# Virtual Signal Processing-Based Integrated Multi-User Detection
## 4 Simulation Results
As the figure shows, for fixed \(K\), the BER using the \(\max(||h||)\) criterion outperforms the random referential symbol selection. This is because random selection is more likely to produce referential symbol errors, leading to an incorrect final symbol set. Figure 4: Variation of the probability of direct applying IMUD with \(K\) under \(N_{T}=1\), \(N_{R}=2\), and various \(\varepsilon\)s.  
Figure 5: Impact of SIC on the BER of signal components (without using IMUD). Figure 7 simulates the system BER of VSP-IMUD under \(K=3\), \(N_{R}=2\), \(\varepsilon=0.1\), and various \(N_{T}\)s. The figure shows that the BER performance improves as \(N_{T}\) increases. This improvement is due to the enhanced signal processing gain achieved by the Tx-side array when equipped with more antennas, thereby improving the overall system BER. Figure 8 compares the system BER of VSP-IMUD, ICom/SIC-ZF, and minimum mean square error (MMSE)-SIC under \(N_{T}=1\), \(N_{R}=2\), \(\varepsilon=0.1\), and \(K\in\{3,5,7\}\). We assume \(K>N_{R}\) in the simulations. In this scenario, conventional ZF and MMSE become inapplicable. To make MMSE applicable when \(K>N_{R}\), we treat \(K-N_{R}+1\) signal components as an equivalent signal, similar to ICom/SIC-ZF. As the figure shows, under VSP-IMUD, the system BER decreases as \(K\) increases. This occurs because, as \(K\) rises, the probability of channel ambiguity occurring also slightly increases. In this situation, although the Rx needs to employ ZF to decode the signals transmitted via the ambiguous channels and subsequently use SIC to remove the detected signals from the mixed signal, the number of SIC applications remains small. That is, the increase in the probability of needing SIC is limited as \(K\) rises, which minimally contributes to the degradation of system BER. Furthermore, as \(K\) increases, the referential symbol can be selected from a larger set of candidate signals, improving the correctness of the referential symbol due to selective diversity gain. Thus, the BER performance of VSP-IMUD improves as \(K\) increases. In contrast, ICom/SIC-ZF and MMSE-SIC require more SIC applications as \(K\) grows, resulting in degraded BER performance as \(K\) increases. Furthermore, VSP-IMUD achieves the lowest system BER,  
Figure 6: BER of VSP-IMUD under \(\varepsilon=0.1\) using various referential symbol selection criteria. Figure 7: System BER of VSP-IMUD under \(K=3\), \(N_{R}=2\), \(\varepsilon=0.1\), and various \(N_{T}\)s.  
followed by MMSE-SIC, while ICom/ZF-SIC exhibits the worst system BER. This is because the reduced reliance on SIC in VSP-IMUD enhances the decoding of multiple user signals, resulting in superior system BER compared to the other SIC-based schemes. Moreover, MMSE-SIC outperforms ICom/ZF-SIC in the medium-\(\gamma\) regime and converges to the performance of ICom/ZF-SIC at low- and high-\(\gamma\) regimes. This is because, when \(\gamma\) is low, noise is the dominant factor affecting the system BER; thereby, even MMSE outperforms ZF in balancing interference suppression and desired signal preservation, and both ICom/ZF-SIC and MMSE-SIC yield similar BER performance. As \(\gamma\) increases, CCI dominates the system BER; therefore, MMSE-SIC outperforms ICom/ZF-SIC in system BER. When \(\gamma\) becomes too large, ZF and MMSE exhibit similar interference suppression performance; thus, ICom/ZF-SIC and MMSE-SIC yield approximately the same system BER. Figure 9 plots the comparison of system SE of VSP-IMUD, ICom/ZF-SIC and MMSE-SIC under \(N_{T}=1\), \(N_{R}=2\), \(\varepsilon=0.1\), and \(K\in\{3,5,7\}\). As the figures show, VSP-IMUD achieves the highest system SE, followed by MMSE-SIC, while ICom/ZF-SIC exhibits the lowest system SE. The analysis is shown in Figure 7. Moreover, MMSE-SIC outperforms ICom/ZF-SIC in the low-\(\gamma\) regime and converges to the performance of ICom/ZF-SIC as \(\gamma\) increases. This observation is consistent with the characteristics of MMSE and ZF.