## 3 Structured Sparse Bayesian Approach for Channel Estimation
### Structured Sparse Bayesian Channel Estimation Problem Formulation
\(C(\mathbf{u})=\frac{\Gamma(\sum_{l=1}^{l_{max}}u_{l})}{\Gamma(u_{1})-\Gamma(u_{l})}\) is the normalization constant. The role of \(\gamma\) is to model the relative difference between different sub-vectors \(\mathbf{u}\). By exploiting the common sparsity structure, the suboptimal solutions in the conventional methods that predict a value of zero for one element of \(\mathbf{h}_{[|\mathbf{l}|}\) and predict a nonzero value for the element of \(\mathbf{h}_{[|\mathbf{l}^{\prime}\
eq|\mathbf{l}|}\) in the same position can be eliminated. Since the expectation of \(\gamma\) with respect to the distribution (19) is given by \(\mathbb{E}[\mathbf{\gamma}]=u_{l}/\sum_{l^{\prime}=1}^{l_{max}}u_{l^{\prime}}\), we can interpret the parameter \(\mathbf{u}\), which provides a initial guess as to the relative difference between the sub-modules of \(\mathbf{h}\). The parameter that provides an initial estimation of the relative difference between the sub-modules of \(\mathbf{h}\) can be interpreted as \(\mathbf{u}\). Note that a default setup for \(\mathbf{u}\) can be \(u_{l}=1\) for \(l=1,\ldots,l_{max}\), which indicates an initial assumption of different delay modules. The maximum posterior (MAP) estimation of \(\mathbf{h}\) can be derived by its posterior mean in Equation (17) as long as \(\alpha_{l,k}\). As a result, in the sections that follow, we concentrate on obtaining the ideal \(\alpha_{l,k}\) and \(\mathbf{\gamma}\) by solving the MAP issue. \[ \begin{split}\Big{\{}\alpha_{l,k}^{opt},\mathbf{\gamma}^{opt}\Big{\}}& =\arg\max_{\alpha_{l,k}\gamma}p(\alpha_{l,k},\mathbf{\gamma}|\mathbf{y}) \\ &=\arg\min_{\alpha_{l,k}\gamma}-\log p(\alpha_{l,k},\gamma,\mathbf{y} )\end{split}  \tag{20}\]  
Since we proposed a new structured prior model for \(\mathbf{h}\), it is clear that the above problems cannot be solved if we continue to use the traditional sparse Bayesian algorithm. Therefore, we provide a solution using the block SMM algorithm. SMM Algorithm for Posterior Channel Estimate Evaluation  
The proposed block SMM algorithm is similar to the expected EM algorithm. It has two main steps. The first step is to replace the optimization step with the expected step,and its purpose is to find the alternative function of the objective function, that is, the local approximate solution. The second step is the minimization step, in which the alternative function found in the first step is minimized. While the block SMM technique divides the parameters to be estimated into many blocks for alternate optimization, the EM algorithm updates all parameters concurrently. This is helpful when solving structured SBL is the only obstacle to successfully minimizing all of the proxy function's parameters. At this point, we assume that \(\mathcal{G}(\alpha_{l,k},\gamma,\mathbf{y}|\hat{\alpha}_{l,k},\hat{\gamma})\) is the alternative function in Step 1, where \((\hat{\alpha}_{l,k},\hat{\gamma})\) is the estimation of \((\alpha_{l,k},\gamma)\). If \(\mathcal{G}(\alpha_{l,k},\gamma,y|\hat{\alpha}_{l,k},\hat{\gamma})\) satisfies the following condition:  
\[ \begin{split}&\mathcal{G}(\alpha_{l,k},\gamma,\mathbf{y}|\hat{ \alpha}_{k},\hat{\gamma})\geq-\log p(\alpha_{l,k},\gamma,\mathbf{y}),\forall\alpha_ {l,k},\gamma\\ &\mathcal{G}(\hat{\alpha}_{l,k},\hat{\gamma},\mathbf{y}|\hat{\alpha}_ {l,k},\hat{\gamma})=-\log p(\hat{\alpha}_{l,k},\hat{\gamma},\mathbf{y})\\ &\frac{\partial\mathcal{G}(\alpha_{l,k},\gamma,\mathbf{y}|\hat{ \alpha}_{l,k},\hat{\gamma})}{\partial\alpha_{l,k}}\Big{|}_{\alpha_{l,k}=\hat{ \alpha}_{l,k}}=\frac{\hat{\alpha}\{-\log p(\alpha_{l,k},\gamma,\mathbf{y})\}}{ \partial\alpha_{l,k}}\Big{|}_{\alpha_{l,k}=\hat{\alpha}_{l,k}}\\ &\frac{\partial\mathcal{G}(\hat{\alpha}_{l,k},\gamma,\mathbf{y}|\hat{ \alpha}_{l,k},\hat{\gamma})}{\partial\gamma}\Big{|}_{\gamma=\hat{\gamma}}= \frac{\hat{\alpha}\{-\log p(\hat{\alpha}_{l,k},\gamma,\mathbf{y})\}}{\partial \gamma}\Big{|}_{\gamma=\hat{\gamma}}\end{split}  \tag{21}\]  
In addition, \(\{\alpha_{l,k},\gamma\}\) follows the following rules in each iteration:  
\[