# Multi-Level Phase Noise Model for CO-OFDM Spatial-Division Multiplexed Transmission

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## Abstract

**:**

## 1. Introduction

## 2. Phase Noise Model for the CO-OFDM SDM System

## 3. Phase Noise Estimation Algorithms

#### 3.1. Blind Phase Noise Estimation Algorithm

^{–3}times greater than LPN and is affected by many parameters [28]. These reasons increase the demand for the phase noise estimation algorithm.

#### 3.2. Pilot-Aided Phase Noise Estimation

## 4. Results

^{−3}(7% HD-FEC threshold). Different pilots (2, 4, and 8) are input instead of the data subcarriers to estimate the phase noise. In general, CPD is about 10

^{−3}times the LPN and is used in this section. As can be seen in Figure 4, the required SNR of BPS with the DPT algorithm is always lower than that of the PA algorithm with insert pilots. The pilots are inserted evenly in this simulation. The required SNR of BPS is about 1.4 dB, 2.7 dB, and 4 dB better than that of eight pilots, four pilots, and two pilots, respectively. The more inserted pilots there are for the PA algorithm, the better the performance of the PN compensation is (as Equation (12) shows). However, the tolerances of the PA are better than those of the BPS. For the 1 dB SNR penalty, the tolerances of the NLW are 1.5 × 10

^{−3}and 3 × 10

^{−3}for the BPS and PA methods. When the tolerance of NLW is reached, the performance of the BPS method decreases greatly compared with PA, which is due to the fact that the PN exceeds the tested phase of BPS.

^{−4}for 64, 128, and 256 subcarriers. Figure 5a–c show that the PN compensation performances are almost the same under different subcarriers. When the HD-FEC threshold is achieved, the required SNRs are about 14.5 dB, 15.9 dB, and 17.3 dB for BPS, four insert pilots’ PA, and eight insert pilots’ PA, respectively. The performance of BPS is better than that of PA (about 1.4 dB for eight insert pilots and 3 dB for four insert pilots) for random subcarriers. The results indicate that the PN compensation performance mainly depends on the PNE algorithm and is little affected by the number of subcarriers. In order to deeply analyze the influence of the subcarriers on the PN compensation, BERs as a function of SNR under different carriers are demonstrated with BPS and PA with eight insert pilots. The required SNR for different subcarriers is almost the same (less than 0.1 dB) at the HD-FEC. Thus, the BPS-DPT with the 32-test-phase algorithm has a better performance compared with the PA algorithm. Additionally, the performance under the same NLW is almost the same under different subcarriers with the same PNE algorithm.

^{2}, 1 × 10

^{3}, 2 × 10

^{3}, and 4 × 10

^{3}, respectively. Figure 7a–c show that the performance of the ratio and the subcarrier is almost the same. The main difference in the performance comes from the algorithm. With different ratios of LPN and CPD, the required SNRs are almost equal. A faint SNR difference in the ratio was shown between different numbers of subcarriers, namely that the required SNR for 256 changed less than that of 64 subcarriers.

## 5. Conclusions

^{−3}to 3 × 10

^{−3}. Furthermore, the ratio of LPN and CPD, the number of subcarriers, and the number of inserted pilots are also demonstrated in this paper. The results show that the traditional phase-noise algorithms can also perform well for multi-carrier SDM systems considering the core phase drift. This means that with the SDM phase noise model, only multi-channel parallel phase noise algorithms can be considered without the difference between the cores for weak-coupled SDMs. These studies can also help to redesign the phase-noise compensation algorithms based on the traditional algorithms and the correlation between the cores of MCFs. The results and discussions may have a better influence on the CPE research for SDM transmission links.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic diagram of phase noise model (

**a**) traditional MCF and (

**b**) multivariate Gaussian model.

**Figure 4.**The comparison for phase noise estimation with PA and BPS for the 128-sub-carrier CO-OFDM SDM system. The required SNR is tested at BER of 3.8 × 10

^{−3}(7% HD-FEC threshold).

**Figure 5.**BER as a function of SNR for the CO-OFDM SDM transmission link under different subcarriers ((

**a**): 64 subcarriers, (

**b**): 128 subcarriers, (

**c**): 256 subcarriers, and (

**d**): performance under different subcarriers). (

**a**) N = 64. (

**b**) N = 128. (

**c**) N = 256.

**Figure 7.**BER vs. SNR with the difference between CPD and LPN. (

**a**) N = 64. (

**b**) N = 128. (

**c**) N = 256.

**Table 1.**Complexity comparison and SE loss between the BPS-DFT algorithm and the PA algorithm in terms of the multiplications.

BPS-DPT (Test Phase: 32 4 × 32 × N) | PA (9 × N_{P}) | ||||
---|---|---|---|---|---|

N | Multiplications | N | N_{p} | Multiplications | |

Complexity | 64 | 8192 | 128 | 2 | 18 |

128 | 16,384 | 4 | 36 | ||

256 | 32,768 | 8 | 72 | ||

N | SE Loss | N | N_{p} | SE Loss | |

SE Loss | 64 | 0 | 128 | 2 | 1.56% |

128 | 0 | 4 | 3.125% | ||

256 | 0 | 8 | 6.25% |

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**MDPI and ACS Style**

Jiang, G.; Yang, L.
Multi-Level Phase Noise Model for CO-OFDM Spatial-Division Multiplexed Transmission. *Photonics* **2023**, *10*, 8.
https://doi.org/10.3390/photonics10010008

**AMA Style**

Jiang G, Yang L.
Multi-Level Phase Noise Model for CO-OFDM Spatial-Division Multiplexed Transmission. *Photonics*. 2023; 10(1):8.
https://doi.org/10.3390/photonics10010008

**Chicago/Turabian Style**

Jiang, Guozhou, and Liu Yang.
2023. "Multi-Level Phase Noise Model for CO-OFDM Spatial-Division Multiplexed Transmission" *Photonics* 10, no. 1: 8.
https://doi.org/10.3390/photonics10010008