# Secure Transmission in mmWave Wiretap Channels: On Sector Guard Zone and Blockages
## 4 Performance Analysis
### AN Assisted Transmission  
For AN assisted transmission, once the eavesdroppers locate in the sector secrecy guard zone, the transmitter emits information-bearing signal along with AN. Thus, substituting Equation (8) into Equation (3), the SOP is derived as  
\[p_{cob} =P\left(\gamma_{D_{B}}\!<\!2^{R_{B}}\!-\!1\right)\!=\!\mathrm{Pr} \left(\frac{P_{S}M_{S}|h_{D}|^{2}L\left(r_{D}\right)}{\sigma_{D}^{2}}\!<\!2^{R _{B}}\!-\!1\right)\] \[=\ \sum_{i\in\{L,N\}}\mathrm{Pr}\left(\left|h_{D}\right|^{2}<\frac{ \left(2^{R_{B}}-1\right)\sigma_{D}^{2}}{P_{S}M_{S}L\left(r_{D}\right)}\right|i \right)P_{i}\left(r_{D}\right) \tag{28}\] \[=\ \sum_{i\in\{L,N\}}\left(\frac{\mathrm{Y}\left(N_{i},\frac{ \left(2^{R_{B}}-1\right)\sigma_{D}^{2}}{P_{S}M_{S}L\left(r_{D}\right)}N_{i} \right)}{\Gamma\left(N_{i}\right)}\right)P_{i}\left(r_{D}\right).\]In the case of \(E\in\Phi_{3}\), i.e., eavesdroppers reside in the sector secrecy guard zone. Substituting Equation (9) into Equation (2), the SOP is derived as  
\[p_{sob}=\Pr\left(\gamma_{E_{B}}>2^{R_{B}-R_{S}}-1\right)=\int_{2^{R_{B}-R_{S}}-1}^ {\infty}f_{\gamma_{E_{B}}}\left(x\right)dx, \tag{29}\]  
where \(f_{\gamma_{E_{B}}}\left(\cdot\right)\) stands for the probability density function of \(\gamma_{E_{B}}\); then, the cumulative distribution function of \(\gamma_{E_{B}}\) is derived as  
\[ \begin{split}& F_{\gamma_{E_{B}}}\left(x\right)=\Pr\left(\gamma_{E_{ B}}<x\right)\\ &=\Pr\left\{\max_{E\in\Phi_{3}}\left\{\frac{\left(P_{S}M_{S}|h_{E }|^{2}L\left(r_{E_{B}}\right)\right)}{P_{A}M_{s}|h_{E}|^{2}L\left(r_{E}\right)+ \sigma_{E}^{2}}\right\}<x\right\}\\ &=\underbrace{\Pr\left\{\max_{E\in\Phi_{3}^{3}}\left\{\frac{ \left(P_{S}M_{S}|h_{E}|^{2}L\left(r_{E_{B}}\right)\right)}{P_{A}M_{s}|h_{E}|^ {2}L\left(r_{E_{B}}\right)+\sigma_{E}^{2}}\right\}<x\right\}}_{Z_{5}}<x\right\} \times\underbrace{\Pr\left\{\max_{E\in\Phi_{3}^{N}}\left\{\frac{\left(P_{S}M_{S }|h_{E}|^{2}L\left(r_{E_{B}}\right)\right)}}{P_{A}M_{s}|h_{E}|^{2}L\left(r_{E_ {B}}\right)+\sigma_{E}^{2}}\right\}<x\right\},\end{split}  \tag{30}\]  
where \(\Phi_{3}^{L}\) and \(\Phi_{3}^{N}\) are the set of LOS and NLOS eavesdroppers, respectively. \(Z_{5}\) and \(Z_{6}\) are calculated by Equations (31) and (32). In \(Z_{5}\), step (g) follows the probability generating functional of the PPP, step (h) is based on [42] (Equation(3.351.2)). In \(Z_{6}\), step (p) follows the probability generating functional of the PPP, step (q) is based on [42] (Equation (3.381.9)):  
\[ \begin{split} Z_{5}&=\Pr\left\{\max_{E\in\Phi_{3}^{ 3}}\left\{\frac{\left(P_{S}M_{S}|h_{E}|^{2}L\left(r_{E_{B}}\right)\right)}{P_{ A}M_{s}|h_{E}|^{2}L\left(r_{E_{B}}\right)+\sigma_{E}^{2}}\right\}<x\right\}\\ &\triangleq U\left(x-\frac{p_{x}}{P_{A}}\right)+U\left(\frac{p_{x}}{P_{A}}-x \right)\exp\left(-\theta_{S}\lambda_{E}\int_{0}^{\infty}\Pr\left(|h_{E}|^{2}> \frac{x\sigma_{E}^{2}}{L\left(r_{E_{B}}\right)\left(P_{S}M_{S}-P_{A}M_{S}x \right)}\right)e^{-\beta r_{E_{B}}}r_{E_{B}}dr_{E_{B}}\right)\\ &\triangleq U\left(x-\frac{p_{x}}{P_{A}}\right)+U\left(\frac{p_{x}}{P_{A}}-x \right)\exp\left(-\theta_{S}\lambda_{E}\left(\frac{\mathrm{Y}\left(2\beta \rho\right)}{\beta^{2}}-\frac{p_{x}^{N_{x}}}{\Gamma\left(N_{x}\right)}\sum \limits_{n=0}^{\infty}\frac{\left(-1\right)^{n}\mathrm{w}}{n!\left(N_{x}+n \right)}\frac{\mathrm{Y}\left(\alpha_{x}\left(N_{x}+n\right)+2,\beta\rho \right)}{\beta^{2}\left(N_{x}+n\right)+2}\right)\right),\end{split}  \tag{31}\]  
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