# Vector Approximate Message Passing Based OFDM-IM Detection for Underwater Acoustic Communications
## 3 Proposed Method
### VAMP Framework for IM Detection
The original variable node \(\mathbf{x}\) is divided as \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\), which are connected by the Dirac delta function \(\delta(\mathbf{x}_{2}-\mathbf{x}_{1})\). Then the joint probability distribution is \(p(\mathbf{y},\mathbf{x}_{2},\mathbf{x}_{1})=p(\mathbf{y}|\mathbf{x}_{2})\delta (\mathbf{x}_{2}-\mathbf{x}_{1})\)\(p(\mathbf{x}_{1})\). The message passing scheduling in the non-loopy graph includes three parts:  
* Approximate belief The expectation propagation (EP) is adopted for belief calculation using the Gaussian approximation and moment matching. For each variable node, the marginal function is the product of all impinged messages. We denote the approximate message for \(\mathbf{x}_{i}\) as \(\mathcal{CN}\big{(}\mathbf{x}_{i};\hat{\mathbf{x}}_{i},\beta_{i}^{-1}\mathbf{ I}\big{)}\), where \(i=1,2\), i.e., \(b_{sp}(\mathbf{x}_{2})=\mu_{f\to x_{2}}(\mathbf{x}_{2})\mu_{\delta\to x_{2}}( \mathbf{x}_{2})\) and \(b_{sp}(\mathbf{x}_{1})=\mu_{p\to x_{1}}(\mathbf{x}_{1})\mu_{\delta\to x_{1}}( \mathbf{x}_{1})\). The approximate belief \(b_{app}(\mathbf{x}_{i})\) for the node \(\mathbf{x}_{i}\) is determined by expectation \(\hat{\mathbf{x}}_{i}\) and covariance \(\beta_{i}^{-1}\) with respect to the marginal function \(b_{sp}(\mathbf{x}_{i})\), i.e., \(\mathbb{E}\big{[}\mathbf{x}_{i}|b_{sp}(\mathbf{x}_{i})\big{]}\) and \(\big{\langle}\text{diag}\left(\text{cov}\big{[}\mathbf{x}_{i}|b_{sp}( \mathbf{x}_{i})\big{]}\right)\big{\rangle}\). * Message from variable node to factor node The message from the variable node to the factor node is \(\mu_{x_{i}\to f}(\mathbf{x}_{i})\propto\frac{b_{app}(\mathbf{x}_{i})}{\mu_{f \to x_{i}}(\mathbf{x}_{i})}\). Because all beliefs are approximated as the Gaussian messages, the messages at \(t\)th iteration are assumed as \(\mu_{\delta\to x_{2}}(\mathbf{x}_{2})\sim\mathcal{CN}\Big{(}\mathbf{r}_{2t}, \gamma_{2t}^{-1}\mathbf{I}\Big{)}\) and \(\mu_{\delta\to x_{1}}(\mathbf{x}_{1})\sim\mathcal{CN}\Big{(}\mathbf{r}_{1t}, \gamma_{1t}^{-1}\mathbf{I}\Big{)}\). * Message from factor node to variable node Message from factor node to variable node is \(\mu_{f\to x_{2}}(\mathbf{x}_{i})\propto\int_{\sim x_{2}}f(\mathbf{x}_{i})\prod \limits_{\mathbf{y}_{j}\in N(f)}\big{/}\mathbf{x}_{i}\mu_{\gamma\to f} \big{(}\mathbf{x}_{j}\big{)}\), where \(\int_{\sim x}\) is to integrate over the other connected factors except \(\mathbf{x}\). \(N(f)\) is the neighboring node of \(f\). According to the sum-product rules, the bidirectional messages for each edge are derived as \(\mu_{f\to x_{2}}(\mathbf{x}_{2})=f(\mathbf{x}_{2})\), \(\mu_{p\to x_{1}}(\mathbf{x}_{1})=p(\mathbf{x}_{1})\), \(\mu_{\delta\to x_{2}}(\mathbf{x}_{2})=\int\delta(\mathbf{x}_{2}-\mathbf{x}_{1} )\mu_{x_{1}\to\delta}(\mathbf{x}_{1})d\mathbf{x}_{1}\) and \(\mu_{\delta\to x_{1}}(\mathbf{x}_{1})=\int\delta(\mathbf{x}_{2}-\mathbf{x}_{1} )\mu_{x_{2}\to\delta}(\mathbf{x}_{2})d\mathbf{x}_{2}\). Due to the equivalent relation in \(\delta(\mathbf{x}_{2}-\mathbf{x}_{1})\),  
Figure 2: Factor graph of VAMP based detection. \(\mu_{x_{1}\to\delta}(\mathbf{x}_{1})=\mu_{\delta\to x_{2}}(\mathbf{x}_{2})\) and \(\mu_{x_{2}\to\delta}(\mathbf{x}_{2})=\mu_{\delta\to x_{1}}(\mathbf{x}_{1})\).