## 5 Performance Evaluations
is shown, where the GS pre-coding and the Richardson pre-coding have a lower BER performance than the ZF pre-coding since the number of iterations are insufficient. Figure 6 shows the BER performances for the conventional GS pre-coding, the Richardson pre-coding, and the ZF pre-coding in a \(100\times 50\) system with respect to distance between the BS and users when an initial solution is \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\) and the 16-QAM is used. For measuring the BER performances, the signal to noise ratio (SNR) value of user \(S_{d}\) is calculated according to given distance \(d\) as follows:  
\[S_{d}=10\log 10(P_{W}P_{R})-N_{0}, \tag{21}\]  
where \(P_{W}\) is total transmit power at the BS, \(P_{R}\) is path loss, and \(N_{0}\) is the thermal noise power of the user. \(P_{R}\) is calculated as follows:  
\[P_{R}=10^{\frac{-20\log\left(\frac{4\pi f_{c}}{10}\right)-10\log d+C_{t}}{10}}, \tag{22}\]  
where \(f_{c}\) is carrier frequency, \(c\) is speed of light, \(n\) is path loss exponent, and \(G_{t}\) is the transmit antenna gain, respectively. Table 4 represents the used simulation parameters in Figure 6. Like the results in Figure 5, the GS pre-coding and the Richardson pre-coding have lower BER performances than the ZF pre-coding since the number of iterations is not enough. Again, an estimation of the number of optimal iterations is important in the iterative pre-coding method. \begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Value** \\ \hline \multirow{2}{*}{\(P_{W}\)} & 37 dBm (6 dBm less than maximum total radiated power in 5G new radio (NR) standards) \\ \hline \(f_{c}\) & 3.5 GHz \\ \hline \(c\) & \(3\times 10^{8}\)m/s \\ \hline \(n\) & 3 \\ \hline \(G_{t}\) & 17 dB \\ \hline \(N_{0}\) & \(10\log B-204\) (\(B\): Bandwidth which is set as 100 MHz) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The used simulation parameters in Figure 6.  
Figure 5: The bit error rate (BER) performances for GS pre-coding, Richardson pre-coding, and ZF pre-coding in \(100\times 50\) system when an initial solution is \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\) and 16-QAM is used. Figure 7 shows the BER performances for the GS pre-coding based on the number of optimal iterations, the GS pre-coding based on the proposed estimation scheme, and the ZF pre-coding in \(100\times 20\), \(100\times 30\), \(100\times 40\), and \(100\times 50\) systems when an initial solution is \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\) and the 16-QAM is used. In Table 5, the number of optimal iterations \(i_{\text{opt}}\) for the GS pre-coding is represented when the initial solutions are \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\) and \(\mathbf{s}_{\text{Diag}}^{(0)}\). The \(i_{\text{opt}}\) is defined as the number of minimum iterations which does not cause variation of the GS solution. In addition, \(i_{\text{opt}}\) is obtained by simulations in advance where the authors find \(i_{\text{opt}}\) when the GS solution in (6) has the same value as the ZF solution. However, \(i_{\text{opt}}\) can be differently calculated according to the channel environment and it is merely used as a performance comparison of the proposed estimation scheme. In Table 5, the initial solution \(\mathbf{s}_{\text{Diag}}^{(0)}\) requires more iterations compared to the initial solution \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\). In Figure 7, \(i_{\text{Avg}}\) is defined as the average value of total estimated \(\hat{i}\)s for 100,000 iterations of Monte-Carlo simulations. The \(i_{\text{Avg}}\) is calculated as 3.08, 4.84, 7.78, and 12.27, respectively, in \(100\times 20\), \(100\times 30\), \(100\times 40\), and \(100\times 50\) systems, respectively. The proposed scheme estimates the number of iterations and finishes an algorithm when the NED between consecutive GS solutions is less than 1%. The estimated results are related to the NEDs in Figure 4 based on the stair matrix with two term NS.