# An Improved Jacobi-Based Detector for Massive MIMO Systems
## 4 Simulation Results  
Numerical simulations are designed to assess the performance of the proposed improved Jacobi iteration algorithm. We provide the simulation results on the BER performance to compare the proposed algorithm with the conventional algorithms. The BER curve of the typical MMSE is used as the benchmark for comparison. The simulation environment is assumed to be the uplink massive MIMO system with the 64-QAM modulation scheme. First, we show simulation results for a massive MIMO system in an uncorrelated Rayleigh fading channel. Figure 4 illustrates the BER performance over a range of signal-to-noise (SNR) between the DJ algorithm and the improved Jacobi-based algorithm, when the complex channels of size \(N_{r}\times N_{t}=128\times 32\). In addition, the number of iteration is denoted by \(K\). As we can see, with the number of iterations is increased, both algorithms can improve the BER performance. However, the DJ algorithm requires massive iterations to meet the BER performance requirement. In comparison, the proposed algorithm can achieve near-optimal performance with a small number of iterations. In addition, compared with the exact MMSE algorithm, the proposed approximated algorithm can achieve a satisfying performance with the relatively small iterations \(K\) (i.e., \(K=3\)). In addition, when \(K=3\), the BER performance of the MMSE algorithm and that of the proposed algorithm has a difference within 0.1 dB. Additionally, the BER performance of the proposed algorithm is compared with the conventional detection algorithms GS and SSOR in Figure 5, when \(N_{r}\times N_{t}=128\times 32\). It is clearly noticed that the BER performances of all algorithms are closer to that of MMSE algorithm when the number of iterations increases. Obviously, the proposed algorithm can achieve the exact performance of the MMSE algorithm when the number of iteration is 3, which requires the computation complexity is \(O\left(8N_{t}^{2}\right).\) The conventional algorithm GS and SSOR algorithm require at least four iterations to achieve the near MMSE performance, which means that the required complexity is \(O\left(9N_{t}^{2}\right)\) and \(O\left(10N_{t}^{2}\right)\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Detectors** & **Complexity (Iteration Times \(K\))** \\ \hline
**MMSE** & \(\frac{5}{6}N_{t}^{3}+\frac{3}{4}N_{t}^{2}+\frac{4}{3}N_{t}\) \\
**GS**[16] & \(\left(K+5\right)N_{t}^{2}\) \\
**SSOR**[17] & \(\left(2N_{t}^{2}+2N_{t}\right)K+2N_{t}^{2}\) \\
**Jacobi**[19] & \(N_{t}^{2}(K+2)\) \\
**Proposed** & \(\left(2N_{t}^{2}+10N_{t}\right)K+2N_{t}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational complexity comparison. Figure 3: Complexity comparison against the number of users with \(K=3\). respectively. As a consequence, the proposed Jacobi-based algorithm outperforms the conventional algorithms in massive MIMO systems. Figure 6 reveals that the advantages of the proposed Jacobi-based algorithm come to an effect when the number of users increases. Note that we can set the number of the receiving antennas at BS as 128 and SNR = 10 dB is applied to the channel. As seen in Figure 6, when the number of single-antenna users increases, the performance of the all algorithms suffers a non-negligible degradation. The proposed algorithm still can achieve almost the same BER performance as the MMSE algorithm with small iterations (i.e., \(K=3\)), regardless of the number of users. In addition, the GS and SSOR algorithms require more iterations (i.e., \(K=4\)) to achieve the near BER performance of MMSE algorithm. Figure 4: Bit error rate (BER) curves of the damped Jacobi (DJ) algorithm and the proposed Jacobi-based algorithm. Figure 5: BER curves of the symmetric successive over-relaxation (SSOR), Gauss Seidel (GS) algorithms and the proposed Jacobi-based algorithm. Finally, for realistic MIMO systems, the spatial correlation plays an important role in the BER performance. Thus, we consider the influence of channel correlation on the performance of BER.