# 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  
The input-output relationship in the system is expressed as in (15). The elements of the received signal vector \(\mathbf{y}_{\text{DD}}\) are denoted by \(\{y[d]\mid 1\leq d\leq NM\}\). Similarly, the elements of the channel matrix \(\mathbf{H}_{\text{DD}}\) are denoted by \(\{H[d,c]\mid 1\leq d,c\leq NM\}\), where \(H[d,c]\) corresponds to the channel response between the \(c\)th transmitted DD domain symbol and the \(d\)th received DD domain symbol. The transmitted signal vector \(\mathbf{x}_{\text{DD}}\) is represented by its elements as \(\{x[c]\mid 1\leq c\leq NM\}\). Furthermore, the noise vector \(\boldsymbol{\eta}_{\text{DD}}\) is denoted by its elements as \(\{\eta[d]\mid 1\leq d\leq NM\}\). The channel matrix \(\mathbf{H}_{\text{DD}}\) is sparse, reflecting the fact that only a limited number of propagation paths contribute to the received signal at each DD domain index. Specifically, \(\mathcal{I}(d)\) and \(\mathcal{J}(c)\) are defined as the sets of indices corresponding to the non-zero entries in the \(d\)th row and \(c\)th column of \(\mathbf{H}_{\text{DD}}\), respectively. Both sets have a cardinality of \(P\), indicating that there are \(P\) significant propagation paths for each transmitted or received symbol. This sparsity significantly reduces the computational complexity required for processing, as only a subset of the matrix elements needs to be considered. Given this sparse structure, the system model described in (15) can be interpreted as a factor graph. The factor graph consists of \(NM\) observation nodes, corresponding to the elements of the received signal vector \(\mathbf{y}_{\text{DD}}\), and \(NM\) variable nodes, corresponding to the elements of the transmitted signal vector \(\mathbf{x}_{\text{DD}}\). In this graph, each observation node \(y[d]\) is directly connected to the variable nodes \(\{x[c]\mid c\in\mathcal{I}(d)\}\), which represent the transmitted symbols contributing to the received signal at index \(d\). Similarly, each variable node \(x[c]\) is connected to the observation nodes \(\{y[d]\mid d\in\mathcal{J}(c)\}\), which represent the received signals that are influenced by the transmitted symbol at the index \(c\). This factor graph representation provides a structured and intuitive way to model the input-output relationship of the system, capturing both the sparsity of the channel and the interactions between transmitted and received symbols. It is particularly useful for designing and analyzing message-passing algorithms, as it directly maps the system model onto a graphical framework that facilitates iterative processing and efficient computation. From (15), the rule for joint MAP detection used to estimate the transmitted signals is formulated as  
\[\widehat{\mathbf{x}}=\operatorname*{arg\,max}_{\mathbf{x}_{\text{DD}}\in A^{ NM\times 1}}\Pr(\mathbf{x}_{\text{DD}}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{ \text{DD}}). \tag{16}\]  
Since the computational complexity of MAP detection increases exponentially with \(NM\), performing full MAP detection for large-scale systems becomes impractical. To address this, we consider performing MAP detection on a symbol-by-symbol basis, an approach that significantly reduces complexity while maintaining reasonable detection performance. The estimation symbol \(\widehat{x}[c]\) can be expressed as follows:  
\[\widehat{x}[c] =\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\Pr\big{(}x[c]=a_{ j}\mid\mathbf{y}_{\text{DD}},\mathbf{H}_{\text{DD}}\big{)} \tag{17}\] \[=\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\frac{1}{Q}\Pr \big{(}\mathbf{y}_{\text{DD}}\mid x[c]=a_{j},\mathbf{H}_{\text{DD}}\big{)}\] \[\approx\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\prod\Pr \big{(}y[d]\mid x[c]=a_{j},\mathbf{H}_{\text{DD}}\big{)}.\]  
The MAP detection in (17) can be effectively realized using the message-passing algorithm. This algorithm operates iteratively, leveraging the structure of the factor graph to exchange information between observation nodes and variable nodes.