# Sum of the Magnitude for Hard Decision Decoding Algorithm Based on Loop Update Detection
## 5 Simulation Results and Statistical Analysis
### Comparison of Algorithm Performance and Average Iteration Numbers  
The performance comparison between Code 1 and Code 2 in five different decoding algorithms under the optimal parameters is shown in Figures 5 and 6, respectively. With an increase in SNR, the coding gain of MSMWSF algorithm is gradually increasing compared with other existing algorithms. At a low SNR, the performance of MSMWSF algorithm is almost the same as other algorithms. According to the simulation diagram of reference [34; 35; 36], a higher number of binary numbers indicates a poorer performance of the WSF algorithm. From Figures 5 and 6, we can see that the performance of Code 1 is better than that of Code 2, which proves the effectiveness of the algorithm proposed in this paper.  
Figure 4: Code 2 weighted factor test.  
Figure 5: Comparison of five algorithms under Code 1.  
Figures 7 and 8 show the performance of Code 1 and Code 2 under the SMWSF algorithm and MSMWSF algorithm, respectively, with the optimal parameters of the weighting factors after introducing the loop update detection algorithm. As seen from Figures 7 and 8, the LUDMSMWSF algorithm improves the decoding performance compared to the MSMWSF algorithm, accelerating the convergence speed. Compared with the WSF algorithm, a coding gain of 2.2 dB is obtained in the case of Code 1 and a coding gain of 2.35 dB in the case of Code 2. Under the same decoding algorithm, the coding gain of about 0.8-1.1 dB can be obtained after introducing the proposed loop update detection algorithm. At the same time, the LUDMSMWSF algorithm is about 0.85-1.05 dB away from FFT-BP decoding at BER of \(10^{-5}\) with a tremendous reduction of computational requirement. In view of the results on Figures 7 and 8, we argue that the proposed symbol flipping algorithm offer the tradeoff points between performance and computational cost.  
We set the SNR to be 3.5 dB with the simulation of 10,000 frames under the condition of Code 1. We then combined the statistical five algorithms for decoding failure frames, as shown in Table 2. Table 2 shows that the WSF algorithm has nearly 99% frame failure. Two decoding algorithms are proposed in this paper. The failure of the frame is about 62% and 42%, which is less than traditional algorithms. It can also be seen that the algorithm proposed in this paper does not increase the complexity of the implementation, but has been reduced to a certain extent.  
The total computation required for decoding this LDPC codewith three various algorithms at 4.5 dB is shown in Table 3. From Table 3, the FFT-BP algorithm is nearly six times the computational requirement of the MSMWSF algorithm only in real addition. Therefore, we can prove that the computational requirement of MSMWSF algorithm is much lower than FFT-BP with no real multiplication and division. The MSMWSF algorithm has the lowest complexity and does not need to consume hardware resources and multiplication and division operations in software overhead.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**SF Algorithm** & **Decoding Failure Frames** \\ \hline WSF algorithm & 9915 \\ MWSF algorithm & 9840 \\ IMWSF algorithm & 6544 \\ SMWSF algorithm & 6200 \\ MSMWSF algorithm & 4184 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Five algorithms for decoding failure frames.  
Figure 10: The average number of addition operations of five algorithms under Code 1.