## 5 Performance Evaluations
In Figure 4, the number of iterations for the NED \(10^{-2}\) as a natural number is about 3, 5, 8, and 12, respectively, in \(100\times 20\), \(100\times 30\), \(100\times 40\), and \(100\times 50\) systems, respectively. For clear comparisons, an estimation error between \(i_{\text{opt}}\) and \(i_{\text{Avg}}\) in a given \(\alpha\) is defined as \(\delta_{\alpha}\), as follows,  
\[\delta_{\alpha}=\big{|}i_{\text{opt}}-i_{\text{Avg}}\big{|}. \tag{23}\]  
Figure 6: The BER performances for for GS pre-coding, Richardson pre-coding and 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 16-QAM is used. The calculated \(\delta_{5}\), \(\delta_{3,3}\), \(\delta_{2,5}\), and \(\delta_{2}\) are 0.08, 0.16, 0.22, and 0.27, respectively, which are less than one and these do not cause serious BER loss compared to the ZF pre-coding. However, accuracy is reduced as the number of users is increased, since the diagonal dominance of \(\mathbf{W}\) is proportional to \(\alpha\). Figure 8 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 the initial solution is \(\mathbf{s}_{\text{Dlag}}^{(0)}\) and the 16-QAM is used. In Figure 8, the calculated \(\delta_{5}\), \(\delta_{3,3}\), \(\delta_{2,5}\), and \(\delta_{2}\) are 0.38, 0.64, 0.74, and 1.21, respectively. The estimation errors are higher than the results in Figure 7 since an initial solution is not robust in low \(\alpha\) systems which have poor MSE performances (See Figure 3). Specifically, \(\delta_{2}\) is larger than one and it causes BER loss compared to the ZF pre-coding. Therefore, the proposed scheme uses \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\) as an initial solution for high accuracy and it does not have high complexity since the number of additional multiplications is not large. The number of required multiplications for the GS pre-coding when \(\mathbf{s}_{\text{Dlag}}^{(0)}\) is an initial solution is \(iN_{u}^{2}+N_{t}N_{u}+N_{u}\) which is slightly higher than \(C_{\text{Pro}}\) in Table 1.  
Figure 9 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 the initial solution is \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\) and the 64-QAM is used. In Figure 9, the calculated \(\delta_{5}\), \(\delta_{3,3}\), \(\delta_{2,5}\), and \(\delta_{2}\) are 0.11, 0.14, 0.23, and 0.28, respectively. The estimation errors are similar to the results in Figure 7 since the different modulation gives the SNR gain and the effect of the modulation does not impact on the estimation of the number of iterations. Figure 7: The BER performances for GS pre-coding based on the number of optimal iterations, GS pre-coding based on the proposed estimation scheme, and 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 16-QAM is used. Figure 8: The BER performances for GS pre-coding based on the number of optimal iterations, GS pre-coding based on the proposed estimation scheme, and ZF pre-coding in \(100\times 20\), \(100\times 30\), \(100\times 40\), and \(100\times 50\) systems when an initial solution is \(\mathbf{s}_{\text{Diag}}^{(0)}\) and 16-QAM is used. Figure 9: The BER performances for GS pre-coding based on the number of optimal iterations, GS pre-coding based on the proposed estimation scheme, and 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 64-QAM is used.