## 3 Closed-Form Approximation of Achievable Spectral Efficiency with Channel Estimation Error
\tag{16}\]  
where \(K^{\prime}\) and \(\gamma^{\prime}_{k}\) represent corresponding constant when overhead is 0.5. We show the reference SINR based on MF precoding. By using channel hardening effect of the massive MIMO systems [4], the reference SINR based on MF precoding, \(\gamma^{ref}_{k,mf}\), can be simplified as follows [38]:  
\[\gamma^{ref}_{k,mf}=\frac{\rho_{r}|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{h}_{k}^{ H}|^{2}}{\rho_{r}\sum\limits_{l\
eq k}|\zeta_{l,mf}\mathbf{h}_{k}\mathbf{h}_{l}^{ H}|^{2}+1}\rightarrow\frac{N_{t}}{K}\left(\frac{\rho_{r}}{\eta_{mf}+1} \right)=\frac{N_{t}}{K}\left(\frac{\rho_{r}}{\rho_{r}+1}\right), \tag{17}\]  
where \(\mathfrak{I}_{mf}\) is the IUI term after MF precoding. For massive MIMO region, \(\mathfrak{I}_{mf}\) can be simplified to \(\rho_{r}\)[4]. Here we can say that \(\frac{Nt}{K}\) is the channel gain of the massive MIMO system. It is obvious that if we increase the number of TX antennas or reduce the number of UEs, we can get a more effective SINR. However, it should be noted that MF precoding does not completely remove the IUI in practical number of TX antennas. It is known that ZF precoding can completely remove IUI with the perfect channel information, but channel gain is reduced to \(N_{t}-K\) as compared with MF precoding, \(N_{t}\). When there is a channel estimation error, the effective SINR based on MF precoding can be represented as follows:  
\[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]where \(\mathbf{\hat{h}}_{k}\) is the \(1\times N_{t}\) estimated channel vector for \(k\)-th UE. Using (11) and statistical approximation, (18) can be simplified as follows:  
\[\hat{\gamma}_{k,mf} = \frac{\rho_{r}\left|\zeta_{k,mf}\xi\mathbf{h}_{k}\mathbf{h}_{k}^{H} +\zeta_{k,mf}\sqrt{1-\xi^{2}}\mathbf{h}_{k}\mathbf{e}_{k}^{H}\right|^{2}}{\rho _{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf}\xi\mathbf{h}_{k}\mathbf{h}_{l}^{H} +\zeta_{l,mf}\sqrt{1-\xi^{2}}\mathbf{h}_{k}\mathbf{e}_{l}^{H}\right|^{2}+1}, \tag{19}\] \[\rightarrow \frac{N_{t}}{K}\left(\frac{\xi^{2}\rho_{r}}{\rho_{r}+1}\right),\]  
where \(\mathbf{e}_{l}\) is the \(1\times N_{t}\) channel estimation error vector. For Equation (19), we use the following approximations:  
\[\left|\zeta_{k,mf}\xi\mathbf{h}_{k}\mathbf{h}_{k}^{H}+\zeta_{k,mf}\sqrt{1- \xi^{2}}\mathbf{h}_{k}\mathbf{e}_{k}^{H}\right|^{2}\approx\xi^{2}\left|\zeta _{k,mf}\mathbf{h}_{k}\mathbf{h}_{k}^{H}\right|^{2}, \tag{20}\]  
\[\sum\limits_{l\
eq k}\left|\zeta_{k,mf}\xi\mathbf{h}_{k}\mathbf{h}_{l}^{H}+ \zeta_{k,mf}\sqrt{1-\xi^{2}}\mathbf{h}_{k}\mathbf{e}_{l}^{H}\right|^{2}\approx \sum\limits_{l\
eq k}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{h}_{l}^{H}\right| ^{2}. \tag{21}\]  
The approximation of (20) and (21) comes from the following equations:  
\[\left|\zeta_{k,mf}\xi\mathbf{h}_{k}\mathbf{h}_{k}^{H}\right|^{2}\gg\left| \zeta_{k,mf}\sqrt{1-\xi^{2}}\mathbf{h}_{k}\mathbf{e}_{k}^{H}\right|^{2}, \tag{22}\]  
\[\sum\limits_{l\
eq k}\left|\zeta_{k,mf}\xi\mathbf{h}_{k}\mathbf{h}_{l}^{H}+ \zeta_{k,mf}\sqrt{1-\xi^{2}}\mathbf{h}_{k}\mathbf{e}_{l}^{H}\right|^{2}\approx \sum\limits_{l\
eq k}\left|\zeta_{k,mf}\xi\mathbf{h}_{k}\mathbf{h}_{l}^{H}+ \zeta_{k,mf}\sqrt{1-\xi^{2}}\mathbf{h}_{k}\mathbf{h}_{l}^{H}\right|^{2}. \tag{23}\]  
By comparing (17) and (19), it can be seen that the SINR based on MF precoding with channel estimation error has the same characteristics, but the signal power is scaled down by the square of the error factor, \(\xi\).