# A Robust Hybrid Iterative Linear Detector for Massive MIMO Uplink Systems
## 5 Numerical Results  
Simulation results are presented in this section as well as a comparison between the proposed algorithm and recently introduced mMIMO UL detectors to show the veracity of proposed algorithm. Simulation results are obtained using MATLAB for various iterations and \(\text{BUAR}=\frac{160}{30}=5.33\), \(\frac{160}{40}=4\), \(\frac{160}{50}=3.2\), \(\frac{160}{60}=2.66\). Results and comparisons are provided in bit-error-rate (BER) with the signal-to-noise ratio (SNR). In addition, the computational complexity comparison is also provided in the number of multiplications. A modulation scheme of 64 QAM and the i.i.d Gaussain channels are utilized. Figure 3 shows the BER comparison of proposed algorithm using joint JA and GS with initialization of the stair matrix and other up-to-date methods for \(30\times 160\) antenna configuration (\(\text{BUAR}=5.33\)). It is clear that the proposed algorithm can considerably speed up the convergence rate compared with existing methods. The proposed algorithm achieved the MMSE performance at \(n=2\), whereas the traditional GS, JA, and NS methods require extra iterations which increases the computational complexity. It is illustrated that the proposed algorithm achieved a BER =\(10^{-4}\) at \(\text{SNR}=18\) dB, whereas the GS method can achieve the target performance at \(\text{SNR}=20\) dB. \begin{table}
\begin{tabular}{l l} \hline \hline
**Method** & **Complexity** \\ \hline NS & \((n-2)K^{3}+NK^{2}+NK\) \\ GS & \(4 nK^{2}\) \\ JA & \(n(4K^{2}-2K)\) \\ Proposed Algorithm & \(K^{2}(1+4n)+K-3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity of the NS, GS, JA, and proposed algorithm. Figure 4 illustrates the BER comparison of proposed algorithm and other methods for \(40\times 160\) antenna configuration (\(\text{BUAR}=4\)). It is illustrated that the proposed algorithm achieved a quasi-MMSE performance at \(n=2\), whereas other methods require high number of iterations. For instance, the proposed algorithm achieved a BER = \(10^{-3}\) at \(\text{SNR}=19\) dB while the GS method achieved the same performance at \(=30\) dB. The performance of the NS and JA methods is very disappointed when \(\text{BUAR}=4\). Figure 5 shows the performance comparison for \(50\times 160\) antenna configuration (\(\text{BUAR}=3.2\)). It is illustrated that the performance of proposed algorithm is better than the GS method. For instance,  
Figure 4: Performance of the GS, JA, NS, and proposed algorithm at \(40\times 160\) mMIMO size and \(n=2\). Figure 3: Performance of the GS, JA, NS, and proposed algorithm at \(30\times 160\) mMIMO size and \(n=2\). at \(\text{SNR}=25\) dB, the \(\text{BER}=10^{-3}\) and \(10^{-2}\) for the proposed algorithm and the GS method, respectively. It is also clear that the JA and NS methods are not desirable when BUAR is small. Figures 6-8 show the performance comparison for \(60\times 160\) antenna configuration (\(\text{BUAR}=2.66\)) and \(n=2,3\), and \(4\), respectively. It is illustrated that the proposed algorithm outperforms the others over all iterations within a few number of iterations. For instance, The proposed algorithm achieved the \(\text{BER}=10^{-3}\) at \(\text{SNR}=26\) dB and \(n=3\), whereas the GS method achieved the target performance at \(n=4\). Figure 5: Performance of the GS, JA, NS, and proposed algorithm at \(50\times 160\) mMIMO size and \(n=2\). Figure 6: Performance of the GS, JA, NS, and proposed algorithm at \(60\times 160\) mMIMO size and \(n=2\). Unlike the GS method, the proposed algorithm can achieve the target performance within a small number of iterations. The JA and NS method are not practical when \(\text{BUAR}\ll 5\). The target performance can be achieved within a small number of iterations when we utilize the proposed algorithm. For instance, \(\text{BER}=10^{-3}\) can be achieved when \(n=2\) and \(n=3\) using the proposed algorithm and the GS method, respectively. In case of the proposed algorithm, the number of multiplications is 32,457, whereas it is 43,200 when the GS method is used.