# Virtual Signal Processing-Based Integrated Multi-User Detection
## 3 Design of VSP-IMUD
### Channel Ambiguity Detection and Elimination  
In the previous subsection, we assumed the structure of the received mixed signal to be unique and employed an MF to recover the data/symbol set sent from multiple TXs. In practice, however, the structure of the mixed signal may arise from various data and channel combinations, resulting in similarities and ambiguities among the characteristics of the equivalent signals. In such cases, the MF cannot achieve accurate data recovery [13]. Therefore, in this subsection, we will design a _channel ambiguity detection (CAD)_ module to identify the channels that contribute to the similarity of the mixed signal's structure, and then decode the signals transmitted via these ambiguous channels successively using SIC. Once the channel ambiguity is completely mitigated, the structure of the remaining mixed signal becomes unique, allowing for exclusive recovery of the dataset from multiple Txs using the IMUD method presented in Section III-A. Below, we will detail the detection and elimination of channel ambiguity. We denote the set of symbols to be detected as \(\Omega\), and initialize it to \(\{x_{0},x_{1},\cdots,x_{K-1}\}\), where \(x_{k}\) (for \(k\in\{0,1,\cdots,K-1\}\)) is the data symbol sent from Tx\({}_{k}\). According to Equation (2), the mixed signal perceived by the Rx is \(\mathbf{y}=\sum_{k=0}^{K-1}\mathbf{h}_{k}x_{k}+\mathbf{n}\). The Rx can then obtain all possible combinations of the aggregated channel matrix \(\mathbf{H}=\left[\mathbf{h}_{0}\ \mathbf{h}_{1}\ \cdots\ \mathbf{h}_{K-1}\right]\) and the symbol set \(\{x_{0},x_{1},\cdots,x_{K-1}\}\). Note that although Rx does not know the desired data sent by the Txs at this time, it can still derive \([\mathbf{r}^{[1]}\ \mathbf{r}^{[2]}\ \cdots\ \mathbf{r}^{[L^{K}]}]\) based on the known modulation scheme, where each element indicates a possible structure or characteristic of the received mixed signal. Next, we use the Euclidean distance between \(\mathbf{r}^{[i]}\) and \(\mathbf{r}^{[j]}\) as the ambiguity coefficient \(\alpha_{[i,j]}\) of \(\mathbf{r}^{[i]}\) and \(\mathbf{r}^{[j]}\), defined in Equation (4):  
\[\alpha_{[i,j]}=\left\|\mathbf{r}^{[i]}-\mathbf{r}^{[j]}\right\|=\left\|\sum_{ k=0}^{K-1}\mathbf{h}_{k}\left(x_{k}^{[i]}-x_{k}^{[j]}\right)\right\| \tag{4}\]  
where \(i,j\in\{1,\cdots,L^{K}\}\) and \(i\
eq j\). Figure 2: Illustration of VSP-IMUD under \(K\) Txs and \(L=2\). We employ a real value \(\varepsilon>0\) as the ambiguity threshold. If \(\alpha_{[i,j]}\leq\varepsilon\), the structures of \(\mathbf{r}^{[i]}\) and \(\mathbf{r}^{[j]}\) are similar (ambiguous), indicating detection of ambiguity between \(\mathbf{r}^{[i]}\) and \(\mathbf{r}^{[j]}\). It should be noted that the value of \(\varepsilon\) affects the processing complexity and BER performance of the Rx. When \(\varepsilon\) is large, more pairs \((\mathbf{r}^{[i]},\mathbf{r}^{[j]})\) will be classified as similar, leading to a greater number of Tx signals that need to be decoded to eliminate ambiguity, which increases the processing burden. Conversely, if \(\varepsilon\) is small, the similarities in features of the mixed signals arising from various datasets may go undetected. In this situation, while IMUD may still be applicable, it could yield an incorrect symbol set, resulting in poor BER performance. It is important to note that \(\varepsilon\) serves to balance decoding complexity and accuracy. Moreover, the degree of ambiguity depends on the \(K\) channel matrices associated with the \(K\) Txs, the \(K\) data symbols, and their modulation order \(L\). Among these factors, the \(K\) channel matrices and data symbols are random and independent, making a quantitative analysis of the relationship among \(\varepsilon\), \(K\) and \(L\) difficult.