# Joint Interference and Phase Alignment among Data Streams in Multicell MIMO Broadcasting
## 5 MIMO-BC Robust Interference Alignment
### Robust Interference Alignment with Power Distribution among User Data Streams
According to matrix theory, Equation (22) can be written as:  
\[I\left(S_{k_{\rm g}}.\hat{S}_{k_{\rm g}}\right)=\log_{2}\det\left\{I_{d_{k_{\rm g }}}+\mathbf{X}_{k_{\rm g}}F\mathbf{X}_{k_{\rm g}}^{H}\right\} \tag{25}\]  
Among them, \(\mathbf{X}_{k_{\rm g}}\) is the matrix that meets the following condition:  
\[\mathbf{X}_{k_{\rm g}}^{H}\mathbf{X}_{k_{\rm g}}=\mathbf{E}^{-1} \tag{26}\]  
At this point, the received symbol \(\hat{S}_{k_{\rm g}}\) can be written as:  
\[\hat{S}_{k_{\rm g}}=\mathbf{X}_{k_{\rm g}}\hat{\mathbf{U}}_{k_{\rm g}}^{H}H_{k_{\rm g},\rm g}\hat{V}_{k_{\rm g}}\mathbf{W}_{k_{\rm g}}\mathbf{S}+\mathbf{n}_{k_{\rm g}} \tag{27}\]  
Among them, \(\mathbf{n}_{k_{\rm g}}\) is the complex Gaussian white noise vector and obeys the distribution \(CN\left(\mathbf{\theta}_{d_{k_{\rm g}}},\,\mathbf{I}_{d_{k_{\rm g}}}\right);\mathbf{Y}_{k_{ \rm g}}=\mathbf{X}_{k_{\rm g}}\hat{\mathbf{U}}_{k_{\rm g}}^{H}\mathbf{H}_{k_{\rm g},\rm g} \hat{V}_{k_{\rm g}}\) is the equivalent MIMO channel with interference alignment. The \(m\)th largest singular value of the equivalent channel is:  
\[\mathbf{\Lambda}_{k_{\rm g},m}=\sqrt{\text{eig}_{m}^{\text{ascond}}\left\{Y_{k_{ \rm g}}\mathbf{Y}_{k_{\rm g}}^{H}\right\}} \tag{28}\]  
Among them, \(\text{eig}_{m}^{\text{ascond}}(\cdot)\) denotes the \(m\)th largest eigenvalue of the matrix. When the power of the user \(k_{\rm g}\) is \(P\), the optimal solution is the water-filling method power allocation of the MIMO channel [23]:  
\[\frac{\mathbf{P}_{k_{\rm g}}^{m}}{P}=\left\{ \begin{array}{rl}\frac{1}{\gamma_{0} }-\frac{1}{\gamma_{m}},\;\gamma_{m}>\gamma_{0}\\ 0,\;\gamma_{m}<\gamma_{0}\end{array} \right. \tag{29}\]  
Among them, \(\gamma_{0}\) is the threshold value; \(\mathbf{P}_{k_{\rm g}}^{m}\) is the power that should be allocated for the \(m\)th data stream of user \(k_{\rm g}\). Suppose the \(d\) singular values of the equivalent channel matrix \(\mathbf{Y}_{k_{\rm g}}\) corresponding to user \(k_{\rm g}\) are \(\mathbf{\Lambda}_{k_{\rm g},1},\;\mathbf{\Lambda}_{k_{\rm g},2},\;\ldots,\;\mathbf{\Lambda }_{k_{\rm g},d}\), and they are arranged in descending order. The SNR corresponding to each subchannel at full power is \(\gamma_{1}=P\left(\mathbf{\Lambda}_{k_{\rm g},1}\right)^{2},\;\gamma_{2}=P\left( \mathbf{\Lambda}_{k_{\rm g},2}\right)^{2},\ldots,\;\gamma_{d_{i}}=P_{i}\left(\mathbf{ \Lambda}_{k_{\rm g},d}\right)^{2}\). Then calculate the threshold \(\gamma_{0}\) as:  
\[\sum_{m=1}^{d}\left(\frac{1}{\gamma_{0}}-\frac{1}{\gamma_{m}}\right)=1 \tag{30}\]  
At this time, the optimal power allocation of user \(k_{\rm g}\) data stream is \(\mathbf{P}_{k_{\rm g}}^{m}=P\left(\frac{1}{\gamma_{0}}-\frac{1}{\gamma_{m}}\right)\). After the data flow of user \(k_{\rm g}\) is allocated for power, it will satisfy Equation (31):  
\[\mathbf{W}_{k_{\rm g}}\mathbf{S}_{k_{\rm g}}\mathbf{S}_{k_{\rm g}}^{H}\mathbf{W}_{k_{\rm g}}^{ H}=\frac{P}{d}\text{diag}\left(\frac{\mathbf{P}_{k_{\rm g}}^{1}}{\left(P/d\right)},\; \frac{\mathbf{P}_{k_{\rm g}}^{2}}{\left(P/d\right)},\ldots,\;\frac{\mathbf{P}_{k_{\rm g} }^{d}}{\left(P/d\right)}\right) \tag{31}\]  
Further, the power allocation matrix \(\mathbf{W}_{k_{\rm g}}\) is:  
\[\mathbf{W}_{k_{\rm g}}=\text{diag}\left(\sqrt{\frac{\mathbf{P}_{k_{\rm g}}^{1}}{\left(P /d\right)}},\;\sqrt{\frac{\mathbf{P}_{k_{\rm g}}^{2}}{\left(P/d\right)}},\ldots, \;\sqrt{\frac{\mathbf{P}_{k_{\rm g}}^{d}}{\left(P/d\right)}}\right) \tag{32}\]