## 5 Noise-Corrected Clustered-Sparse Channel Estimation
### Data Detection and Noise Measurement  
To obtain the MAP estimate in (32), the hyperparameters related to the cluster distribution, \(\mathbf{\theta}\triangleq[\gamma_{1},\gamma_{2},\cdots,\gamma_{C},\beta_{1},\beta _{2},\cdots,\beta_{C}]\), can be iteratively updated, as shown in the proposed PB-CSBL algorithm, however, the detection of data signal \(\mathbf{x}_{\mathbf{d}}^{(m)}\) and the noise distribution \(\mathbf{\lambda}\) are still unresolved. Using the notations \(\mathbf{x}_{\mathbf{d}}\triangleq\left[(\mathbf{x}_{\mathbf{d}}^{(1)})^{T},( \mathbf{x}_{\mathbf{d}}^{(2)})^{T},\cdots,(\mathbf{x}_{\mathbf{d}}^{(M)})^{T}\right]\), \(\mathbf{\varphi}\triangleq[\mathbf{\lambda},\mathbf{x}_{\mathbf{d}}]\), and \(\mathbf{\zeta}\triangleq[\mathbf{\theta},\mathbf{\varphi}]\), the unknown hyperparameters set \(\mathbf{\zeta}\) can be estimated by maximizing the marginal likelihood function \(p(\mathbf{y};\mathbf{\zeta})\). This is equivalent to minimizing \(-log\,p(\mathbf{y};\mathbf{\zeta})\), giving the effective cost function:  
\[\mathcal{L}(\mathbf{\zeta})=\mathbf{y}^{H}\mathbf{\Sigma}_{\mathbf{y}}^{-1}(\mathbf{\zeta}) \mathbf{y}+log|\mathbf{\Sigma}_{\mathbf{y}}(\mathbf{\zeta})|, \tag{34}\]  
where \(\mathbf{\Sigma}_{\mathbf{y}}(\mathbf{\zeta})=\mathbf{\Lambda}(\mathbf{\lambda})+\mathbf{U}( \mathbf{x}_{\mathbf{d}})\mathbf{\Sigma}_{0}(\mathbf{\theta})\mathbf{U}^{H}(\mathbf{x} _{\mathbf{d}})\). Similar to the cluster distribution learning, the expectation maximization (EM) method [41] can be employed to solve (34) iteratively. While \(\mathbf{h}\) is considered as the nuisance variable, \(\mathbf{\zeta}\) can be estimated by maximizing:  
\[Q(\mathbf{\zeta}|\mathbf{\zeta}^{(old)})=E_{\mathbf{h}|\mathbf{y};\mathbf{\zeta}^{(old)}}[ log\,p(\mathbf{y},\mathbf{h};\mathbf{\zeta})]=E_{\mathbf{h}|\mathbf{y};\mathbf{\zeta}^{(old)}}[ log\,p(\mathbf{y}|\mathbf{h};\mathbf{\varphi})]+E_{\mathbf{h}|\mathbf{y};\mathbf{\zeta}^{(old)}}[ log\,p(\mathbf{h};\mathbf{\theta})], \tag{35}\]where \(\xi^{(old)}\) denotes the estimated hyperparameters in the previous iteration. As shown in (35), the joint maximization simplifies into two independent maximizations over \(\theta\) and \(\varphi\). On optimizing the latter function in (35) with respect to \(\theta\), one obtains the learning rules about \(\theta\) as (26) and (28). Maximizing the former over \(\mathbf{x}_{\mathbf{d}}^{(m)}\) yields the minimum mean squared error (MMSE) equalizer:  
\[\widetilde{\mathbf{x}}_{\mathbf{d}}^{(m)}=\Big{[}diag(\Lambda_{\mathbf{d}}) \mathbf{C}_{\mathbf{X}}^{-1}+Pdiag\Big{(}\mathbf{F}_{\mathbf{D}}\mathbf{ \Xi}_{\mathbf{h}}^{(m)}\mathbf{F}_{\mathbf{D}}^{H}\Big{)}\Big{]}^{-1}diag \Big{[}\Big{(}\boldsymbol{\overline{\mu}}_{\mathbf{h}}^{(m)}\Big{)}^{H} \mathbf{F}_{\mathbf{D}}^{H}\Big{]}\mathbf{y}_{\mathbf{d}}^{(m)}, \tag{36}\]  
where \(\mathbf{y}_{\mathbf{d}}^{(m)}\) and \(\lambda_{\mathbf{d}}\) are the subvectors of \(\mathbf{y}^{(m)}\) and \(\lambda\) at the location of data subcarriers, respectively. \(\mathbf{C}_{\mathbf{X}}\) is the covariance matrix of \(\mathbf{x}_{\mathbf{d}}^{(m)}\), \(\mathbf{F}_{\mathbf{D}}\) is the \(K_{d}\times L\) DFT matrix at the location of data subcarriers. \(\mathbf{\Xi}_{\mathbf{h}}^{(m)}\) is defined as \(\mathbf{\Xi}_{\mathbf{h}}^{(m)}\triangleq\overline{\mu}_{\mathbf{h}}^{(m)} \Big{(}\overline{\mu}_{\mathbf{h}}^{(m)}\Big{)}^{H}+\overline{\mathbf{\Xi}}_{ \mathbf{h}}^{(m)}\), where \(\overline{\mu}_{\mathbf{h}}^{(m)}\in\mathbb{C}^{L\times 1}\) and \(\overline{\mathbf{\Xi}}_{\mathbf{h}}^{(m)}\in\mathbb{C}^{L\times L}\) denote the posterior mean and covariance matrix of the \(m\)th OFDM block, respectively.