## 3 FTN Signal Processing Based on the UEP Algorithm
### UEP Setting Algorithm for the HSS-Based LDPC Decoding Method  
For the UEP-based FTN signals to be generated, it is necessary to divide encoded bits using the priority algorithm and to allot different FTN interference ratios accordingly. In this process, the priority algorithm determines the importance of each bit. In this section, we explain the HSS-based LDPC decoding algorithm briefly and how to apply it to the UEP setting algorithm. The \(\rm(N,K)\) LDPC code is  
Figure 3: The transceiver structure of the FTN model based on the UEP algorithm: (**a**) Transmitter; (**b**) receiver. defined based on a sparse parity-check matrix, H, where N bit nodes and \((\text{N}-\text{K})\) check nodes are connected by edges. In the H matrix, the bit nodes/check nodes are mapped to the columns/rows, where one element in a row is connected to another in a column by an edge. Bit nodes consist of information and parity bit nodes, while information bit nodes are connected by edges with check nodes by features of the matrix, H. The LDPC codes are shown in the small example (6, 3), shown in Figure 4.  
Figure 4b is a Tanner graph, which can be deduced from the parity-check matrix of Figure 4a. In Figure 4b, each bit is represented by a bit node (shown as a circle), and each check is represented by a check node (shown as a square). An edge exists between the bit node, \(j\), and the check node, \(i\), and only if \(\mathbf{H}(\mathbf{i},\mathbf{j})=\mathbf{1}\). The LDPC decoder performs an iterative exchange between the bit nodes and the check nodes, until the maximum number of iterations has been reached. The decoding algorithm produces the probabilities of each bit being \"0\" or \"1\" for every iteration. These probabilities are used in the log likelihood ratio (LLR) and are computed in the following way:  
\[\mathbf{LLR}_{i}(\mathbf{bit}_{i})=\mathbf{log}\Bigg{(}\frac{\mathbf{P}(\mathbf{bit}_{i}=\mathbf{0})}{ \mathbf{P}(\mathbf{bit}_{i}=\mathbf{1})}\Bigg{)}. \tag{7}\]  
The Tanner graph of the (6, 3) LDPC code, as shown in Figure 5, is an example to explain the HSS decoding process. Figure 4: Representation of an LDPC code using (a) a parity-check matrix \(\mathbf{H}\) and (b) Tanner graph. Figure 5: _Cont._  
Each bit node updates all edges of \(\mathbf{u}_{ij}\), which are linked from the \(\mathbf{i}^{th}\) check node to the \(j^{th}\) bit node, \(S_{j}\). The final decision on a bit is taken at the end of the decoding process by computing the sign of the local a posteriori log density ratio:  
\[S_{j}=LLR_{j}+\sum\
olimits_{k=\mathbf{i}}^{d_{\mathbf{v}}}u_{ij}, \tag{8}\]  
where \(\mathbf{d}_{\mathbf{v}}\) means the number of edges linked to the \(j^{th}\) bit node. In Equation (8), as shown in Figure 5a, if \(j\) is 1, \(i\) is equal to 1 and 3. Depending on whether \(S_{j}\) is positive or negative, the bit, \(j\), is considered as a \"0\" or \"1\". As shown in Figure 5, the process of updating a check node and bit node is defined from Step 1 to Step 3. Computing the probability of edge, \(\mathbf{u}_{i,j}\), which is linked from the \(\mathbf{i}^{th}\) check node to the \(j^{th}\) bit node for a degree, \(\mathbf{d}_{\mathbf{c}}\), is as follows Equation (9):  
\[\mathbf{u}_{i,j}=\oplus_{k=\mathbf{1},\mathbf{n}\
eq j}^{\mathbf{d}_{\mathbf{c}}}\mathbf{v}_{i,\mathbf{n}},\mathbf{n}\in\mathbf{CN}_{i}, \tag{9}\]  
where \(\mathbf{d}_{\mathbf{c}}\) denotes the number of edges linked to the check node. \(\mathbf{v}_{i,\mathbf{n}}\) is the probability of the edge, which is linked from the \(j^{th}\) bit node to the \(\mathbf{i}^{th}\) check node, and \(\mathbf{n}\) is the index of the bit node group, which is linked to the check node, \(\mathbf{CN}_{i}\).