## 4 Proposed Estimation Scheme
### Detailed Expression for Proposed Scheme  
For estimating the number of optimal iterations, the proposed scheme deals with an initial solution \(\mathbf{s}^{(0)}\) and normalized Euclidean distance (NED) between \(\mathbf{s}^{(i)}\) and \(\mathbf{s}^{(i-1)}\). The GS solution \(\mathbf{s}^{(i)}\) is converged to the ZF solution \(\mathbf{s}_{\text{ZF}}\) regardless of an initial solution. The \(\mathbf{s}_{\text{ZF}}\) is calculated as follows,  
\[\mathbf{s}_{\text{ZF}}=\mathbf{W}^{-1}\mathbf{x}. \tag{7}\]  
However, an initial solution is an important parameter for accelerating the convergence rate of the GS pre-coding. One of the widely used initial solutions in the GS is the zeros vector, since it is not complex. The authors in [14; 23] used the zeros vector as an initial solution. However, the convergence rate is very slow since diagonal entries of \(\mathbf{W}\) are approximated to \(N_{t}\) in the massive MIMO system. For efficient usage of the massive MIMO system, the authors in [15; 22; 25] used an initial solution \(\mathbf{s}^{(0)}_{\text{Diag}}\) as follows,  
\[\mathbf{s}^{(0)}_{\text{Diag}}=\mathbf{D}^{-1}\mathbf{x}. \tag{8}\]  
The \(\mathbf{s}^{(0)}_{\text{Diag}}\) is a robust initial solution when the diagonal dominance of \(\mathbf{W}\) is large. In [27], the authors proved that \(\mathbf{W}\) is a fully diagonal dominant matrix when \(\alpha\) is larger than \(5.83\). Therefore, the authors in [14; 16; 17; 18; 19; 20; 22; 23] evaluated performances in the massive MIMO systems where \(\alpha\) is larger or equal to six. In addition, the required complexity for calculating \(\mathbf{s}^{(0)}_{\text{Diag}}\) is low since \(\mathbf{D}\) is a diagonal matrix. However, \(\mathbf{s}^{(0)}_{\text{Diag}}\) is not robust when \(\alpha\) is less than \(5.83\). For accelerating the convergence rate of the GS solution, the authors in [20; 21; 24] used an initial solution \(\mathbf{s}^{(0)}_{\text{NS,Diag}}\) using a two term-NS based on \(\mathbf{D}\) as follows,  
\[\mathbf{s}^{(0)}_{\text{NS,Diag}}=\left(\sum_{n=0}^{1}\Bigl{(}\mathbf{I}_{N_{ u}}-\mathbf{D}^{-1}\mathbf{W}\Bigr{)}^{n}\right)\mathbf{D}^{-1}\mathbf{x}, \tag{9}\]  
where \(\mathbf{I}_{m}\) is \(m\times m\) identity matrix. Therefore, the authors in [21; 24] evaluated performances in the massive MIMO systems, where \(\alpha\) is larger or equal to four, unlike the authors in [14; 16; 17; 18; 19; 20]. Recently, the stair matrix is a newly emerging initial matrix since the robustness of the stair matrix is higher than the diagonal matrix with slight additions of the multiplications. The stair matrix \(\mathbf{S}\) is satisfied with one of the following conditions:  
\[ \begin{array}{l}-\mathbf{S}_{m\ m-1}=0,\ \mathbf{S}_{m\ m+1}=0,\\ \text{where}\ m=2,4,\cdots,2\Bigl{\lfloor}\frac{N_{u}}{2}\Bigr{\rfloor}.\\ -\mathbf{S}_{m\ m-1}=0,\ \mathbf{S}_{m\ m+1}=0,\\ \text{where}\ m=1,3,\cdots,2\Bigl{\lfloor}\frac{N_{u}-1}{2}\Bigr{\rfloor}+1. \end{array} , \tag{10}\]  
where \(\lfloor\cdot\rfloor\) is floor operator. The authors in [17; 18; 19] used an initial solution \(\mathbf{s}^{(0)}_{\text{Stair}}\) for accelerating the convergence rate of the GS solution as follows,  
\[\mathbf{s}^{(0)}_{\text{Stair}}=\mathbf{S}^{-1}\mathbf{x}. \tag{11}\]  
However, in [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25], the trials for an initial solution as a two term NS based on the stair matrix in the GS were not revealed. In addition, a robust initial solution is required for a high accuracy of estimation and fast convergence of the GS solution in the massive MIMO systems where \(\alpha\) is less than \(5.83\).