## 4 Bayesian Learning-Based Clustered-Sparse Channel Estimation
### Partition-Based Clustered-Sparse Bayesian Learning Algorithm  
When the cluster partition information is known, the hyperparameters \(\mathbf{\theta}\), the posterior mean \(\mathbf{\mu}_{\mathbf{h}}\) and the posterior covariance \(\mathbf{\Sigma}_{\mathbf{h}}\) can be iteratively obtained through the EM method. In other words, cluster partition and cluster distribution jointly determine the clustered-sparse channel distribution. However, it is usually difficult to obtain the cluster partition in advance. As shown in Figure 1, several significant clusters with high energy in CIRs are scattered over a long time spread, and usually do not overlap each other. As shown in (28), the boundary information of these clusters is important for calculating \(\mathbf{B}_{d}\), because different clusters have different coherence times (i.e., different AR coefficients). In particular, some weak clusters that can be ignored will cause the hyperparameter estimation error of significant clusters when these clusters are not distinguished. Therefore, the PB-CSBL algorithm is proposed to update cluster partition by pruning the paths with very small energy (i.e., \(\gamma_{d,i}\to 0\)). As shown in Algorithm 1, the PB-CSBL algorithm mainly includes three sub-parts: Channel Estimation, Cluster Partition, Cluster Evolution, and they will loop until the iteration stop condition is reached. In the sub-part of Cluster Partition,only the dense paths with high energy are retained to form clusters and the paths with negligible energy are pruned. For a cluster with some pruned paths, it can further split into more clusters according to the positions of the pruned paths. To reduce the impact of an inaccurate clustering structure at the beginning of the iteration, the maximum possible number of discrete paths in one cluster is given. In the sub-part of Cluster Evolution, the derived cluster distribution learning rules are included. A good initial estimation of the unknown hyperparameters is significant for achieving the global maximum instead of a local maximum. For practical implementation, it was found that an initial estimation given by \(\mathbf{\Gamma}_{d}=\mathbf{I}_{L_{d}}\) and \(\mathbf{B}_{d}=\mathbf{I}_{M}\) was sufficient for the proposed PB-CSBL algorithm. ```
1:the received pilot signal \(\mathbf{y}_{\mathbf{p}}\); the dictionary matrix \(\mathbf{U}_{\mathbf{P}}\); the noise variance \(\lambda\); the length of discrete paths \(L\); the maximum number of iterations \(r_{max}\); the maximum number of discrete paths in one cluster \(LC_{max}\); the threshold for prunning small hyperparameters \(\gamma_{th}\); the threshold to stop the whole algorithm \(\epsilon\). 2:the list of path power \(\boldsymbol{\gamma}_{list}=\mathbf{1}_{L}\); the number of clusters \(C=1\); the list of cluster structure \(\mathbf{CLS}:\mathbf{\Gamma}_{1}=\mathbf{I}_{L}\), \(\mathbf{B}_{1}=\mathbf{I}_{M}\) and the delay range of the 1 st cluster \(R_{d}=[0,L-1]\); the iteration counter \(r=0\). 3:Channel Estimation:
4:\(\mathbf{\Sigma}_{\mathbf{0}}\gets diag(\mathbf{\Gamma}_{1}\otimes \mathbf{B}_{1},\mathbf{\Gamma}_{2}\otimes\mathbf{B}_{2},\ldots,\mathbf{\Gamma }_{C}\otimes\mathbf{B}_{C})\). 5:\(\mathbf{\Sigma}_{\mathbf{h}}=\mathbf{\Sigma}_{0}-\mathbf{\Sigma}_{0}\mathbf{U }_{\mathbf{P}}^{H}{\left(\lambda I+\mathbf{U}_{\mathbf{P}}\mathbf{\Sigma}_{0} \mathbf{U}_{\mathbf{P}}^{H}\right)^{-1}}\mathbf{U}_{\mathbf{P}}\mathbf{\Sigma }_{0}\). 6:\(\boldsymbol{\mu}_{\mathbf{h}}\leftarrow\lambda^{-1}\mathbf{\Sigma}_{\mathbf{h}} \mathbf{U}_{\mathbf{P}}^{H}\mathbf{V}_{\mathbf{P}}\). 7:Cluster Evolution:
8:for\(d=1,2,\ldots,C\)do
9:for\(i=1,2,\ldots,L_{d}\)do
10:\(\gamma_{d,i}\leftarrow\frac{1}{M}Tr\left[\mathbf{B}_{d}^{-1}\left(\mathbf{ \Sigma}_{\mathbf{h}}^{d,j}+\boldsymbol{\mu}_{\mathbf{h}}^{d,j}\left(\boldsymbol {\mu}_{\mathbf{h}}^{d,j}\right)^{H}\right)\right]\), and update \(\boldsymbol{\gamma}_{list}\) with \(\gamma_{d,j}\). 11:endfor
12:\(L_{d}^{\prime}\leftarrow min(L_{d},LC_{max})\).