# Comparative Analysis of Data Detection Techniques for 5G Massive MIMO Systems
## 5 Numerical Results  
In this paper, we utilize independent and identically distributed (i.i.d) Gaussian channels, the MIMO size is \(32\times 128\), and \(32\times 256\), the modulation order is 64 QAM, and several iterations are considered to attain the target. The performance of the proposed initialized AMP-based detector is presented in BER versus the signal-to-noise ratio (SNR). Figure 3 illustrates the performance of a detector based on the initialized AMP detector using the SOR, the GS, and the JA methods when \(n=2\) and MIMO size is \(32\times 128\). The AMP detector initialized by JA, GS, and SOR methods outperforms the conventional AMP-based detector. In other words, a large number of iterations is required to achieve the target performance if the efficient initialization is absent. For instance, at SNR = 30 dB, the BER is close to \(10^{-3}\) and \(10^{-2}\) for the proposed AMP-GS and conventional AMP, respectively.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Method** & **Number of Multiplications** \\ \hline AMP & \(2 KNn\) \\ \hline AMP-SOR & \(2 KNn+4K^{2}+4K\) \\ \hline AMP-GS & \(2 KNn+4K^{2}\) \\ \hline AMP-JA & \(2 KNn+4K^{2}-2K\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Complexity comparison among proposed detectors.  
When \(n=3\), the performance of the proposed detectors is significantly improved, and it is still better than the original AMP detector (Figure 4). For instance, at SNR = 26 dB, the BER is \(10^{-4}\) and \(10^{-3}\) for the initialized AMP-based detector (AMP-GS and AMP-SOR) and the AMP-based detector without initialization, respectively. It is also clear that all proposed initialization methods achieve better performance than the original AMP-based detector.  
To study the impact of MIMO size in proposed algorithms, the performance of a \(32\times 256\) MIMO size is also presented. Figure 5 presents the performance of 32 MIMO system when \(n=2\). The proposed initialized detectors are outperforming the original AMP detector. The BER is \(10^{-4}\) at SNR = 24 dB and  
Figure 4: Performance of initialized AMP-based detector for \(32\times 128\) massive MIMO system, 64 QAM, and \(n=3\).  
Figure 3: Performance of initialized approximate message passing (AMP)-based detector for \(32\times 128\) massive multiple-input multiple-output (MIMO) system, 64 QAM, and \(n=2\).  
26 dB in proposed detectors and AMP detectors, respectively. It is also clear that extra iterations are required in the AMP to achieve the MMSE performance.  
Figure 6 compares the performance-complexity profile of proposed detectors and the original AMP detector to attain Bit-error-rate \(\text{BER}=10^{-4}\). Initialized AMP-based detector achieves the target BER with a small number of iterations, and hence, low computational complexity. The AMP-Jacobi (AMP-JA)-based detector achieved the target performance with the lowest computational complexity. In contrast, the AMP-based detector without initialization requires a high number of iterations, and thus, high complexity is required to achieve the target performance. As shown in Section 2, iterative methods (JA, GS, and SOR) require a small number of iterations to achieve a high performance.  
Figure 5: Performance of initialized AMP-based detector for \(32\times 256\) massive MIMO system, 64 QAM, and \(n=2\).  
Figure 6: Performance–complexity trade-off to obtain \(\text{BER}=10^{-4}\) in \(32\times 128\).