# A Universal Low-Complexity Demapping Algorithm for Non-Uniform Constellations
## 4 Proposed Low-Complexity Demapping Algorithm
### Create the Subsets  
Taking 2D-NUC as an example, and assume that the received symbol is in the first quadrant, the creation process of the subsets is as described in Algorithm 1. Since 2D-NUC is quadrant symmetric, the subsets can be derived from the subsets of the first quadrant when the received symbol is in other quadrants. To facilitate the understanding of the creation process of the subsets, we take 2D-256 NUC designed for CR = 6/15 as an example. Assuming that the most significant bit of 2D-256 NUC is \(b_{7}\) and the least significant bit is \(b_{0}\). Then, \(b_{7}\) and \(b_{6}\) determine the quadrant where the constellation point is located, and \(b_{5}\) to \(b_{0}\) determine the position of the constellation point in a specific quadrant. The subsets creation process of \(b_{7}\) and \(b_{6}\) is similar. Taking \(b_{6}\) as an example, the constellation points contained in \(\chi^{0}_{6}\) and \(\chi^{1}_{6}\) of the Max-Log-MAP algorithm are shown in Figure 3a. Where the red circles represent the constellation points contained in \(\chi^{0}_{6}\), and the blue circles represent the constellation points contained in \(\chi^{1}_{6}\). It can be seen that the constellation points contained in \(\chi^{0}_{6}\) are all located in the first and fourth quadrants, and the constellation points contained in \(\chi^{1}_{6}\) are all located in the second and third quadrants. When the received symbol is in the first quadrant, it is obvious that the constellation points in the fourth quadrant in \(\chi^{0}_{6}\) can never have the minimum ED with the received symbol. Therefore, only the constellation points in the first quadrant in \(\chi^{0}_{6}\) need to be retained and then obtain the subset \(S^{0}_{6}\). Since \(\chi^{1}_{6}\) does not contain the constellation points in the first quadrant, it is necessary to keep the constellation points closest to the boundary of the first quadrant in \(\chi^{1}_{6}\), and then obtain the subset \(S^{1}_{6}\). The constellation points contained in \(S^{0}_{6}\) and \(S^{1}_{6}\) are shown in Figure 3b. Figure 3: The subsets of ED comparison to calculate LLR(\(b_{6}\)) when the received symbol is in the first quadrant. (**a**) Subsets of Max-Log-MAP algorithm; (**b**) Subsets obtained after some constellation points are removed. The subsets creation process of \(b_{5}\) to \(b_{0}\) is simpler. Since both \(\chi_{i}^{0}\) and \(\chi_{i}^{1}\)(\(i\) = 0, 1,..., 5) contain constellation points in the first quadrant, the subsets \(S_{i}^{0}\) and \(S_{i}^{1}\) can be obtained by retaining the constellation points in the first quadrant in \(\chi_{i}^{0}\) and \(\chi_{i}^{1}\) respectively. Taking \(b_{3}\) as an example, the constellation points contained in \(\chi_{3}^{0}\) and \(\chi_{3}^{1}\) are shown in Figure 4a, and the constellation points contained in \(S_{3}^{0}\) and \(S_{3}^{1}\) are shown in Figure 4b. Calculate the union of \(S_{i}^{k}\), i.e., \(S=U_{i=0}^{7}S_{i}^{k}\), \((k=0,\ 1)\). Then, \(S\) is the subset used for ED calculation when the received symbol is in the first quadrant, as shown in Figure 5a. The constellation points contained in \(S\) are represented by green circles. To further reduce the number of constellation points contained in \(S\) and \(S_{i}^{k}\), the condensation characteristic of NUC can be used. A constellation cluster containing several constellation points can be regarded as a virtual point, and its coordinate is set to the mean value of these constellation points. Using virtual points to replace the corresponding constellation points in \(S\) and \(S_{i}^{k}\), the reduced subset \(C\) used for ED calculation and the reduced subsets \(C_{i}^{k}\) used for ED comparison are obtained. The subset \(C\) is shown in Figure 5b. The above is the process of creating the subsets for 2D-NUC. Since 1D-NUC can be decomposed into two independent NU-PAMs, the process of creating its subsets is simpler.