## 3 Proposed CE Method
### Fast SBL-Based CE  
In the EM algorithm, at the \(E\) step, the posterior covariance matrix \(\mathbf{\Sigma}\) is computed with \(\mathbf{\Gamma}\), and the posterior mean \(\mu\) is computed with the \(\mathbf{\Sigma}\). Then, \(\mathbf{\Gamma}\) is updated based on \(\mathbf{\Sigma}\) and \(\mu\) at the \(M\) step. These updates are performed alternatively at every iteration until convergence. Upon convergence, many of the diagonal components of \(\mathbf{\Gamma}\), \(\gamma_{l}\) converge to zero and so do the corresponding \([\mu]_{l}\), resulting in sparse estimates of the CIR vector \(\mathbf{h}\)[17]. To investigate the convergence behavior of the SBL-based CE, \(\mu\), diagonal elements of \(\mathbf{\Sigma}\), and \(\mathbf{\Gamma}\) at the first, second, and fifth iterations are represented in Figure 1b-d, respectively, for a given time-invariant CIR depicted in Figure 1a. It can be observed that the diagonal elements of \(\mathbf{\Sigma}\) decrease while \(\mathbf{\Gamma}\) converges to \(\mu\) as the iteration progresses. It is also noted that \(\mu\) and \(\mathbf{\Gamma}\) become more sparse, in which distinct components become more dominant while insignificant components diminish substantially.  
For faster convergence, the observations from Figure 1 suggest the following conditions:  
_Condition 1:_ diagonal elements of \(\mathbf{\Sigma}\) need to have small values.  
_Condition 2:_\(\mathbf{\Gamma}\) should be determined more by \(\mu\) rather than the diagonal elements of \(\mathbf{\Sigma}\).  
To achieve the above conditions, we propose a new LR for \(\mathbf{\Gamma}\) as  
\[\gamma_{l}^{(k)}=w_{1}\big{[}\mathbf{\Sigma}^{(k)}\big{]}_{l,l}+w_{2}\big{|} \big{[}\mu^{(k)}\big{]}_{l}\big{|}^{2} \tag{21}\]  
where \(w_{1}\) and \(w_{2}\) control the weights of the diagonal elements of \(\mathbf{\Sigma}\) and \(\mu\) in the update of \(\mathbf{\Gamma}\).  
The judicious selection of weights \(w_{1}\) and \(w_{2}\) can achieve the above conditions rapidly, accelerating the convergence, which will be discussed in the following subsection. The proposed F-SBL algorithm is summarized in Algorithm 1.  
```
0:\(\mathbf{y}_{i}\) and \(\mathbf{\Phi}_{i}\) for \(\forall i\in I_{t}\), stopping parameter \(\epsilon_{0}\), the maximum number of iterations \(N_{iter}\), weights \(w_{1}\) and \(w_{2}\), noise variance \(\sigma_{n}^{2}\)
0:\(\mathbf{\hat{h}}_{i}\) for \(\forall i\in I_{t}\)
1:for\(i=0\) to \(|I_{t}|-1\)do
2: Initialization: \(\mathbf{\Gamma}^{(0)}=\frac{1}{N_{path}}\mathbf{I}_{N_{path}}\) and \(k=0\)
3:while\(||\mathbf{\Gamma}^{(k)}-\mathbf{\Gamma}^{(k-1)}||_{F}^{2}>\epsilon_{0}\) and \(k<N_{iter}\)do
4:\(k\gets k+1\)
5:\(\mathbf{\Sigma}^{(k)}\leftarrow\left(\frac{1}{\sigma_{n}^{2}}\mathbf{\Phi}_{i} ^{H}\mathbf{\Phi}_{i}+\left(\mathbf{\Gamma}^{(k-1)}\right)^{-1}\right)^{-1}\)
6:\(\mu^{(k)}\leftarrow\frac{1}{\sigma_{n}^{2}}\mathbf{\Sigma}^{(k)}\mathbf{\Phi }_{i}\mathbf{y}_{i}\)
7:\(\gamma_{l}^{(k)}\gets w_{1}\big{[}\mathbf{\Sigma}^{(k)}\big{]}_{l,l}+w_{ 2}\big{|}\big{[}\mu^{(k)}\big{]}_{l}\big{|}^{2},l=0,\cdots,\)\(N_{path}-1\)
8:endwhile
9:\(\mathbf{\hat{h}}_{i}\leftarrow\mu^{(k)}\)
10:endfor
```  
**Algorithm 1** F-SBL algorithm.