# 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  
Due to the asymmetry of the MIMO-BC channel, the performance of the algorithm [1] is not ideal. Therefore, in this paper, when designing the receiving filter, only the intercell interference is concerned; when designing the transmitting filter, intercell and intracell interference (excluding interfering signals between user data streams) are handled. In the downlink communication, for the user \(k_{g}\), the interference suppression matrix is designed by maximizing the ratio of the desired signal power to the intercell interference power:  
\[\hat{\mathbf{U}}_{k_{g}}=\arg\quad\max_{\hat{\mathbf{U}}_{k_{g}}\in\mathbb{C}^{M\times d }}\quad\frac{\hat{\mathbf{U}}_{k_{g}}^{H}Z_{k_{g}}\hat{\mathbf{U}}_{k_{g}}}{\hat{\mathbf{ U}}_{k_{g}}^{H}Q_{k_{g}}\hat{\mathbf{U}}_{k_{g}}} \tag{14}\]  
where \(\mathbf{Z}_{k_{g}}\) and \(\mathbf{Q}_{k_{g}}\) are vectors which are obtained by:  
\[\mathbf{Z}_{k_{g}}=\mathbf{H}_{k_{g}g}\hat{V}_{k_{g}}\mathbf{S}_{k_{g}}\mathbf{S}_{k_{g}}^{H} \hat{\mathbf{V}}_{k_{g}}^{H}\hat{H}_{k_{g},\,g}^{H} \tag{15}\]  
\[\mathbf{Q}_{k_{g}}=\sum_{ \begin{array}{c}j=1\\ j\
eq\mathrm{g}\end{array} }^{\mathrm{G}}\sum_{l=1}^{K}\mathbf{H}_{k_{g}j}\hat{V}_{l _{g}}\hat{S}_{l_{g}}\hat{V}_{l_{l}}^{H}\hat{H}_{k_{g},\,j}^{H}+\delta_{n}^{2} \mathbf{I}_{N}\]  
\[= \sum_{ \begin{array}{c}j=1\\ j\
eq\mathrm{g}\end{array} }^{\mathrm{G}}\sum_{l=1}^{K}\frac{P\rho^{2}}{d\left(1 +\delta_{e}^{2}\right)^{2}}\hat{\mathbf{H}}_{k_{g}j\tau}\hat{\mathbf{V}}_{l_{g}}\hat{V }_{l_{g}}^{H}\hat{H}_{k_{g},\,j\tau}^{H}+\left[(G-1)KP\left(1-\frac{|\rho|^{2} }{1+\delta_{e}^{2}}\right)+\delta_{n}^{2}\right]\mathbf{I}_{N} \tag{16}\]  
According to Rayleigh entropy [21], the optimal interference suppression matrix is expressed as:  
\[\hat{\mathbf{U}}_{k_{g}}=\mathbf{v}_{\max}^{d}\bigg{\{}\left(\mathbf{Q}_{k_{g}}\right)^{-1 }\mathbf{Z}_{k_{g}}\bigg{\}} \tag{17}\]where \(\mathbf{v}_{\max}^{d}(\cdot)\) denotes the unit eigenvector corresponding to the first \(d\) largest eigenvalues of the evaluation matrix. In the uplink communication, for the user \(k_{\text{g}}\), the precoding matrix is designed by maximizing the SNR of the user \(k_{\text{g}}\) as follows:  
\[\hat{V}_{k_{\text{g}}}=\arg \begin{array}{c}\max\\ \hat{V}_{k_{\text{g}}}\in\mathbb{C}^{M\times d}\end{array} \frac{\hat{\mathbf{V}}_{ k_{\text{g}}}^{H}\Big{(}\mathbf{A}_{k_{\text{g}}}\Big{)}\hat{V}_{k_{\text{g}}}}{\hat{V}_{k_{ \text{g}}}^{H}\Big{(}\mathbf{B}_{k_{\text{g}}}\Big{)}\hat{\mathbf{U}}_{k_{\text{g}}}} \tag{18}\]  
where vectors \(\mathbf{A}_{k_{\text{g}}}\) and \(\mathbf{B}_{k_{\text{g}}}\) are calculated as:  
\[ \begin{array}{c}\mathbf{A}_{k_{\text{g}}}=\mathbf{H}_{k_{\text{g}},\text{g}}^{H} \hat{\mathbf{U}}_{k_{\text{g}}}\mathbf{S}_{k_{\text{g}}}\mathbf{S}_{k_{\text{g}}}^{H} \hat{\mathbf{U}}_{k_{\text{g}}}^{H}\mathbf{H}_{k_{\text{g}},\text{g}}\\ =\frac{P}{d}\bigg{(}\frac{\rho}{1+\delta_{\text{c}}^{2}}\bigg{)}^{2}\hat{\mathbf{H }}_{k_{\text{g}},\,\text{jtr}}^{H}\hat{\mathbf{U}}_{k_{\text{g}}}\mathbf{\Omega}_{k_{ \text{g}}}^{H}\hat{\mathbf{H}}_{k_{\text{g}},\,\text{jtr}}+P\bigg{(}1-\frac{|\rho| ^{2}}{1+\delta_{\text{c}}^{2}}\bigg{)}\mathbf{I}_{M}\end{array}  \tag{19}\]  
\[\begin{array}{c}\mathbf{B}_{k_{\text{g}}}= \begin{array}{c}\sum\limits_{l=1}^{ K}\\ l\
eq k\end{array} \sum\limits_{m=1}^{d}\frac{p\rho^{2}}{d\big{(}1+\delta_{\text{c}}^{2} \big{)}^{2}}\hat{\mathbf{H}}_{\text{g},\,\text{jtr}}^{H}\mathbf{u}_{\text{j}}^{m}\mathbf{ u}_{\text{j}}^{mH}\hat{\mathbf{H}}_{\text{g},\,\text{jtr}}\\ \hskip 14.226378 pt\begin{array}{c}\hskip 14.226378 pt\begin{array}{c} \hskip 14. The diagonal matrix \(W_{k_{\rm g}}\) is the power distribution matrix and satisfies \(I\left(W_{k_{\rm g}}W_{k_{\rm g}}^{H}\right)=d\).