## 3 Problem Formulation and Solution
We define  
\[\mathbf{w}=\left[\underbrace{\underbrace{\mathbf{w}_{11},\mathbf{w}_{12}, \ldots\ldots,\mathbf{w}_{1j},\ldots\ldots,\mathbf{w}_{1N},\underbrace{\mathbf{ w}_{21},\ldots\ldots,\mathbf{w}_{2N},}_{N\cdot N_{T}},\ldots\ldots,\mathbf{w}_{ ij},\ldots\ldots,\mathbf{w}_{MN}}_{M\cdot N\cdot N_{T}}}_{M\cdot N\cdot N_{T}} \right]^{T}. \tag{16}\]6  
and  
\[\mathbf{w}=\left[\underbrace{\underbrace{\mathbf{w}_{11},\mathbf{w}_{12}, \ldots\ldots,\mathbf{w}_{1j},\ldots\ldots,\mathbf{w}_{1N},\underbrace{\mathbf{ w}_{21},\ldots\ldots,\mathbf{w}_{2N},}_{N\cdot N_{T}},\ldots\ldots,\mathbf{w}_{ ij},\ldots\ldots,\mathbf{w}_{MN}}_{M\cdot N\cdot N_{T}}}_{M\cdot N\cdot N_{T}} \right]^{T}. \tag{16}\]7  
where \(\mathbf{\theta}_{ij}=(\theta_{ij1},\theta_{ij2},\ldots.\ldots,\theta_{ijN_{T}})^{T}\), as defined in (7). Every \(\mathbf{\theta}_{j}\) is a vector with \(M\cdot N_{T}\) dimensions, and \(\mathbf{\theta}\) is a vector with \(M\cdot N\cdot N_{T}\) dimensions, which consists of \(\theta_{j}\) for all \(j\in\mathcal{N}\). We use \(\theta_{j}\) as one of the optimization variables and seek the partial difference for it. Let  
\[\mathbf{w}=\left[\underbrace{\underbrace{\mathbf{w}_{11},\mathbf{w}_{12}, \ldots\ldots,\mathbf{w}_{1j},\ldots\ldots,\mathbf{w}_{1N},\underbrace{\mathbf{ w}_{21},\ldots\ldots,\mathbf{w}_{2N},}_{N\cdot N_{T}},\ldots\ldots,\mathbf{w}_{ ij},\ldots\ldots,\mathbf{w}_{MN}}_{M\cdot N\cdot N_{T}}}_{M\cdot N\cdot N_{T}} \right]^{T}. \tag{16}\]8 \[\mathbf{w}=\left[\underbrace{\underbrace{\mathbf{w}_{11},\mathbf{w}_{12}, \ldots\ldots,\mathbf{w}_{1j},\ldots\ldots,\mathbf{w}_{1N},\underbrace{\mathbf{ w}_{21},\ldots\ldots,\mathbf{w}_{2N},}_{N\cdot N_{T}},\ldots\ldots,\mathbf{w}_{ ij},\ldots\ldots,\mathbf{w}_{MN}}_{M\cdot N\cdot N_{T}}}_{M\cdot N\cdot N_{T}} \right]^{T}. \tag{16}\]9  
and  
\[\mathbf{\hat{H}}_{ij}=\mathbf{h}_{ij}\mathbf{h}_{ij}^{H},i\in\mathcal{M},j \in\mathcal{N},\text{ and then, let}\]0  
Then, we define  
\[\mathbf{\hat{H}}_{ij}=\mathbf{h}_{ij}\mathbf{h}_{ij}^{H},i\in\mathcal{M},j \in\mathcal{N},\text{ and then, let}\]1  
and  
\[\mathbf{\hat{H}}_{ij}=\mathbf{h}_{ij}\mathbf{h}_{ij}^{H},i\in\mathcal{M},j \in\mathcal{N},\text{ and then, let}\]2Note that the only difference between \(\mathbf{\tilde{\Omega}}_{ij}(\mathbf{w})\) and \(\mathbf{\Omega}_{ij}(\mathbf{w})\) is that \(\mathbf{\Omega}_{ij}(\mathbf{w})\) has one more summation term than \(\mathbf{\tilde{\Omega}}_{ij}(\mathbf{w})\). Similar to (19), we can transform (8) into  
\[ \begin{split} R_{ij}&=\ \log_{2}\left(\frac{\theta_{j}^{ \mathrm{H}}\mathbf{\Omega}_{ij}(\mathbf{w})\mathbf{\theta}_{j}+\sigma^{2}}{ \theta_{j}^{\mathrm{H}}\mathbf{\tilde{\Omega}}_{ij}(\mathbf{w})\mathbf{\theta }_{j}+\sigma^{2}}\right)\\ &=\ \log_{2}\left(\theta_{j}^{\mathrm{H}}\mathbf{\Omega}_{ij}( \mathbf{w})\mathbf{\theta}_{j}+\sigma^{2}\right)-\log_{2}\left(\theta_{j}^{ \mathrm{H}}\mathbf{\tilde{\Omega}}_{ij}(\mathbf{w})\mathbf{\theta}_{j}+ \sigma^{2}\right).\\ &\qquad\qquad i\in\mathcal{M},\ j_{1},j_{2}\in\mathcal{N}.\end{split}  \tag{27}\]  
The partial difference of \(R_{ij}\) for \(\mathbf{\theta}\) can be calculated as  
\[\
abla_{\mathbf{\theta}}R_{ij}(\mathbf{\theta},\mathbf{w})=\left(\frac{2 \theta_{j}^{\mathrm{H}}\mathbf{\Omega}_{ij}(\mathbf{w})}{\theta_{j}^{ \mathrm{H}}\mathbf{\Omega}_{ij}(\mathbf{w})\mathbf{\theta}_{j}+\sigma^{2}}- \frac{2\theta_{j}^{\mathrm{H}}\mathbf{\tilde{\Omega}}_{ij}(\mathbf{w})}{ \theta_{j}^{\mathrm{H}}\mathbf{\tilde{\Omega}}_{ij}(\mathbf{w})\mathbf{\theta }_{j}+\sigma^{2}}\right)/\ln 2,\ i\in\mathcal{M},\ j\in\mathcal{N}. \tag{28}\]  
The first-order Taylor expansion for \(f_{ij}(\mathbf{w}_{ij},\mathbf{\theta}_{ij})\) at \(\mathbf{w}^{[\ell]}\mathbf{\theta}^{[\ell]}\) is denoted as \(F_{ij}(\mathbf{w},\mathbf{\theta};\mathbf{w}^{[\ell]},\mathbf{\theta}^{[\ell]})\):  
\[