# Dual Threshold Self-Corrected Minimum Sum Algorithm for 5G LDPC Decoders
## 4 Performance Analysis and Comparisons  
### Experiments and Simulations  
Some experiments are conducted at the code rates of 1/3 and 2/3 in the 5G LDPC coding scheme. The coded bits are transmitted by Binary Phase Shift Keying (BPSK) modulation with the Additive White Gaussian Noise (AWGN) channel on the Matlab platform. The algorithms of BP, MS, NMS, SCMS, DT-SCMS are simulated with the parameter settings in Table 2.  
The simulation results are shown in Figures 3 and 4.  
As can be seen from Figures 3 and 4, the DT-SCMS algorithm is superior to the SCMS algorithm both in error characteristics and convergence characteristics. Especially in the areas of high signal-to-noise ratio (SNR), the DT-SCMS algorithm performs more prominently than the SCMS algorithm in terms of convergence. The number of iterations of the SCMS algorithm and the DT-SCMS algorithm in different SNRs in Figure 4b are listed in Table 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **R = 1/3** & **R = 2/3** \\ \hline Code length (bits) & 4224 & 1584 \\ SNR (dB) & [\(-\)0.5:1.4] & [1.6:3.4] \\ Maximum number of iterations & 30 & 30 \\ Correction factor \(\alpha\) & 1.2 & 1.2 \\ Adjustment factor \(\theta 1\) & 0.1 & 0.125 \\ Adjustment factor \(\theta 2\) & \(-\)1.2 & \(-\)1.125 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameters.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**SNR (dB)** & **2.8** & **3** & **3.1** & **3.2** & **3.3** & **3.4** \\ \hline SCMS [31] & 3311 & 2811 & 2607 & 2467 & 2246 & 2214 \\ DT-SCMS & 2748 & 2311 & 2124 & 2016 & 1797 & 1761 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Iterations of SCMS and DT-SCMS under different SNRs (R = 2/3).  
From Figure 3(b) and Table 3, it can be seen that, as the channel SNR increases, the number of iterations of LDPC decoding is gradually reduced. Obviously, the convergence characteristics of the DT-SCMS algorithm are better than those of the SCMS algorithm, especially when the SNR is high. This is because the dynamic judgment of the DT-SCMS algorithm on the erased messages is more effective in suppressing the unreliable messages passing, thus speeding up the convergence of decoding. According to Table 3, by calculation, when the code rate is 2/3, the DT-SCMS algorithm can reduce the number of iterations by up to 20.46% compared with the SCMS algorithm, and the average reduction proportion is 18.68%.  
Figure 3: Simulations of belief propagation (BP), minimum sum (MS), normalized minimum sum (NMS), self-corrected minimum sum (SCMS), DT-SCMS, when R=1/3.  
In the experiment with R = 2/3, several intermediate values of \(L^{(l-1)}\big{(}q_{ij}\big{)}\) are taken out from the process of iteration, and the values of \(th1\) and \(th2\) are calculated, as shown in Table 4.  
From Table 4, we can see that \(th1\) and \(th2\) change dynamically with the value of \(L^{(l-1)}\big{(}q_{ij}\big{)}\). In addition, experiments show that this dynamic adjustability improves the decoding performance of the proposed algorithm.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(L^{(l-1)}\big{(}q_{ij}\big{)}\) & \(-\)**1.1247** & **5.2952** & \(-\)**3.9980** & **8.1010** & **3.2369** \\ \hline \(th1\) & \(-\)0.1406 & 0.6619 & \(-\)0.4998 & 1.0126 & 0.4046 \\ \(th2\) & 1.2653 & \(-\)5.9571 & 4.4977 & \(-\)9.1136 & \(-\)3.6415 \\ \hline \end{tabular}
\end{table}
Table 4: Some values of \(th1\) and \(th2\).  
Figure 4: Simulations of BP, MS, NMS, SCMS, DT-SCMS, when R=2/3.