# Histogram Based Clustering for Nonlinear Compensation in Long Reach Coherent Passive Optical Networks

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## Abstract

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## 1. Introduction

## 2. Histogram Based Clustering

- Calculation of the 2D histogram of the in-phase and quadrature components of the ${N}_{S}$ received distorted symbols.
- Find the lowest contour line in the histogram that results in M isolated islands, M being the number of clusters to be identified.
- Assign a class ID to the values of the boundary for each island.
- For each received symbol, find the closest boundary point and associate it with its class ID.

## 3. Simulation Setup

## 4. Results and Discussion

#### 4.1. Performance Analysis

#### 4.2. Block Size and Complexity Analysis

^{th}complex symbol $s\left[k\right]={s}_{i}\left[k\right]+j\xb7{s}_{q}\left[k\right]$ were:

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PON | Passive optical network |

LR | Long reach |

DSP | Digital signal processor |

SPM | Self-phase modulation |

XPM | Cross-phase modulation |

FWM | Four wave mixing |

QAM | Quadrature amplitude modulation |

DBP | Digital back-propagation |

IVSTF | Inverse Volterra series transfer function |

SVM | Support vector machine |

DBSCAN | Density based spatial clustering of applications with noise |

HBC | Histogram based clustering |

LD | Laser diode |

CW | Continuous-wave |

DP-MZM | Dual parallel Mach–Zehnder modulator |

EDFA | Erbium doped fibre amplifier |

ASE | Amplified spontaneous emission |

DAC | Digital to analogue converter |

SSMF | Standard single mode fibre |

DPT | Dynamic polarization tracker |

ADC | Analogue-to-digital converter |

BER | Bit error rate |

SNR | Signal-to-noise ratio |

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**Figure 1.**Flow diagram of the proposed clustering algorithm. (

**a**) Distorted input 16-QAM constellation. (

**b**) Calculated 2D histogram. (

**c**) Optimum boundaries superimposed on the 2D histogram. (

**d**) Boundaries for each cluster on top of the received distorted constellation. (

**e**) Classified constellation.

**Figure 2.**Block diagram of the simulated coherent LR-PON system. S/P: serial-to-parallel conversion. DAC: digital-to-analogue converter. LD: laser diode. DP-MZM: dual parallel Mach–Zehnder modulator. EDFA: erbium doped fibre amplifier. VOA: variable optical attenuator. SSMF: standard single mode fibre. DPT: Dynamic polarization tracker. LPF: low pass filter. ADC: analogue-to-digital converter.

**Figure 3.**Performance of the proposed. HBC compared to that of maximum likelihood and k-means in terms of (

**a**) BER and (

**b**) effective Q-factor derived from BER. (

**c**,

**d**,

**e**) Classified constellations using maximum likelihood, k-means, and HBC, respectively, for a launch optical power of 3 mW. Classified constellations at optimum launch optical powers for the different detection schemes: (

**f**) 5 mW for maximum likelihood, (

**g**) 6 mW for k-means, and (

**h**) 7 mW for HBC. (

**i**,

**j**,

**k**) Classified constellations at elevated launched optical power (10 mW) for maximum likelihood detection, k-means, and HBC, respectively.

**Figure 4.**Analysis of the decision regions (data are for a launch optical power of 10 mW). (

**a**) Histogram of the received constellation. (

**b**) Rectangular shaped decision regions obtained using maximum likelihood. (

**c**) Linear decision regions after k-means clustering and (

**d**) after HBC. In (

**b**–

**d**), the received constellations are superimposed on the decision regions. In addition, for all cases, we included a detailed section of the lower left constellation, corresponding to the white rectangle in (

**a**).

**Figure 5.**BER in terms of the processed block size alongside with the efficiency of HBC for a power level of 7 mW at which optimum performance is achieved for HBC. For comparison purposes, the BER obtained using maximum likelihood is also included.

System Parameters | |||
---|---|---|---|

Laser linewidth | 0.5 MHz | Fibre lengths (L${}_{1}$,L${}_{2}$) | 80 km, 0–20 km |

Laser power | 1 mW | Fibre attenuation | 0.2 dB |

MZM insertion loss | 6 dB | Fibre chromatic dispersion | 16 ps/nm/km |

Amplifier gain | 20 dB | Fibre PMD | 3.16 $\mathrm{fs}/\sqrt{\mathrm{km}}$ |

Amplifier noise figure | 4 dB | Nonlinear coefficient ($\gamma $) | 1.3·W${}^{-1}$·km${}^{-1}$ |

Attenuator | 20 dB | Fibre effective area | 80 $\mathsf{\mu}$m${}^{2}$ |

PD thermal noise density | 10 pA/$\sqrt{Hz}$ | Electrical filter bandwidth | 10.5 GHz |

PD responsivity | 1 W/A | Electrical RX filter order | 4 |

Signal parameters | |||

Modulation format | 16-QAM | No. of synchronization symbols | 64 |

Electrical TX filter | 4^{th}-order Bessel | Bit rate | 56 Gbps |

Simulation parameters | |||

Number of simulated symbols | 16,384 | Sampling rate | $8.96\times {10}^{11}$s^{−1} |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aldaya, I.; Giacoumidis, E.; de Oliveira, G.; Wei, J.; Pita, J.L.; Marconi, J.D.; Fagotto, E.A.M.; Barry, L.; Abbade, M.L.F. Histogram Based Clustering for Nonlinear Compensation in Long Reach Coherent Passive Optical Networks. *Appl. Sci.* **2020**, *10*, 152.
https://doi.org/10.3390/app10010152

**AMA Style**

Aldaya I, Giacoumidis E, de Oliveira G, Wei J, Pita JL, Marconi JD, Fagotto EAM, Barry L, Abbade MLF. Histogram Based Clustering for Nonlinear Compensation in Long Reach Coherent Passive Optical Networks. *Applied Sciences*. 2020; 10(1):152.
https://doi.org/10.3390/app10010152

**Chicago/Turabian Style**

Aldaya, Ivan, Elias Giacoumidis, Geraldo de Oliveira, Jinlong Wei, Julián Leonel Pita, Jorge Diego Marconi, Eric Alberto Mello Fagotto, Liam Barry, and Marcelo Luis Francisco Abbade. 2020. "Histogram Based Clustering for Nonlinear Compensation in Long Reach Coherent Passive Optical Networks" *Applied Sciences* 10, no. 1: 152.
https://doi.org/10.3390/app10010152