## 3 Optimization Problem Solution  
The problem in (9) is highly nonconvex which makes obtaining a global optimal solution intractable, so an iterative algorithm is presented. This solution is based on an alternating optimization that updates \(\mathbf{\Theta}_{k}\), \(\mathbf{\Gamma}\) and \(\mathbf{M}_{k}\) individually with the other parameters kept fixed to solve the three convex subproblems. First, given \(\mathbf{\Theta}_{k}\) and \(\mathbf{\Gamma}\), the optimal combiner at the \(k\)th destination \(\mathbf{M}_{k}\) is obtained by solving the unconstrained convex problem for \(\mathbf{M}_{k}\), which is independent of the constraints in (9b) and (9c). The optimal solution can then be obtained by taking the derivative of (8) with respect to \(\mathbf{M}_{k}\) and setting it to zero. Solving \(\frac{\partial}{\partial\mathbf{M}_{k}}\text{tr}(\text{MSE}_{k})=0\) gives  
\[\mathbf{M}_{k}=\mathbf{C}_{n_{k}}^{-1}\mathbf{G}_{k}\mathbf{\Gamma}\mathbf{H }_{k}\mathbf{\Theta}_{k}, \tag{10}\]  
where  
\[\mathbf{C}_{n_{k}}=(\mathbf{G}_{k}\mathbf{\Gamma}\mathbf{H}\mathbf{\Theta}_{ k}\mathbf{\Theta}_{k}^{H}\mathbf{H}_{k}^{H}\mathbf{\Gamma}^{H}\mathbf{G}_{k}^{H}+ \mathbf{C}_{n_{k}}+\mathbf{E}_{n_{k}})^{-1}\mathbf{G}_{k}\mathbf{\Gamma} \mathbf{H}_{k}\mathbf{\Theta}_{k},\]  
which is known as the Wiener filter. Second, with \(\mathbf{M}_{k}\) from (10) and given \(\mathbf{\Theta}_{k}\), \(\mathbf{\Gamma}\) can be obtained by solving the following problem. The objective function in (9) can be expressed as  
\[\begin{split}\text{SMSE}=&\sum_{k=1}^{K}\text{tr}(( \mathbf{M}_{k}^{H}\mathbf{G}_{k}\mathbf{\Gamma}\mathbf{H}\mathbf{\Theta}_{k}- \mathbf{I}_{N_{k}})(\mathbf{M}_{k}^{H}\mathbf{G}_{k}\mathbf{\Gamma}\mathbf{H} \mathbf{\Theta}_{k}\\ &-\mathbf{I}_{N_{k}})^{H}+\mathbf{M}_{k}^{H}(\sigma_{l}^{2}\mathbf{ G}_{k}\mathbf{\Gamma}\mathbf{R}_{Ll}\mathbf{R}_{Ll}^{H}\Note that since \(\mathbf{\Theta}_{k}\) is known, the constraint in (9c) is eliminated. The optimization problem then becomes  
\[ \begin{split}\underset{\mathbf{\Gamma}}{\text{min}}& \text{SMSE}\\ s.t.&\text{tr}(\mathbf{\Gamma}(\sum_{k=1}^{K}\mathbf{H}_{k }\mathbf{\Theta}_{k}\mathbf{\Theta}_{k}^{H}\mathbf{H}_{k}^{H}+\sigma_{t}^{2}\mathbf{R} _{LI}\mathbf{R}_{LI}^{H}+\mathbf{I}_{N_{t}})\mathbf{\Gamma}^{H})\leq P_{r}.\end{split}  \tag{12}\]  
Consider the singular-value decomposition (SVD) of the equivalent source-to-relay and relay-to-destination channels:  
\[\mathbf{H}=[\mathbf{H}_{1}\mathbf{\Theta}_{1},\dots,\mathbf{H}_{K}\mathbf{\Theta}_{K} ]=\mathbf{U}_{h}\Lambda_{h}\mathbf{V}_{h}^{H}, \tag{13}\]  
and  
\[\mathbf{G}=[\mathbf{G}_{1}^{T},\dots,\mathbf{G}_{K}^{T}]^{T}=\mathbf{U}_{g} \Lambda_{g}\mathbf{V}_{g}^{H}. \tag{14}\]  
The optimal relay precoding matrix is similar to that in [7]:  
\[\mathbf{\Gamma}=\mathbf{V}_{g}\mathbf{A}\mathbf{U}_{h}^{H}, \tag{15}\]  
and from (13) and (14), we have  
\[\mathbf{H}_{k}\mathbf{\Theta}_{k}=\mathbf{U}_{h}\Lambda_{h}\mathbf{V}_{h_{k}}^{H}, \tag{16}\]  
and  
\[\mathbf{G}_{k}=\mathbf{U}_{g_{k}}\Lambda_{g}\mathbf{V}_{g}^{H}. \tag{17}\]  
Note that  
\[\mathbf{V}_{h}=[\mathbf{V}_{h1}^{T},\dots,\mathbf{V}_{h_{K}}^{T}]^{T},\]  
and  
\[\mathbf{U}_{g}=[\mathbf{U}_{g1}^{T},\dots,\mathbf{U}_{gK}^{T}]^{T},\]  
which have dimensions \(d\times L_{1}\) and \(N_{i}\times L_{2}\), respectively. Substituting (13)-(15) in (11) gives  
\[ \begin{split}\text{SMSE}=&\sum_{k=1}^{K}\text{tr}(( \mathbf{M}_{k}^{H}\mathbf{U}_{g_{k}}\Lambda_{g}\mathbf{A}\Lambda_{h}\mathbf{V} _{h_{k}}^{H}-\mathbf{I}_{N_{k}})\times\\ &(\mathbf{M}_{k}^{H}\mathbf{U}_{g_{k}}\Lambda_{g}\mathbf{A} \Lambda_{h}\mathbf{V}_{h_{k}}^{H}-\mathbf{I}_{N_{k}})^{H}\\ &+\mathbf{M}_{k}^{H}(\sigma_{t}^{2}\mathbf{U}_{g_{k}}\Lambda_{g} \mathbf{A}\mathbf{U}_{h}^{H}\mathbf{R}_{LI}\mathbf{R}_{LI}^{H}\mathbf{U}_{h} \mathbf{A}^{H}\Lambda_{g}\mathbf{U}_{g_{k}}^{H}\\ &+\mathbf{U}_{g_{k}}\Lambda_{g}\mathbf{A}\mathbf{A}^{H}\Lambda_{ g}\mathbf{U}_{g_{k}}^{H}+\mathbf{I}_{N_{k}}+\mathbf{U}_{g_{k}}\Lambda_{g}\mathbf{A} \end{split}  \tag{18}\]  
Using the matrix identities  
\[