# Joint Interference and Phase Alignment among Data Streams in Multicell MIMO Broadcasting
## 5 MIMO-BC Robust Interference Alignment
### Phase Alignment  
In the time delay error CSI, the phase alignment is used to rotate the interference between the data streams into the signal space of the target data stream to enhance the received power of the target data stream [24].  
After processing in Section 5.1, the signal for user \(k_{\text{g}}\) is:  
\[\hat{S}_{k_{\text{g}}}=\overline{\boldsymbol{H}}_{k_{\text{g}}}S_{k_{\text{g}}}+ \sum_{\begin{array}{c}j=1\\ j\
eq k\end{array}}^{K}\overline{\boldsymbol{H}}_{\text{j}}S_{\text{j}}+\sum_{ \begin{array}{c}j=1\\ j\
eq k\end{array}}^{G}\overline{\boldsymbol{H}}_{\text{j}}S_{\text{j}}+ \overline{\boldsymbol{n}}_{k_{\text{g}}} \tag{33}\]  
where \(\overline{\boldsymbol{H}}_{k_{\text{g}}}=\boldsymbol{\hat{H}}_{k_{\text{g}}}^ {H}\boldsymbol{H}_{k_{\text{g}},\text{g}}\hat{V}_{k_{\text{g}}}W_{k_{\text{g}}}\); \(\overline{\boldsymbol{H}}_{\text{j}}=\boldsymbol{\hat{U}}_{k_{\text{g}}}^{H} \boldsymbol{H}_{k_{\text{g}},\text{g}}\hat{V}_{\text{j}}\boldsymbol{W}_{\text {j}}\); \(\overline{\boldsymbol{H}}_{\text{j}}=\boldsymbol{\hat{U}}_{k_{\text{g}}}^{H} \boldsymbol{H}_{k_{\text{g}},\text{j}}\hat{V}_{\text{j}}\boldsymbol{W}_{\text {j}}\); \(\overline{\boldsymbol{n}}_{k_{\text{g}}}=\boldsymbol{\hat{U}}_{k_{\text{g}}}^ {H}\boldsymbol{n}_{k_{\text{g}}}\).  
Here, the rate is given by:  
\[\boldsymbol{R}_{\text{g}}=E\left(\overline{\boldsymbol{H}}_{k_{\text{g}}} \right)E\left(\overline{\boldsymbol{H}}_{k_{\text{g}}}^{H}\right) \tag{34}\]  
Based on the MMSE criterion, the transmitted signal and the received signal are phase aligned [18], and the interference signal is reconstructed. At this time, the resulting constraint expression is:  
\[l_{k_{\text{g}}}=\text{min}E\bigg{\{}\left\|\boldsymbol{R}_{k_{\text{g}}}^{ \theta}\boldsymbol{c}_{k_{\text{g}}}-\left(\overline{\boldsymbol{H}}_{k_{ \text{g}}}\boldsymbol{\Psi}_{k_{\text{g}}}^{\theta}\boldsymbol{c}_{k_{\text{g }}}+\overline{\boldsymbol{n}}_{k_{\text{g}}}\right)\right\|^{2}\bigg{\}}\]  
\[\text{s.t.}\boldsymbol{R}_{k_{\text{g}}}^{\theta}=\left|\boldsymbol{R}_{k_{ \text{g}}}\right|\odot\left(\epsilon_{k_{\text{g}}}c_{k_{\text{g}}}^{H}\right) \tag{35}\]  
Among them, \(\boldsymbol{\epsilon}_{k_{\text{g}}}\in\mathbb{C}^{d_{k_{\text{g}}}\times 1}\) consists of MPSK constellation points and satisfies \(\boldsymbol{\epsilon}_{k_{\text{g}}}^{H}\boldsymbol{\epsilon}_{k_{\text{g}}}= d_{k_{\text{g}}}\), and \(\odot\) denotes the Hadamard product of the matrix. Solving Equation (35) we get:  
\[\boldsymbol{\Psi}_{k_{\text{g}}}^{\theta}=E\left(\overline{\boldsymbol{H}}_{ k_{\text{g}}}^{H}\right)\boldsymbol{R}_{k_{\text{g}}}^{-1}\boldsymbol{R}_{k_{ \text{g}}}^{\theta} \tag{36}\]  
where \(\boldsymbol{\Psi}_{k_{\text{g}}}^{\theta}\) is the estimated rate matrix of the MPSK constellation. Thus, the transmit signal of the user \(k_{\text{g}}\) can be expressed as:  
\[\boldsymbol{S}_{k_{\text{g}}}=\mathsf{g}_{k_{\text{g}}}^{\theta}\boldsymbol{ \Psi}_{k_{\text{g}}}^{\theta}\boldsymbol{\epsilon}_{k_{\text{g}}}\]  
\[\text{s.t.}\mathsf{g}_{k_{\text{g}}}^{\theta}=\frac{1}{\sqrt{trace\bigg{[} \left(\boldsymbol{R}_{k_{\text{g}}}^{\theta}\right)^{2}\boldsymbol{R}_{k_{ \text{g}}}^{-1}\bigg{]}}} \tag{37}\]  
From such results, it is possible to minimize the constraint (35).