# Precoding for RIS-Assisted Multi-User MIMO-DQSM Transmission Systems
## 2 System Model
### Transmission  
In order to generate the DQSM signal, the sequence of input bits \(\mathbf{a}_{k}=\{d_{n}\}_{n=1}^{m}\), with \(d_{n}\in\{0,1\}\) is split in two flows. Each data flow is composed of a sequence \(\mathbf{b}_{k}=\{d_{n}\}_{n=1}^{m/2},k=\{1,2\}\), which is fed into a quadrature spatial modulation (QSM) block. Figure 2 shows the DQSM modulation block used for the proposed scheme. The QSM signals are denoted by \(\mathbf{x}_{i}^{\prime}\in\mathbb{C}^{L\times 1},i=\{1,2\}\), with components \(x^{\prime}_{l}\in\{0,s_{\mathcal{R}},s_{\mathcal{S}}\}\) where \(s_{\mathcal{R}}\) and \(s_{\mathcal{S}}\) represent the real and the imaginary parts of the quadrature amplitude modulation of \(M\)-th order (\(M\)-QAM) symbol \(s\), respectively. The output vector of the \(i\)-th QSM block \(\mathbf{x}_{i}^{\prime}\) is defined as  
\[\mathbf{x}_{i}^{\prime}=(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{L}^{\prime} )^{T},i=\{1,2\}, \tag{1}\]  
where the index in \(\vec{x_{i}}\) denotes the transmitted signal in the \(k\)-th position. In order to generate the QSM vector \(\mathbf{x}_{i}^{\prime}\), the input bit sequence is divided into three streams. One  
Figure 1: RIS-Assisted MU-DQSM system model. stream is used to modulate a \(M\)-QAM signal and the other two streams (spatial bits) are used to modulate the position in the QSM output vector. For an input bit sequence of length \(m_{QSM}=m_{DQSM}/2\), the first \(\log_{2}(M)\) bits modulate a \(M\)-QAM symbol, the remaining \(2\log_{2}(L)=m_{QSM}-\log_{2}(M)\) bits are divided into two streams with \(\log_{2}(L)\) spatial bits each. These spatial bits modulate the position of the \(M\)-QAM symbol in the output vector \(\mathbf{x}^{\prime}_{i}\in\mathbb{C}^{L\times 1}\) using an SM block as follows: the real part of the \(M\)-QAM symbol is assigned to a specific position in the output vector, while the remaining \(L-1\) positions are set to zero. The imaginary part of the \(M\)-QAM symbol is assigned to another or even the same position in the output vector. Finally, these two signals are combined to obtain the QSM output vector \(\mathbf{x}^{\prime}_{i}\)[29]. The number of bits that can be transmitted using QSM is \(m_{QSM}=log_{2}(M)+2 log_{2}(L)\), The two QSM signals \(\mathbf{x}^{\prime}_{1}\) and \(\mathbf{x}^{\prime}_{2}\) are then weighted by the factors \(B_{1}\) and \(B_{2}\), which guarantee the maximal Euclidean distance between symbols. These two signals are combined to generate the DQSM output vector \(\tilde{\mathbf{x}}_{k}\) intended for the \(k\)-th user as [19]  
\[\tilde{\mathbf{x}}_{k}=B_{1}\mathbf{x}^{\prime}_{1}+B_{2}\mathbf{x}^{\prime}_ {2}. \tag{2}\]  
The output vector of the \(k\)-th DQSM block \(\tilde{\mathbf{x}}_{k}\in\mathcal{A}\) can be written as  
\[\tilde{\mathbf{x}}_{k}=(\tilde{x}_{1},\tilde{x}_{2},\ldots,\tilde{x}_{L})^{T}, \tag{3}\]  
where the index in \(\tilde{x}_{l}\) denotes the transmitted signal in the \(l\)-th position. The complete transmission block for \(K\) users in the system before precoding is composed as  
\[[\tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2},\ldots,\tilde{\mathbf{x}}_{j}, \tilde{\mathbf{x}}_{j+1},\ldots,\tilde{\mathbf{x}}_{K}]. \tag{4}\]  
where the first \(j\) users are the users under multiuser interference (MUI). The other \(K-j\) users are the users free from MUI. The number of bits that can be transmitted to each user using DQSM is  
\[m_{DQSM}=2(log_{2}(M)+2 log_{2}(L)). \tag{5}\]  
DQSM is mainly used because it has good BER performance and low detection complexity [19]. As shown in Figure 1, the precoder is composed of two stages. Users under MUI are first taken to Precoder 1. The Precoder 1 is intended to pre-cancel the MUI in the second hop (from the RIS to the MS). The output of Precoder 1 is the vector \(\mathbf{x}_{1}\) defined as  
\[\mathbf{x}_{1}=\sum_{k=1}^{j}\mathbf{W}_{k}\tilde{\mathbf{x}}_{k}=(x_{1},x_{2 },\ldots,x_{N_{k}^{\prime}})^{T}. \tag{6}\]  
Figure 2: DQSM modulation block.