# 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
In the message-passing algorithm, each observation node \(y[d]\) transmits to the connected variable nodes \(\{x[c]\mid c\in\mathcal{I}(d)\}\) a message consisting of the mean and variance of the Gaussian interference term. These messages encapsulate statistical information about the received signal and interference; this information is crucial for refining the estimates of the transmitted symbols. Conversely, each variable node \(x[c]\) exchanges information with its connected observation nodes \(y[d]\mid d\in\mathcal{J}(c)\). The message transmitted from a variable node takes the form of a probability mass function (PMF) denoted as \(\mathbf{P}_{c,d}=\{p_{c,d}(a_{j})\mid a_{j}\in\mathbb{A}\}\). Here, \(\mathbf{P}_{c,d}\) represents the likelihood of the transmitted symbol \(x[c]\) being equal to each candidate symbol \(a_{j}\) in the constellation set \(\mathbb{A}\). This probabilistic representation enables the algorithm to incorporate uncertainty and refine the detection process iteratively. The entire message-passing process, which involves the exchange of messages between variable nodes and observation nodes, is visually represented in Figure 2. In this figure, the connections between nodes are shown as edges, illustrating the flow of information. For each observation node \(y[d]\), the set of edges linking it to variable nodes is denoted by \(e_{1},e_{2},\cdots,e_{S}=\mathcal{I}(d)\), where \(S\) represents the number of connected variable nodes. Similarly, for each variable node \(x[c]\), the edges connecting it to observation nodes are represented by \(e_{1},e_{2},\cdots,e_{S}=\mathcal{J}(c)\). This iterative exchange of messages between nodes ensures that the algorithm systematically updates its estimates of the transmitted symbols. The use of the MP algorithm is particularly advantageous in sparse systems in which the factor graph has a limited number of connections per node. This sparsity reduces the computational complexity, making the algorithm both efficient and scalable for large-scale systems. Moreover, by exploiting the statistical properties of the messages, the MP algorithm achieves robust performance, even in challenging scenarios with noise and interference. The message-passing algorithm obtains the final detected signal through iterative computation. Assume that in the \(i\)th iteration, the messages passed from observation nodes to variable nodes are the mean \(\mu_{d,c}^{i}\) and variance \(\left(\sigma_{d,c}^{i}\right)^{2}\) of the interference term \(\zeta_{d,c}^{i}\). This interference term is approximated as a Gaussian random variable, as described by [13]  
\[y[d]=x[c]H[d,c]+\underbrace{\sum_{e\in\mathcal{I}(d),e\
eq c}x[e]H[d,e]+z[d]} _{\zeta_{d,c}^{(i)}}, \tag{18}\]  
The mean \(\mu_{d,c}^{i}\) and variance \(\left(\sigma_{d,c}^{i}\right)^{2}\) of \(\zeta_{d,c}^{i}\) are calculated as  
\[\mu_{d,c}^{(i)} =\sum_{e\in\mathcal{I}(d),e\
eq c}\sum_{j=1}^{Q}p_{e,d}^{(i-1)} \big{(}a_{j}\big{)}a_{j}H[d,e], \tag{19}\] \[\left(\sigma_{d,c}^{(i)}\right)^{2} =\sum_{e\in\mathcal{I}(d),e\
eq c}\left(\sum_{j=1}^{Q}p_{e,d}^{(i- 1)}\big{(}a_{j}\big{)}\big{|}a_{j}\big{|}^{2}|H[d,e]|^{2}\right.\] \[\left.-\left|\sum_{j=1}^{Q}p_{e,d}^{(i-1)}\big{(}a_{j}\big{)}a_{j} H[d,e]\right|^{2}\right)+\sigma^{2}.\]  
Message Updating from the Variable Node to the Observation Node via LSE Normalization and Adaptive Damping  
The ADL-MP algorithm is detailed in Algorithm 1, wherein a step-by-step procedure for its implementation is presented. This algorithm operates iteratively over variable nodes  
Figure 2: Messages in factor graph. and observation nodes within the factor graph structure to efficiently estimate transmitted symbols.