# Virtual Signal Processing-Based Integrated Multi-User Detection
## 3 Design of VSP-IMUD
### Integrated Multi-User Detection  
As Figure 1 plots, when multiple Txs simultaneously send signals to the common Rx, the mixed signals received by the Rx can be expressed as  
\[\mathbf{y}=\sum_{k=0}^{K-1}\mathbf{h}_{k}x_{k}+\mathbf{n}=\mathbf{r}+\mathbf{n} \tag{1}\]  
where \(\mathbf{n}\) denotes the Additive White Gaussian Noise (AWGN) vector whose elements have zero mean and variance \(\sigma_{n}^{2}\). For clarity, we denote the mixed signal \(\sum_{k=0}^{K-1}\mathbf{h}_{k}x_{k}\) as \(\mathbf{r}\). Figure 1: System model. Under IMUD, the received mixed signal \(\mathbf{r}\) is treated as a structured equivalent signal. Based on the characteristics of such an equivalent signal, the data carried in multiple signal components can be decoded simultaneously, thereby realizing MUD. Given \(K\) and \(L\), there will be \(L^{K}\) forms of \(\mathbf{r}\). Assuming that all Txs adopt \(L\)-order modulation, we can represent the modulation symbol set as \(S=\{s_{1},\cdots,s_{L}\}\), with each Tx\({}_{k}\)'s desired data \(x_{k}\in S\). Based on the composite channel matrix \(\mathbf{H}\) and all possible combinations of data from the \(K\) Txs, the Rx can derive \(L^{K}\) possible forms of \(\mathbf{r}\), as presented in Equation (2). \[ \begin{bmatrix}\mathbf{r}^{[1]}&\mathbf{r}^{[2]}&\cdots&\mathbf{r}^{[L^{K}]} \end{bmatrix} =[\mathbf{h}_{0}&\mathbf{h}_{1}&\cdots&\mathbf{h}_{K-1}] \begin{bmatrix} x_{0}^{[1]}&x_{0}^{[2]}&\cdots&x_{0}^{[L^{K}]}\\ x_{1}^{[1]}&x_{1}^{[2]}&\cdots&x_{1}^{[L^{K}]}\\ \vdots&\vdots&\ddots&\vdots\\ x_{K-1}^{[1]}&x_{K-1}^{[2]}&\cdots&x_{K-1}^{[L^{K}]}\end{bmatrix} . \tag{2}\]  
As Equation (2) shows, there are \(L^{K}\) possible combinations of the symbol set \(\{x_{0},x_{1},\cdots,x_{K-1}\}\). Superscripts \([1],[2],\cdots,[L^{K}]\) are used to index these symbol combinations. We denote the data combination indexed by \([m]\) as \(\mathbf{x}^{[m]}=[x_{0}^{[m]}\quad x_{1}^{[m]}\quad\cdots\quad x_{K-1}^{[m]} ]^{T}\), where \(m\in\{1,2,\cdots,L^{K}\}\) and \(x_{k}^{[m]}\) (\(k\in\{0,1,\cdots,K-1\}\)) represents the data symbol sent from Tx\({}_{k}\). Consequently, the \(m\)-th form of the received mixed signal \(\mathbf{r}\) can be expressed as \(\mathbf{r}^{[m]}=\mathbf{H}\mathbf{x}^{[m]}\). If \(\mathbf{x}^{[m]}\) is uniquely related to \(\mathbf{r}^{[m]}\), we can obtain \(\mathbf{x}^{[m]}\) by detecting the structured \(\mathbf{r}^{[m]}\) in an integrated manner. We define the spatial feature of \(\mathbf{r}^{[m]}\) as  
\[\mathbf{g}^{[m]}\mathbf{=}\frac{\mathbf{r}^{[m]}}{\left\|\mathbf{r}^{[m]} \right\|}. \tag{3}\]  
According to Equations (2) and (3), given \(\mathbf{H}\) and \(\mathbf{x}^{[m]}\), the structure and spatial features of \(\mathbf{r}^{[m]}\) are determined. Based on Equation (3), we can employ a matched filter (MF) \(\mathbf{g}^{[m]}\) to process \(\mathbf{r}^{[m]}\). Since \(\mathbf{x}^{[m]}\) is the target that the Rx aims to detect, we need \(L^{K}\) filter chains to attempt to decode the data vector \(\mathbf{x}^{[m]}\) from \(\mathbf{r}^{[m]}\). For each filter chain, we calculate the modulus of its output and select the one with the largest modulus as the correct chain. Then, we can determine the receiving filter as \(\mathbf{g}^{[M]}\) where \(M\) is computed in terms of \(M=\underset{m\in\{1,\cdots,L^{K}\}}{\arg\max}\ \|[\mathbf{g}^{[m]}]^{H}\mathbf{r}^{[m]}\|\). In this way, \(\mathbf{x}^{[m]}\) is decoded integrally. In practical applications, as \(K\) and/or \(L\) increase, the structure of SVP-IMUD becomes more complicated. To mitigate this complexity, we can divide the \(K\) Txs into multiple smaller groups, effectively reducing \(K\) before applying VSP-IMUD. Furthermore, we can increase \(N_{R}\) to enhance the hardware processing capabilities at the Rx, achieving a trade-off between the number of matched filter chains and the number of receiving antennas.