# Mitigation of Nonlinear Impairments by Using Support Vector Machine and Nonlinear Volterra Equalizer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. Support Vector Machine

#### SVM-based Detection

#### 2.2. Nonlinear Volterra Equalizer

## 3. Simulation Setup

**a**) is used to test a back-to-back (B2B) scenario, that is, no transmission link was simulated. The setup (

**b**) is used to examine a dispersion uncompensated link, where the dispersion is compensated by DSP at the end of the transmission. The parameters for the SSMF are given by the attenuation coefficient $\alpha $ = 0.2 dB/km, the dispersion coefficient $D\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}17\phantom{\rule{0.166667em}{0ex}}$ps/(nm·km), dispersion slope $S\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.06$ ps/(nm${}^{2}$·km) and the nonlinear coefficient $\gamma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.3$ (W·km)${}^{-1}$. For a complete compensation of span loss an EDFA (NF $=5$ dB) is applied. After transmission a Gaussian optical filter with 90 GHz bandwidth is used to reduce ASE noise. The received signal is detected by a coherent receiver and downsampled to 128 GS/s. After matched filtering an ideal EDC is used to compensate for dispersion. After the equalization stage, which consists of either an FFE, an NLVE or no equalizer at all (w/o), the signal is downsampled to symbol frequency and detected. Detection and demodulation is performed either linear by using conventional linear decision thresholds and demapping, here called linear detection (LD), or by machine learning algorithms such as SVM or KMA [8,9]. System performance is evaluated by BER. The hard-decision forward error correction (HD-FEC) limit is assumed to be $3.7\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$. We examine the suitability of the SVM as a classifier and combine the mentioned equalizer schemes with the SVM to achieve the maximum gain of the machine learning algorithm.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ASE | Amplified Spontaneous Emission |

BCSVM | Binary Coding Support Vector Machine |

BER | Bit Error Ratio |

B2B | Back-to-Back |

DSP | Digital Signal Processing |

EDC | Electronic Dispersion Compensation |

EDFA | Erbium Doped Fiber Amplifier |

HD-FEC | Hard-Decision Forward Error Correction |

KMA | K-Means Algorithm |

LD | Linear Detection |

MMSE | Minimum Mean Square Error |

NF | Noise Figure |

NLPN | Nonlinear Phase Noise |

OVA | One versus All |

OVO | One versus One |

OSNR | Optical Signal-to-Noise Ratio |

QAM | Quadrature Amplitude Modulation |

RBF | Radial Basis Function |

SSMF | Standard Single Mode Fiber |

SVM | Support Vector Machine |

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**Figure 1.**Binary Coding Support Vector Machine (BCSVM) nonlinear classification using four support vector machines (SVMs) [3]: (

**a**) Coding and classification scheme for BCSVM based detection and (

**b**) the processing structure for BCSVM used for 16-QAM signal detection.

**Figure 2.**Illustration of one iteration during training for (

**a**) OVO, (

**b**) OVA and (

**c**) BCSVM methods in case of 16-QAM transmission. The opposite classes are marked in red and blue and the corresponding hyperplane is indicated by the dashed line.

**Figure 3.**Simulation setup of the 64 Gbd 64-QAM single-polarization coherent optical simulation system including two different setups for the link. By using the setup (

**a**) a B2B transmission with noise loading is examined. The setup (

**b**) consists of a 100 km SSMF transmission with subsequent electronic dispersion compensation (EDC) to investigate a dispersion uncompensated link.

**Figure 4.**Simulations results in case of transmitter I/Q imbalances. BER vs. phase mismatch for an amplitude mismatch of 0.125 and 28 dB OSNR.

**Figure 5.**BER as a function of the launch power at 100 km dispersion unmanaged transmission: (

**a**) shows equalization by FFE[1] and NLVE[4,2,5] combined with LD and only SVM detection. (

**b**) shows the combined structure FFE[1] and SVM and (

**c**) shows the combination NLVE[4,2,5] with SVM.

**Figure 6.**BER as a function of the launch power at 100 km for NLVE equalization with optimal and reduced number of coefficients in combination with SVM based detection.

**Figure 7.**(

**a**) Shows the BER as function of the number of training symbols for 100 km dispersion unmanaged transmission at 5 dBm launch power. (

**b**) Shows the the corresponding constellation diagram for NLVE[4,2,5] & SVM trained with 1024 symbols and (

**c**) Shows the constellation diagram for NLVE[4,2,5] & SVM trained with 3072 symbols.

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**MDPI and ACS Style**

Weixer, R.; Koch, J.; Plany, P.; Ohlendorf, S.; Pachnicke, S. Mitigation of Nonlinear Impairments by Using Support Vector Machine and Nonlinear Volterra Equalizer. *Appl. Sci.* **2019**, *9*, 3800.
https://doi.org/10.3390/app9183800

**AMA Style**

Weixer R, Koch J, Plany P, Ohlendorf S, Pachnicke S. Mitigation of Nonlinear Impairments by Using Support Vector Machine and Nonlinear Volterra Equalizer. *Applied Sciences*. 2019; 9(18):3800.
https://doi.org/10.3390/app9183800

**Chicago/Turabian Style**

Weixer, Rebekka, Jonas Koch, Patrick Plany, Simon Ohlendorf, and Stephan Pachnicke. 2019. "Mitigation of Nonlinear Impairments by Using Support Vector Machine and Nonlinear Volterra Equalizer" *Applied Sciences* 9, no. 18: 3800.
https://doi.org/10.3390/app9183800