# A Partitioned IRS-Aided Transmit SM Scheme for Wireless Communication
## 2 System Model  
### Transmission  
The proposed PIRS-TSM system model is illustrated in Figure 1, where \(N_{t}\) transmit antennas are deployed at the Tx and a single antenna is equipped at the Rx. In addition, an IRS is composed of \(N\) reflecting elements which make phase shifts independently. Only one of the \(N_{t}\) transmit antennas is activated to transmit information at each time. We follow the hypothesis that the information from Tx to Rx can be transmitted only through the IRS due to the physical barrier. Let vector \(\mathbf{a}=[0\quad 0\dots 1\quad 0\dots 0]\) represent the transmit antennas, where only one non-zero element is used to represent the activated antenna. In contrast to the PIRS-TGSK system [20] where the incoming data bits exclusively map the activated antenna index, the PIRS-TSM system transmits both antenna index and modulated symbol. During each transmission time slot, the incoming data bits \(S\) are divided into two parts: \(S=S_{1}+S_{2}\), in which \(S_{1}=\log_{2}N_{t}\) bits denote the index of the activated antenna and \(S_{2}=\log_{2}M\) bits denote the \(M\)-ary constellation symbols. Let \(x\) represent the data symbol selected from \(M\)-ary constellations with normalized power, i.e., \(\left|x\right|^{2}=1\). The channel from the Tx to the IRS is denoted by \(\mathbf{H}\in\mathbb{C}^{N_{t}\times N}\), while the channel from the IRS to the Rx is denoted by \(g\)\(\in\)\(\mathbb{C}^{N\times 1}\). Assuming \(\mathbf{H}\) and \(g\) follow independent Rayleigh fading \(\mathcal{CN}(0,1)\), each entry of \(\mathbf{H}\) is represented as \(h_{t,i}=\alpha_{t,i}e^{-\beta_{t,i}(t}=1,\dots,N_{t},i=1,\dots,N)\), while each entry of \(g\) is represented as \(g_{i}=\beta_{i}e^{-\beta_{t}}\). The phase-shift matrix of the IRS is denoted by  
Figure 1: PIRS-TSM system model. a diagonal matrix \(\mathbf{\Phi}\in\mathbb{C}^{N\times N}\), where each element represents the phase shift of a reflecting element, written as \(e^{i\phi_{i}}(i=1,\dots,N)\). We assume perfect channel state information (CSI) at the Tx. The IRS obtains phase-shift information from the controller linked to the Tx. In this system, the IRS partitioning method is the same as that of the PIRS-TSSK scheme, i.e., the IRS is divided into groups that have a one-to-one correspondence with the transmit antennas. Specifically, a total number of \(N\) reflecting elements is divided into \(N_{t}\) groups, each of which consists of \(n=\frac{N}{N_{t}}\) elements. Each group of reflecting elements can be regarded as a sub-IRS, which maximizes the signal energy from the corresponding transmit antenna to the Rx by phase adjustment. Therefore, during data transmission of each time slot, no matter which transmit antenna is activated, all reflecting elements on the IRS work at predetermined phase-shift angles based on the CSI. In other words, all reflecting elements on the IRS work at the \"ON\" state, with one of the IRS groups able to fully offset the channel phases, while the other groups are not fully offset. In fact, this IRS partitioning method is a trade-off between widening the differences in the channels corresponding to different transmit antennas and maintaining the beamforming of the IRS. The partitioning method is validated to provide more diversity when activating different transmit antennas [22] so as to improve the BER performance. For convenience, the transmit energy is assumed to be 1, and the received signal can be expressed as follows:  
\[y=\mathbf{a}\mathbf{H}\mathbf{\Phi}gx+w=Lx+w, \tag{1}\]  
where \(w\) denotes the additive white Gaussian noise (AWGN) with mean 0 and variance \(N_{0}\), and  
\[L=\sum_{i=(t-1)n+1}^{tn}\alpha_{t,i}\beta_{i}+\sum_{ \begin{subarray}{c}k=1\\ k\
eq t\end{subarray} }^{N_{t}}\sum_{i=(k-1)n+1}^{kn}\alpha_{t,i}\beta_{i}e^{i \left(\theta_{k,i}-\theta_{ij}\right)}, \tag{2}\]  
with \(k\) representing the index of the sub-IRS. The phase shift of the reflecting element on the \(k\)-th sub IRS is \(\phi_{k,i}=\theta_{k,i}+\psi_{i}(k=1,\dots,N_{t})\).