## 3 Closed-Form Approximation of Achievable Spectral Efficiency with Channel Estimation Error  
In this Section, we present the closed-form approximation of achievable SE with channel estimation error. We should keep in mind that antenna grouping-based RS overhead reduction scheme causes channel estimation error. As \(N_{g}\) increases, channel estimation error increases. The estimated channel, \(\hat{\mathbf{H}}\), is modeled as follows:  
\[\hat{\mathbf{H}}=\xi\mathbf{H}+\sqrt{1-\xi^{2}}\mathbf{E}, \tag{11}\]  
where \(\xi\in[0,\,1]\) is the error factor, which reflects the degree of channel estimation error, and \(\mathbf{E}\in\mathbb{C}^{K\times N_{t}}\) is the error matrix with the same statistical characteristic but independent of the channel, \(\mathbf{H}\). \(\xi\) can also be modeled as follows:  
\[\xi=\sqrt{\xi\left(1-\frac{N_{g}}{\varepsilon}\right)}, \tag{12}\]  
where \(\xi\) is the channel estimation error regardless of antenna group, and \(\varepsilon\) is the adjustable factor to reflect the seriousness of the channel estimation error depending on the applied technique. For the performance analysis, the maximum achievable SE can be derived by using i.i.d. Rayleigh channel with zero mean and unit variance. From (1), and (7), the symbol received by \(k\)-th user is given by  
\[y_{k}=\sqrt{p_{t}}\xi_{k}\left(\mathbf{h}_{k}\circ\mathbf{b}_{k}\right)\mathbf{ f}_{k}x_{k}+n_{k}+\sqrt{p_{t}}\xi_{l}\sum_{l\
eq k}\left(\mathbf{h}_{k}\circ \mathbf{b}_{k}\right)\mathbf{f}_{l}x_{l}, \tag{13}\]  
where \(\mathbf{h}_{k}\) and \(\mathbf{b}_{k}\) are the \(1\times N_{t}\) channel vectors for \(k\)-th user, and \(\mathbf{f}_{k}\) is the \(N_{t}\times 1\) precoding vector for \(k\)-th UE. The last term of (13) is the IUI. \(\zeta_{k}\) is a normalized factor for the precoding process of \(k\)-th UE, and it is approximated as \(\zeta_{k,m\
eq}\approx\frac{1}{\sqrt{N_{t}K}}\) for MF precoding and \(\zeta_{k,ZF}\approx\sqrt{\frac{N_{t}-K}{K}}\) for ZF precoding [35] as we mentioned in the previous section. Assuming all the path loss component to UE \(k\) is the same,which is true in a real system, the effective SINR at the receiver (RX) for user \(k\), \(\gamma_{k}\), can be expressed as follows:  
\[\gamma_{k}=\frac{\rho_{r}|\zeta_{k}\mathbf{h}_{k}\mathbf{f}_{k}|^{2}}{\rho_{r}| \sum\
olimits_{l\
eq k}\zeta_{l}\mathbf{h}_{k}\mathbf{f}_{l}|^{2}+1}, \tag{14}\]  
where \(\rho_{r}=\frac{\rho_{r}\cdot\beta_{k}}{N_{0}B}\) is the received signal-to-noise ratio at RX, \(\beta_{k}\) is the large scale fading or path loss component between TX and \(k\)th UE, and \(N_{0}B\) is the noise power in the given bandwidth, \(B\). As was shown in the previous section, path loss components from 1 st, 2 nd, 3 rd, \(\cdots\), \(N_{t}\)th antenna in BS to antenna in \(k\)th UE are the same (\(\beta_{i}=\beta_{i,1}=\beta_{i,2},\cdots,\beta_{i,N_{t}}\)). This is because the distances among antennas in BS are much shorter than the distances between antennas in BS and antennas in UEs. The data rate, \(R\) of single isolated cell can be represented as follows:  
\[R=\alpha\cdot B\cdot\sum\limits_{k=1}^{K}\mathbb{E}\left[\log_{2}\left(1+ \gamma_{k}\right)\right], \tag{15}\]  
where \(\alpha\) is the scaling factor for RS overhead [4], and \(\mathbb{E}[\cdot]\) is the expectation operation. In the ideal case, i.e., perfect blind channel estimation with no channel estimation error, \(\alpha\) is equal to 1. In a real situation, however, by using (10), we can say that \(\alpha=\left(1-\frac{\chi_{r}}{100}\right)\), when a RS overhead reduction scheme is applied. Since the scaling factor should be more than 0.5 in the system design perspective, the spectral efficiency (SE) based on capacity analysis can be represented as follows:  
\[SE\approx\left\{ \begin{array}{l}\alpha\cdot\sum\limits_{k=1}^{K}\mathbb{E} \left[\log_{2}\left(1+\gamma_{k}\right)\right],\ \alpha>0.5.\\ 0.5\cdot\sum\limits_{k=1}^{K^{\prime}}\mathbb{E}\left[\log_{2}\left(1+\gamma^ {\prime}_{k}\right)\right],\ \alpha\leq 0.5.\end{array} \right.