# Kramers–Kronig Transmission with a Crosstalk-Dependent Step Multiple-Input Multiple-Output Volterra Equalizer in a Seven-Core Fiber

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## Abstract

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## 1. Introduction

## 2. Principle of KK Transmission System

## 3. Experimental Setup and Analysis of Seven-Core Transmission System with KK Receiver

## 4. XT-MIMO Nonlinear Equalization Algorithm Based on Volterra Series

- Assume that the 16-QAM signal data matrix for all cores input to the equalizer is represented as$$\begin{array}{cc}\hfill \mathit{D}& =\left[\begin{array}{cccc}{y}_{1}\left(1\right)& {y}_{1}\left(2\right)& \dots & {y}_{1}\left(n\right)\\ {y}_{2}\left(1\right)& {y}_{2}\left(2\right)& \dots & {y}_{2}\left(n\right)\\ \dots & \dots & \dots & \dots \\ {y}_{m}\left(1\right)& {y}_{m}\left(2\right)& \dots & {y}_{m}\left(n\right)\end{array}\right]=\left[\begin{array}{c}{\mathit{y}}_{1}\\ {\mathit{y}}_{2}\\ \dots \\ {\mathit{y}}_{m}\end{array}\right].\hfill \end{array}$$
- Calculate the third-order nonlinear length as$$\begin{array}{ccc}\hfill {L}_{nl}& =& \sum _{{v}_{3}={v}_{1}}^{{L}_{3}}\sum _{{v}_{2}={v}_{3}}^{{L}_{3}}\sum _{{v}_{1}=1}^{{L}_{3}}\hfill \\ & =& \frac{1}{12}[3{L}_{3}\left({L}_{3}+1\right)+{L}_{3}\left({L}_{3}+1\right)\left(2{L}_{3}+1\right)].\hfill \end{array}$$
- Calculate the total channel length as$$\begin{array}{c}\hfill {L}_{c}={L}_{1}+{L}_{nl}.\end{array}$$
- Initialize the butterfly equalizer, and the tap coefficient of the equalizer can be expressed as$$\begin{array}{cc}\hfill \mathit{w}\left(k\right)& =\left[\begin{array}{cccc}{w}_{1}^{0}\left(k\right)& {w}_{2}^{0}\left(k\right)& \dots & {w}_{m}^{0}\left(k\right)\\ {w}_{1}^{1}\left(k\right)& {w}_{2}^{1}\left(k\right)& \dots & {w}_{m}^{1}\left(k\right)\\ \dots & \dots & \dots & \dots \\ {w}_{1}^{m{L}_{c}}\left(k\right)& {w}_{2}^{m{L}_{c}}\left(k\right)& \dots & {w}_{m}^{m{L}_{c}}\left(k\right)\end{array}\right]=\left[\begin{array}{cccc}{\mathit{w}}_{11}\left(k\right)& {\mathit{w}}_{12}\left(k\right)& \dots & {\mathit{w}}_{1m}\left(k\right)\\ {\mathit{w}}_{21}\left(k\right)& {\mathit{w}}_{22}\left(k\right)& \dots & {\mathit{w}}_{2m}\left(k\right)\\ \dots & \dots & \dots & \dots \\ {\mathit{w}}_{m1}\left(k\right)& {\mathit{w}}_{m2}\left(k\right)& \dots & {\mathit{w}}_{mm}\left(k\right)\end{array}\right].\hfill \end{array}$$Let ${w}_{i}^{\u2308{H}_{1}\u2309}\left(0\right)(i=1,2,\dots ,m)$ is 1, where ${H}_{1}=\frac{{L}_{1}}{2}$ and the other terms of $\mathit{w}\left(0\right)$ are 0. ${\mathit{w}}_{ij}\left(k\right)(i,j=1,2,...,m)$ is a column vector of length ${L}_{c}$.
- Intercept the $(k+1)\u2308{H}_{1}\u2309-\u230a{H}_{1}\u230b$ to $(k+1)\u2308{H}_{1}\u2309+\u230a{H}_{1}\u230b$ signals of each core from $k=1$ and arrange them in reverse order to obtain the linear input sequence of the equalizer for each channel as$$\begin{array}{cc}\hfill {\mathit{y}}_{i}^{1}\left(k\right)& =\left[\begin{array}{c}{y}_{i}((k+1)\u2308{H}_{1}\u2309+\u230a{H}_{1}\u230b)\\ \dots \\ {y}_{i}((k+1)\u2308{H}_{1}\u2309-\u230a{H}_{1}\u230b)\end{array}\right].\hfill \end{array}$$Intercept the signals from $(k+1)\u2308{H}_{1}\u2309-\u230a{H}_{3}\u230b$ to $(k+1)\u2308{H}_{3}\u2309+\u230a{H}_{3}\u230b$ of each core and arrange them in reverse order, using (12) to find the cubic product of the input signals corresponding to the third-order Volterra kernel for each channel.$$\begin{array}{cc}\hfill {\mathit{y}}_{i}^{nl}\left(k\right)& =\left[\begin{array}{c}{y}_{i}\left(A\right){y}_{i}\left(A\right){y}_{i}\left(A\right)\\ \dots \\ {y}_{i}(A-{v}_{3}){y}_{i}(A-{v}_{2}){y}_{i}(A-{v}_{1})\\ \dots \\ {y}_{i}\left(B\right){y}_{i}\left(B\right){y}_{i}\left(B\right)\end{array}\right],\hfill \end{array}$$$$\begin{array}{cc}\hfill A& =(k+1)\u2308{H}_{3}\u2309+\u230a{H}_{3}\u230b\hfill \\ \hfill B& =(k+1)\u2308{H}_{3}\u2309-\u230a{H}_{3}\u230b.\hfill \end{array}$$
- Splice the linear sequence with a third order to obtain the input sequence for each channel, shown as$$\begin{array}{cc}\hfill {\mathit{y}}_{i}\left(k\right)& =\left[\begin{array}{c}{\mathit{y}}_{i}^{1}\left(k\right)\\ {\mathit{y}}_{i}^{nl}\left(k\right)\end{array}\right].\hfill \end{array}$$
- Splice the MIMO equalizer input sequence of each channel into a MIMO input sequence, as shown in (14).$$\begin{array}{cc}\hfill \mathit{y}\left(k\right)& =\left[\begin{array}{cccc}{\mathit{y}}_{1}\left(k\right)& {\mathit{y}}_{2}\left(k\right)& \dots & {\mathit{y}}_{m}\left(k\right)\\ {\mathit{y}}_{2}\left(k\right)& {\mathit{y}}_{1}\left(k\right)& \dots & {\mathit{y}}_{1}\left(k\right)\\ \dots & \dots & \dots & \dots \\ {\mathit{y}}_{m}\left(k\right)& {\mathit{y}}_{m}\left(k\right)& \dots & {\mathit{y}}_{m-1}\left(k\right)\end{array}\right]\hfill \end{array}$$
- Calculate the MIMO output signal as$$\begin{array}{c}\hfill \tilde{\mathit{x}}\left(k\right)={\mathit{w}}^{T}(k-1)\mathit{y}\left(k\right)=\left[\begin{array}{c}{\tilde{x}}_{1}\left(k\right)\\ {\tilde{x}}_{2}\left(k\right)\\ \dots \\ {\tilde{x}}_{m}\left(k\right)\end{array}\right].\end{array}$$
- Calculate the error as$$\begin{array}{c}\hfill \epsilon \left(k\right)=\mathit{d}\left(k\right)-\tilde{\mathit{x}}\left(k\right)=\left[\begin{array}{c}{\epsilon}_{1}\left(k\right)\\ {\epsilon}_{2}\left(k\right)\\ \dots \\ {\epsilon}_{m}\left(k\right)\end{array}\right].\end{array}$$
- Update the inverse of the correlation matrix as [34]:$$\begin{array}{cc}\hfill {\mathbf{R}}_{D,i}^{-1}\left(k\right)& =\frac{1}{\lambda}[{\mathbf{R}}_{D,i}^{-1}(k-1)-\frac{{\mathbf{R}}_{D,i}^{-1}(k-1){\mathit{y}}_{i}\left(k\right){\mathit{y}}_{i}^{T}\left(k\right){\mathbf{R}}_{D,i}^{-1}(k-1)}{\lambda +{\mathit{y}}_{i}^{T}\left(k\right){\mathbf{R}}_{D,i}^{-1}(k-1){\mathit{y}}_{i}\left(k\right)}].\hfill \end{array}$$
- Calculate the crosstalk coefficient matrix according to the measured crosstalk matrix as$$\begin{array}{cc}\hfill {\mathit{XT}}_{opt}\left(k\right)& =\left[\begin{array}{cccc}X{T}_{opt,11}\left(k\right)& X{T}_{opt,12}\left(k\right)& \dots & X{T}_{opt,1m}\left(k\right)\\ X{T}_{opt,21}\left(k\right)& X{T}_{opt,22}\left(k\right)& \dots & X{T}_{opt,2m}\left(k\right)\\ \dots & \dots & \dots & \dots \\ X{T}_{opt,m1}\left(k\right)& X{T}_{opt,m2}\left(k\right)& \dots & X{T}_{opt,mm}\left(k\right)\end{array}\right],\hfill \end{array}$$$$\begin{array}{c}\hfill X{T}_{opt,ij}\left(k\right)=\rho \frac{X{T}_{ij}\left(k\right)-X{T}_{min}\left(k\right)}{X{T}_{max}\left(k\right)-X{T}_{min}\left(k\right)}(i,j=1,2,...,m).\end{array}$$$X{T}_{ij}\left(k\right)$ represents the value of the measured crosstalk level of k-th time interval, $X{T}_{max}\left(k\right)$ and $X{T}_{min}\left(k\right)$ are the maximum and minimum values in the crosstalk matrix, respectively. $\rho $ is the scaling factor. For example, in Figure 3b, $X{T}_{12}\left(k\right)={10}^{-46.7/10}$, $X{T}_{max}\left(k\right)={10}^{-4.67/10}$ and $X{T}_{min}\left(k\right)={10}^{-50.3/10}$, which is negligible for less than −50 dB (gray cells in the crosstalk matrix), corresponding to $X{T}_{opt,ij}\left(k\right)$ of 0. Since the inter-core XT changes relatively slowly, the crosstalk matrix is collected once in one minute. We use the average crosstalk power in one minute for the calculation of the algorithm crosstalk coefficient, i.e., $X{T}_{ij}\left(1\right)=X{T}_{ij}\left(2\right)=...=X{T}_{ij}\left(k\right)$. The transmitted signal is processed with the crosstalk matrix in the corresponding time interval. Considering that ${\mathit{XT}}_{opt}\left(k\right)$ is a singular matrix, the inverse matrix of ${\mathit{XT}}_{opt}\left(k\right)$ is$$\begin{array}{cc}\hfill {\mathit{XT}}_{opt}^{-1}\left(k\right)& =\left[\begin{array}{cccc}X{T}_{opt,11}^{\prime}\left(k\right)& X{T}_{opt,12}^{\prime}\left(k\right)& \dots & X{T}_{opt,1m}^{\prime}\left(k\right)\\ X{T}_{opt,21}^{\prime}\left(k\right)& X{T}_{opt,22}^{\prime}\left(k\right)& \dots & X{T}_{opt,2m}^{\prime}\left(k\right)\\ \dots & \dots & \dots & \dots \\ X{T}_{opt,m1}^{\prime}\left(k\right)& X{T}_{opt,m2}^{\prime}\left(k\right)& \dots & X{T}_{opt,mm}^{\prime}\left(k\right)\end{array}\right],\hfill \end{array}$$
- Update the tap coefficient of the equalizer using (20) and (21):$$\begin{array}{c}\hfill {\mathit{w}}_{ij}\left(k\right)={\mathit{w}}_{ij}(k-1)+X{T}_{opt,ij}^{\prime}\frac{{\mathit{R}}_{D,i}^{-1}(k-1){\mathit{y}}_{i}\left(k\right)}{\lambda +{\mathit{y}}_{i}^{T}\left(k\right){\mathit{R}}_{D,i}^{-1}(k-1){\mathit{y}}_{i}\left(k\right)}{\epsilon}_{i}\left(k\right).\end{array}$$See Appendix A for a detailed explanation of the derivation process.
- Repeat steps 5 to steps 12 until the end of the training signal is updated and converged.
- Extract the payload signal and repeat steps 5 to steps 8 until the end of the payload signal is updated, and complete the MIMO equalization.

## 5. Experimental Results and Performance Analysis of XT-MIMO Nonlinear Equalization

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 2.**Experimental setup for the seven-core KK 16-QAM coherent optical communication system (TLS: tunable laser source; AWG: arbitrary waveform generator; IQM: I/Q modulator; EA: electric amplifier; OC: optical coupler; VOA: variable optical attenuator; DL: delay line; EDFA: erbium-doped fiber amplifier; OS: optical switch; PD: photodetector; DPO: digital phosphor oscilloscope). Inset (

**i**) shows the spectrum of transmitted signals. Inset (

**ii**,

**iii**) show the transmitter and receiver DSP, respectively.

**Figure 6.**BER as a function of frequency interval when carrier power is 10 dBm and the signal power is 0 dBm.

**Figure 7.**BER as a function of frequency interval when carrier power is 10 dBm and the signal power is 0 dBm.

**Figure 9.**Flow diagram of the XT-MIMO nonlinear equalization algorithm based on the Volterra series.

**Figure 10.**Experimental results: (

**a**) the constellation diagram and (

**b**) the mean square error curve of the signal after passing through the SISO-Volterra algorithm, the MIMO-Volterra algorithm, and the XT-MIMO-Volterra algorithm, respectively.

**Figure 11.**Experimental results. (

**a**) BER as a function of signal power when the carrier power is 10 dBm, (

**b**) BER as a function of CSPR when the carrier power is 10 dBm, (

**c**) the relationship between BER with frequency interval (

**c**) and received power (

**d**), respectively.

Parameters | Values | |
---|---|---|

Loss@1550 nm (dB/km) | 0.25 | |

Trench-assisted | Mode field diameter@1550 nm ($\mathsf{\mu}$m) | 9.5 |

homogeneous | Core layer diameter ($\mathsf{\mu}$m) | 7.9 |

seven-core fiber | Core spacing ($\mathsf{\mu}$m) | 41.5 ± 1.5 |

Cladding diameter ($\mathsf{\mu}$m) | 150 ± 2 | |

FIFO modules | Maximum insertion loss (dB) | 1.5 |

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**MDPI and ACS Style**

Tian, F.; Wu, T.; Yu, C.; Wang, C.; Yue, M.; Gao, R.; Zhang, Q.; Li, Z.; Tian, Q.; Wang, F.;
et al. Kramers–Kronig Transmission with a Crosstalk-Dependent Step Multiple-Input Multiple-Output Volterra Equalizer in a Seven-Core Fiber. *Photonics* **2023**, *10*, 1017.
https://doi.org/10.3390/photonics10091017

**AMA Style**

Tian F, Wu T, Yu C, Wang C, Yue M, Gao R, Zhang Q, Li Z, Tian Q, Wang F,
et al. Kramers–Kronig Transmission with a Crosstalk-Dependent Step Multiple-Input Multiple-Output Volterra Equalizer in a Seven-Core Fiber. *Photonics*. 2023; 10(9):1017.
https://doi.org/10.3390/photonics10091017

**Chicago/Turabian Style**

Tian, Feng, Tianze Wu, Chao Yu, Chuxuan Wang, Mohai Yue, Ran Gao, Qi Zhang, Zhipei Li, Qinghua Tian, Fu Wang,
and et al. 2023. "Kramers–Kronig Transmission with a Crosstalk-Dependent Step Multiple-Input Multiple-Output Volterra Equalizer in a Seven-Core Fiber" *Photonics* 10, no. 9: 1017.
https://doi.org/10.3390/photonics10091017