## 3 Structured Sparse Bayesian Approach for Channel Estimation
### Structured Sparse Bayesian Channel Estimation Problem Formulation
Because \(\alpha_{l,k}\) is coupled with the adjacent terms \(\alpha_{l,k-1}\) and \(\alpha_{l,k+1}\) in the term \(\xi_{k}\), we are unable to acquire \(\alpha_{l,k}\) by setting (26) to zero. Here, we have \(\alpha_{l,k}\), \(\alpha_{l,k+1}\), \(\alpha_{l,k-1}\). The expression for Equation (26) is  
\[\frac{\partial\mathcal{G}(\alpha_{l,k},\hat{\gamma}^{(i)},\mathbf{y}|\hat{\alpha}_{ l,k}^{(i)},\hat{\gamma}^{(i)})}{\partial\alpha_{l,k}}=\frac{a}{\alpha_{l,k}}-b+ \frac{1}{\alpha_{l,k}}-\hat{\gamma}^{(i)}\mathbf{g} \tag{29}\]  
By setting (29) to zero, we can obtain  
\[\hat{\alpha}_{l,k}^{(i+1)}=\frac{a+1}{\hat{\gamma}^{(i)}\mathbf{g}+b} \tag{30}\]  
Here, we do not address special cases (when \(k=1\) and \(k=\max\)). To update \(\gamma\), by leaving out the terms that are unrelated to \(\gamma\) and inserting (13) and (19) into (24), we obtain  
\[ \begin{split}&\mathcal{G}(\hat{\mathbb{a}}_{l,k}^{(i+1)},\gamma, \gamma|\hat{\mathbb{a}}_{l,k}^{(i)},\hat{\gamma}^{(i)})=\\ &-\sum_{l=0}^{l_{\text{max}}}\mathbf{u}\log\gamma-\log\lvert diag( \gamma)\times diag(\mathbf{D}_{l})\rvert\\ &+tr((diag(\gamma)\times diag(\mathbf{D}_{l}))\times(\hat{\mathbf{ \sum}}+\hat{\mu}\hat{\mu}^{H}))\\ &+const\end{split}  \tag{31}\]  
where \(\alpha_{l,k}\) is fixed at its latest estimation \(\alpha_{l,k}=\alpha_{l,k}^{(i+1)}\). In relation to \(\gamma\), the derivative of (31) can be written as  
\[ \begin{split}&\frac{\partial\mathcal{G}(\hat{\mathbb{a}}_{l,k}^{(i+1 )},\gamma,\gamma|\hat{\mathbb{a}}_{l,k}^{(i)},\hat{\gamma}^{(i)})}{\partial \alpha_{l,k}}\\ &=\frac{l_{\text{max}}+\mathbf{u}}{\gamma}\\ &-\left(\hat{\mathbb{a}}_{l,k}^{(i+1)}+\hat{\mathbb{a}}_{l,k-1}^{ (i+1)}+\hat{\mathbb{a}}_{l,k+1}^{(i+1)}\right)\times\left(\sum_{j=1}^{L} \left[\hat{\mathbf{\Sigma}}\right]_{[j,j]}+\left|\hat{\mathbf{\mu}}_{j]}\right|^{2} \right)\end{split}  \tag{32}\]  
The best value of \(\gamma\) can be attained by setting (32) to zero as  
\[ \begin{split}\hat{\gamma}^{(i+1)}&=\frac{l_{\text{ max}}+\mathbf{u}}{\hat{\mathbb{a}}_{l,k}^{(i+1)}+\hat{\mathbb{a}}_{l,k-1}^{(i+1)}+ \hat{\mathbb{a}}_{l,k+1}^{(i+1)}}\\ &\times\left(\sum_{j=1}^{L}\left[\hat{\mathbf{\Sigma}}\right]_{[j,j] }+\left|\left[\hat{\mathbf{\mu}}\right]_{j}\right|^{2}\right)^{-1}\end{split}  \tag{33}\]  
Step 1 and Step 2 are performed iteratively until convergence. Algorithm 1 is a summary of the SMM algorithm presented in this paper. ```
0: threshold \(\varepsilon\), maximum iterations \(iter_{max}\)
0: Setting the initial value \(\alpha_{k}\) and \(\gamma\) as the first loop value. 0: Channel estimation \(\mathbf{h}\). 1: Step 1: the \(i\)-th iteration \(\alpha_{l,k}^{(i)},\hat{\gamma}^{(i)}\) as fixed parameters \(\alpha_{l,k},\gamma\). Update the posterior mean and covariance matrix of \(\mathbf{h}\) using (16). 2: Step 2: Update \(\alpha_{l,k}\) by (29). 3: Update \(\gamma\) by (32). 4: Convergence of judgment:Let \(\mathbf{h}^{i+1}\) denote the estimation of \(\mathbf{h}\) after the (i+1)th iteration. 5: Output \(\mathbf{\hat{h}}\)
6:else
7: Set \(i=i+1\) and runtun Step 1
8:endif
```  
**Algorithm 1** SMM algorithm for structured SBL  
Afterwards, the algorithm will be further studied and introduced into the MIMO system. The dimensions of the channel change from the SISO system to the MIMO system, and the channel dimension increases linearly with the increase of the number of antennas. The sparse structure of the channel will therefore expand from one-dimensional to multi-dimensional. The calculation of the posterior \(\mathbf{\mu}\) and covariance matrix \(\mathbf{\Sigma}\) of \(\mathbf{h}\) in the MA-step in Algorithm 1 accounts for the majority of the computing effort for each iteration \(iter_{max}\). Therefore, the maximum computational complexity order of the structured SBL-based channel estimation algorithm is \(\mathcal{O}\big{(}iter_{\max}L^{3}S^{3}\big{)}\).