## 4 Proposed Estimation Scheme
### Detailed Expression for Proposed Scheme
Therefore, this paper uses an initial solution \(\mathbf{s}^{(0)}_{\text{NS,Stair}}\) as a two term NS based on the stair matrixfor improving the accuracy of estimation and accelerating the convergence rate of the GS solution compared to the exiting studies as follows,  
\[\mathbf{s}_{\text{NS,Stair}}^{(0)}=\left(\sum_{n=0}^{1}\left(\mathbf{I}_{N_{n}}- \mathbf{S}^{-1}\mathbf{W}\right)^{n}\right)\mathbf{S}^{-1}\mathbf{x}. \tag{12}\]  
The \(\mathbf{s}^{(i)}\) is converged to \(\mathbf{s}_{\text{ZF}}\) as the number of iterations is increased. Figure 2 shows the GS pre-coding symbol \(s_{m}^{(i)}\) in complex plane with respect to the number of increased iterations in \(100\times 20\), \(100\times 30\), \(100\times 40\), and \(100\times 50\) systems. The case (a) and (b) are results for an initial matrix of \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\) and \(\mathbf{s}_{\text{Diag}}^{(0)}\), respectively. In Figure 2, the number of maximum iterations and the symbol index \(m\) are set as eight and one, respectively. The generality is not lost due to the fixed symbol index since the statistical properties of \(x_{m}\) and \(\mathbf{W}\) are not varied. In both cases (a) and (b), \(s_{m}^{(i)}\) is converged to \(s_{m,\text{ZF}}\) as the number of iterations is increased where \(s_{m,\text{ZF}}\) is the \(m\)-th entry of \(\mathbf{s}_{\text{ZF}}\) since \(\mathbf{W}\) has the property of a Hermitian positive definite. In addition, the ED between \(s_{m}^{(i)}\) and \(s_{m}^{(i-1)}\) is decreased as \(i\) is increased. Therefore, the ED is a monotonic decrease function with respect to \(i\). Finally, the convergence rate of \(s_{m}^{(i)}\) accelerates more quickly when \(\alpha\) is increased, since the diagonal dominance of \(\mathbf{W}\) is proportional to \(\alpha\). Additionally, the convergence rate for the case (a) is faster than for case (b) since \(\mathbf{S}\) is closer to \(\mathbf{W}\) than \(\mathbf{D}\). However, \(s_{m}^{(i)}\) in the \(100\times 50\) system is not perfectly converged to \(s_{m,\text{ZF}}\) in both cases (a) and (b) since the number of maximum iterations is insufficient. For estimating the number of optimal iterations, one can consider the absolute method which is a mathematical analysis for obtaining closed form solutions of \(E\big{[}\mathbf{s}^{(i)}\big{]}\). However, an approach of the closed form solution has several disadvantages as follows:  
First, the mathematical analysis is very complex since (5) includes an iterative method with respect to \(i\). Second, the closed form solution should be analyzed according to \(\mathbf{s}^{(0)}\). Third, the main advantages for the massive MIMO systems, such as the channel hardening effect and favorable propagation, are not gradually valid as \(\alpha\) is decreased, and it causes severe errors for several random matrix theories. Therefore, this paper proposes the relative method using the NED which is a relative value of the ED for estimating the number of optimal iterations. The \(i\)-th NED \(\overline{e}^{(i)}\) is calculated at each \(i\) and \(i\) is estimated when the \(\hat{i}\)-th NED is lower than a threshold which should be set as a small number for optimal error performance before an algorithm is performed. The proposed estimation scheme which uses the relative method does not require complex analysis and an algorithm is not varied regardless of an initial solution and \(\alpha\). For general expressions, the ED \(e^{(i)}\) is calculated as follows,  
\[e^{(i)}=\big{|}\mathbf{s}^{(i)}-\mathbf{s}^{(i-1)}\big{|},\ i=1,2,\cdots, \tag{13}\]  
Figure 2: The \(s_{m}^{(i)}\) in complex plane with respect to the number of iterations in \(100\times 20\), \(100\times 30\), \(100\times 40\), and \(100\times 50\) systems: (**a**) An initial solution for \(\mathbf{s}_{\text{NS,Stair}}^{(0)}\); (**b**): An initial solution for \(\mathbf{s}_{\text{Diag}}^{(0)}\). where \(\mathbf{s}^{(0)}\) is initialized as \(\mathbf{s}^{(0)}_{\text{NS,Stair}}\) for improving the estimation accuracy and accelerating convergence rate of the GS solution.