## 5 Performance Evaluations  
For performance evaluations of the proposed estimation scheme, the bit error rate (BER) performances and the complexity are measured. However, for a better understanding of the BER performances, the average magnitude for the diagonal dominance of \(\mathbf{W}\) and mean square error (MSE) performances between the ZF pre-coding symbols and the GS pre-coding symbols with respect to various initial solutions are shown in advance. In addition, the NEDs with respect to the number of users are shown for clear visualization of the results in Figure 2. Table 3 represents the used parameters for performance evaluations. In Table 3, the uplink transmit power for channel estimation is set as 3 dB less than the downlink transmit power \(P\). The proposed scheme uses an initial solution as \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\). However, the additional performances are evaluated by using an initial solution \(\mathbf{s}_{\text{Diag}}^{(0)}\) which is used in [15; 22; 25] for showing high accuracy of the proposed estimation scheme. The average magnitude for the diagonal dominance of \(\mathbf{W}\) is calculated as follows,  
\[v=\frac{1}{N_{u}}\sum\limits_{m=1}^{N_{u}}\frac{w_{mm}}{\sum\limits_{k=1,k\
eq m }^{N_{u}}|w_{mk}|}, \tag{20}\]  
where \(w_{mk}\) is the entry for the \(m\)-th row and the \(k\)-th column of \(\mathbf{W}\). The average magnitude for the diagonal dominance of \(\mathbf{W}\) decreases as the number of users increases since the law of large numbers is weak in a low \(\alpha\) system. Figure 3 shows the MSE performances between the ZF pre-coding symbols and the GS pre-coding symbols with respect to various initial solutions in (8), (9), (11), and (12). An initial solution based on the two term NS has a better MSE performance than the initial solution without the NS since the NS accelerates the inversion of the initial matrix to \(\mathbf{W}^{-1}\). However, the MSE performances for an initial solution based on the two term NS are converged to the MSE performances for an initial solution without the NS since the diagonal dominance of \(\mathbf{W}\) decreases as \(\alpha\) is decreased. In addition, an initial solution based on the stair matrix has a better MSE performance than an initial solution based  
on the diagonal matrix since the stair matrix includes non-diagonal entries additionally unlike the diagonal matrix. Figure 4 shows the NEDs in (17) with respect to the number of iterations when \(\mathbf{s}^{(0)}_{\text{NS,Stair}}\) and \(\mathbf{s}^{(0)}_{\text{Dlag}}\) are initial solutions. The NED using \(\mathbf{s}^{(0)}_{\text{NS,Stair}}\) as an initial solution is lower than the NED using \(\mathbf{s}^{(0)}_{\text{Dlag}}\) as an initial solution since the stair matrix includes additional non-diagonal entries of \(\mathbf{W}\) and NS is applied. In addition, the NED decreases as the number of iterations increases since the diagonal dominance of \(\mathbf{W}\) is proportional to \(\alpha\). Figure 5 shows the BER performances for the GS pre-coding, the Richardson pre-coding--which is one of popular low-complexity ZF methods in [11; 12; 13]--and the ZF pre-coding in \(100\times 50\) systems when an initial solution is \(\mathbf{s}^{(0)}_{\text{NS,Stair}}\) and the 16- quadrature amplitude modulation (QAM) is used. The Richardson solution is perfectly approximated to the ZF solution in any \(\alpha\) system like the GS pre-coding. In Figure 5, parameters \(L_{\text{G}}\) and \(L_{\text{R}}\) are the number of iterations in the GS pre-coding and the Richardson pre-coding, respectively. The importance of estimation of the number of optimal iterations  
Figure 4: The normalized Euclidean distance (NEDs) with respect to the number of iterations when the initial solutions are \(\mathbf{s}^{(0)}_{\text{NS,Stair}}\) and \(\mathbf{s}^{(0)}_{\text{Dlag}}\). Figure 3: The mean square error (MSE) performances between ZF pre-coding symbols and GS pre-coding symbols with respect to different initial solutions.