# Adaptive Threshold-Aided K-Best Sphere Decoding for Large MIMO Systems
## 4 Simulation Results  
The proposed AKSD and the conventional KSD were tested with various values of \(K\) for comparison. The improved K-best SD (IKSD) algorithm [7] was also considered for comparison. In [7], three threshold rules were proposed, namely, the adaptive, normalized, and fixed thresholds. In this study, for comparison, we tested the adaptive threshold-based IKSD and the fixed threshold-based IKSD. In addition, the schemes in [9; 14] were tested for comparison. The radius adaptive \(K\)-best decoder (RAKSD) in [9] employs tree decomposition and adaptive pruning to reduce the complexity, whereas the dynamic threshold-aided \(K\)-best sphere decoder (DKSD) in [14] exploits a dynamic threshold to adjust \(K\). For the threshold in the proposed AKSD algorithm, the constant \(\alpha\) was set to 1. For the initial radius, we set \(\gamma=5\), as in [8]. The complexity was measured by the average number of visited nodes. Furthermore, the mean values in the simulation results were obtained by averaging over \(10^{7}\) independent channel realizations. For simulations, we considered two large MIMO configurations: \((N_{r},N_{t},\Omega)=(48,48,16)\) and \((64,64,16)\). In Figure 1, the BER comparison results for a MIMO system with \(N_{r}=48\), \(N_{t}=48\), and 16-QAM are shown. We can observe that the AKSD scheme with relatively small \(K\) achieves comparable or better BER performance with respect to the conventional KSD scheme with large \(K\). Specifically, the performance of the proposed AKSD with \(K=50\) is nearly the same as that of the conventional KSD with \(K=512\) at high SNRs, where the AKSD achieves an SNR gain of 0.7 dB over the conventional KSD with \(K=256\) at \(\text{BER}=10^{-4}\). Figure 1 also shows that the AKSD scheme achieves better BER performance compared to the adaptive threshold-based IKSD and fixed threshold-based IKSD. The SNR gains of the AKSD at \(\text{BER}=~{}10^{-4}\) over the adaptive threshold-based IKSD with \(K=200\) and the fixed threshold-based IKSD with \(K=138\) are approximately 0.7 dB and 0.25 dB, respectively. Moreover, the AKSD scheme outperforms both the RAKSD and DKSD by approximately 2.0 dB and 2.5 dB, respectively, at a BER of \(10^{-3}\). Figure 2 compares the BER performance for \(N_{r}=64\), \(N_{t}=64\), and 16-QAM. Similar to Figure 1, it demonstrates that the AKSD scheme achieves nearly the same BER performance with that of the conventional KSD scheme with larger \(K\) at high SNRs. It is also worth noting that the proposed AKSD outperforms the fixed threshold-based IKSD with \(K=400\), achieving an approximate SNR gain of 0.3 dB for \(\text{BER}=10^{-4}\). In Figure 2, it is also observed that the AKSD scheme substantially outperforms the DKSD and RAKSD schemes. In Figure 3, the complexity comparison for a MIMO system with \(N_{r}=48,N_{t}=48\) and 16-QAM is shown. Figure 3 shows that the RAKSD scheme requires lower complexity than the other compared schemes; however, its BER performance is significantly worse than those of the other schemes, as shown in Figure 1. A similar observation can also be made for the DKSD scheme. Figure 3 also shows that the proposed scheme requires lower complexity than the other schemes providing comparable BER performances. Specifically, the complexity-reduction ratio of the AKSD at SNR = 20 dB, compared with the conventional KSD with \(K=512\) is approximately 70%, whereas they achieve comparable BER performance, as shown in Figure 1. Furthermore, the AKSD scheme provides 20% and 31% complexity savings compared with the fixed threshold-based IKSD and the adaptive threshold-based IKSD, respectively. We note that the AKSD scheme outperforms the fixed threshold-based IKSD and adaptive threshold-based IKSD in terms of BER performance. In Figure 3, it is also observed that the complexity reduction of the proposed scheme becomes larger as the SNR increases.