## 3 FTN Signal Processing Based on the UEP Algorithm
### UEP Setting Algorithm for the HSS-Based LDPC Decoding Method
The \(\oplus\) operator is defined as follows:  
\[v_{i}\oplus v_{j}=sign(v_{i})\times sign\big{(}v_{j}\big{)}\times(\min\big{(} \big{|}v_{i}\big{|},\big{|}v_{j}\big{|}\big{)}-\text{offset}, \tag{10}\]  
\[\text{offset}=\log\left(\frac{1+e^{-(|v_{i}|+|v_{j}|)}}{1+e^{-(|v_{i}|-|v_{j}|) }}\right). \tag{11}\]  
Figure 5: Tanner graph and the HSS-based LDPC decoding process: (**a**) Step 1: first check of the node update; (**b**) Step 2: second check of the node update; (**c**) Step 3: third check of the node update. The edge, \(\mathbf{v}_{ij}\), is updated for a degree, \(\mathbf{d}_{\mathbf{v}}\), which can be expressed as follows (Equation (12)):  
Figure 4.0  
where \(\mathbf{d}_{\mathbf{v}}\) denotes the number of edges linked to the \(j^{th}\) bit node. \(\mathbf{n}\) is the index of the check node group, which is linked to the bit node, \(S_{j}\). In the conventional LDPC decoding process, the \(\mathbf{u}_{ij}\) coming out of the check node is updated using Equations (9)-(11). Then, all \(\mathbf{v}_{ij}\) messages coming out of the bit nodes linked to this check node are updated with Equation (12). In the \(\mathbf{l}^{th}\) iteration, the update of the bit node is based only on messages updated in the \((\mathbf{I-1})^{th}\) iteration. However, in the HSS-based LDPC decoding process, the already updated bit node and check node in the \(\mathbf{l}^{th}\) iteration move to a different bit and check node. As an example, in Figure 5, \(\mathbf{v}_{3,\mathbf{I}}\) was already updated two times, before being processed by the third check node. This higher update frequency per iteration allows the decoder to converge faster, because there are fewer iterations. The HSS-based LDPC decoding algorithm, proposed in [15; 16], is used to accelerate the LDPC decoding speed, because it needs fewer iterations to maintain the same performance. However, in this study, it was used to configure the priority algorithm. In deciding which group has the higher priority, the HSS-based LDPC decoding algorithm, which is explained above, was applied. Unlike conventional LDPC decoding processing, which calculates bit nodes' and check nodes' updating sequentially based only on messages updated in the previous iteration, in the HSS-based LDPC decoding process, the already updated bit node and check node in the same iteration move to a different bit and check node. Therefore, the bit node groups linked to the check nodes, which are arranged on the left, are assigned a higher priority. For example, as shown in Figure 5, the bit node group, which is linked to the first check node in Step 1, is the most important bit group. By classifying the group depending on the importance of the encoded bits, based on the decoding process, the UEP algorithm applies different FTN interference ratios according to the importance of the encoded bits, and improves the performance in the same overall interference to a higher degree than the conventional FTN method. Figure 6 shows the arrangement of the bit node and check node groups for the priority algorithm of the UEP setup. \(\mathbf{N}_{\mathbf{G}_{\mathbf{c}}}\) and \(\mathbf{N}_{\mathbf{G}_{\mathbf{b}}}\) mean the number of check node groups and bit node groups, respectively. In the half rate LDPC codes in the Digital Video Broadcasting-Satellite-Second Generation (DVB-S2) standard with N = 64,800, the number of check nodes is 32,400 and one check node group has 360 data, therefore, \(\mathbf{N}_{\mathbf{G}_{\mathbf{c}}}=90\). Additionally, the number of bit nodes is 64,800 and one bit node group has 360 data, therefore, \(\mathbf{N}_{\mathbf{G}_{\mathbf{b}}}=180\). In Figure 6, the left part shows the sequence of bit nodes, transmitted in the conventional way. For instance, without the UEP setting, signals are transmitted in the order from the 1 st bit node group to the 180 th bit node group (see the left part of Figure 6). However, we rearranged the bit node group linked to the check node group, which was calculated in advance (see the right part of Figure 6).