# A Universal Low-Complexity Demapping Algorithm for Non-Uniform Constellations
## 5 Performance Evaluation
### Complexity Comparison  
We compare the complexity of three algorithms from two aspects, i.e., the number of one-dimensional EDs to be calculated and the number of EDs to be compared when calculating the LLRs of each received symbol. Table 1 shows the complexity comparison when demapping 2D-256 NUC. When CR = 2/15, 2D-256 NUC becomes 2D-16 NUC after condensation characteristic is used, so the complexity of QCSR and SCSR algorithms is greatly reduced compared with the Max-Log-MAP algorithm. Moreover, the subsets of the QCSR algorithm are obtained based on statistical results, and the lower the SNR, the more constellation points are contained in the subsets. Therefore, when CR = 2/15, the complexity of the SCSR algorithm is lower than that of the QCSR algorithm. As CR increases, the number of constellation points that can be condensed gradually decreases. When CR = 6/15 and 10/15, using the condensation characteristic only reduces the number of constellation points in NUC, but cannot turn it into a low-order constellation. Therefore, the complexity of QCSR and SCSR algorithms is increased, but the complexity of the SCSR algorithm is still lower than that of the QCSR algorithm. When CR is increased to 13/15, there are no condensed constellation points in 2D-256 NUC, and neither the QCSR algorithm nor the SCSR algorithm can use condensation characteristics. Further, because of the high SNR, the number of constellation points contained in the subsets of the QCSR algorithm is greatly reduced. Therefore, the number of EDs that the QCSR algorithm needs to calculate is slightly less than that of the SCSR algorithm. However, as described in Section 4, \(C_{i}^{k}\) is a subset of \(C\). Therefore, the number of EDs that the SCSR algorithm needs to compare is still far less than the QCSR algorithm. For all CRs, the number of EDs to be calculated for the SCSR algorithm is reduced by 67.58% to 96.48%, compared with the Max-Log-MAP algorithm. Then, the number of EDs to be compared is reduced by 74.07% to 98.97%. Table 2 shows the complexity comparison when demapping 1D-1024 NUC. Since the QCSR algorithm is not designed for 1D-NUC, only the Max-Log-MAP algorithm and SCSR algorithm are compared. 1D-NUC can be decomposed into two independent NU-PAMs, and only one-dimensional demapping is required. Therefore, the demapping complexity of 1D-NUC is greatly reduced compared with that of 2D-NUC. Similar to 2D-NUC, 1D-NUC becomes a low-order constellation in low CR, so the complexity of the SCSR algorithm is greatly reduced. With the CR increases, the number of constellation points that can be condensed gradually decreases, so the complexity of the SCSR algorithm gradually increases, but the complexity of the SCSR algorithm is always lower than that of the Max-Log-MAP algorithm. For all CRs, the number of EDs to be calculated for the SCSR algorithm is reduced by 46.88% to 90.63%, compared with the Max-Log-MAP algorithm. Then, the number of EDs to be compared is reduced by 49.38% to 96.88%. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**CR**} & **Demapping Algorithm** & **Number of One-Dimensional EDs to be** & **Number of EDs to be Compared** \\  & **Algorithm** & **Calculated and Reduction Percentage** & **and Reduction Percentage** \\ \hline All & Max-Log-MAP & 512 & 2048 \\  & QCSR & 32 (93.75\%) & 64 (96.88\%) \\
2/15 & SCSR & 16 (96.48\%) & 21 (98.97\%) \\  & QCSR & 132 (74.22\%) & 528 (74.22\%) \\  & SCSR & 84 (83.59\%) & 274 (86.62\%) \\
10/15 & QCSR & 160 (68.75\%) & 640 (68.75\%) \\  & SCSR & 150 (70.70\%) & 475 (76.81\%) \\
13/15 & QCSR & 160 (68.75\%) & 640 (68.75\%) \\  & SCSR & 166 (67.58\%) & 531 (74.07\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity comparison when demapping 2D-256 NUC.