# Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels

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## Abstract

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## 1. Introduction

## 2. Notation

## 3. Long-Haul Optical Link MIMO Channel Model

## 4. SDM Communication System Model

#### 4.1. Transmitter

#### 4.2. Linear MMSE MIMO Receiver

#### 4.3. Matched Filter-Based Receiver for SDM

## 5. Numerical Simulation of Linear MIMO FSE Receiver for MDL-Impaired Optical Channel

#### 5.1. Channel Model

#### 5.2. Transmitter and Linear MIMO FSE Receiver Parameters

#### 5.3. Signal-to-Noise at the Input of the Receiver

#### 5.4. Performance Loss Metric for FSE-Based MIMO Receiver

- $M{L}_{95}$ is defined as the 95th percentile of the $L\left(i\right)$ distribution obtained for any optical channel realization and mode;
- $AM{L}_{95}$ is defined as the 95th percentile of the $AL$ distribution obtained for any optical channel realization.

#### 5.5. Numerical Simulation Results

- The distribution of $AL$ is not Gaussian, as we can observe by comparing the difference between upper and lower outliers for higher ${\sigma}_{g}$ values and their asymmetry;
- There are no negative values of $AL$, since the FSE MIMO receiver cannot improve on average the ${\overline{SNR}}_{in}$. However, by taking all values of $L\left(i\right)$ for any received mode i, we can find that, for certain channels and modes, the FSE MIMO receiver can locally improve the $SN{R}_{in}\left(i\right)$ of a particular mode i, but always at the cost of another mode of the receiver;
- The performance degradation measured as $AL$ is milder than measured as $L\left(i\right)$ when more than 95% coverage of the channels is considered. Note that the difference is negligible when median values are taken into account.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Spatial division multilpexing (SDM) communication system model with linear multiple-input multiple-output (MIMO) receiver.

**Figure 2.**SDM communication system model with linear minimum mean square error (MMSE) MIMO receiver.

**Figure 3.**SDM communication system model with matched filter-based receiver (

**a**) and its reordered version (

**b**).

**Figure 4.**MIMO coherent optical communication system model with matched filter-based receiver in the absence of mode-dependent loss (MDL) (

**a**) and its equivalent discrete-time system model (

**b**).

**Figure 5.**SDM communication system model with linear fractionally-spaced equalizer (FSE) MIMO receiver and integer oversampling rate ${r}_{ov}$.

**Figure 6.**$M{L}_{95}$ (up) and $AM{L}_{95}$ (down) as defined in Section 5.4 for ${\overline{SNR}}_{in}$ = 5, 6.2, and 10 dB (

**a**) and ${\overline{SNR}}_{in}$ = 15, 30, and 60 dB (

**b**) for different values of the transmitter roll-off factor $\alpha $. Note that ${\sigma}_{g}=0$ corresponds to a channel without MDL.

**Figure 7.**Probability distribution for $AL$ as defined in (43) (up) and $L\left(i\right)$ as defined in (42) (down) for ${\overline{SNR}}_{in}$ = 5, 6.2, and 10 dB (

**a**) and ${\overline{SNR}}_{in}$ = 15, 30, and 60 dB (

**b**). $\alpha $ = 0.9 for all graphs. Note that ${\sigma}_{g}=0$ corresponds to a channel without MDL.

Parameter | Symbol | Value and Reference |
---|---|---|

Span length | ${\ell}_{span}$ | 50 km |

Number of spans | ${K}_{amp}$ | 100 |

Number of spatial and polarization modes | D | 6 |

Center wavelength | ${\lambda}_{c}$ | 1469 nm [41] |

Modal dispersion | ${\sigma}_{\tau}/\sqrt{{\ell}_{span}}$ | 3.1 $\mathrm{ps}/\sqrt{\mathrm{km}}$ [41] |

Dispersion coefficient | ${D}_{CD}=-\frac{2\pi c}{{\lambda}_{c}^{2}}\overline{{\beta}_{2}}$ | 20.1 $\mathrm{ps}/(\mathrm{nm}$·$\mathrm{km})$ [41] |

Underestimation dispersion factor | ${U}_{CD}$ | $2\%$ [9] |

Amplifier gain STD | ${\sigma}_{g}$ | 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 dB |

Symbol rate | ${R}_{s}=1/{T}_{s}$ | 64 GBaud |

Oversampling factor | ${r}_{ov}$ | 2 |

Roll off factor | $\alpha $ | 0.1, 0.5, 0.7, 0.9 |

Number of channel realizations | ${N}_{ch}$ | 10000 |

Signal to noise ratio at the receiver input | ${\overline{SNR}}_{in}$ | 60 30 15 10 6.2 5 dB [27] |

Number of taps | ${N}_{taps}$ | 1000 |

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**MDPI and ACS Style**

Torres, L.M.; Cañete, F.J.; Díez, L.
Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels. *Sensors* **2022**, *22*, 798.
https://doi.org/10.3390/s22030798

**AMA Style**

Torres LM, Cañete FJ, Díez L.
Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels. *Sensors*. 2022; 22(3):798.
https://doi.org/10.3390/s22030798

**Chicago/Turabian Style**

Torres, Luis M., Francisco J. Cañete, and Luis Díez.
2022. "Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels" *Sensors* 22, no. 3: 798.
https://doi.org/10.3390/s22030798