# Polar-Coded Transmission over 7.8-km Terrestrial Free-Space Optical Links

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## Abstract

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## 1. Introduction

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- Long-distance transmission of polar and LDPC codes over a 7.8-km terrestrial FSO link, demonstrating that the characteristics, especially block error performance, of polar codes are better than those of the regular LDPC codes;
- -
- Investigating factors that cause differences in the characteristics of polar and LDPC codes in FSO communications;
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- Comparing the performance of LDPC codes used in the recent standardization of the fifth-generation mobile communications system (5G) numerically, and clarifying the effectiveness of polar code transmission.

## 2. CA-SCLD Polar-Coded FSO Transmission System

## 3. Experimental Setup

#### 3.1. Tokyo FSO Testbed

#### 3.2. Error-Correcting Codes

_{b}/N

_{0}= 2 dB. The relationship between these measured execution times approximately follows that of the theoretical values, corroborating the superiority of the CA-SCLD polar code. Here, the relatively long LDPC transmission execution time was due to the program not being particularly well-optimized.

#### 3.3. Data Frame Format

## 4. Experimental Results

#### 4.1. Lena Image

#### 4.2. Error-correcting Performance

#### 4.3. Effect of Interleaving

#### 4.4. Comparison with Numerical Results

## 5. Discussion on Experimental Results

#### 5.1. Different BER and BLER Tendency of Polar and LDPC Codes

#### 5.2. Comparison with 5G LDPC Codes

_{b}/N

_{0}as the characteristics of channel codes are generally evaluated in a Gaussian noise channel and its relative characteristics are preserved in a fading environment. Table 4 shows the simulation parameters of the compared codes, where the basic parameters match those used for the experiments in this study (shown in Table 1). The decoding method for 5G NR LDPC was the offset min-sum algorithm with a maximum iteration number of 20. This decoding method and the iteration number are commonly used in 5G systems and it also considers the computational complexity and delay time of decoding [36,37,38]. In addition, we investigated the performance of the 5G NR LDPC code based on sum-product decoding with 50 iterations, which was the best decoding method.

_{b}/N

_{0}. The 5G NR LDPC code based on the sum-product algorithm exhibited the best performance among these codes. The relationship between CA-SCLD polar and regular LDPC codes is approximately the same in Figure 6a and Figure 8a. The former is better than the latter in the higher SNR region, and these curves cross a certain threshold. However, the threshold SNR was much lower than that shown in previous figures. This can be attributed to differences in the channel model. The AWGN channel is shown in Figure 10, whereas the gamma–gamma fading channel is shown in Figure 6 and Figure 8. Furthermore, the gamma–gamma distributions in Figure 8 are far from ideal because they were obtained through a real-field experiment. These factors deteriorate the performance of the CA-SCLD polar code more than that of the regular LDPC code; hence, the crossover point moves toward the higher-SNR region.

^{−4}. However, it was found that the polar code has better characteristics, which is consistent with the results of a previous study [37]. In addition, [37,39] showed that implementation of the 5G NR LDPC code is much more complex that of CA-SCLD polar code. The LDPC code used in CCSDS [30] utilizes an irregular check matrix generated using a protograph. The structure of this check matrix is similar to that of the 5G NR LDPC; therefore, the BER characteristics were considered comparable. The application of nonbinary LDPC has been discussed; however, it is not currently implemented due to the increased decoding complexity compared to the gain obtained [40]. It can be concluded that the application of polar code is effective compared to recent practical LDPC codes.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Distribution of LLR ${L}_{\mathrm{A}\mathrm{W}\mathrm{G}\mathrm{N}}({y}_{i},{h}_{i})$ for OOK scheme with and without channel equalizer. In the simulation, the signal-to-noise ratio (SNR) = 10.0 dB and the intensity variation follows a gamma–gamma distribution with scintillation index of 0.2. For the distribution without equalization, we set ${h}_{i}=1$ for all bits.

**Figure 5.**Decoding result of Lena image at SNR = 6.0 dB. Note that each pixel of the Lenna image has three values for red, green, and blue (RGB).

**Figure 6.**(

**a**) BER and (

**b**) BLER of experimental data. The solid and dotted lines are for the cases with and without channel equalization, respectively.

**Figure 7.**BLER performance with block interleaving (dashed lines) and without block interleaving (solid lines). The solid lines are the same ones shown in Figure 6b.

**Figure 8.**Performance comparison of experiment (solid lines) and simulation results (dashed lines): (

**a**) BER and (

**b**) BLER. BLER performance for experimental result is the same as shown in Figure 7.

CA-SCLD Polar | SCLD Polar | Regular LDPC | |
---|---|---|---|

Code length $N$ | 2048 | ||

Code rate $R$ | 0.5 | ||

CRC length $k$ | 24 | - | - |

List size ${L}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ | 32 | - | |

Column and row weights $({d}_{v},{d}_{c})$ | - | - | (6,3) |

Decoding iterations ${I}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ | - | - | 50 |

CA-SCLD Polar | SCLD Polar | Regular LDPC | |
---|---|---|---|

Theoretical value [a.u.] | 589,056 | 573,440 | 793,600 |

Normalized by CA-SCLD polar | 1 | 0.973 | 1.35 |

Execution time [ms] | 6064 | 5985 | 13,687 |

Error-Correcting Code | Computational Cost for Decoding |
---|---|

CA-SCLD polar code [26,27] | ${L}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(N\left(1+{\mathrm{log}}_{2}N\right)+K\left(2{\mathrm{log}}_{2}\left({L}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)+4\right)-k\right)$ |

SCLD polar code [26,27] | ${L}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(N\left(1+{\mathrm{log}}_{2}N\right)+K\left(2{\mathrm{log}}_{2}\left({L}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)+3\right)\right)$ |

Regular LDPC code [28,29] | ${I}_{\mathrm{m}\mathrm{a}\mathrm{x}}(2{d}_{v}N+\left(2{d}_{c}+1\right)\left(N-K\right))$ |

5G NR LDPC | Regular LDPC | ||
---|---|---|---|

Code length | 2048 | ||

Code rate | 0.5 | ||

Column and row weights | Variable, base-graph 2 | (6,3) | |

Decoding algorithm | Offset min-sum | Sum-product | Sum-product |

Decoding iterations | 20 | 50 | 50 |

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**MDPI and ACS Style**

Fujita, S.; Okamoto, E.; Takenaka, H.; Endo, H.; Fujiwara, M.; Kitamura, M.; Shimizu, R.; Sasaki, M.; Toyoshima, M.
Polar-Coded Transmission over 7.8-km Terrestrial Free-Space Optical Links. *Photonics* **2023**, *10*, 462.
https://doi.org/10.3390/photonics10040462

**AMA Style**

Fujita S, Okamoto E, Takenaka H, Endo H, Fujiwara M, Kitamura M, Shimizu R, Sasaki M, Toyoshima M.
Polar-Coded Transmission over 7.8-km Terrestrial Free-Space Optical Links. *Photonics*. 2023; 10(4):462.
https://doi.org/10.3390/photonics10040462

**Chicago/Turabian Style**

Fujita, Shingo, Eiji Okamoto, Hideki Takenaka, Hiroyuki Endo, Mikio Fujiwara, Mitsuo Kitamura, Ryosuke Shimizu, Masahide Sasaki, and Morio Toyoshima.
2023. "Polar-Coded Transmission over 7.8-km Terrestrial Free-Space Optical Links" *Photonics* 10, no. 4: 462.
https://doi.org/10.3390/photonics10040462