## 4 Hybrid Message Passing Detector  
We will formulate the probability model for the proposed symbol detection algorithm in this section. Subsequently, we provide an in-depth explanation of the hybrid MP algorithm. The DD domain input-output relationship takes on the vectorized form:  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{w}. \tag{28}\]  
In the context at hand, we have several key variables. Firstly, the received signal vector, denoted as \(\mathbf{y}\), belongs to the complex vector space \(\mathbb{C}^{M_{\mathrm{S}}N_{T}\times 1}\). Secondly, the EC matrix, denoted as \(\mathbf{H}\) (which is considered as known at the receiver when performing the symbol detection), belongs to the complex matrix space \(\mathbb{C}^{M_{\mathrm{S}}N_{T}\times M_{\mathrm{S}}N_{T}}\). It is noteworthy that \(\mathbf{H}\) is sparse and characterized by its elements \(H[d,c]\). Moving on, the information vector \(\mathbf{x}\) is an element of the complex vector space \(\mathbb{C}^{M_{\mathrm{S}}N_{T}\times 1}\), with each component \(x[c]\) taking values from the set \(\mathcal{A}\) and satisfying \(1\leq c\leq M_{\mathrm{S}}N_{T}\). Finally, we introduce the additive white Gaussian noise vector \(\mathbf{w}\), which is an element of the complex vector space \(\mathbb{C}^{M_{\mathrm{S}}N_{T}\times 1}\). To elaborate further, the structure of \(\mathbf{H}\) is expressed as follows:  
\[\mathbf{H}=\sum_{i=0}^{p-1}\sum_{j=-N_{i}}^{N_{i}}\mathbf{I}_{N}\big{(}-[j-k_{ \tau_{i}}]_{N}\big{)}\otimes\Bigg{[}\mathbf{I}_{M}(l_{\tau_{i}})\times\frac{1} {N}\Bigg{(}\frac{e^{-j2\pi\left(-j-\kappa_{\tau_{i}}\right)}-1}{e^{-j\frac{2 \pi}{N}\left(-j-\kappa_{\tau_{i}}\right)}-1}\Bigg{)}h_{i}e^{-j2\pi\left(\tau_{ i}\tau_{i}\right)}\Bigg{]}. \tag{29}\]  
In our framework, we have two circular shifting matrices denoted as \(\mathbf{I}_{N}\) and \(\mathbf{I}_{M}\). \(\mathcal{R}_{d}\) and \(\mathcal{C}_{c}\) are defined as the sets whose the \(d\)-th row and \(c\)-th column are non-zero elements in the EC \(\mathbf{H}\), respectively. By examining the composition of the channel matrix, it becomes evident that the cardinality of both sets denoted as \(|\mathcal{R}_{d}|\) and \(|\mathcal{C}_{c}|\) equals a constant value, which we denote as \(S\). The transmitted signals are estimated by the MAP rule. Referring to Equation (10), this estimation procedure can be expressed as:  
\[\widehat{\mathbf{x}}=\operatorname*{arg\,max}_{\mathbf{x}\in\mathcal{A}^{N_{T} M_{\mathrm{S}}}}\mathrm{P}(\mathbf{x}\mid\mathbf{y},\mathbf{H}). \tag{30}\]  
Consider that each of the transmitted symbols in the vector \(\mathbf{x}\) is assumed to be independently drawn from the constellation set \(\mathcal{A}\) with equal probability, which is equivalent to a uniform prior for \(\mathbf{x}\). Furthermore, given the approximate independence of the components of vector \(\mathbf{y}\), we can approximate the a posteriori probability, denoted as \(\mathrm{P}(x[c]=a_{j}|\mathbf{y},\mathbf{H})\), as follows:  
\[\mathrm{P}\big{(}x[c]=a_{j}\mid\mathbf{y},\mathbf{H}\big{)}=\frac{1}{Q} \mathrm{P}\big{(}\mathbf{y}\mid x[c]=a_{j},\mathbf{H}\big{)}\approx\prod_{d \in\mathcal{C}_{c}}\mathrm{P}\big{(}\mid y[d]\mid x[c]=a_{j},\mathbf{H}\big{)}. \tag{31}\]  
The elements \(\widehat{x}[c]\) and \(x[c]\), which belong to the vectors \(\widehat{\mathbf{x}}\) and \(\mathbf{x}\), respectively, satisfy the condition \(1\leq c\leq M_{\mathrm{S}}N_{T}\). The representation encapsulated in Equation (31) can be interpreted via a factor graph, as illustrated in Figure 3. This schematic elucidation provides a sophisticated methodology for ascertaining the marginal distribution, capitalizing on the fact that the variables are independent conditionally. Within this graph, each rectangular node, termed a function node (FN), denotes a specific function, such as \(f_{d}=p(y_{d}|\mathbf{x}_{d})\), where \(\mathbf{x}_{d}=\{x[e]|e\in\mathcal{R}_{d}\}\). On the flip side, every circular node, embematic of a variable node (VN), aligns with a unique variable.