# Secure Transmission in mmWave Wiretap Channels: On Sector Guard Zone and Blockages
## 4 Performance Analysis
### Direct Transmission
In \(Z_{2}\), step (c) follows the probability generating functional of the PPP, step (d) is based on [42] (Equation (3.381.9)):  
\[Z_{1} =\Pr\left\{\frac{\max\limits_{E\in\Phi_{1}^{\mathrm{L}}}\left(PM_ {S}|h_{E}|^{2}L\left(r_{E}\right)\right)}{\sigma_{E}^{2}}<x\right\}\overset{a}{ =}\exp\left(-\theta_{S}\lambda_{E}\int_{r}^{\infty}\Pr\left(\left|h_{E}\right|^ {2}>\frac{v\sigma_{E}^{4}t_{E}^{4}}{PM_{S}\zeta_{E}}\right)e^{-\beta r_{E}}r_{ E}dr_{E}\right) \tag{14}\] \[\overset{\mathrm{b}}{=}\exp\left(-\theta_{S}\lambda_{E}\left( \frac{\Gamma\left(2,\beta r\right)}{\beta^{2}}-\frac{\left(\frac{\Gamma_{N} \mu_{E}}{PM_{S}\zeta_{E}}\right)^{N_{L}}}{\Gamma\left(N_{L}\right)}\sum \limits_{n=0}^{\infty}\frac{\left(-1\right)^{n}\left(\frac{N_{n}\
u_{E}^{2}}{ PM_{S}\zeta_{E}^{2}}\right)^{n}}{n!(N_{L}+n)}\frac{\Gamma\left(\alpha_{L} \left(N_{L}+n\right)+2,\beta r\right)}{\beta^{4}\left(N_{L}+n\right)+2}\right) \right),\]\[Z_{2} =\Pr\left\{\frac{\max\limits_{E\in\Phi_{1}^{N}}\left(PM_{S}|h_{E}|^{2} L(r_{E_{A}})\right)}{\sigma_{E}^{2}}<x\right\}=E\left\{\prod\limits_{E\in\Phi_{1}^{N}} \Pr\left(\left|h_{E}\right|^{2}<\frac{x\sigma_{E}^{2}}{PM_{S}L(r_{E_{A}})} \right)\left|\Phi_{1}^{N}\right\}\] \[\overset{c}{=}\exp\left(-\theta_{S}\lambda_{E}\int_{r}^{\infty} \Pr\left(\left|h_{E}\right|^{2}>\frac{x\sigma_{E}^{2}\kappa_{A}}{PM_{S}\varsigma _{N}}\right)\left(1-e^{-\beta r_{E_{A}}}\right)r_{E_{A}}dr_{E_{A}}\right)\] \[\overset{d}{=}\exp\left(-\theta_{S}\lambda_{E}\left(\left(\frac{ \left(N_{N}-1\right)!}{\Gamma(N_{N})}\right)\sum\limits_{m=0}^{N_{N}-1}\frac{ \left(\frac{x\sigma_{E}^{2}N_{N}}{PM_{S}\varsigma_{N}}\right)^{m}}{m!}\times \frac{\Gamma\left(\frac{x\sigma_{E}^{2}N_{N}}{PM_{S}\varsigma_{N}}\right)^{m} }{\kappa_{N}\left(\frac{x\sigma_{E}^{2}N_{N}}{PM_{S}\varsigma_{N}}\right)^{m }\frac{x\sigma_{E}^{2}}{PM_{S}\varsigma_{N}}}\right) \tag{15}\] \[=\sum\limits_{i\in\left\{L,N\right\}}\left(\frac{\mathrm{Y}\left( N_{i},\frac{\left(2^{R_{B}}-1\right)\sigma_{D}^{2}}{PM_{S}L\left(r_{D}\right)}N_{i} \right)}{\Gamma\left(N_{i}\right)}\right)P_{i}\left(r_{D}\right), \tag{11}\]0  
For the sake of simplicity, \(F_{\gamma_{E_{A}}}\left(x\right)\) can be simplified as  
\[=\sum\limits_{i\in\left\{L,N\right\}}\left(\frac{\mathrm{Y}\left( N_{i},\frac{\left(2^{R_{B}}-1\right)\sigma_{D}^{2}}{PM_{S}L\left(r_{D}\right)}N_{i} \right)}{\Gamma\left(N_{i}\right)}\right)P_{i}\left(r_{D}\right), \tag{11}\]1  
where \(B=\left(\frac{\left(N_{N}-1\right)! }{\Gamma(N_{N})}\right)\sum\limits_{m=0}^{ N_{N}-1}\frac{q^{m}}{m! }\times\frac{\Gamma\left(\frac{x\sigma_{E}^{2}}{N_{N}}q^{ \sigma_{E}N}\right)}{\kappa_{N}q^{\frac{\max_{N}+2}{\kappa_{N}}}}\); here, \(\Gamma\left(\cdot,\cdot\right)\) is the upper incomplete gamma function [42] (Equation (8.350)), \(A_{N}=\frac{q^{N_{N}}}{\Gamma(N_{N})}\sum\limits_{n=0}^{\infty}\frac{\left(-1 \right)^{n}q^{n}}{n!\left(N_{N}+n\right)}\frac{\Gamma(\kappa_{N}(N_{N}+n)+2, \beta r)}{\beta^{n}(N_{N}+n)+2}\), \(A_{L}=\frac{p^{N_{L}}}{\Gamma(N_{L})}\sum\limits_{n=0}^{\infty}\frac{\left(-1 \right)^{n}p^{n}}{n!\left(N_{L}+n\right)}\frac{\left(\kappa_{A}(N_{L}+n)+2, \beta r\right)}{\beta^{n}\left(N_{L}+n\right)+2}\), \(q=\frac{x\sigma_{E}^{2}N_{N}}{PM_{S}\varsigma_{N}}\), \(p=\frac{N_{L}x\sigma_{E}^{2}}{PM_{S}\varsigma_{L}}\).