## 3 Optimization Problem Solution
Using the matrix identities, (27) can be written as a function of \(\theta_{k}=\operatorname{vec}(\mathbf{\Theta}_{k})\) which gives\[\text{SMSE}= \sum_{k=1}^{K}[(\mathbf{S}_{k}\theta_{k}-\text{vec}(\mathbf{I}_{N_{k }}))^{H}(\mathbf{S}_{k}\theta_{k}-\text{vec}(\mathbf{I}_{N_{k}})) \tag{28}\] \[+\sum_{m=1,m\
eq k}^{K}\theta_{m}^{H}(\mathbf{I}_{d}\otimes\mathbf{ H}_{m}^{H}\mathbf{F}^{H}\tilde{\mathbf{G}}_{k}^{H}\mathbf{G}_{k}\mathbf{F}\mathbf{H} _{m})\theta_{m}]\] \[+\mathbf{t}_{2}\] \[= \sum_{k=1}^{K}[(\mathbf{S}_{k}\theta_{k}-\text{vec}(\mathbf{I}_ {N_{k}}))^{H}(\mathbf{S}_{k}\theta_{k}-\text{vec}(\mathbf{I}_{N_{k}}))\] \[+\theta_{k}^{H}\mathbf{T}_{k}\theta_{k}]+\mathbf{t}_{2},\]  
where  
\[\mathbf{S}_{k}=\mathbf{I}_{N_{k}}\otimes(\tilde{\mathbf{G}}_{k}\mathbf{F} \mathbf{H}),\]  
\[\mathbf{T}_{k}=\mathbf{I}_{N_{k}}\otimes\sum_{m=1,m\
eq k}^{K}(\mathbf{H}^{H} \mathbf{T}^{H}\tilde{\mathbf{G}}_{k}^{H}\tilde{\mathbf{G}}_{k}\mathbf{F} \mathbf{H}),\]  
and  
\[\mathbf{t}_{2}=\sum_{k=1}^{K}\text{tr}(\mathbf{M}_{k}^{H}\mathbf{C}_{n_{k}} \mathbf{M}_{k}),\]  
which is independent of \(\mathbf{\Theta}_{k}\) and so can be ignored. Introducing \(\mathbf{T}=\text{bd}(\mathbf{T}_{1},\ldots,\mathbf{T}_{K})\), where \(\text{bd}(\cdot)\) denotes a block-diagonal matrix, and \(\tilde{\mathbf{S}}_{k}=[\mathbf{S}_{k1},\ldots,\mathbf{S}_{kk}]\), where \(\mathbf{S}_{kk}=\mathbf{S}_{k}\) and \(\mathbf{S}_{jk}=0,j\
eq k\), (27) can be written as a function of \(\theta=[\theta_{1}^{T},\theta_{1}^{T},\ldots,\theta_{K}^{K}]^{T}\), which gives  
\[\Phi_{1}(\theta)=\sum_{k=1}^{K}(\tilde{\mathbf{S}}_{k}\theta-\text{vec}( \mathbf{I}_{N_{k}}))^{H}(\tilde{\mathbf{S}}_{k}\theta-\text{vec}(\mathbf{I} _{N_{k}}))+\theta^{H}\mathbf{T}\theta \tag{29}\]  
Now, defining  
\[\mathbf{E}_{j}=\mathbf{I}_{N_{k}}\otimes(\mathbf{H}_{j}^{H}\mathbf{T}^{H} \mathbf{F}\mathbf{H}_{j}),\]  
\[+\sum_{m=1,m\
eq k}^{K}\theta_{m}^{H}(\mathbf{I}_{d}\otimes\mathbf{ H}_{m}^{H}\mathbf{F}^{H}\tilde{\mathbf{G}}_{k}^{H}\mathbf{G}_{k}\mathbf{F}\mathbf{H} _{m})\theta_{m}]\]0  
and  
\[+\sum_{m=1,m\
eq k}^{K}\theta_{m}^{H}(\mathbf{I}_{d}\otimes\mathbf{ H}_{m}^{H}\mathbf{F}^{H}\tilde{\mathbf{G}}_{k}^{H}\mathbf{G}_{k}\mathbf{F}\mathbf{H} _{m})\theta_{m}]\]1  
where \(\tilde{\mathbf{E}}_{iK}=\mathbf{I}_{dN_{k}}\) and \(\tilde{\mathbf{E}}_{ij}=0,i\
eq j\), the optimal \(\theta\) can be obtained by solving the following problem  
\[+\sum_{m=1,m\
eq k}^{K}\theta_{m}^{H}(\mathbf{I}_{d}\otimes\mathbf{ H}_{m}^{H}\mathbf{F}^{H}\tilde{\mathbf{G}}_{k}^{H}\mathbf{G}_{k}\mathbf{F}\mathbf{H} _{m})\theta_{m}]\]2 \[+\sum_{m=1,m\
eq k}^{K}\theta_{m}^{H}(\mathbf{I}_{d}\otimes\mathbf{ H}_{m}^{H}\mathbf{F}^{H}\tilde{\mathbf{G}}_{k}^{H}\mathbf{G}_{k}\mathbf{F}\mathbf{H} _{m})\theta_{m}]\]3 \[+\sum_{m=1,m\
eq k}^{K}\theta_{m}^{H}(\mathbf{I}_{d}\otimes\mathbf{ H}_{m}^{H}\mathbf{F}^{H}\tilde{\mathbf{G}}_{k}^{H}\mathbf{G}_{k}\mathbf{F}\mathbf{H} _{m})\theta_{m}]\]4  
This is a QCQP problem which can be solved using the CVX toolbox for disciplined convex programming. The symbols employed in this work are summarized in \[+\sum_{m=1,m\
eq k}^{K}\theta_{m}^{H}(\mathbf{I}_{d}\otimes\mathbf{ H}_{m}^{H}\mathbf{F}^{H}\tilde{\mathbf{G}}_{k}^{H}\mathbf{G}_{k}\mathbf{F}\mathbf{H} _{m})\theta_{m}]\]5 The proposed iterative algorithm is given in Algorithm 1. The convergence of this algorithm is guaranteed as follows. As the three subproblems are convex, each update of \(\mathbf{\Theta}_{k}\), \(\mathbf{\Gamma}\), and \(\mathbf{M}_{k}\) may decrease or at least not increase the value of the objective function. Thus, the iterative algorithm will converge to an optimum solution. ```
1:Initialize the algorithm with \(\mathbf{\Theta}^{(0)}=\sqrt{\frac{P_{r}}{L}\mathbf{I}_{L}}\) and \(\mathbf{\Gamma}^{(0)}=\sqrt{\frac{P_{r}}{\text{tr}(\mathbf{H}_{S\text{SR}}\mathbf{ \Theta}^{(0)}_{k}(\mathbf{H}_{S\text{SR}}\mathbf{\Theta}^{(0)}_{k})^{H}+\sigma_{t}^{2} \mathbf{R}_{Ll}\mathbf{R}_{Ll})^{H}+\mathbf{I}_{N_{r}}}}\mathbf{I}_{N_{r}}\), and set \(n=0\).