## 4 Digital Baseband Precoding and Combining
Using (48), the estimated symbol vector \(\mathbf{s}_{k}^{-}\) (46) can be written as  
\[\mathbf{s}_{k}^{-}=\left(\mathbf{H}\mathbf{V}_{BB,k}\right)^{H}\Big{\{}\left(\mathbf{H}\mathbf{V}_ {BB,k}\right)\left(\mathbf{H}\mathbf{V}_{BB,k}\right)^{H}+\mathbf{R}_{N}\Big{\}}^{-1}\mathbf{ y}_{D}^{-}. \tag{49}\]  
Substituting (48) into (47) leads to the following MMSE formulation as  
\[M_{min}=tr(\mathbf{D})=tr\Big{\{}\left(\mathbf{I}_{N_{t}}+\mathbf{E}\right)^{-1}\Big{\}}, \tag{50}\]  
where \(\mathbf{E}=\mathbf{V}_{BB,k}^{H}\mathbf{H}_{\alpha\beta k}^{H}\mathbf{G}_{BB,k}^{H}\mathbf{G}_{ \alpha\beta k}^{H}\Big{\{}\sigma_{r}^{2}\mathbf{G}_{\alpha\beta k}\mathbf{G}_{BB,k}^{H} \mathbf{G}_{\alpha\beta k}^{H}+\sigma_{d}^{2}\mathbf{I}_{N_{r}}\Big{\}}^{-1}\times\mathbf{ G}_{\alpha\beta k}\mathbf{G}_{BB,k}\mathbf{H}_{\alpha\beta k}\mathbf{V}_{BB,k}\) and this expression depends on \(\mathbf{V}_{BB,k}\) and \(\mathbf{G}_{BB,k}\). The design objective here is to find these two precoding matrices that would be able to minimize MSE. Therefore, the optimization problem can be formulated as  
\[\min_{\mathbf{V}_{BB,k},\mathbf{G}_{BB,k}}tr(\mathbf{D})=\sum_{n=1}^{N_{t}}\mathbf{D}_{n,n}\]  
\[\mathbf{G}_{\alpha k}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\in\mathbb{C}^{N_{r}^{ RF}\times N_{rt}^{RF}}. \tag{44}\]0  
\[\mathbf{G}_{\alpha k}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\in\mathbb{C}^{N_{r}^{ RF}\times N_{rt}^{RF}}. \tag{44}\]1  
The problem (51) is also non-convex, and hence, mathematically intractable. Therefore, a sub-optimal solution is expected to maximize the spectral efficiency (16). Furthermore, the baseband precoding matrices \(\mathbf{V}_{BB,k}\), \(\mathbf{G}_{BB,k}\) must fulfill the transmit power constraint at the source and each relay node. This curtailment leads to the immediate conclusion that the optimal solution is difficult to derive. In addition, the objective function involves a series of matrix inversions and multiplications, and it shows a complex and non-linear function of baseband precoding matrices, which needs to be designed. Sub-optimal solution of \(\mathbf{G}_{BB,k}\) can be obtained analytically through MSE diagonalization procedure [48]. Considering this motivation factor, \(\mathbf{G}_{BB,k}\) can be evaluated by performing diagonalization of the equivalent channels \(\mathbf{H}_{\alpha\beta k}\), \(\mathbf{G}_{\alpha\beta k}\), when observed from the corresponding baseband processing units. It is worth highlighting that SVD, of these two equivalent channels, facilitates in deriving \(\mathbf{G}_{BB,k}\), \(\mathbf{V}_{BB,k}\) as  
\[\mathbf{G}_{\alpha k}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\in\mathbb{C}^{N_{r}^{ RF}\times N_{rt}^{RF}}. \tag{44}\]2  
\[\mathbf{G}_{\alpha k}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\in\mathbb{C}^{N_{r}^{ RF}\times N_{rt}^{RF}}. \tag{44}\]3It is obvious from (52) and (53) that diagonalization of \(\mathbf{D}\) in (51) can be obtained by setting \(\mathbf{G}_{BB,k}=\mathbf{V}_{2}(;1:N_{s})\mathbf{U}_{1}(;1:N_{s})^{H}\) and \(\mathbf{V}_{BB,k}=\mathbf{V}_{1}(;1:N_{s})\) where \(\mathbf{U}_{1}\in\mathbb{C}^{N_{H}^{RF}\times N_{tot}^{RF}}\), \(\mathbf{V}_{1}\in\mathbb{C}^{N_{t}^{RF}\times N_{tot}^{RF}}\), \(\mathbf{V}_{2}\in\mathbb{C}^{N_{rel}^{RF}\times N_{rel}^{RF}}\) are unitary matrices. The proposed algorithm is summarized in \[\mathbf{G}_{\alpha k}=\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\mathbf{F}_{2k}\in\mathbb{C}^{N_{r}^{ RF}\times N_{rt}^{RF}}. \tag{44}\]4