# A Partitioned IRS-Aided Transmit SM Scheme for Wireless Communication
## 3 Performance Analysis
Specifically, for the independent RVs \(X\) and \(Y\), the mean and variance follow the rules: \(\text{E}[XY]=\text{E}[X]\text{E}[Y],\text{Var}[XY]=\text{Var}[X]\text{Var}[Y]+ \text{E}^{2}[X]\text{Var}[Y]+\text{E}^{2}[Y]\text{Var}[X],\text{E}[X+Y]=\text {E}[X]+\text{E}[Y],\text{Var}[X\pm Y]=\text{Var}[X]+\text{Var}[Y]\), while for the un-independent RVs \(X\) and \(Y\), the property of covariance is \(\text{Cov}(X,Y)=\frac{1}{2}(\text{Var}[X+Y]-\text{Var}[X]-\text{Var}[Y])\). Then the mean and variance of the following terms can be figured out as follows:  
\[\text{E}[\alpha_{t,i}\beta_{i}]=\frac{\pi}{4},\]  
\[\text{Var}[\alpha_{t,i}\beta_{i}]=\frac{16-\pi^{2}}{16},\]  
\[\text{E}[\left(\alpha_{t,i}\beta_{i}e^{i\left(\theta_{k,i}-\theta_{t,i}\right) }\right)_{\mathbb{R}^{I}}]=\text{E}[\left(\alpha_{t,i}\beta_{i}e^{i\left( \theta_{k,i}-\theta_{t,i}\right)}\right)_{\mathbb{I}}]=0,\]  
\[\text{Var}[\left(\alpha_{t,i}\beta_{i}e^{i\left(\theta_{k,i}-\theta_{t,i} \right)}\right)_{\mathbb{R}^{I}}]=\text{Var}[\left(\alpha_{t,i}\beta_{i}e^{i \left(\theta_{k,i}-\theta_{t,i}\right)}\right)_{\mathbb{I}^{\prime}}]=\frac{1 }{2}.\]  
After some mathematical calculations, we have \(\text{E}[L_{\mathbb{R}}]=\frac{n\pi}{4},\text{Var}[L_{\mathbb{R}}]=\frac{n(16- \pi^{2}+8N_{l}-8)}{16}\). It can be seen that \(\lambda_{1}=L_{\mathbb{R}}^{2}|x-\hat{x}|^{2}\) is a non-central chi-square RV with one degree of freedom. Since the number of reflecting elements \(N\gg 1\), the Central Limit Theorem (CLT) is applicable. Assuming \(\mu_{r}=\text{E}[L_{\mathbb{R}}],\sigma_{r}^{2}=\text{Var}[L_{\mathbb{R}}]\), according to the genetic characteristic function (CF) of the non-central \(\chi^{2}\) distribution, the MGF of \(\lambda_{1}\) can be obtained as follows [5]:  
\[M_{\lambda_{1}}(s)=\left(\frac{1}{1-2s\sigma_{r}^{2}|x-\hat{x}|^{2}}\right)^{ \frac{1}{2}}\exp\left(\frac{s\mu_{r}^{2}|x-\hat{x}|^{2}}{1-2s\sigma_{r}^{2}|x -\hat{x}|^{2}}\right). \tag{11}\]  
Similarly, it can be calculated that \(\text{E}[L_{\mathbb{I}}]=0\), and \(\sigma_{l}^{2}=\text{Var}[L_{\mathbb{I}}]=\frac{n(N_{l}-1)}{2}\). Therefore, \(\lambda_{2}=L_{\mathbb{I}}^{2}|x-\hat{x}|^{2}\) is a central chi-square RV with one degree of freedom. The MGF of \(\lambda_{2}\) can be obtained as follows:  
\[M_{\lambda_{2}}(s)=\left(\frac{1}{1-2s\sigma_{l}^{2}|x-\hat{x}|^{2}}\right)^{ \frac{1}{2}}. \tag{12}\]  
Then, the MGF of \(\lambda\) can be obtained by substituting (11) and (12) to (10). Finally, substitute the MGF of \(\lambda\) to Equation (8) to calculate APEP. It is worth noting that the APEP is independent of \(t\) and \(\hat{t}\). _(2) Second case \((t\
eq\hat{t})\)_: Assume that the antenna index is detected erroneously. Then, we have \(B=Lx-\hat{L}\hat{x}\). Consequently, we have  
\[\lambda=|B|^{2}=|Lx-\hat{L}\hat{x}|^{2}=B_{\mathbb{R}}^{2}+B_{\mathbb{I}}^{2}, \tag{13}\]  
while \(B_{\mathbb{R}}\sim\mathcal{N}\left(\mu_{B_{\mathbb{R}}},\sigma_{B_{\mathbb{R} }}^{2}\right)\) and \(B_{\mathbb{I}}\sim\mathcal{N}\left(\mu_{B_{\mathbb{I}}},\sigma  
\[\mathbf{C}= \begin{bmatrix}\sigma_{1}^{2}&\sigma_{1,2}\\ \sigma_{1,2}&\sigma_{2}^{2}\end{bmatrix}