## 4 Proposed Estimation Scheme
### Detailed Expression for Proposed Scheme
The proposed algorithm initializes \(i_{\text{Max}}\) and the algorithm outputs \(i_{\text{Max}}\) when the number of optimal iterations is not estimated at the \((i_{\text{Max}}-1)\)-th loop. Algorithm 1 represents an algorithm for the GS pre-coding based on the proposed estimation scheme. For another initial matrix, only \(i_{\text{Max}}\) and \(\mathbf{s}^{(0)}\) are varied. In Table 3, \(\beta=N_{u}/(N_{t}-N_{u})\) is an approximate normalization factor for the transmit power using the distribution of Wishart matrix in [30] as follows,  
\[\mathbf{s}_{\text{NS,Stair}}^{(0)}=\left(\sum_{n=0}^{1}\left(\mathbf{I}_{N_{n}}- \mathbf{S}^{-1}\mathbf{W}\right)^{n}\right)\mathbf{S}^{-1}\mathbf{x}. \tag{12}\]1  
where \(\text{tr}(\cdot)\) is trace operator. The \(\beta\) is perfectly the same as \(\text{tr}\Big{(}\mathbf{W}^{-1}\Big{)}\) when \(\alpha\) is larger or equal to 2 [14]. Finally, the proposed scheme sets \(\eta\) as 0.01 where the proposed algorithm outputs \(\hat{i}\) when the NED between consecutive GS solutions is less than 1%. The \(\eta=0.01\) is the same as 1% since the total summation for the NEDs is one. However, the set threshold is not an optimal value, and slight errors between GS solutions based on the number of optimal iterations and estimated iterations occurred due to the threshold. However, its errors almost vanish by using digital demodulation. The calculation of an optimal threshold requires further study in future work. \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{1}{c}{**Scheme**} & \multicolumn{1}{c}{**The Number of Multiplications**} \\ \hline GS pre-coding without estimation \((C_{\text{GS}})\) & \(C_{\text{GS}}=(i+4)N_{u}^{2}+(N_{t}+4)N_{u}-4\) \\ \cline{2-3}  & \(\mathbf{s}^{(0)}_{\text{NS}_{\text{Max}}}:\ 4N_{u}^{2}+4N_{u}-4\)[12] \\  & \(\mathbf{s}^{(i)}:\ N_{u}^{2}\) \\  & \(\mathbf{H}^{*}\mathbf{s}^{(0)}:N_{t}N_{u}\) \\ \hline GS pre-coding based on proposed estimation & \(C_{\text{Pro}}=(i+4)N_{u}^{2}+(i+N_{t}+4)N_{u}-4\) \\ scheme \((C_{\text{Pro}})\) & \(\cdot\ \text{GS}:(i+4)N_{u}^{2}+(N_{t}+4)N_{u}-4\) \\  & \(\cdot\ \text{g}^{(1)}\), \(\text{g}^{(2)}\), \(\cdots\), \(\text{g}^{(0)}:\ N_{u}\) \\ \hline ZF pre-coding \((C_{\text{ZF}})\) & \(C_{\text{ZF}}=N_{u}^{3}+2N_{u}^{2}+N_{t}N_{u}\) \\ \cline{2-3}  & \(\cdot\ \mathbf{W}^{-1}:\ N_{u}^{3}+N_{u}^{2}\) (Gauss - Jordan) [28,29] \\  & \(\cdot\ \mathbf{H}^{\text{H}}\Big{(}\mathbf{W}^{-1}\mathbf{x}\Big{)}:\ N_{u}^{2}+N_{t }N_{u}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number of required multiplications for conventional GS pre-coding, GS pre-coding based on the proposed estimation scheme, and ZF pre-coding. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{1}{c}{**Value or Scheme**} \\ \hline Digital modulation & 16-quadrature amplitude modulation (QAM), 64-QAM \\ \hline Channel coding & 1/2 rate low density parity check (LDPC) \\ \hline \(N_{t}\) & 100, 256 \\ \hline \(N_{u}\) & 20, 30, 40, 50, 64, 128 \\ \hline The number of cells & Single cell \\ \hline Channel model & 8 multi-path Rayleigh fading \\ \hline AWGN (Gaussian noise) & Zero mean and unit power \\  & (Base station (BS) controls transmit power \(P\)) \\ \hline Duplexing mode & Time division duplexing (TDD) \\ \hline Channel estimation & Least square (LS) based on uplink pilot \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The simulation parameters used.