# Relaxation of the Radio-Frequency Linewidth for Coherent-Optical Orthogonal Frequency-Division Multiplexing Schemes by Employing the Improved Extreme Learning Machine

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## Abstract

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## 1. Introduction

- We propose a modified ELM under supervised learning for maximizing the system performance (the BER minimization) of a phase-uncorrelated OFDM signal in the optical domain based on the adoption of the pilot subcarriers as training samples, as well as the consideration of the regularization parameter in the learning stage.
- Taking into account the RF phase error as well as the subcarrier modulation format, we find the sub-optimal ELM parameters (the number of hidden nodes, penalty coefficient, and activation function) that yield the best BER via extensive simulations. This result is explained by the evaluation of the error vector magnitude (EVM) metric in the training as well as testing steps, which can properly quantify the root mean square error for complex numbers in the telecommunication industry.
- We verify that when the Moore–Penrose generalized inverse of the hidden layer output matrix takes into account the regularization parameter, the ELM significantly improves in terms of stability and precision. As a result, the distortion induced by the laser oscillators is less within the constellation symbols.
- For several signal to noise ratio (SNR) levels and RF-linewidth values, we respectively observe the superiority, and competitiveness of the novel ELM algorithm in terms of the BER metric among the benchmark PAE and a fully-real ELM, and the sophisticated ELM defined in the complex plane and non-effective bandwidth CPE compensator for binary phase-shift keying (BPSK) and QPSK modes.

## 2. Background

#### 2.1. Extreme Learning Machine

- Setting the hidden neurons L, the activation function $g(\xb7)$, and the regularization parameter C.
- Randomly choosing the input weights ${w}_{j}$ as well as biases ${b}_{j}$.
- Finding the output layer weights $\beta $ via Equation (2), where N samples $({x}_{i},{t}_{i})$ are known.

#### 2.2. Coherent Optical OFDM Network

## 3. Proposed Extreme Learning Machine Algorithm for Laser Phase-Noise Reduction Purposes

## 4. Results and Discussion

#### 4.1. Parameters’ Optimization of the Extreme Learning Machine

#### 4.2. Performance Evaluation

#### 4.3. Complexity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ADC | Analog-to-digital converter |

AWGN | Additive white Gaussian noise |

BER | Bit error rate |

BPF | Band pass filter |

BPSK | Binary phase shift keying |

C-ELM | Extreme learning machine in the complex plane |

CO | Coherent optical |

COS | Cosine |

CP | Cyclic prefix |

CPE | Common phase error |

DAC | Digital-to-analog converter |

ELMs | Extreme learning machines |

EVM | Error vector magnitude |

FEC | Forward error correction |

FFT | Fast Fourier transform |

HL | Hard limit |

HT | Hyperbolic tangent |

ICI | Inter-carrier interference |

OFDM | Orthogonal frequency division multiplexing |

PAE | Pilot-assisted equalization |

QAM | Quadrature amplitude modulation |

QPSK | Quadrature phase-shift keying |

RC | Real-complex |

RF | Radio frequency |

SIG | Sigmoid |

SLFNs | Single-hidden layer feedforward networks |

SNR | Signal to noise ratio |

RB | Radial basis |

R-ELM | Fully-real extreme learning machine |

TB | Triangular basis |

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**Figure 1.**(

**a**) Block diagram of a coherent optical orthogonal frequency division multiplexing (CO-OFDM) system, where its five parts are noticed. BPF: band pass filter. (

**b**) Structure of the generic OFDM modem whose differences occur in the pilots’ information as well as the equalization stage. ADC: analog-to-digital converter.

**Figure 3.**According to diverse radio-frequency (RF) linewidths and (

**a**–

**c**) binary phase-shift keying (BPSK), (

**d**–

**f**) quadrature phase-shift keying (QPSK), and (

**g**–

**i**) 16QAM subcarrier modulation formats, the BER in terms of the regularization parameter and number of hidden neurons by considering the SIG activation function.

**Figure 4.**According to (

**a**) BPSK, (

**b**) QPSK, and (

**c**) 16QAM constellations, the bit error rate (BER) in terms of the number of hidden nodes with the combined linewidth as parameter. The penalty coefficient is not considered.

**Figure 5.**For the different modulations: (

**a**) BPSK, (

**b**) QPSK, and (

**c**) 16QAM, the system performance as a function of the RF linewidth with the activation function as parameter.

**Figure 6.**For (

**i**) SIG, (

**ii**) COS, (

**iii**) HL, (

**iv**) TB, (

**v**) RB, and (

**vi**) HT activation functions, histogram constellations together EVM values at training and testing stages. The subcarrier modulation format and combined linewidth correspond to 16QAM and 100 kHz, respectively.

**Figure 8.**By considering diverse phase-noise compensation techniques for the studied subcarrier modulation formats, the BER against the SNR with the RF linewidth as parameter. For an SNR of 20 dB and different combined linewidths, Subfigures show constellations.

**Figure 9.**Taking into account a phase-noise free QPSK-OFDM signal, its positive spectrum as a function of the SNR values.

**Figure 10.**By considering an unmodulated QPSK-OFDM signal without CP and AWGN, its frequency profile in the last subcarriers as the combined linewidth increases.

Function | $g\left(x\right)$ |

Sigmoid (SIG) | $\frac{1}{1+exp(-x)}$ |

Cosine (COS) | $cos\left(x\right)$ |

Hard limit (HL) | $\left\{\begin{array}{c}1\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}x\ge 0\hfill \\ 0\phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}\hfill \end{array}\right.$ |

Triangular basis (TB) | max$(1-|x|,0)$ |

Radial basis (RB) | $exp(-{x}^{2})$ |

Hyperbolic tangent (HT) | $\frac{1-exp(-x)}{1+exp(-x)}$ |

Parameter | Value |
---|---|

Subcarrier modulation format | BPSK, QPSK, and 16QAM |

Bit rate | 10 Gbps |

Number of data | 112 |

Number of pilots | 16 |

FFT size | 128 |

CP length | 1/10 |

Sample rate | 100 Gsps |

Number of bits | ${10}^{6}$ |

Technique | Training Time | Testing Time | Total Time |
---|---|---|---|

PAE | - | - | $1.3581\times {10}^{-4}$ |

R-ELM | $2.0836\times {10}^{-4}$ | $8.4786\times {10}^{-5}$ | $2.9315\times {10}^{-4}$ |

C-ELM | $6.9722\times {10}^{-5}$ | $2.7325\times {10}^{-5}$ | $9.7047\times {10}^{-5}$ |

RC-ELCM | $1.5905\times {10}^{-4}$ | $1.1450\times {10}^{-4}$ | $2.7355\times {10}^{-4}$ |

CPE compensation | - | - | $2.2054\times {10}^{-5}$ |

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## Share and Cite

**MDPI and ACS Style**

Zabala-Blanco, D.; Mora, M.; Azurdia-Meza, C.A.; Dehghan Firoozabadi, A.; Palacios Játiva, P.; Soto, I.
Relaxation of the Radio-Frequency Linewidth for Coherent-Optical Orthogonal Frequency-Division Multiplexing Schemes by Employing the Improved Extreme Learning Machine. *Symmetry* **2020**, *12*, 632.
https://doi.org/10.3390/sym12040632

**AMA Style**

Zabala-Blanco D, Mora M, Azurdia-Meza CA, Dehghan Firoozabadi A, Palacios Játiva P, Soto I.
Relaxation of the Radio-Frequency Linewidth for Coherent-Optical Orthogonal Frequency-Division Multiplexing Schemes by Employing the Improved Extreme Learning Machine. *Symmetry*. 2020; 12(4):632.
https://doi.org/10.3390/sym12040632

**Chicago/Turabian Style**

Zabala-Blanco, David, Marco Mora, Cesar A. Azurdia-Meza, Ali Dehghan Firoozabadi, Pablo Palacios Játiva, and Ismael Soto.
2020. "Relaxation of the Radio-Frequency Linewidth for Coherent-Optical Orthogonal Frequency-Division Multiplexing Schemes by Employing the Improved Extreme Learning Machine" *Symmetry* 12, no. 4: 632.
https://doi.org/10.3390/sym12040632