## 5 Numerical Results and Discussion  
In this Section, we verify the proposed scheme using monte-carlo (MC) simulations with \(10^{4}\) iterations, and present the related discussions. The simulation parameter for this section is shown in Table 3.  
We used \(K=0.1N_{t}\) to meet the minimum requirement of channel hardening effect. The noise power \(N_{0}B\) is normalized to unity with \(\rho_{t}=80\) W and \(B=20\) MHz bandwidth. Assuming we use a macro-cell type setup with 2 GHz carrier frequency, the path loss in dB is modeled as \(128.1+37.6\log(d)\) with distance \(d\) in kilometers [41]. Then, the setup is equivalent to average user distance of 1.194 km. We set \(\xi=1\) and \(\epsilon=300\) in Equation (12). Obviously, the parameter can be changed depending on the applied technique and circumstances. Before showing the numerical results, we present the seriousness of channel estimation error in Figure 5.  
Figure 5: Seriousness of channel estimation error: normalized mean square error (NMSE) versus \(N_{g}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Coherence Time, \(\tau_{c}\) & 5 ms \\ Coherence Bandwidth, \(BW_{c}\) & 180 kHz \\ max RS time, \(\tau_{p}\) & 2.5 ms \\ Total symbols in coherence interval, \(S_{tot}\) & 840 \\ TX power, \(\rho_{t}\) & 80 W \\ Signal Bandwidth, \(B\) & 20 MHz \\ Carrier Frequency, \(f_{c}\) & 2 GHz \\ Path Loss Model & ETSI \\ Number of TX antennas, \(N_{t}\) & 100\(\sim\)1000 \\ Max. number of serviced UEs for downlink, & 100 \\ Error factor without RS overhead reduction, \(\xi\) & 1 \\ Precoding & ZF, MF \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation Parameters. We show the seriousness of channel estimation error using normalized mean square error (NMSE) which is defined as:  
\[NMSE=\frac{\left\|\mathbf{H}-\mathbf{\hat{H}}\right\|_{2}^{2}}{\left\|\mathbf{H }\right\|_{2}^{2}}. \tag{34}\]  
\(\mathbf{\hat{H}}\) is the estimated channel and it is a function of \(N_{g}\) as we shown in the previous section. NMSE is linearly increased as \(N_{g}\) increases. When \(N_{g}\) is 40, \(NMSE\) has reached around 14%, while we can get bandwidth benefit thanks to the RS overhead reduction. Figure 6 shows the numerical results of SE versus \(N_{t}\). Red \"*'s and black 'x' indicate the simulation results, and lines are plotted based on theoretical analysis. Our theoretical analysis is well-matched with the simulation result. When \(N_{g}\) is small which is less than 4, due to the excessive RS overhead which reaches the maximum allowable RS overhead, SE is not increased, even though \(N_{t}\) is increased. However, when \(N_{g}\) is larger than 8, the SE increases, as \(N_{t}\) increases up to 1000. RS overhead reduction technique is definitely necessary to increase \(N_{t}\) and SE, and fully enjoy the benefit of massive MIMO. However, increasing \(N_{g}\) causes channel estimation error as we have seen from previous Section. Increasing \(N_{g}\) gives a bandwidth advantage by giving more room for data, but due to channel estimation error, SINR becomes worse. To show the relationship between SE and \(N_{g}\), we present Figure 7, SE versus \(N_{g}\). As observed, if \(N_{g}\) is larger than 16, the SE is reduced due to worsened SINR. It is noticeable in ZF precoding because as \(N_{g}\) grows, IUI terms grows. The IUI term in ZF precoding is almost negligible when there is little channel estimation error. In the case of MF precoding, there is no SE reduction, but SE is fixed to a certain point even though \(N_{g}\) increases. This is because IUI is already a dominant term in MF precoding even when there is little channel estimation error. When we use MF precoding, if \(N_{g}\) is larger than 16, the system is already working in a power limited regime, thus increasing \(N_{g}\) gives little help in increasing SE. This phenomenon is maintained even if we increase \(N_{t}\). Now, we show the effectiveness of the closed-form equation for \(N_{g}\) determination. Figures 8 and 9 show the determined \(N_{g}\) versus SE.