# Adaptive Damping Log-Domain Message-Passing Algorithm for FTN-OTFS in V2X Communications
## 3 Adaptive Damping Log-Domain Message-Passing Algorithm
### An Overview of the Message-Passing Algorithm
For each variable node \(x[c]\) and each observation node \(y[e]\), the log-likelihood associated with the constellation point \(a_{k}\) is defined as  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\]0  
The normalized probability transmitted from the variable node \(x[c]\) to the observation node \(y[d]\) is determined using the LSE normalization technique, which ensures numerical stability and avoids computational issues related to underflow or overflow. The expression for the normalized log-probability is given as  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\]1  
The message passed from the variable node \(x[c]\) to its connected observation node \(y[d]\) is represented by a PMF vector, which is denoted as \(\mathbf{p}^{(i)}_{c,d}\), The update process for \(\mathbf{p}^{(i)}_{c,d}\) is defined as  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\]2  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\]3  
where \(\Delta^{(i)}\) represents the adaptive damping factor, while \(\eta^{(i)}\) serves as the convergence indicator. The value of \(\eta^{(i)}\) can be computed as  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\]4  
where \(\mathbb{I}(\cdot)\) is the indicator function, returning 1 when the condition holds and 0 otherwise; \(\gamma\) is a tolerance threshold for determining convergence confidence; and \(p^{(i)}_{c}(a_{j})\) denotes the posterior probability of symbol \(a_{j}\) at variable node \(x[c]\) in the \(i\) th iteration, which can be calculated as  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\]5  
If \(\eta^{(i)}>\eta^{(i-1)}\), the decision for the transmitted symbol is updated as follows:  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\]6  
In the proposed algorithm, the decisions for the transmitted symbols are updated only when the estimates obtained in the current iteration provide an improvement over those from the previous iteration. This ensures that the algorithm consistently refines its results and avoids unnecessary updates that do not enhance the quality of the symbol detection. The algorithm is designed to terminate when any of the following conditions is satisfied:  
1. Convergence is achieved. Specifically, the convergence indicator \(\eta^{(i)}=1\), signifying that the algorithm has fully converged to a stable solution where no further significant updates are necessary. 2. Significant degradation in performance. Specifically, the convergence indicator \(\eta^{(i)}<\eta^{(j)}-\epsilon\), where \(\eta^{(j)}\) is the maximum convergence indicator observed across all previous iterations \(j=1,\ldots,i-1\). Here, \(\epsilon\) is a predefined threshold, set to 0.2. This condition ensures that the algorithm halts if the current iteration shows a substantial decline in performance compared to the best result achieved so far, preventing unnecessary computations and avoiding divergence. 3. Maximum iteration limit. Specifically, the algorithm reaches the predefined maximum number of iterations, which acts as a safeguard to ensure computational efficiency and prevent the algorithm from running indefinitely. This condition is particularly useful in scenarios where convergence is slow or cannot be achieved due to challenging system conditions.