## 3 Solution of the Optimization Problem
### Optimality Conditions for SDP Relaxation
\begin{split}& Tr\big{[}\big{(}\tilde{\mathbf{W}}^{*}-r\tilde{\mathbf{V}}^{* }-r\tilde{\mathbf{V}}^{*}_{0}\big{)}\mathbf{H}\big{]}\\ &=Tr\big{[}\big{(}\mathbf{W}^{*}-r\mathbf{V}^{*}-r\mathbf{V}^{*}_{0}-(r+1) \sum_{n=1}^{N_{T}-\tilde{\mathbf{r}}}b_{n}\mathbf{\pi}_{1,n}\mathbf{\pi}_{1,n}^{H}\mathbf{\pi} _{1,n}^{H}\big{)}\mathbf{H}\big{]}\\ &=Tr\big{[}\big{(}\mathbf{W}^{*}-\mathbf{V}^{*}-\mathbf{V}^{*}_{0}\big{)}\mathbf{ H}\big{]}\geq\frac{\sigma_{I}^{2}}{\tilde{\mathbf{r}}^{*}},\end{split}  \tag{44}\]  
\[ \begin{split}& Tr\big{[}\mathbf{G}_{k}^{H}\big{(}\tilde{\mathbf{W}}^{*}-r_{k} \tilde{\mathbf{V}}^{*}_{0}\big{)}\mathbf{G}_{k}\big{]}\\ &=Tr\big{[}\mathbf{G}_{k}^{H}\big{(}\mathbf{W}^{*}-\sum_{n=1}^{N_{T}- \tilde{\mathbf{r}}}b_{n}\mathbf{\pi}_{1,n}\mathbf{\pi}_{1,n}^{H}-r_{k}\mathbf{V}^{*}\big{)}\mathbf{G }_{k}\big{]}\\ &\leq Tr\big{[}\mathbf{G}_{k}^{H}\big{(}\mathbf{W}^{*}-r_{k}\mathbf{V}^{*} \big{)}\mathbf{G}_{k}\big{]}\leq r_{k}\sigma_{E,k}^{2},\end{split}  \tag{45}\]\[ \begin{split}& Tr\big{[}\big{(}\tilde{\mathbf{W}}^{*}+\tilde{\mathbf{V}}^{*}+ \tilde{\mathbf{V}}_{0}^{*}\big{)}\mathbf{H}\big{]}\\ &=Tr\big{[}\big{(}\mathbf{W}^{*}+\mathbf{V}^{*}+\mathbf{V}_{0}^{*}\big{)}\mathbf{H }\big{]}\geq\frac{\omega_{I}}{1-\bar{\rho}^{*}},\\ & Tr\big{[}\mathbf{G}_{k}^{H}\big{(}\tilde{\mathbf{W}}^{*}+\tilde{\mathbf{V}}^ {*}+\tilde{\mathbf{V}}_{0}^{*}\big{)}\mathbf{G}_{k}\big{]}\\ &=Tr\big{[}\mathbf{G}_{k}^{H}\big{(}\mathbf{W}^{*}+\mathbf{V}^{*}+\mathbf{V}_{0}^ {*}\big{)}\mathbf{G}_{k}\big{]}\geq\bar{t}^{*},\ \forall k,\\ & Tr\big{(}\tilde{\mathbf{W}}^{*}+\tilde{\mathbf{V}}^{*}+\tilde{\mathbf{V}}_ {0}^{*}\big{)}=Tr\big{(}\mathbf{W}^{*}+\mathbf{V}^{*}+\mathbf{V}_{0}^{*}\big{)}\leq P,\\ &\tilde{\mathbf{W}}^{*}\succeq\mathbf{0},\ \ \tilde{\mathbf{V}}^{*}\succeq\mathbf{0},\ \ \tilde{\mathbf{V}}_{0}^{*}\succeq\mathbf{0}.\end{split}  \tag{46}\]  
The properties from (44) to (49) indicate that the new solution \(\big{\{}\tilde{\mathbf{W}}^{*},\tilde{\mathbf{V}}^{*},\tilde{\mathbf{V}}_{0}^{*},\tilde{ \mathbf{V}}^{*},\tilde{t}^{*}\big{\}}\) can achieve the same optimal value as \(\big{\{}\mathbf{W}^{*},\mathbf{V}^{*},\mathbf{V}_{0}^{*},\rho^{*},t^{*}\big{\}}\) while (44)-(49) demonstrate that the new optimal solution satisfies all the constraints of primal problem (20) with \(\text{rank}(\tilde{\mathbf{W}}^{*})=1\). Theorem 1 is thus proved. With Theorem 1, we can achieve the global optimal solution of the optimization problem (13) with \(\text{rank}(\mathbf{W})=1\) as follows. First, we solve the SDP relaxation optimization problem (20) via CVX and obtain the solution \(\big{\{}\mathbf{W}^{*},\mathbf{V}^{*},\mathbf{V}_{0}^{*},\rho^{*},t^{*}\big{\}}\). If the rank of the obtained \(\mathbf{W}^{*}\) equals one, then \(\big{\{}\mathbf{W}^{*},\mathbf{V}^{*},\mathbf{V}_{0}^{*},\rho^{*},t^{*}\big{\}}\) will be the optimal solution to problem (20). Otherwise, if the rank of \(\mathbf{W}^{*}\) has a larger rank than one, a new optimal solution \(\big{\{}\tilde{\mathbf{W}}^{*},\tilde{\mathbf{V}}^{*},\tilde{\mathbf{V}}_{0}^{*},\rho^{*},\tilde{t}^{*}\big{\}}\) with \(\text{rank}(\tilde{\mathbf{W}}^{*})=1\) can be constructed according to (23)-(26). Then, \(\big{\{}\tilde{\mathbf{W}}^{*},\tilde{\mathbf{V}}^{*},\tilde{\mathbf{V}}_{0}^{*},\tilde{ \mathbf{V}}^{*},\tilde{t}^{*}\big{\}}\) will be the optimal solution to (20). Hence, dropping the rank-one constraint in (15) results in no loss of optimality to (13).