# Polar Code BP Decoding Optimization for Green 6G Satellite Communication: A Geometry Perspective
## 2 BP Decoding Algorithm for Polar Codes
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{E_{b}/N_{0}}\) & **ESBP-R** & **ESBP-D** \\ \hline
1 & 2.8\% & 4.3\% \\
1.5 & 11.0\% & 8.8\% \\
2 & 5.8\% & 9.9\% \\
2.5 & 4.7\% & 15.9\% \\
3 & 2.0\% & 7.6\% \\
3.5 & 2.0\% & 5.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The ratio of reduced iteration times compared to three existing early convergence algorithms. ## Appendix A Deduction of Equation (25)

For a binary random variable \(X\in\{0,1\}\), we can assume its probability distribution is controlled by the logit parameter \(\lambda\) as follows:

\[p(x;\lambda)=\frac{e^{\lambda x}}{1+e^{\lambda}}.\]

That is

- When \(x=1\), \(p(1;\lambda)=\frac{e^{\lambda}}{1+e^{\lambda}}\). - When \(x=0\), \(p(0;\lambda)=\frac{1}{1+e^{\lambda}}\). The KL divergence is defined as

\[D_{KL}(p(x;\lambda_{1})\|p(x;\lambda_{2}))=\sum_{x}p(x;\lambda_{1})\log\frac{p( x;\lambda_{1})}{p(x;\lambda_{2})}.\]

Substituting the probability mass function of the Bernoulli distribution

\[D_{KL}=p(1;\lambda_{1})\log\frac{p(1;\lambda_{1})}{p(1;\lambda_{2})}+p(0; \lambda_{1})\log\frac{p(0;\lambda_{1})}{p(0;\lambda_{2})}.\]

Expanding the calculation

\[=\frac{e^{\lambda_{1}}}{1+e^{\lambda_{1}}}\log\biggl{(}\frac{\frac{e^{ \lambda_{1}}}{1+e^{\lambda_{1}}}}{\frac{e^{\lambda_{2}}}{1+e^{\lambda_{2}}}} \biggr{)}+\frac{1}{1+e^{\lambda_{1}}}\log\biggl{(}\frac{\frac{1}{1+e^{ \lambda_{1}}}}{\frac{1}{1+e^{\lambda_{2}}}}\biggr{)}.\]

Simplifying

\[=\frac{e^{\lambda_{1}}}{1+e^{\lambda_{1}}}\log\biggl{(}e^{(\lambda_{1}- \lambda_{2})}\frac{1+e^{\lambda_{2}}}{1+e^{\lambda_{1}}}\biggr{)}+\frac{1}{1+ e^{\lambda_{1}}}\log\biggl{(}\frac{1+e^{\lambda_{2}}}{1+e^{\lambda_{1}}} \biggr{)}.\]

Decomposing the logarithmic terms

\[=\frac{e^{\lambda_{1}}}{1+e^{\lambda_{1}}}\biggl{[}(\lambda_{1}-\lambda_{2})+ \log\frac{1+e^{\lambda_{2}}}{1+e^{\lambda_{1}}}\biggr{]}+\frac{1}{1+e^{\lambda _{1}}}\log\frac{1+e^{\lambda_{2}}}{1+e^{\lambda_{1}}}.\]

Extracting the common logarithmic term

\[=(\lambda_{1}-\lambda_{2})\frac{e^{\lambda_{1}}}{1+e^{\lambda_{1}}}+\log \frac{1+e^{\lambda_{2}}}{1+e^{\lambda_{1}}}\biggl{(}\frac{e^{\lambda_{1}}}{1+ e^{\lambda_{1}}}+\frac{1}{1+e^{\lambda_{1}}}\biggr{)}.\]

Since the sum of the denominator and numerator equals 1, taking the base as \(e\),we obtain the final expression:

\[D_{KL}=(\lambda_{1}-\lambda_{2})\frac{e^{\lambda_{1}}}{1+e^{\lambda_{1}}}-\ln \biggl{(}\frac{1+e^{\lambda_{1}}}{1+e^{\lambda_{2}}}\biggr{)}.\]

## References

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* [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON] Early stopping of BP polar decoding based on parity-check sums. In Proceedings of the 2022 IEEE 95 th Vehicular Technology Conference:(VTC2022-Spring), Helsinki, Finland, 19-22 June 2022; pp. 1-5. [CrossRef]
* [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON]; [PERSON] Efficient early termination schemes for belief-propagation decoding of polar codes. In Proceedings of the 2015 IEEE 11 th International Conference on ASIC (ASICON), Chengdu, China, 3-6 November 2015; pp. 1-4.