# Precoding for RIS-Assisted Multi-User MIMO-DQSM Transmission Systems
## 4 Interference Cancellation
### Strategy II. Joint-BD Precoding  
In order to improve the flexibility in the design of the systems with \(N_{s}\geq N_{t}\), a joint-precoding strategy is proposed in this subsection. Strategy II combines the channel matrix \(\mathbf{G}\) and all channel matrices \(\mathbf{H}_{k}\) in order to generate an equivalent channel matrix for each user. Then, the BD technique can be used to precancel the interference in both hops at the same time. Figure 5 shows a block diagram of the strategy II utilized. Strategy II can be used when all users are under MUI or when all users see the complete RIS. Let us consider an equivalent matrix defined as \(\mathbf{H}_{k}^{Eq}\triangleq(\mathbf{H}_{k}\mathbf{\Theta}\mathbf{G})\in \mathbb{C}^{N_{t}\times N_{t}}\). Then, the received signal for the \(k\)-th user in (10) can be rewritten as  
\[\mathbf{y}_{k}=\sqrt{\gamma_{k}}\mathbf{H}_{k}^{Eq}\mathbf{t}_{\mathbf{x}}+ \mathbf{n}_{k}, \tag{25}\]  
Considering \(\mathbf{P}=\mathbf{I}\) in the Precoder 2, it follows that \(\mathbf{t}_{\mathbf{x}}=\mathbf{x}\). Now, considering a precoding matrix \(\mathbf{Z}_{k}^{Eq}\in\mathbb{C}^{N_{t}\times N_{t}}\) for the \(k\)-th user, the transmission vector in Precoder 1 can be written as  
\[\mathbf{t}_{\mathbf{x}}=\sum_{k=1}^{K}\mathbf{Z}_{k}^{Eq}\tilde{\mathbf{x}}_{k}. \tag{26}\]  
The received signal by the \(k\)-th user can be written as  
\[\mathbf{y}_{k}=\sqrt{\gamma_{k}}\mathbf{H}_{k}^{Eq}\sum_{k=1}^{K}\mathbf{Z}_{ k}^{Eq}\tilde{\mathbf{x}}_{k}+\mathbf{n}_{k}, \tag{27}\]  
which can be also expressed as  
\[\mathbf{y}_{k}=\sqrt{\gamma_{k}}\mathbf{H}_{k}^{Eq}\mathbf{Z}_{k}^{Eq}\tilde{ \mathbf{x}}_{k}+\sqrt{\gamma_{k}}\mathbf{H}_{k}^{Eq}\sum_{i=1,j\
eq k}^{K} \mathbf{Z}_{i}^{Eq}\tilde{\mathbf{x}}_{i}+\mathbf{n}_{k}, \tag{28}\]  
where the first term in (28) is the signal sent to the \(k\)-th user, the second term is the interference produced by the other users in the system, whereas the third term is the noise. In order to remove the interference term, we require that \(\mathbf{H}_{k}^{Eq}\mathbf{Z}_{i}^{Eq}=\mathbf{0},\forall i\
eq k\). This condition can be written as  
\[\overline{\mathbf{H}}_{k}^{Eq}\mathbf{Z}_{k}^{Eq}=\mathbf{0},\quad k=1,2,\ldots,K, \tag{29}\]  
where the matrix \(\overline{\mathbf{H}}_{k}^{Eq}\) contains all users matrices in the system except that of the \(k\)-th user, i.e.,  
\[\overline{\mathbf{H}}_{k}^{Eq}=\Big{[}(\mathbf{H}_{1}^{Eq})^{H},\ldots,( \mathbf{H}_{k-1}^{Eq})^{H},(\mathbf{H}_{k+1}^{Eq})^{H},\ldots,(\mathbf{H}_{K }^{Eq})^{H}\Big{]}^{H}. \tag{30}\]  
The matrix \(\mathbf{Z}_{k}^{Eq}\) is obtained by decomposing \(\overline{\mathbf{H}}_{k}^{Eq}\) into its singular values as  
\[\overline{\mathbf{H}}_{k}^{Eq}=\mathbf{U}_{k}[\boldsymbol{\Sigma}_{k}, \mathbf{0}]\Big{[}\mathbf{V}_{k}^{(1)}\mathbf{V}_{k}^{(0)}\Big{]}^{H}. \tag{31}\]  
Similar to (22), \(\boldsymbol{\Sigma}_{k}\) is a diagonal matrix containing the non-negative singular values of \(\overline{\mathbf{H}}_{k}^{Eq}\) and the matrix \(\mathbf{V}_{k}^{(0)}\) contains the last \(N_{r}\) columns of \(\mathbf{V}_{k}\), which form an orthogonal basis that is in the null space of \(\overline{\mathbf{H}}_{k}^{Eq}\). Eliminating the interference term in (28), the received signal can be rewritten as  
\[\mathbf{y}_{k}=\sqrt{\gamma_{k}}\mathbf{H}_{k}^{Eq}\mathbf{Z}_{k}^{Eq} \bar{\mathbf{s}}_{k}+\mathbf{n}_{k}, \tag{32}\]  
Figure 5: Strategy II (joint-BD precoder). which is an interference-free signal. Finally, the complete system can be represented as  
\[ \begin{bmatrix}\mathbf{y}_{1}\\ \mathbf{y}_{2}\\ \vdots\\ \mathbf{y}_{K}\end{bmatrix} = \begin{bmatrix}\mathbf{H}_{1}^{Eq}\mathbf{Z}_{1}^{Eq }&\cdots&\mathbf{0}\\ \mathbf{0}&\cdots&\mathbf{0}\\ \vdots&\ddots&\vdots\\ \mathbf{0}&\cdots&\mathbf{H}_{K}^{Eq}\mathbf{Z}_{K}^{Eq}\end{bmatrix} \begin{bmatrix}\mathbf{\tilde{x}}_{1}\\ \mathbf{\tilde{x}}_{2}\\ \vdots\\ \mathbf{\tilde{x}}_{K}\end{bmatrix} + \begin{bmatrix}\mathbf{n}_{1}\\ \mathbf{n}_{2}\\ \vdots\\ \mathbf{n}_{K}\end{bmatrix} . \tag{33}\]  
where \(\sqrt{\gamma}_{k}=1\) without loss of generality.