## 4 Hybrid Message Passing Detector
Given the inherent sparsity of the EC matrix \(\mathbf{H}\), the interconnections between variables and FCs remain notably finite. Owing to the impact of the fractional Doppler, it is discernible that the quantity of VNs linked to each FN significantly surpasses the available path count. As a result, the detection procedure for fractional Doppler systems attains a notably elevated level of complexity. We introduce the concept of \(\mathcal{I}_{d^{\prime}}^{i}\), which represents the non-zero elements in \(\mathcal{R}_{d}\) that are related to \(h_{i}\). Similarly, we define \(\mathbf{x}_{d}^{i}\) as the VNs from \(\mathbf{x}_{d}\) that are associated with \(h_{i}\). There are \(2N_{T_{i}}+1\) elements for both \(\mathcal{I}_{d}^{i}\) and \(\mathbf{x}_{d}^{i}\). To mitigate the complexity of OTFS symbol detection, we select the element whose absolute value is the highest from \(\mathcal{I}_{d}^{i}\) as the main path, as it affects symbol detection most. The main paths within \(\mathcal{R}_{d}\) are denoted as a set \(\mathcal{I}_{d}\), and the remaining elements of \(\mathcal{R}_{d}\) as \(\overline{\mathcal{I}_{d}}\). The non-approximate MP algorithm is employed for \(\mathcal{I}_{d}\), while a GA-based MP algorithm is used for \(\overline{\mathcal{I}_{d}}\) in the context of symbol detection. Consequently, the collections of VNs are referred to as \(\bar{\mathbf{x}}_{d}\) and \(\bar{\mathbf{x}}_{d}\), respectively. The partial interconnections between the VNs and FNs are delineated in Figure 4.  
The detection algorithm is utilized to realize the factor graph model for calculating the marginal distribution. We begin by considering \(\psi_{e/d}(x)=\mu_{x\to f_{e/d}}(x)\) and \(\phi_{e/d}(x)=\mu_{f_{d}\to x}(x)\) as the iteration's value from node \(x[c]\) to node \(f_{d}\) and from node \(f_{d}\) to node \(x[c]\), respectively. Here, define \(\tilde{\phi}_{e/d}(x)=\bar{\eta}_{f_{d}\to x[c]}(x[c])\) and \(\tilde{\psi}_{e/d}(x)=\bar{\eta}_{x[c]\to f_{d}}(x[c])\) as the previous iteration's value of \(\mu_{f_{d}\to x[c]}(x[c])\) and \(\mu_{x[c]\to f_{d}}(x[c])\), respectively. Specifically, the message transmitted towards a VN \(x_{c}\) from an FN \(f_{d}\) and the message passing procedure from \(x[c]\), a VN, to \(f_{d}\), an associated factor node, can be established as follows:  
Figure 4: The connections between function and VNs within the local context. Figure 3: Factor graph depiction of the analyzed OTFS system. \[ \begin{cases}\phi_{e/d}(x[c])=\sum_{\mathbf{x}_{d}\sim x[c]}p(y[d]\mid\mathbf{x}_{ d})\prod_{c\in\mathcal{R}_{d}\sim c}\psi_{e/d}(x[e]).\\ \psi_{e/d}(x[c])=p(x[c])\prod_{c\in\mathcal{C}_{c}\sim d}\phi_{e/d}(x[c]).\end{cases}  \tag{32}\]  
In order to enhance the performance, we incorporate a \"damping factor\", denoted as \(\Delta\), where \(\Delta\) falls within the range of (0,1), during the message update process \(\mu_{f_{d}\sim x[c]}(x[c])\). Consequently, Equation (32) can be reformulated as:  
\[ \begin{split}\phi(x[c])=&\sum_{\mathbf{x}_{d}\sim x[c]}p (y[d]\mid\mathbf{x}_{d})\prod_{c\in\mathcal{R}_{d}\sim c}\psi(x[e])\\ &+\Delta\cdot\hat{\phi}(x).\end{split}  \tag{33}\]  
We can see that the damping factor has obvious effects on the detection performance in the simulation outcomes. Likewise, the message passing procedure from \(x[c]\), a VN, to \(f_{d}\), an associated factor node, is established as follows:  
Being estimated through an approximate distribution, \(b(x[c])\), which is referred to as the \"belief\", \(x[c]\) can be reformulated as:  
\[b(x[c])=p(x[c])\prod_{e\in\mathcal{C}_{c}}\phi_{e/d}(x[c]).\] (  
Following this, we employ the message \(\mu_{f_{d}\to x[c]}(x[c])\) to adapt the message \(\mu_{x[c]\to f_{d}}(x[c])\), \(a=y[d]-\sum_{c\in\mathcal{\bar{T}}_{d}\sim c}H[d,e]x[e]-H[d,c]x[c]\). Consequently, we establish the belief \(b(x[c])\).