## 4 Simulation Results  
### BER Performance Evaluation  
The BER performance of the proposed algorithm was evaluated and compared with the ones of other detection algorithms by numerical simulations. The simulation parameters are listed in Table 2. Several typical detection algorithms were selected for comparison, which were introduced in Section 1 and listed as LMMSE, AltMin, and ADMM.  
We first considered the number of iterations for the antenna configurations of \(N_{t}\times N_{r}=16\times 128\), \(32\times 128\) and \(64\times 128\). Figure 2 illustrates the BER performance of the proposed MIDE detection algorithm against the number of iterations, where the SNR was set as 3 dB. It is observed that with the increase of \(N_{r}/N_{t}\), the convergence number of the iterations required by the proposed MIDE was almost the same and a very small one, e.g., 10 in all simulations. This demonstrated a reliable BER performance and a fast detection convergence in our proposed MIDE algorithm. Moreover, with the increase of the transmitting antennas, the diversity gain of the MIMO system decreased, leading to a degradation of the BER performance of the proposed algorithm and the compared MIMO detectors.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Algorithm** & **Complexity** \\ \hline
**LMMSE**[8] & \(\frac{5}{6}N_{t}^{3}+\frac{3}{4}N_{t}^{2}+\frac{4}{3}N_{t}+N_{r}N_{t}^{2}+N_{r} N_{t}\) \\
**AltMin**[18] & \(\left(4K+2\right)N_{r}N_{t}\) \\
**ADMM**[20] & \(\frac{1}{2}N_{t}^{3}+N_{t}^{2}+N_{r}N_{t}^{2}+N_{r}N_{t}+K\left(N_{t}^{2}+N_{r} N_{t}\right)\) \\
**MIDE** & \(N_{t}N_{r}+N_{t}^{2}+K\left(2N_{r}N_{t}+3N_{t}\right)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Complexity comparison in terms of complex multiplication operations. LMMSE, linear minimum mean squared error; AltMin, alternating minimization; ADMM, alternating direction method of multipliers; MIDE, modified iterative discrete estimation.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Channel Model** & **Uncorrelated Rayleigh Flat Fading** \\ \hline Modulation scheme & 16-QAM \\ Number of transmitting antennas (\(N_{t}\)) & 16, 32, 64 \\ Number of receiving antennas (\(N_{r}\)) & 128 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameters.  
Figure 3 depicts the BER performance comparison of the IDE-based detection algorithm with the fixed damping factor \(\alpha~{}=~{}0.05\) and the proposed MIDE detection with the self-updated damping factor. It is clear that the proposed MIDE-based detection with the self-updated damping factor showed better BER performance than the conventional IDE-based detection algorithm with all antenna configurations.  
Moreover, Figure 4 compares the BER performances of the proposed MIDE algorithm and the conventional ADMM algorithm. From the figure, it is clear that the BER performance of both algorithms degraded when the number of users increased. However, it is observed that the proposed MIDE-based detection performed better than the conventional ADMM detection in terms of BER performance with all antenna configurations. Furthermore, we can observe from Figure 4 that when the target of BER was set as \(10^{-3}\), the SNR required by the proposed algorithm was at least 0.5 dB less than the one of the conventional ADMM algorithm.  
Figure 3: BER performance of the conventional damping factor and the proposed damping factor.  
Figure 2: BER performance versus proposed iterations with SNR = 3 dB.  
Finally, we show the BER performance comparison of the conventional AltMin algorithm, the LMMSE algorithm, and the proposed MIDE algorithm in Figure 5.  
It is clear that with any ratio of \(N_{r}/N_{t}\), the proposed MIDE algorithm performed better than all the compared algorithms in terms of BER performance.