# Secrecy Enhancing Scheme for Spatial Modulation Using Antenna Selection and Artificial Noise
## 4 Secrecy Rate Analysis  
The received signal at the legitimate receiver Bob, \(y_{b}\), can be rearranged as follows by inserting the result in (11) into (9). \[y_{b}= \sqrt{1-\rho^{2}}\hat{h}_{j}{}^{\prime}x_{m}+\sqrt{\rho^{2}}\hat{h}_{j}{ }^{\prime}(x_{m}+\beta_{1}\
u)+\sqrt{\rho^{2}}\tilde{h}_{i}{}^{\prime}\beta_{ 2}\
u+n_{b}. \tag{18}\]  
If we let:  
\[\tilde{n}_{b}= \sqrt{\rho^{2}}\tilde{h}_{j}{}^{\prime}(x_{m}+\beta_{1}\
u)+ \sqrt{\rho^{2}}\tilde{h}_{i}{}^{\prime}\beta_{2}\
u+n_{b}, \tag{19}\]then it can be approximated as zero-mean complex Gaussian noise with variance of \(\eta_{b}=E[\hat{n}_{b}(\hat{n}_{b})^{H}]\). In addition, letting \(\theta=1-\rho^{2}\), then the normalized value of \(y_{b}\) with noise variance of \(\eta_{b}\) can be approximated by:  
\[\tilde{y}_{b}=\frac{y_{b}}{\sqrt{\eta_{b}}}\simeq\sqrt{\frac{\theta}{\eta_{b}}} \hat{h_{j}}^{\prime}x_{m}+\hat{n}_{b}, \tag{20}\]  
where \(\hat{n}_{b}=\hat{n}_{b}/\sqrt{\eta_{b}}\). After choosing the optimum matrices \(\mathbf{T}_{k}\) and \(\mathbf{T}_{q}\), the probability of selecting each transmit antenna and each symbol \(x_{m}\) is \(\frac{1}{N_{t}}\) and \(\frac{1}{M}\), respectively. Thus, the received signal at Bob follows a complex Gaussian distribution, which can be represented as:  
\[P(\tilde{y}_{b}|\hat{h_{j}}^{\prime},x_{m})=\frac{1}{\pi}\exp\left(-\mid\tilde {y}_{b}-\sqrt{\frac{\theta}{\eta_{b}}}\hat{h_{j}}^{\prime}x_{m}\mid^{2}\right), \tag{21}\]  
and:  
\[P(\tilde{y}_{b})=\frac{1}{N_{t}M}\sum_{j=1}^{N_{t}}\sum_{m=1}^{M}\frac{1}{\pi }\exp\left(-\mid\tilde{y}_{b}-\sqrt{\frac{\theta}{\eta_{b}}}\hat{h_{j}}^{ \prime}x_{m}\mid^{2}\right). \tag{22}\]  
Then, the ergodic rate of Bob can be written as,  
\[R_{b} =E_{\hat{\mathbf{h}}}\left[\int\sum_{j=1}^{N_{t}}\sum_{m=1}^{M}p( \tilde{y}_{b},\hat{h_{j}}^{\prime},x_{m})\log_{2}\frac{p(\tilde{y}_{b},\hat{h_ {j}}^{\prime},x_{m})}{p(\tilde{y}_{b})p(\hat{h_{j}}^{\prime},x_{m})}d\tilde{y }_{b}\right] \tag{23}\] \[=\log_{2}N_{t}M-\frac{1}{N_{t}M}\sum_{j=1}^{N_{t}}\sum_{m=1}^{M}E _{\hat{\mathbf{h}}}\left[E_{h_{b}}\left[\log_{2}\sum_{j^{\prime}=1}^{N_{t}} \sum_{m^{\prime}=1}^{M}\right.\right.\] \[\times\left.\left.\exp\left(-\left(\mid\sqrt{\frac{\theta}{\eta_{ b}}}\hat{g_{j,m}^{\prime}}^{\prime,m^{\prime}}+\hat{n}_{b}\mid^{2}-\mid\hat{n}_{b} \mid^{2}\right)\right)\right]\right],\]  
in which \(\hat{g_{j,m}^{\prime}}^{\prime,m^{\prime}}\) is given by:  
\[\hat{g_{j,m}^{\prime}}^{\prime,m^{\prime}}=\hat{h_{j}}^{\prime}x_{m}-\hat{h_{ j^{\prime}}}^{\prime}x_{m^{\prime}}. \tag{24}\]  
Using [PERSON]'s inequality, the lower bound of \(R_{b}\) is obtained as:  
\[R_{b}^{low} =\log_{2}N_{t}M-\frac{1}{N_{t}M}\sum_{j=1}^{N_{t}}\sum_{m=1}^{M} \log_{2}\sum_{j^{\prime}=1}^{N_{t}}\sum_{m^{\prime}=1}^{M} \tag{25}\] \[\tilde{n}_{b}= \sqrt{\rho^{2}}\tilde{h}_{j}{}^{\prime}(x_{m}+\beta_{1}\
u)+ \sqrt{\rho^{2}}\tilde{h}_{i}{}^{\prime}\beta_{2}\
u+n_{b}, \tag{19}\]0 \[\tilde{n}_{b}= \sqrt{\rho^{2}}\tilde{h}_{j}{}^{\prime}(x_{m}+\beta_{1}\
u)+ \sqrt{\rho^{2}}\tilde{h}_{i}{}^{\prime}\beta_{2}\
u+n_{b}, \tag{19}\]1 \[\tilde{n}_{b}= \sqrt{\rho^{2}}\tilde{h}_{j}{}^{\prime}(x_{m}+\beta_{1}\
u)+ \sqrt{\rho^{2}}\tilde{h}_{i}{}^{\prime}\beta_{2}\
u+n_{b}, \tag{19}\]2In order to derive the ergodic rate at the passive eavesdropper, we first represent (10) using \(\theta\) as follows:  
\[ \begin{split} y_{e}=\sqrt{\theta}\hat{g}_{j}^{\prime}x_{m}+\\ \sqrt{\theta}(\hat{g}_{j}^{\prime}\beta_{1}+\hat{g}_{i}^{\prime} \beta_{2})v+\sqrt{\rho^{2}}(\hat{g}_{j}^{\prime}x_{m}+\hat{g}_{j}^{\prime}\beta _{1}v+\hat{g}_{i}^{\prime}\beta_{2}v)+n_{e}.\end{split}  \tag{26}\]  
If we let:  
\[ \begin{split}\tilde{n}_{e}=&\sqrt{\theta}(\hat{g}_{ j}^{\prime}\beta_{1}+\hat{g}_{i}^{\prime}\beta_{2})v+\sqrt{\rho^{2}}(\hat{g}_{j}^{ \prime}x_{m}+\hat{g}_{j}^{\prime}\beta_{1}v+\hat{g}_{i}^{\prime}\beta_{2}v)+n_ {e},\end{split}