# Beam Grouping Based RS Resource Reuse and De-Contamination in Large Scale MIMO Systems
## 5 Numerical Results
\right),\]  
By using Label (13) and Label (14), \(\mathbf{y}\) can be rewritten as follows:  
\[\mathbf{y} = \sqrt{\rho_{t}}\zeta\left(\begin{array}{ccccc}\mathbf{g}_{1} \mathbf{A}_{1}\mathbf{a}^{H}(\vec{\theta}_{1})&\mathbf{g}_{1}\mathbf{A}_{1} \mathbf{a}^{H}(\vec{\theta}_{2})&\cdots&\mathbf{g}_{1}\mathbf{A}_{1}\mathbf{a }^{H}(\vec{\theta}_{i})&\cdots&\mathbf{g}_{1}\mathbf{A}_{1}\mathbf{a}^{H}( \vec{\theta}_{K})\\ \mathbf{g}_{2}\mathbf{A}_{2}\mathbf{a}^{H}(\vec{\theta}  
Figure 8 shows the SE (bps/Hz) versus SNR (dB) with \(N_{t}=320,K=32\), \(\chi_{DM-RS}=10\), \(v_{BGRR}=32,16,8\), and MF group precoding. We assume that one cell has a 180-degree angle.  
We only show the case of WCR because the case of WOCR would show the similar characteristics, while it gives more improvement of performance with the proposed BGRR scheme. In the case without the BGRR scheme, the number of TX antennas is limited to 273 as shown in the previous section. It is obvious that the DUD gives better performance than the SUD. In the case of the SUD, the UE based grouping scheme (red solid line with 'O' marks) shows better performance than the angle based grouping scheme (blue solid'marks). Note that the distribution of UEs is not even and MF cannot remove the interference completely in the angle based grouping scheme. Some of the groups could have a lot of UEs and the inter-user interference due to the MF precoding can result in performance loss.  
Figure 9 shows the SE (bps/Hz) versus SNR (dB) performance, when the ZF group precoding is used with the same parameters in Figure 8.  
As shown in the figure, the performance is different in the case of SUD. In the high SNR region, especially, the performance of the angle based grouping scheme is superior to that of the UE based grouping scheme. This is due to the fact that ZF is near optimum in the high SNR region and can remove inter-user interference completely, even when more UEs are in the same group. In the case of UE based grouping scheme, all of the groups have the same UEs, which means that group angle can be very small, and it causes bad channel conditions. This means that, to get enhanced performance, we should choose between UE based grouping and angle based grouping schemes depending on the situations. Note that the performance trend is maintained as the number of TX antennas and/or UEs is increased.  
Figures 10 and 11 show the same case with Figures 8 and 9 except that the number of TX antenna is increased from 320 to 640. The trend is the same as the previous case, but the performance difference between \"with BGRR\" and \"without BGRR\" becomes more than the previous case due to higher beamforming effect.  
Figure 8: spectral efficiency (SE) (bps/Hz) versus signal-to-noise ratio (SNR) (dB), when \(N_{t}=320\), \(K=32\), \(\chi_{DM-RS}=10\), \(v_{BGRR}=32,16,~{}8\), matched filtering (MF) group precoding. Red solid lines with ‘O’ marks are the UE based grouping and blue solid ‘ marks are the angle based grouping.  
Figure 10: SE versus SNR (dB), when \(N_{t}=640,K=32\), \(\chi_{DM-RS}=10\), \(v_{BGRR}=32,16,8\), MF group precoding. Red solid lines with ’O’ marks are the UE based grouping and blue solid ’ marks are the angle based grouping.  
Figure 9: SE (\(\mathrm{bps}/\mathrm{Hz}\)) versus SNR (dB), when \(N_{t}=320,K=32\), \(\chi_{DM-RS}=10\), \(v_{BGRR}=32,16,8\), zero-forcing (ZF) group precoding. Red solid lines with ’O’ marks are the UE based grouping and blue solid ’ marks are the angle based grouping.  
Since the LS-MIMO is usually used for the DUD case, we can expect drastic performance gain by the proposed BGRR scheme. From Figures 12 and 13, the expected improvement of performance is 444.7% (\(85\to 378\) (\(\text{bps}/\text{Hz}\))) in the case of the MF group precoding scheme, and 340.5% (\(111\to 378\) (\(\text{bps}/\text{Hz}\))) in the case of the ZF group precoding scheme.  
As another analysis, we present the case of the channel estimation error of the sounding reference signals. We model the estimated channel as follows:  
\[\mathbf{\hat{H}}=\xi\mathbf{H}+\sqrt{1-\xi^{2}}\mathbf{E}, \tag{17}\]  
where \(\mathbf{\hat{H}}\) is the estimated channel matrix, \(\xi\in[0,\,1]\) is the error factor which reflects the degree of channel estimation error, and \(\mathbf{E}\in\mathbb{C}^{K\times\mathbb{N}_{t}}\) is the error matrix with the same statistical characteristic but independent of the channel.