# Vector Approximate Message Passing Based OFDM-IM Detection for Underwater Acoustic Communications
## 3 Proposed Method
### VAMP Framework for IM Detection  
According to the Equation (10), the data detection problem is defined as  
\[\hat{\mathbf{x}}^{(\delta)}=\operatorname*{arg\,min}_{\mathbf{x}^{(\delta)} \in\mathcal{S}}\biggl{\{}\left\|\mathbf{y}^{(\delta)}-\mathbf{H}^{(\delta)} \mathbf{x}^{(\delta)}\right\|_{2}^{2}\biggr{\}}. \tag{12}\]  
The problem means to recover the vector \(\mathbf{x}^{(\delta)}\) from noisy linear observation \(\mathbf{y}^{(\delta)}\) with noise \(\mathbf{w}^{(\delta)}\). \(\mathbf{w}^{(\delta)}\) follows Gaussian distribution with the element \(w^{(\delta)}\sim\mathcal{CN}\bigl{(}0,\gamma_{w}^{-1}\bigr{)}\), and \(\gamma_{w}\) is the noise precision. Once the problem is solved, the indices are detected and the symbols are equalized jointly. The detection is executed for each group, and the superscript \(g\) is omitted for simplification. In the field of digital communications, many problems such as channel estimation, data detection or user detection, are involved as Bayesian estimation problems. The key to the Bayesian message-passing graph is appropriate assumptions about the prior distribution and Gaussian noise. The standard linear regression problem corresponding to the problem in (12) is given by  
\[\hat{\mathbf{x}}=\operatorname*{arg\,min}_{\mathbf{x}}\frac{1}{2}\|\mathbf{y} -\mathbf{H}\mathbf{x}\|_{2}^{2}+f(\mathbf{x}), \tag{13}\]  
where \(f(\mathbf{x})\) is the penalty function. \(\mathbf{y}\in\mathbb{C}^{\tilde{M}\times 1}\) and \(\mathbf{H}\in\mathbb{C}^{\tilde{M}\times\tilde{N}}\) is the measurement matrix, which is the diagonal matrix with the channel components. Relate the measurement matrix \(\mathbf{H}\) with the system model in Section 2, \(\tilde{M}=\tilde{N}=n\). VAMP is first proposed in [15] to solve the problem (13), which has been proved robust to a broader class of large random matrices \(\mathbf{\Phi}\) compared with conventional AMP method [9]. Assuming known prior function \(p(\mathbf{x})\) and likelihood function \(p(\mathbf{y}|\mathbf{x})\), the posterior function is calculated through Bayesian rule as  
\[p(\mathbf{x}|\mathbf{y})=\frac{p(\mathbf{y}|\mathbf{x})p(\mathbf{x})}{p( \mathbf{y})}, \tag{14}\]  
where \(p(\mathbf{y})=\int p(\mathbf{y}|\mathbf{x})p(\mathbf{x})d\mathbf{x}\). According to different estimate criteria, the minimum mean square (MMSE) estimation is \(\hat{\mathbf{x}}_{\text{mmse}}=\operatorname*{arg\,min}_{\hat{\mathbf{x}}} \int\|\mathbf{x}-\hat{\mathbf{x}}\|p(\mathbf{x}|\mathbf{y})d\mathbf{x}= \mathbb{E}[\mathbf{x}|\mathbf{y}]\) and the maximum a posteriori (MAP) estimation is \(\hat{\mathbf{x}}_{\text{m}}=\operatorname*{arg\,max}_{\mathbf{x}}p(\mathbf{x}| \mathbf{y})\). The basic VAMP framework is given as in Algorithm 1. In Algorithm 1, \(\eta(\cdot)\) is the denoising function which is parameterized by \(\mathbf{r}\) and \(\gamma\). \(\left\langle\eta^{\prime}(\mathbf{r}_{k},\gamma_{k})\right\rangle\) is the divergence at \(\mathbf{r}_{k}\) and \(\eta^{{}^{\prime}}(\mathbf{r}_{k},\gamma_{k})=\text{diag}\Bigl{[}\frac{\partial \eta(\mathbf{r}_{k},\gamma_{k})}{\partial\mathbf{r}_{k}}\Bigr{]}\). The Onsager term \(\alpha_{k}\mathbf{r}_{k}\) cancels the correlation between \(\mathbf{\Phi}\) and \(\mathbf{x}\) and guarantees the Gaussianity of estimated errors. To involve the problem (12) into the VAMP framework, the factor graph and message scheduling of the VAMP-based detection are shown in Figure 2.  
In Figure 2, the leftmost node denotes the likelihood function defined as \(p\big{(}\mathbf{y};\mathbf{H}\mathbf{x}_{2},\gamma_{w}{}^{-1}\mathbf{I}\big{)}= \prod\limits_{m=1}^{n}\big{(}\frac{\gamma_{w}}{2\pi}\big{)}\exp\left(-\frac{ \gamma_{w}\big{\|}\mathbf{y}_{m}-\mathbf{H}_{m}^{T}\mathbf{x}_{n}^{T}\big{\|}_{ 2}^{2}}{2}\right)\), where \(\mathbf{H}_{m}^{T}\) is the \(m\)th row of the measurement matrix \(\mathbf{H}\). The rightmost factor node denotes the prior distribution \(p(\mathbf{x}_{1})\).