## 3 CB and Power Control
### CB with Instantaneous CSI  
With CB precoding, the \(m\)th AP transmits the signal  
\[x_{m}=\sqrt{\rho_{ap}}\sum_{i=1}^{K}\sqrt{\eta_{mi}}\hat{g}_{mi}^{*}\hat{s}_{i}, \tag{13}\]  
Figure 3: The pilot is in a block within a \(\tau_{c}\) and \(B_{c}\), and in different blocks, pilot will be multiplexing.  
where \(\rho_{ap}\) is the transmit power of each AP, and \(s_{i}\) is the data signal to the \(i\)th terminal, with \(\mathbb{E}(|s_{i}|^{2})=1\). \(\eta_{mi}\) is the power control coefficient. The signal received by the \(k\)th terminal is  
\[y_{k}=\sum_{m=1}^{M}g_{mk}x_{m}+w_{k}=DS_{k}+UI_{kk^{\prime}}+ES+w_{k}, \tag{14}\]  
where \(DS_{k}\), \(UI_{kk^{\prime}}\), and \(ES\) represent the strength of the desired signal, the interference caused by the \(k^{\prime}\) terminal, and the error signal caused by the channel estimation error, respectively. \(DS_{k}\), \(UI_{kk^{\prime}}\), and \(ES\) are expressed as follows:  
\[DS_{k}=\sqrt{\rho_{ap}}\sum_{m=1}^{M}\sqrt{\eta_{mk}}\hat{g}_{mk}\hat{g}_{mk}^ {*}s_{k}, \tag{15}\]  
\[UI_{kk^{\prime}}=\sqrt{\rho_{ap}}\sum_{i\
eq k}^{K}\sum_{m=1}^{M}\sqrt{\eta_{ mi}}\hat{g}_{mk}\hat{g}_{mi}^{*}s_{i}, \tag{16}\]  
\[ES=\sqrt{\rho_{ap}}\sum_{i=1}^{K}\sum_{m=1}^{M}\sqrt{\eta_{mi}}\hat{g}_{mi} \hat{g}_{mi}^{*}s_{i}. \tag{17}\]  
The gain, \(\rho_{ap}\left(\sum_{m=1}^{M}\sqrt{\eta_{mk}}\hat{g}_{mk}\hat{g}_{mk}^{*} \right)^{2}\), obtained by the terminal via downlink is used for reception. Unlike the scheme in the previous literature [8; 13; 14; 15], the channel hardening is no longer exploited, and channel statistical information, \(\sqrt{\rho_{ap}}\mathbb{E}[\sum_{m=1}^{M}\sqrt{\eta_{mk}}\hat{g}_{mk}\hat{g}_{ mk}^{*}]\), is not used for reception. However, accurate estimated CSI participates in receiving processing. An achievable rate of the \(k\)th terminal with CB is \(SE_{k}=log_{2}(1+\text{SINR}_{k})\), and the expression of \(\text{SINR}_{k}\) is shown as follows:  
\[\text{SINR}_{k}=\frac{\rho_{ap}\left(\sum_{m=1}^{M}\sqrt{\eta_{mk}}\hat{g}_{ mk}\hat{g}_{mk}^{*}\right)^{2}}{\rho_{ap}\sum_{i\
eq k}^{K}\left(\sum_{m=1}^{M} \sqrt{\eta_{mi}}\hat{g}_{mk}\hat{g}_{mi}^{*}\right)^{2}+\rho_{ap}\sum_{i=1}^{K }\sum_{m=1}^{M}\eta_{mi}\hat{g}_{mi}\hat{g}_{mi}^{*}\hat{g}_{mk}\hat{g}_{mk}^ {*}+\sigma_{w}^{2}}. \tag{18}\]  
\(\hat{g}_{mk}\) is the channel estimation error and cannot be obtained accurately. Therefore, substitute \(\mathbb{E}[\hat{g}_{mk}\hat{g}_{mk}^{*}]\) for \(\hat{g}_{mk}\hat{g}_{mk}^{*}\) as follows:  
\[\text{SINR}_{k}=\frac{\rho_{ap}\left(\sum_{m=1}^{M}\sqrt{\eta_{mk}}\hat{g}_{mk }\hat{g}_{mk}^{*}\right)^{2}}{\rho_{ap}\sum_{i\
eq k}^{K}\left(\sum_{m=1}^{M} \sqrt{\eta_{mi}}\hat{g}_{mk}\hat{g}_{mi}^{*}\right)^{2}+\rho_{ap}\sum_{i=1}^{ K}\sum_{m=1}^{M}\eta_{mi}\hat{g}_{mi}\hat{g}_{mi}^{*}\mathbb{E}[\hat{g}_{mk} \hat{g}_{mk}^{*}]+\sigma_{w}^{2}}. \tag{19}\]  
The interference term in (19), \((\sum_{m=1}^{M}\sqrt{\eta_{mi}}\hat{g}_{mk}\hat{g}_{mi}^{*})^{2}\), appears complex in power control calculations, and optimization tools, such as convex programming (CVX), cannot process such terms directly. Thus, the real part and the imaginary part are handled separately.  
\[\left(\sum_{m=1}^{M}\sqrt{\eta_{mi}}\hat{g}_{mk}\hat{g}_{mi}^{*}\right)^{2}= \left(\sum_{m=1}^{M}\sqrt{\eta_{mi}}\hat{g}_{mk}\hat{g}_{mi}^{*}\right)^{2}+ \left(\sum_{m=1}^{M}\sqrt{\eta_{mi}}\hat{\Im}(\hat{g}_{mk}\hat{g}_{mi}^{*}) \right)^{2}. \tag{20}\]