## 4 Digital Baseband Precoding and Combining  
After finding the analog RF processing components at all communicating nodes, it is required to derive the corresponding digital baseband precoding/combining matrices. For this purpose, the equivalent baseband channels corresponding to \(\mathbf{H}_{k}\) and \(\mathbf{G}_{k}\) are defined as  
\[\mathbf{H}_{\alpha k}=\mathbf{F}_{1k}^{H}\mathbf{H}_{k}\mathbf{V}_{RF}\in\mathbb{C}^{N_{rel}^{ RF}\times N_{t}^{RF}}, \tag{43}\]  
\[\mathbf{G}_{\alpha k}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\in\mathbb{C}^{N_{r}^{ RF}\times N_{rt}^{RF}}. \tag{44}\]  
where \(\mathbf{H}_{\alpha k}\) denotes the baseband equivalent channel from the source to the \(k\)th relay node, and \(\mathbf{G}_{\alpha k}\) represents the baseband equivalent channel from the \(k\)th relay node to the destination. The received signal at the output of destination RF combiner \(\mathbf{W}_{RF}\) (31) using (43) and (44) can be written as  
\[y_{D}^{-}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\mathbf{G}_{BB,k}\mathbf{F}_{1k}^{H}\bm {H}_{k}~{}\mathbf{V}_{RF}\mathbf{V}_{BB,k}\mathbf{s}_{k}+\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_ {2k}\mathbf{G}_{BB,k}\mathbf{F}_{1k}^{H}\mathbf{z}_{k}+\mathbf{W}_{RF}^{H}\mathbf{z}_{d}\in~{} \mathbb{C}^{N_{r}^{RF}\times 1}\]  
\[=\mathbf{G}_{\alpha k}\mathbf{G}_{BB,k}\mathbf{H}_{\alpha k}\mathbf{V}_{BB,k}\mathbf{s}_{k}+\mathbf{n} =\mathbf{H}\mathbf{V}_{BB,k}\mathbf{s}_{k}+\mathbf{n}, \tag{45}\]  
where \(\mathbf{H}=\mathbf{G}_{\alpha k}\mathbf{G}_{BB,k}\mathbf{H}_{\alpha k}\in\mathbb{C}^{N_{r}^{ RF}\times N_{r}^{RF}}\) is the equivalent baseband channel from the source to the destination, and \(\mathbf{n}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\mathbf{G}_{BB,k}\mathbf{F}_{1k}^{H}\mathbf{z}_ {k}+\mathbf{W}_{RF}^{H}\mathbf{z}_{d}\in~{}\mathbb{C}^{N_{r}^{RF}\times 1}\) is the equivalent noise vector at the destination. It is noteworthy that \(\mathbf{H}\) is a function of combined baseband processing unit \(\mathbf{G}_{BB,k}\) at the \(k\)th relay node. The baseband combiner \(\mathbf{W}_{BB,k}\) at the destination is used to estimate \(\mathbf{s}_{k}\), which is given as  
\[\mathbf{s}_{k}^{-}=\mathbf{W}_{BB,k}^{H}\mathbf{y}_{D}^{-}. \tag{46}\]where \(\mathbf{s}_{k}^{-}\) is the estimated information symbol vector at the destination when data transmission takes place through the \(k\)th relay node. To enhance the system performance, it is desired to design the baseband processing units that enable end-to-end SNR maximization. This goal can be achieved by formulating a problem based on MSE, which can be expressed as  
\[M=E\Big{\{}\mathbf{s}_{k}^{-}-\mathbf{s}_{k}{}^{2}\Big{\}}. \tag{47}\]  
The minimization of (47) provides the optimal baseband combiner \(\mathbf{W}_{BB,k}^{H}\)[48] as  
\[\mathbf{W}_{BB,k}^{H}=\left(\mathbf{H}\mathbf{V}_{BB,k}\right)^{H}\Big{\{}\left(\mathbf{H}\mathbf{ V}_{BB,k}\right)\left(\mathbf{H}\mathbf{V}_{BB,k}\right)^{H}+\mathbf{R}_{N}\Big{\}}^{-1}, \tag{48}\]  
where \(\mathbf{R}_{N}=E\big{[}\mathbf{m}\mathbf{m}^{H}\big{]}=\sigma_{r}^{2}\Big{(}\mathbf{G}_{ \alpha\beta k}\mathbf{G}_{BB,k}\Big{)}\Big{(}\mathbf{G}_{\alpha\beta k}\mathbf{G}_{BB,k} \Big{)}^{H}+\sigma_{d}^{2}\mathbf{I}_{N_{r}}\) is the equivalent noise covariance matrix.