## 3 Proposed Algorithm
### GAMP-SBL-Based Channel Estimation  
GAMP is a very low-complexity Bayesian iteration technique [25] to approximate the a posteriori distribution \(p(\mathbf{c}1\mathbf{y},\mathbf{\alpha},\sigma^{2})\). As a result, it can be embedded in the EM framework to approximate the a posteriori distribution of the basis coefficients in place of the true a posteriori distribution and greatly reduce the complexity of computation. Define \(\mathbf{\theta}\triangleq\left\{\mathbf{\alpha},\sigma^{2}\right\}\) as hyperparameter and assume a posteriori independence between hidden variables \(\left\{c_{n}\right\}\). As a result, the true a posteriori distribution can be approximated as follows:  
\[\hat{p}(c_{n}|\mathbf{y},\hat{r}_{n},\tau_{n}^{r},\mathbf{\theta})=\frac{p(c_ {n}|\mathbf{\theta})\mathcal{N}(c_{n}|\hat{r}_{n},\tau_{n}^{r})}{\int_{c}p(c_ {n}|\mathbf{\theta})\mathcal{N}(c_{n}|\hat{r}_{n},\tau_{n}^{r})} \tag{32}\]  
where \(\hat{r}_{n}\) and \(\tau_{n}^{r}\) are the scalars updated during the iteration of the GAMP algorithm. In combination with the Gaussian prior distribution of the basis coefficients, it can be easily shown that the approximate posterior also obeys a Gaussian distribution, and its mean and variance are respectively given as follows:  
\[\mu_{n}^{c}=\frac{\hat{r}_{n}}{1+\alpha_{n}\tau_{n}^{r}} \tag{33}\]  
\[\phi_{n}^{c}=\frac{\tau_{n}^{r}}{1+\alpha_{n}\tau_{n}^{r}} \tag{34}\]  
The noiseless output is the other approximation \(z_{m}=\varphi_{m}^{\mathrm{T}}\mathbf{c}\), where \(\varphi_{m}^{\mathrm{T}}\) is the \(m\)-th row of \(\mathbf{\Phi}\). The true posterior distribution \(p(z_{m}|\mathbf{y},\mathbf{\theta})\) can also be approximated as follows:  
\[\hat{p}\Big{(}z_{m}\Big{|}\mathbf{y},\hat{p}_{m},\tau_{m}^{p},\mathbf{\theta }\Big{)}=\frac{p(y_{m}|z_{m},\mathbf{\theta})\mathcal{N}\Big{(}z_{m}\Big{|} \hat{p}_{m},\tau_{m}^{p}\Big{)}}{\int_{z}p(y_{m}|z_{m},\mathbf{\theta}) \mathcal{N}\Big{(}z_{m}\Big{|}\hat{p}_{m},\tau_{m}^{p}\Big{)}} \tag{35}\]  
where \(\hat{p}_{m}\) and \(\tau_{m}^{p}\) are the scalars updated during the iteration of the GAMP algorithm. According to the assumption of Gaussian white noise, we can obtain \(p(y_{m}|z_{m},\mathbf{\theta})=\mathcal{N}\big{(}z_{m},\sigma^{2}\big{)}\). Therefore, \(\hat{p}\Big{(}z_{m}\Big{|}\mathbf{y},\hat{p}_{m},\tau_{m}^{p},\mathbf{\theta }\Big{)}\) also obeys a Gaussian distribution, and its mean and variance are respectively given as follows:  
\[\mu_{m}^{z}=\frac{\tau_{m}^{p}\sigma^{-2}y_{m}+\hat{p}_{m}}{1+\sigma^{-2}\tau_ {m}^{p}} \tag{36}\]  
\[\phi_{m}^{z}=\frac{\tau_{m}^{p}}{1+\sigma^{-2}\tau_{m}^{p}} \tag{37}\]  
Using the above approximation, we can define the scalar functions: \(g_{\mathrm{in}}(\cdot)\) and \(g_{\mathrm{out}}(\cdot)\). In the MMSE criterion, the input scalar function \(g_{\mathrm{in}}(\cdot)\) is directly defined as the posterior mean \(\mu_{n}^{c}\)[26], i.e.,  
\[g_{\mathrm{in}}(\hat{r}_{n},\tau_{n}^{r},\mathbf{\theta})=\mu_{n}^{c}=\frac{ \hat{r}_{n}}{1+\alpha_{n}\tau_{n}^{r}} \tag{38}\]The partial derivative of \(\tau_{n}^{r}g_{\text{in}}(\hat{r}_{n},\tau_{n}^{r},\theta)\) with respect to \(\hat{r}_{i}\) is the posterior variance, i.e.,  
\[\tau_{n}^{r}\frac{\partial}{\partial\hat{r}_{n}}g_{\text{in}}(\hat{r}_{n},\tau_{ n}^{r},\theta)=\phi_{n}^{c}=\frac{\tau_{n}^{r}}{1+\alpha_{n}\tau_{n}^{r}} \tag{39}\]  
Similarly, the relation of the output scalar function \(g_{\text{out}}(\cdot)\) to the posterior mean \(\mu_{m}^{z}\) and its partial derivation are derived as follows:  
\[g_{\text{out}}\Big{(}\hat{p}_{m},\tau_{m}^{p},\theta\Big{)}=\frac{1}{\tau_{m}^ {p}}(\mu_{m}^{z}-\hat{p}_{m}) \tag{40}\]  
\[\tau_{m}^{p}\frac{\partial}{\partial\hat{p}_{m}}g_{\text{out}}\Big{(}\hat{p}_{ m},\tau_{m}^{p},\theta\Big{)}=\frac{\phi_{m}^{z}-\tau_{m}^{p}}{\tau_{m}^{p}} \tag{41}\]  
Given the definitions \(g_{\text{in}}(\cdot)\) and \(g_{\text{out}}(\cdot)\), an efficient algorithm can be derived to generate an approximate posterior distribution of the variables \(\mathbf{c}\) and \(\mathbf{z}\).