# Virtual Signal Processing-Based Integrated Multi-User Detection
## 3 Design of VSP-IMUD
### Channel Ambiguity Detection and Elimination
For this reason, we utilize simulations to investigate the impact of different values of \(\varepsilon\) on the probability of direct decoding with IMUD, as illustrated in Figure 4. Consequently, we can empirically determine the value of \(\varepsilon\) to achieve an optimal balance between signal processing complexity and system performance. In order to detect the ambiguity among various pairs \((\mathbf{r}^{[i]},\mathbf{r}^{[j]})\), we define an \(L^{K}\times L^{K}\) ambiguity matrix \(\mathbf{F}\) as shown in Equation (5). As depicted, the element in the \(i\)-th row and \(j\)-th column of \(\mathbf{F}\) is either 1 or 0, indicating whether the spatial features of \(\mathbf{r}^{[i]}\) and \(\mathbf{r}^{[j]}\) are ambiguous. Specifically, if the Euclidean distance between \(\mathbf{r}^{[i]}\) and \(\mathbf{r}^{[j]}\) is less than \(\varepsilon\), then \(\mathbf{F}\)'s element \(\mathbf{F}(i,j)\) is set to 1; otherwise, it is set to 0. It should be noted that under \(i=j\), we set \(\mathbf{F}(i,j)=0\). \[\mathbf{F}= \begin{bmatrix}0&1&0&0&\cdots&0\\ 1&0&0&0&\cdots&1\\ 0&0&0&0&\cdots&0\\ 0&0&0&0&\cdots&1\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&1&0&1&\cdots&0\end{bmatrix}  \tag{5}\]  
Next, we calculate the sum of the elements in each row of \(\mathbf{F}\), and record the result of the \(i\)-th row as \(\xi_{\mathbf{F}(i)}=\sum_{j=1}^{L^{K}}\mathbf{F}(i,j)\). The value \(\xi_{\mathbf{F}(i)}\) reflects the number of mixed signals that are similar to \(\mathbf{r}^{[i]}\). The larger \(\xi_{\mathbf{F}(i)}\) is, the more ambiguous the structure of \(\mathbf{r}^{[i]}\) is in relation to other mixed signals. In this case, if one performs IMUD on the received signal, s/he will have to choose among multiple possible symbol sets, resulting in a high BER due to this ambiguity. The ambiguity arises from the fact that the same aggregated channel matrix \(\mathbf{H}=[\mathbf{h}_{0}\ \mathbf{h}_{1}\ \cdots\ \mathbf{h}_{K-1}]\) and various symbol sets \(\{x_{0}^{[m]},x_{1}^{[m]},\cdots,x_{K-1}^{[m]}\}\) where \(m\in\{1,\cdots,L^{K}\}\) can produce identical structures for the mixed signals. To avoid ambiguity, we identify the channels in \(\mathbf{H}\) that contribute to the ambiguity of the signal structure and subsequently remove them from \(\mathbf{H}\). In what follows, we will establish the criterion for determining the contribution of channel \(\mathbf{h}_{k}\) (\(k\in\{0,1,\cdots,K-1\}\)) to the similarity between different mixed signals. We use \(c_{k}\) to denote the ambiguity coefficient of \(\mathbf{h}_{k}\) and initialize \(c_{k}\) to 0, indicating that \(\mathbf{h}_{k}\) does not contribute to ambiguity. We assume that the \(i^{*}\)-th row of \(\mathbf{F}\) that outputs the largest \(\xi_{\mathbf{F}(i^{*})}\) has been determined. If \(\mathbf{F}(i^{*},j)=1\), the structures of \(\mathbf{r}^{[i^{*}]}\) and \(\mathbf{r}^{[j]}\) are similar. Then, according to Equation (4), we have \(\alpha_{[i^{*},j]}=\left\|\mathbf{r}^{[i^{*}]}-\mathbf{r}^{[j]}\right\|= \left\|\sum_{k=0}^{K-1}\mathbf{h}_{k}\left(x_{k}^{[i^{*}]}-x_{k}^{[j]}\right) \right\|\leq\varepsilon\). Under \(\mathbf{F}(i^{*},j)=1\), if \(x_{k}^{[i^{*}]}=x_{k}^{[j]}\) (\(i^{*}\
eq j\)), we can have \(\mathbf{h}_{k}(x_{k}^{[i^{*}]}-x_{k}^{[j]})=0\). In this case, we cannot determine the contribution of \(\mathbf{h}_{k}\) to the ambiguity of the mixed signal structures; thus, the value of \(c_{k}\) remains unchanged. Conversely, if \(x_{k}^{[i^{*}]}\
eq x_{k}^{[j]}\) (\(i^{*}\
eq j\)), \(\mathbf{h}_{k}(x_{k}^{[i^{*}]}-x_{k}^{[j]})\
eq 0\) holds, indicatingthat \(\mathbf{h}_{k}\) contributes to the structure ambiguity of the received signals. Therefore, we increment \(c_{k}\) as \(c_{k}\gets c_{k}+1\). If \(\mathbf{F}(i^{*},j)=0\), \(\mathbf{h}_{k}\) does not contribute to the ambiguity of the received signals' structures, and the value of \(c_{k}\) remains unchanged. After checking each \(\mathbf{F}(i^{*},j)=1\) in the \(i^{*}\)-_th_ row, we can obtain the ambiguity coefficient \(c_{k}\) for each \(\mathbf{h}_{k}\).