# Joint Interference and Phase Alignment among Data Streams in Multicell MIMO Broadcasting
## 6 Algorithm Performance Analysis
### Time Delay Error CSI System and Rate and Bit Error Rate Analysis  
The actual channel is denoted by \(\mathbf{H}\) to analyze the performance of the system. From Section 5.2, it can be seen that in the actual channel environment, the actual received signal of the user \(k_{\text{g}}\) is:  
\[\hat{\mathbf{S}}_{k_{\text{g}}}=\mathbf{H}_{k_{\text{g}},\mathbf{\xi}}\mathbf{\Psi}_{k_{\text{ g}}}^{\theta}\mathbf{c}_{k_{\text{g}}}+\hat{\mathbf{U}}_{k_{\text{g}}}^{H}\times \left\{ \begin{array}{ll}\sum\limits_{l=1}^{K}\mathbf{H}_{k_{\text{g}},\mathbf{\xi}}\hat{\mathbf{V}}_{l_{\text{g}}}\mathbf{W}_{l_{\text{g}}}\mathbf{S}_{l_{\text{g} }}+\sum\limits_{\begin{array}{c}j=1\\ j\
eq\text{g}\end{array}}^{G}\sum\limits_{l=1}^{K}\mathbf{H}_{k_{\text{g}},\hat{j}} \hat{\mathbf{V}}_{l_{\text{l}}}\mathbf{W}_{l_{\text{l}}}\mathbf{S}_{l_{\text{l}}}\\ \end{array} \right\}+\hat{\mathbf{U}}_{k_{\text{g}}}^{H}\mathbf{n}_{k_{\text{g}}} \tag{38}\]  
The true \(\mathbf{R}_{k_{\text{g}}}^{\theta}\) corresponding to Equation (34) is:  
\[\overline{\mathbf{R}}_{k_{\text{g}}}^{\theta}=\overline{\mathbf{H}}_{k_{\text{g}}}\bm {\Psi}_{k_{\text{g}}}^{\theta}=\overline{\mathbf{H}}_{k_{\text{g}}}\Big{(}E\Big{(} \overline{\mathbf{H}}_{k_{\text{g}}}^{H}\Big{)}\mathbf{R}_{k_{\text{g}}}^{-1}\mathbf{R}_{ k_{\text{g}}}^{\theta}\Big{)} \tag{39}\]  
From Equation (13) we know that:  
\[E\Big{[}\overline{\mathbf{R}}_{k_{\text{g}}}^{\theta}\Big{]}=\mathbf{R}_{k_{\text{g}}}^ {\theta} \tag{40}\]  
So, the received signal of the \(w\)th freedom of user \(k\) is:  
\[y_{k_{\text{g}}}^{w}\approx\mathcal{g}_{k_{\text{g}}}^{\theta}c_{w}\sum \limits_{x=1}^{d_{k}}|\rho_{w,x}|+\hat{\mathbf{u}}_{k_{\text{g}}}^{wH}\times \left\{ \begin{array}{ll}\sum\limits_{\begin{array}{c}k\\ l=1\\ l\
eq k\end{array}}^{K}\mathbf{H}_{k_{\text{g}},\hat{j}}\mathbf{W}_{\text{g}}\mathbf{S}_{l_{ \text{g}}}+\sum\limits_{\begin{array}{c}j=1\\ j\
eq\text{g}\end{array}}^{G}\sum\limits_{l=1}^{K}\mathbf{H}_{k_{\text{g}},\hat{j}} \hat{\mathbf{V}}_{l_{\text{l}}}\mathbf{W}_{l_{\text{l}}}\mathbf{S}_{l_{\text{l}}}\\ \end{array} \right\} \tag{41}\]  
where \(\rho_{w,x}\) is the \((w,x)\) element of \(\mathbf{R}_{k_{\text{g}}}\). The residual interference can be expressed as:  
\[ \begin{split} J_{\text{{k}}_{\text{g}}}\approx\hat{\mathbf{u}}_{\text{{k} }_{\text{g}}}^{\text{{i}}\text{{i}}\text{{r}}\text{{i}}}\left\{\begin{array}{ ll}\sum\limits_{l=1}^{K}\mathbf{H}_{\text{{k}}_{\text{g}},\text{{g}}}\hat{V}_{\text{{g}}_{ \text{{i}}}}\mathbf{W}_{\text{{g}}_{\text{{i}}}}\mathbf{S}_{\text{{g}}_{\text{{i}}}} \left(\mathbf{H}_{\text{{k}}_{\text{g}},\text{{g}}}\hat{V}_{\text{{g}}_{\text{{i}} }}\mathbf{W}_{\text{{g}}_{\text{{i}}}}\mathbf{S}_{\text{{g}}_{\text{{i}}}}\right)^{H}\\ l\
eq k\end{array}\right.\\ \left.+\begin{array}{ll}\sum\limits_{l=1}^{G}\mathbf{H}_{\text{{k}}_{\text{g}}, \text{{i}}}\hat{V}_{\text{{i}}}\mathbf{W}_{\text{{i}}_{\text{{i}}}}\mathbf{S}_{\text{{ i}}}\left(\mathbf{H}_{\text{{k}}_{\text{g}},\text{{i}}}\hat{V}_{\text{{i}}_{\text{{i}} }}\mathbf{W}_{\text{{i}}_{\text{{i}}}}\mathbf{S}_{\text{{i}}_{\text{{j}}}}\right)^{H} \\ j\
eq\text{g}\end{array}\right\}\hat{\mathbf{u}}_{\text{{k}}_{\text{g}}}^{w}\end{split}  \tag{42}\]  
In the case of progressively higher SNR, Equation (13) is brought into Equation (42), therefore:  
\[E\left(\mathbf{J}_{\text{{k}}_{\text{g}}}\right)\approx P(GK-1)\left(1-\frac{\left| \rho\right|^{2}}{1+\delta_{\text{e}}^{2}}\right) \tag{43}\]  
The corresponding output average SINR is:  
\[ \begin{split}\text{SINR}_{\text{{k}}_{\text{g}}}^{w}=\frac{E \bigg{\{}\left(\text{g}_{\text{{k}}_{\text{g}}}^{\text{{g}}}\right)^{2}\bigg{\}} }{\left(\delta_{\text{n}}\right)^{2}+\mathbf{J}_{\text{{k}}_{\text{g}}}}\times E \Bigg{\{}\left(\sum\limits_{x=1}^{d_{\text{{k}}_{\text{g}}}}\left|\rho_{w,x} \right|\right)^{2}\Bigg{\}}\end{split}  \tag{44}\]  
The sum rate of the user \(k_{\text{g}}\) can be expressed as:  
\[