## 3 Structured Sparse Bayesian Approach for Channel Estimation
### Structured Sparse Bayesian Channel Estimation Problem Formulation  
We present the channel coefficient characteristics in the DD domain in Figure 4. According to the characteristics of channel sparsity, we propose a new estimation algorithm. We provide an example of the DD domain channel corresponding to (7). As shown in Figure 4, the DD domain channel only has 4 non-zero responses over the whole DD channel, whose coordinates align with the associated delay and Doppler shifts. Therefor, estimating channel \(\mathbf{h}\) can be viewed as a sparse recovery problem. Although the traditional sparse Bayesian algorithm can be used here, there are only values on each path according to the channel sparsity in the DD domain, and when a value is detected, it means that there are values before and after the value, so the traditional algorithm does not reflect this feature. According to the sparse characteristics of the channel, we propose an improved sparse Bayesian algorithm, as follows:  
\[p(\mathbf{h}|\alpha_{l,k},\gamma)=\mathcal{N}c(\mathbf{h}|0,diag(\gamma)\times diag (\boldsymbol{D_{l}})) \tag{13}\]  
where \(\boldsymbol{\gamma}=\{\gamma_{1},\gamma_{2},\ldots,\gamma_{l_{max}+1}\}\), \(\boldsymbol{D_{l}}\) is a diagonal matrix whose n-th diagonal element is given by  
\[[\boldsymbol{D_{l}}]_{k,k}=\left\{ \begin{array}{cc}\alpha_{l,1}\!+\!\alpha_{ l,2}&k=1\\ \alpha_{l,k-1}\!+\!\alpha_{l,k}\!+\!\alpha_{l,k+1}&2\leq k\leq 2k_{\max}+2Q\\ \alpha_{l,k-1}\!+\!\alpha_{l,k+1}&k=2k_{\max}+2Q+1\end{array} \right. \tag{14}\]  
where \(\boldsymbol{\alpha}_{l}=\{\alpha_{l,1},\alpha_{l,2},\ldots,\alpha_{l,2k_{\max} +2Q+1}\}\), \(\boldsymbol{D_{l}}=\{\boldsymbol{D_{l}},\boldsymbol{D_{2}},\ldots,\boldsymbol {D_{l_{max}+1}}\}\)]. As above, in the design of the diagonal matrix \(\boldsymbol{D_{l}}\), the design of the parameter \(\alpha_{l,k}\) not only affects the \(k\)th element of channel \(\mathbf{h}\) to be estimated; it also affects its neighbouring elements. As \(\alpha_{l,k}\) tends to infinity, the \(k\)-th, \((k+1)\)th, and \((k-1)\)th elements are simultaneously driven to zero. The distribution of zero elements exhibits a blocky distribution. On the other hand, by dividing the whole h into \(l_{max}\) column vectors, the function of \(\boldsymbol{\gamma}\) is used to distinguish the difference between the different column vectors. According to Equation (9), the posterior distribution of \(\mathbf{h}\) can be expressed as  
\[p(\mathbf{h}|\boldsymbol{y},\alpha_{l,k},\boldsymbol{\gamma})\propto p( \mathbf{y}|\mathbf{h})p(\mathbf{h}|\alpha_{l,k},\boldsymbol{\gamma}) \tag{15}\]  
Figure 4: Characteristics of sparsity. where the likelihood function of \(y\) is \(p(\mathbf{y}|\mathbf{h})\), which is  
\[p(\mathbf{y}|\mathbf{h})=\mathcal{N}c\Big{(}\mathbf{y}|\mathbf{\Phi}\mathbf{h},\sigma^{2} \mathbf{I}\Big{)} \tag{16}\]  
By substituting Equations (13) and (16) into Formula (15), it is not difficult to prove that this is a posterior distribution that follows a complex Gaussian distribution through mathematical calculation, and the mean and covariance are as follows:  
\[ \begin{split}&\mathbf{\mu}=\frac{1}{\sigma^{2}}\sum\mathbf{\Phi}^{H}\mathbf{y} \\ &\sum=(diag\{\mathbf{\gamma}\}\times diag\{\mathbf{D}_{\mathbf{l}}\}+\frac{1 }{\sigma^{2}}\mathbf{\Phi}^{H}\mathbf{\Phi})^{-1}\end{split}  \tag{17}\]  
Here, as in the traditional sparse Bayesian algorithm, we express the Gamma hyperprior with respect to \(\alpha\) as follows:  
\[ \begin{split} p(\mathbf{\alpha})&=\prod_{k=1}^{2k_{ max}+2Q+1}Gam(\alpha_{l,k}|a,b)\\ &=\prod_{k=1}^{2k_{max}+2Q+1}\frac{b^{a}}{\Gamma(a)}\alpha_{l,k} ^{a}\exp(-b\alpha_{l,k})\end{split}  \tag{18}\]  
where \(a\) and \(b\) are fixed parameters. In this section, we introduce the small positive numbers a and b (\(a=b=10^{-4}\)). Such a configuration encourages a flat prior for \(\alpha\). We also select a Dirichlet hyperprior for \(\gamma\),  
\[p(\mathbf{\gamma})=C(\mathbf{u})\prod_{l=1}^{l_{max}+1}\mathbf{\gamma}^{u_{l}} \tag{19}\]  
where \(\mathbf{u}=[u_{1},u_{2},\ldots,u_{l_{max}}]\) is a fixed parameter.