# Precoding for RIS-Assisted Multi-User MIMO-DQSM Transmission Systems
## 4 Interference Cancellation
### Strategy I. ZF-BD Precoding  
In this strategy, firstly a BD technique is used in Precoder 1 at the BS to generate the signals with appropriate phases required by the RIS in order to cancel the interference in the second hop. Then, a ZF technique is used in Precoder 2 to pre-cancel the interference in the first hop. Figure 4 shows a block diagram of Strategy I. Strategy I can be used when some users are MUI-free. Let us consider a ZF precoding in Precoder 2. The precoding matrix is defined as \(\mathbf{F}=\mathbf{G}^{+}\). Then, the output can be written as [32]  
\[\mathbf{t}_{\mathbf{x}}=\mathbf{F}\mathbf{x}, \tag{14}\]  
where \(\mathbf{G}^{+}\) represent the inverse of the channel matrix \(\mathbf{G}\). Substituting (14) in (10) and considering the blind RIS, we obtain  
\[\mathbf{y}_{k}=\sqrt{\gamma}_{k}\mathbf{H}_{k}\mathbf{x}+\mathbf{n}_{k}. \tag{15}\]  
Equation (15) represents the signals received by all users. For users that are free from MUI, the received signal is  
\[\mathbf{y}_{k}^{\prime}=\sqrt{\gamma}_{k}\mathbf{H}_{k}\mathbf{x}_{2}+\mathbf{ n}_{k}. \tag{16}\]  
Since \(\mathbf{y}_{k}^{\prime}\) is free from MUI, only Precoder 2 is used to precancel the interference. For users affected by MUI, and substituting (6) in (15), the received signal \(\mathbf{y}_{k}\) by the \(k\)-th user is  
\[\mathbf{y}_{k}=\sqrt{\gamma}_{k}\mathbf{H}_{k}\sum_{k=1}^{j}\mathbf{W}_{k} \tilde{\mathbf{x}}_{1}^{k}+\mathbf{n}_{k}, \tag{17}\]  
where \(j\) is the number of users under MUI. Equation (17) can be rewritten as  
Figure 3: Users under MUI are illuminated by two or more beams. \[\mathbf{y}_{k}=\sqrt{\gamma_{k}}\mathbf{H}_{k}\mathbf{W}_{k}\tilde{\mathbf{x}}_{1}^ {k}+\sqrt{\gamma_{k}}\mathbf{H}_{k}\sum_{i=1,i\
eq k}^{j}\mathbf{W}_{i}\tilde{ \mathbf{x}}_{1}^{i}+\mathbf{n}_{k}. \tag{18}\]  
The first term in (18) is the signal intended for the \(k\)-th user, the second term is the interference produced by the other users in the system, whereas the third term is noise. The mathematical model of the complete system can be represented as  
\[ \begin{bmatrix}\mathbf{y}_{1}\\ \mathbf{y}_{2}\\ \vdots\\ \mathbf{y}_{j}\end{bmatrix} = \begin{bmatrix}\mathbf{H}_{1}&\mathbf{H}_{1}& \cdots&\mathbf{H}_{1}\\ \mathbf{H}_{2}&\mathbf{H}_{2}&\cdots&\mathbf{H}_{2}\\ \vdots&&\ddots&\vdots\\ \mathbf{H}_{j}&\mathbf{H}_{j}&\cdots&\mathbf{H}_{j}\end{bmatrix}\begin{bmatrix} \mathbf{W}_{1}\tilde{\mathbf{x}}_{1}^{1}\\ \mathbf{W}_{2}\tilde{\mathbf{x}}_{1}^{2}\\ \vdots\\ \mathbf{W}_{j}\tilde{\mathbf{x}}_{1}^{j}\end{bmatrix} + \begin{bmatrix} \mathbf{n}_{1}\\ \mathbf{n}_{2}\\ \vdots\\ \mathbf{n}_{j}\end{bmatrix} , \tag{19}\]  
where we set \(\sqrt{\gamma}_{k}=1\) without loss of generality. In order to eliminate the interference term, we require that \(\mathbf{H}_{k}\mathbf{W}_{i}=\mathbf{0},\forall i\
eq k\), where \(\mathbf{0}\) denotes an all-zero matrix. This equation can be written as  
\[\overline{\mathbf{H}}_{k}\mathbf{W}_{k}=\mathbf{0},\quad k=1,2,\ldots,j, \tag{20}\]  
where the matrix \(\overline{\mathbf{H}}_{k}\) contains all user's matrices in the system except that of the \(k\)-th user, i.e.,  
\[\overline{\mathbf{H}}_{k}=\Big{[}(\mathbf{H}_{1})^{H},\ldots,(\mathbf{H}_{k-1} )^{H},(\mathbf{H}_{k+1})^{H},\ldots,(\mathbf{H}_{K})^{H}\Big{]}^{H}. \tag{21}\]  
Figure 4: Strategy I (ZF-BD precoding technique). The matrix \(\mathbf{W}_{k}\) can be obtained by decomposing \(\overline{\mathbf{H}}_{k}\) into its singular values as  
\[\overline{\mathbf{H}}_{k}=\mathbf{U}_{k}[\mathbf{\Sigma}_{k},\mathbf{0}]\Big{[} \mathbf{V}_{k}^{(1)}\mathbf{V}_{k}^{(0)}\Big{]}^{H}, \tag{22}\]  
where \(\mathbf{U}_{k}\) is a unitary matrix, \(\mathbf{\Sigma}_{k}\) is a diagonal matrix containing the non-negative singular values of \(\overline{\mathbf{H}}_{k}\) with dimension equals to the rank of \(\overline{\mathbf{H}}_{k}\), \(\mathbf{0}\) is an all-zero matrix, \(\mathbf{V}_{k}^{(1)}\) contains vectors corresponding to the nonzero singular values and \(\mathbf{V}_{k}^{(0)}\) contains vectors corresponding to the zero singular values.