# Efficient Massive MIMO Detection for M-QAM Symbols
## 4 Simulation Results  
In this section, numerical results obtained by MATLAB simulations are presented. We consider a large-system limit in which \(K_{r}\) and \(K_{t}\) are tens to hundreds of antennas at a proportional rate \(\rho=\frac{K_{t}}{K_{r}}\leq 1\). To assess the error-rate performance for the proposed INL-BOXDCD algorithm, we perform Monte Carlo simulations of a MIMO system adopting different forms of QAM under a Rayleigh flat fading channel. The simulation trials are \(10^{6}\). In this scenario, \(M_{b}=15\), \(\eta=0.1\). The simulation results are plotted as the bit-error-rate (BER) vs. the signal-to-Gaussian noise ratio (SNR). Figure 1 illustrates the detection performances of the INL-BOXDCD detector for 16 QAM symbols for \(\rho=60/120,60/80\) and \(60/60\) with \(V=1\). We can alter the condition number of the channel correlation matrix \(\mathbf{R}\) by changing the relationship between \(K_{r}\) and \(K_{t}\). When \(\rho=1\), we address fully loaded MIMO systems, and the noise is magnified by  
a factor of a relatively large condition number. Under such conditions, the convergence property of INL-BOXDCD deteriorates. With the decreases in \(\rho\), the performance of the INL-BOXDCD detector significantly increases. Increasing \(V\) might mitigate the negative effects of approximation error. Figure 2 depicts the effect of iterations \(V\) on the BER performance of the INL-BOXDCD detector under 4 QAM, 16 QAM and 64 QAM for a 60 \(\times\) 60 MIMO system with \(N_{u}=300\). For 4 QAM symbol transmission (Figure 2a), the INL-BOXDCD detector with one iteration provides a significant accuracy improvement over zero. However, after one iteration, the BER performance of the INL-BOXDCD detector can no longer be improved. The INL-BOXDCD detector for 16 QAM and 64 QAM signals with \(V=1\) shows poor performance. When \(V\) increases to 10, 20 and 30, the BER gradually decreases. With a given \(N_{u}\), Figure 2b,c show that the INL-BOXDCD exhibits a BER floor that is reduced as \(V\) increases. Figure 3 shows the effect of \(N_{u}\) on the BER performance of the INL-BOXDCD detector for 20 \(\times\) 20, 60 \(\times\) 60 and \(100\times 100\) MIMO systems. With a fixed \(N_{u}\), the INL-BOXDCD displays a BER floor and the BER floor decreases when \(N_{u}\) increases. For a small value of \(N_{u}=300\), the INL-BOXDCD detector for the \(20\times 20\) MIMO system offers better performance than the 60 \(\times\) 60 and \(100\times 100\) MIMO systems. When \(N_{u}\) increases to \(10^{3}\), the INL-BOXDCD detector with large-scale MIMO systems, such as the 60 \(\times\) 60 and 100 \(\times\) 100 systems, shows significantly better performance than the 20 \(\times\) 20 MIMO system. When \(N_{u}\) further increases to \(10^{5}\), the BER performance of the INL-BOXDCD detector for the 60 \(\times\) 60 MIMO system is slightly worse than that of the \(100\times 100\) MIMO system but significantly better than that of the \(20\times 20\) MIMO system. Figure 4 plots the BER performance of the INL-BOXDCD detector for a given SNR for a \(100\times 100\) antenna configuration. The performance of the MMSE, iterated MMSE and BOXDCD are also examined. The figure shows that the performance of the INL-BOXDCD detector improves significantly with increasing \(N_{u}\) and \(V\). The detection performance of the classical MMSE detector is inferior to that of the INL-BOXDCD detector. The INL-BOXDCD (\(N_{u}=10^{3},V=1\)) detector provides a gain of approximately 3 dB over iterated MMSE (\(V=10\)) when BER is less than \(10^{-4}\). The BOXDCD detector with \(N_{u}=300\) provides poor performance. With increasing \(N_{u}\), the detection performance of the BOXDCD detector improves. As \(V\) increases from 1 to 3, the BER performance curve of the INL-BOXDCD detector with \(N_{u}=10^{3}\) gradually approaches that of the BOXDCD detector with \(N_{u}=10^{5}\).