## 4 Proposed Estimation Scheme
### Detailed Expression for Proposed Scheme
After the calculation of \(\mathbf{s}^{(0)}\), the proposed algorithm is initiated with \(i=1\) and the first NED \(\overline{e}^{(1)}\) is calculated as follows,  
\[\overline{e}^{(1)}=e^{(1)}=\big{|}\mathbf{s}^{(1)}-\mathbf{s}^{(0)}\big{|}, \tag{14}\]  
where \(\overline{e}^{(1)}\) is the same as \(e^{(1)}\) since \(\mathbf{s}^{(0)}\) is obtained without the GS pre-coding and the ED between \(\mathbf{s}^{(0)}\) and \(\mathbf{s}_{\text{ZF}}\) is large in low \(\alpha\) like the results shown in Figure 2. However, the result \(\overline{e}^{(1)}\) is used in the next loop for the normalization factor. The second NED \(\overline{e}^{(2)}\) is calculated as follows,  
\[\overline{e}^{(2)}=\frac{e^{(2)}}{\gamma^{(2)}}=\frac{\big{|}\mathbf{s}^{(2)} -\mathbf{s}^{(1)}\big{|}}{\gamma^{(2)}}, \tag{15}\]  
where \(\gamma^{(i)}\) is the \(i\)-th normalization factor and is calculated by summation from \(e^{(1)}\) to \(e^{(i)}\) except for \(\gamma^{(1)}\) as follows,  
\[\gamma^{(1)}=1, \tag{16}\] \[\gamma^{(i)}=\sum\limits_{k=1}^{i}e^{(k)},\;i=2,3,\cdots.\]  
The NED which is represented as the ratio stands for the relative expression of the ED since \(\gamma^{(i)}\) is the result of the summation of the EDs from \(e^{(1)}\) to \(e^{(i)}\). Then, \(\overline{e}^{(2)}\) is compared with a threshold \(\eta\). The number of optimal iterations \(\hat{i}\) is estimated as two when \(\overline{e}^{(2)}\) is lower than \(\eta\). On the other hand, the next loop for \(i=3\) is performed when \(\overline{e}^{(2)}\) is larger than \(\eta\) since \(\mathbf{s}^{(i)}\) is not converged to \(\mathbf{s}_{\text{ZF}}\). In this way, the \(i\)-th NED \(\overline{e}^{(i)}\) is calculated as follows,  
\[\overline{e}^{(i)}=\frac{e^{(i)}}{\gamma^{(i)}}=\frac{\big{|}\mathbf{s}^{(i)} -\mathbf{s}^{(i-1)}\big{|}}{\gamma^{(i)}},\;i=1,2,\cdots. \tag{17}\]  
An algorithm ends when \(e^{(\hat{i})}\) is lower than \(\eta\) and the number of estimated iterations is \(\hat{i}\). Finally, the GS pre-coding symbols vector \(\mathbf{s}\) in (6) is generated by using the matched filter to the GS solution \(\mathbf{s}^{(\hat{i})}\). Table 1 represents the number of required multiplications for the conventional GS pre-coding, the GS pre-coding based on the proposed estimation scheme, and the ZF pre-coding. In Table 1, only the number of multiplications is considered since the main operation is multiplication. The GS pre-coding based on the proposed estimation scheme has nearly the same complexity as conventional GS pre-coding since the additional operations for the GS pre-coding based on the proposed estimation scheme are only calculations of \(\overline{e}^{(i)},\;i=1,2,\cdots,\hat{i}\) and it requires only \(\hat{i}N_{u}\) multiplications where an order of the complexity with respect to \(N_{u}\) is merely one. However, the GS pre-coding based on the proposed estimation scheme has a greater complexity than the ZF when the number of estimated iterations is high. The massive MIMO systems where \(\alpha\) is low and an inappropriate initial matrix is used require a high number of iterations for optimal error performance. Therefore, the number of maximum iterations \(i_{\text{Max}}\) should be previously limited. The difference of the number of multiplications between the ZF pre-coding and the GS pre-coding based on the proposed estimation scheme is defined as \(C_{\text{D}}=C_{\text{ZF}}-C_{\text{Pro}}\), and \(C_{\text{D}}\), calculated as follows,  
\[\mathbf{s}_{\text{NS,Stair}}^{(0)}=\left(\sum_{n=0}^{1}\left(\mathbf{I}_{N_{n}}- \mathbf{S}^{-1}\mathbf{W}\right)^{n}\right)\mathbf{S}^{-1}\mathbf{x}. \tag{12}\]0  
In (18), \(i_{\text{Max}}\) is initialized as the maximum \(i\) which is satisfied with the condition for \(C_{\text{D}}>0\) using given \(N_{u}\). Table 2 represents \(i_{\text{Max}}\) with respect to the number of users when \(N_{t}\) is 100 and an initial matrix is \(\mathbf{s}^{(0)}_{\text{NS,Stair}}\). In (18), \(i_{\text{Max}}\) is calculated as \(i_{\text{Max}}=N_{u}-4\).