital coherent receivers," IEEE J. Sel. Top. Quantum Electron., vol. 16, no. 5, pp. 1180-1192, 2010.
    [101] E. Ip and J. M. Kahn, "Compensation of dispersion and nonlinear impairments using digital backpropagation," J. Lightw. Technol., vol. 26, no. 20, pp. 3416-3425, 2008.
    [102] O. Sidelnikov, A. Redyuk, and S. Sygletos, "Nonlinear Equalization in Long Haul Transmission Systems Using Dynamic Multi-Layer Perceptron Networks," pp. 1-3: IEEE.
    [103] M. Schetzen, The Volterra and Wiener theories of nonlinear systems. Wiley, 1980.
    [104] N. Hurley and S. Rickard, "Comparing measures of sparsity," IEEE Trans. Inf Theory, vol. 55, no. 10, pp. 4723-4741, 2009.
    [105] G. Hirano and T. Shimamura, "A modified IPNLMS algorithm using system sparseness," presented at the 2012 International Symposium on Intelligent Signal Processing and Communications Systems, Taipei, Taiwan, 2012,
    [106] E. J. Candès and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21-30, 2008.
    [107] S.-Y. Lu, C.-C. Wei, C.-Y. Chuang, Y.-K. Chen, and J. Chen, "81.7% complexity reduction of Volterra nonlinear equalizer by adopting L1 regularization penalty in an OFDM long-reach PON," in European Conference on Optical Communication, Göteborg ,Sweden, pp. 1-3.
    [108] C.-Y. Chuang et al., "Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect," in Optical Fiber Communication Conference, San Diego, CA ,USA,Paper ,M1F.2.
    [109] Q. Lin, X. Chen, and J. Peña, "A sparsity preserving stochastic gradient methods for sparse regression," Comput. Optim. Appl., vol. 58, no. 2, pp. 455-482, 2014.
    [110] L. Liu et al., "High performance and cost effective CO-OFDM system aided by polar code," Opt. Express, vol. 25, no. 3, pp. 2763-2770, 2017.
    [111] J. Luo, J. Li, Q. Sui, Z. Li, and C. Lu, "40 Gb/s mode-division multiplexed DD-OFDM transmission over standard multi-mode fiber," IEEE Photon. J., vol. 8, no. 3, pp. 1-7, 2016.
    [112] T. A. Eriksson, H. Bülow, and A. Leven, "Applying neural networks in optical communication systems: possible pitfalls," IEEE Photon. Technol. Lett., vol. 29, no. 23, pp. 2091-2094, 2017.
    [113] W. Zhou et al., "135-GHz D-Band 60-Gbps PAM-8 Wireless Transmission Employing a Joint DNN Equalizer With BP and CMMA," J. Lightw. Technol., vol. 38, no. 14, pp. 3592-3601, 2020.
    [114] W.-H. Huang et al., "Nonlinear Equalization Based on Artificial Neural Network in DML-Based OFDM Transmission Systems," J. Lightw. Technol., vol. 39, no. 1, pp. 73-82, 2021.
    [115] S. An, Q. Zhu, J. Li, Y. Ling, and Y. Su, "112-Gb/s SSB 16-QAM signal transmission over 120-km SMF with direct detection using a MIMO-ANN nonlinear equalizer," Opt. Express, vol. 27, no. 9, pp. 12794-12805, 2019.
    [116] G. sharan Yadav, C.-Y. Chuang, K.-M. Feng, J.-H. Yan, J. Chen, and Y.-K. Chen, "Reducing computation complexity by using elastic net regularization based pruned Volterra equalization in a 80 Gbps PAM-4 signal for inter-data center interconnects," Opt. Express, vol. 28, no. 26, pp. 38539-38552, 2020.

    QR CODE