# Virtual Signal Processing-Based Integrated Multi-User Detection
## 4 Simulation Results  
In this section, we use MATLAB simulations to demonstrate the performance of VSP-IMUD. The system consists of \(K\) Txs and one Rx. Since we focus on a multi-user communication scenario, we set \(K>1\) in our simulations. Although reducing \(K\) to 1 can eliminate CCI among multiple Txs, it also limits the system's ability to accommodate only one user's data transmission at a time, thereby degrading the system's spectral efficiency (SE). Therefore, we omit the study of the single-user scenario in this work. Each Tx is equipped with \(N_{T}\) antennas and employs BPSK modulation to transmit a single data stream to the Rx with power \(P_{T}\). The common Rx has \(N_{R}\) antennas. If \(N_{T}=1\), the Tx transmits omnidirectionally; otherwise, when \(N_{T}>1\), the Tx employs singular value decomposition (SVD)-based precoding to obtain the transmitted signal. In what follows, we will evaluate both the probability of direct decoding using IMUD and BER performance under VSP-IMUD. In our simulations, we assume perfect CSI for simplicity. When imperfect CSI is considered, \(\mathbf{h}_{k}\) can be re-modeled as \(\mathbf{\hat{h}}_{k}=\rho\mathbf{h}_{k}+\sqrt{1-\rho^{2}}\mathbf{e}\)[18], where \(\mathbf{h}_{k}\) and \(\mathbf{\hat{h}}_{k}\) denote accurate and inaccurate channel matrices, respectively. The coefficient \(\rho\in(0,1]\) indicates the degree of CSI imperfection, with \(\rho=1\) representing perfect CSI. The matrix \(\mathbf{e}\) is an \(N_{R}\times N_{T}\) complex diagonal Gaussian matrix with zero mean and unit variance. It is important to note that all of the investigated methods are susceptible to inaccuracies in CSI. However, since the modeling of CSI imperfections and their influence on data transmission, as well as efforts to improve the algorithms' robustness to these imperfections, have been extensively studied [19], we have omitted this discussion in the present work. Figure 4 plots the variation in the probability of directly applying IMUD with \(K\) under \(N_{T}=1\), \(N_{R}=2\), and various \(\varepsilon\)s. As the figure shows, for a fixed \(K\), the probability of direct decoding with IMUD increases as \(\varepsilon\) grows. When \(\varepsilon\) is held constant, the probability increases as \(K\) reduces. This trend is due to the fact that a reduction in either \(\varepsilon\) or \(K\) decreases the detection probability of channel ambiguity, which in turn increases the likelihood of directly applying IMUD. Conversely, if \(\varepsilon\) or \(K\) increases, the detection probability of channel ambiguity rises, thereby reducing the probability of direct decoding with IMUD. The probability curve plotted in Figure 4 indicates that IMUD can be directly applied when  
no channel ambiguity is detected. Under very small \(\varepsilon\) (e.g., 0.1), the probability of directly using IMUD remains unchanged as \(K\) varies, as no channel ambiguity is detected. We define the transmit power normalized by noise power \(\sigma_{n}^{2}\) as \(\gamma=10\lg(P_{T}/\sigma_{n}^{2})\), and set \(\gamma\in[-10,20]\)dB in the simulation. Figure 5 simulates the impact of SIC on the BER of signal components under \(K=7\), \(N_{T}=1\), and \(N_{R}=2\), where the Rx does not apply IMUD to decode the Txs' data. Given \(N_{R}=2\), the Rx must perform SIC five times to decode all \(K=7\) signals until the last two signal components can be directly recovered. As illustrated in the figure, due to the error propagation issue associated with SIC, the BER for later decoded data is worse than that for previously decoded data. Figure 6 displays the BER of VSP-IMUD under \(\varepsilon=0.1\) using various referential symbol selection criteria. We use \(\max(||h||)\) to indicate that the data transmitted via the channel with highest channel gain is selected as the referential symbol to determine the final symbol set. For comparison, we denote random referential symbol selection as _random_.