## 3 Closed-Form Approximation of Achievable Spectral Efficiency with Channel Estimation Error
Using (11) and statistical approximation, The effective SINR based on ZF precoding with channel estimation error can be written as follows [39]:  
\[\hat{\gamma}_{k,ZF} = \frac{\left|\zeta_{k,ZF}\mathbf{h}_{k}(\mathbf{\hat{H}}\mathbf{ \hat{H}}^{H})^{-1}\mathbf{\hat{h}}_{k}^{H}\right|^{2}}{\mathbf{h}_{k}(\mathbf{ \hat{H}}\mathbf{\hat{H}}^{H})^{-1}\mathbf{\hat{H}}_{[k]}\mathbf{\hat{H}}_{[k] }^{H}(\mathbf{\hat{H}}\mathbf{\hat{H}}^{H})^{-1}\mathbf{h}_{k}^{H}\sum \
olimits_{l\
eq k}\zeta_{l,ZF}+1}, \tag{24}\] \[\rightarrow \frac{N_{t}-K}{K}\left(\frac{\xi^{2}\rho_{r}}{(1-\xi^{2})\rho_{r }+1}\right),\]  
where \(\mathbf{\hat{H}}_{[k]}^{H}=[\mathbf{\hat{h}}_{1}^{H},\mathbf{\hat{h}}_{2}^{H },\cdots,\mathbf{\hat{h}}_{k-1}^{H},\mathbf{\hat{h}}_{k+1}^{H},\cdots,\mathbf{ \hat{h}}_{K}^{H}]\in\mathbb{C}^{N_{t}\times(K-1)}\). For (24), the following approximations are used:  
\[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]0 \[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]1\[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]2 \[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]3 \[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]4 \[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]5  
From (17) and (24), we can see that the SINR based on ZF precoding with channel estimation error is scaled down by \(\xi^{2}\) and \((1-\xi^{2})\) for the signal and the interference power from reference SINR based on MF precoding, respectively. Obviously, the residual IUI term for ZF precoding is due to channel estimation error. The SINRs for the cases of reference MF precoding, reference ZF precoding, MF precoding with channel estimation error, and ZF precoding with channel estimation error are summarized in \[\hat{\gamma}_{k,mf}=\frac{\rho_{r}\left|\zeta_{k,mf}\mathbf{h}_{k}\mathbf{ \hat{h}}_{k}^{H}\right|^{2}}{\rho_{r}\sum\limits_{l\
eq k}\left|\zeta_{l,mf} \mathbf{h}_{k}\mathbf{\hat{h}}_{l}^{H}\right|^{2}+1}, \tag{18}\]6  
If we put the SINR in the Table 2 into the Equation (16), we can get the theoretical achievable SE of RS overhead reduction scheme with channel estimation error. In a real situation, the channel estimation error due to any kind of interference can be reflected in error factor \(\xi\). The RS overhead reduction performance can be also reflected in RS overhead reduction factor and/or the number of antenna in each group, \(N_{g}\). \(N_{g}\) increment guarantees \(N_{t}\) increment. However, increasing \(N_{g}\) is not always beneficial. As we can see from Equation (12), \(N_{g}\) increment also causes \(\xi\) increment. Obviously, \(\xi\) increment can cause performance reduction. In the case of RZF precoding, the SINR is shown in Equation (19) of [39] and Equation (17) of [40]. Substituting the SINR of RZF, \(\gamma_{k,RZF}\) into Equation (16), one can obtain the achievable SE of RZF precoding. Since the expressions in [39; 40] are too complex to use in real system analysis, we show the simplified expression using the SINRs of MF and ZF precoding. Basically in low power regime, the SINR of RZF precoding is similar to MF precoding.