# Multi-Antenna Jammer-Assisted Secure Short Packet Communications in IoT Networks
## 3 Secrecy Performance Analysis
### Secrecy Throughput
#### 3.1.2 NAN Scheme  
The closed-form expression for the secrecy throughput of the IoT downlink short packet communications system with the NAN scheme can be approximated as (here, we assume that \(N_{j}\geq 3\); when \(N_{j}\leq 3\), the result can be directly derived by similar procedures):\[T^{NAN} \approx \frac{M_{1}B}{2N}\sum_{m=1}^{M_{2}}\Bigg{(}\frac{\pi}{M_{2}}\sqrt{1 -t_{m}^{2}}e^{-\frac{\alpha_{m}}{\lambda_{a}\tilde{\gamma}_{ae}}}\left(\Bigg{(} \frac{M_{1}\lambda_{j}(t_{m}+1)}{2\lambda_{a}\tilde{\gamma}_{ae}(N_{j}-1)}+ \frac{1}{\tilde{\gamma}_{je}}\Bigg{)}^{1-N_{j}}\right. \tag{12}\] \[\times\frac{e^{-\frac{M_{1}(t_{m}+1)}{2\lambda_{a}\tilde{\gamma}_ {ae}}}}{\lambda_{a}\tilde{\gamma}_{ae}\tilde{\gamma}_{je}^{N_{j}-1}}\!+\!\frac{ \lambda_{j}e^{-\frac{M_{1}(t_{m}+1)}{2\lambda_{a}\tilde{\gamma}_{ae}}}}{ \lambda_{a}\tilde{\gamma}_{ae}\tilde{\gamma}_{je}^{N_{j}-1}}\Bigg{(}\frac{M_{ 1}\lambda_{j}(t_{m}+1)}{2\lambda_{a}\tilde{\gamma}_{ae}(N_{j}-1)}+\frac{1}{ \tilde{\gamma}_{je}}\Bigg{)}^{-N_{j}}\Bigg{)}\Bigg{)}\] \[+\frac{B_{a}^{N_{j}-2}\tilde{\gamma}_{ae}^{N_{j}-2}e^{-\frac{ \alpha_{1}-1}{\lambda_{a}\tilde{\gamma}_{ab}}}}{N\lambda_{j}^{N_{j}-1}\tilde {\gamma}_{je}^{N_{j}-1}}\left(\sum_{k=1}^{N_{j}-2}\frac{\left(N_{j}-1\right)^{ N_{j}-1}(k-1)\lambda_{j}^{k}\tilde{\gamma}_{je}^{k}e^{-\frac{M_{1}\alpha_{1}}{ \lambda_{a}\tilde{\gamma}_{ab}}-\frac{M_{1}}{\lambda_{a}\tilde{\gamma}_{ae}}} }{\left(N_{j}-2\right)!\left(M_{1}\lambda_{j}\tilde{\gamma}_{je}+\lambda_{a} \tilde{\gamma}_{ae}\left(N_{j}-1\right)\right)^{k}}\right.\] \[\times\left(-\frac{\varpi_{1}}{\lambda_{a}\tilde{\gamma}_{ab}}- \frac{1}{\lambda_{a}\tilde{\gamma}_{ae}}\right)^{N_{j}-2-k}-\frac{e^{\frac{ \varpi_{1}\gamma_{ae}\left(N_{j}-1\right)}{2\lambda_{a}\tilde{\gamma}_{je}^{N _{j}}\varpi_{ab}}}+\frac{N_{j}^{-1}}{\lambda_{j}\tilde{\gamma}_{je}}}{\left( N_{j}-2\right)!\left(-\frac{\varpi_{1}}{\lambda_{a}\tilde{\gamma}_{ab}}-\frac{1}{ \lambda_{a}\tilde{\gamma}_{ae}}\right)^{N_{j}-2}}\] \[\times\left(N_{j}-1\right)^{N_{j}-1}\!Ei\!\left(-\frac{\varpi_{1 }}{\lambda_{a}\tilde{\gamma}_{ab}}+\frac{1}{\lambda_{a}\tilde{\gamma}_{ae}} \right)\!\left(M_{1}+\frac{\lambda_{a}\tilde{\gamma}_{ae}\left(N_{j}-1\right)} {\lambda_{j}\tilde{\gamma}_{je}}\right)\right)\Bigg{)}\] \[+\frac{B_{a}^{N_{j}-1}\tilde{\gamma}_{ae}^{N_{j}-1}\!\left(N_{j} -1\right)^{N_{j}}e^{-\frac{\varpi_{1}-1}{\lambda_{a}\tilde{\gamma}_{ab}}}}{N \lambda_{j}^{N_{j}-1}\gamma_{je}^{N_{j}-1}\!\left(N_{j}-1\right)!}\left(\sum_ {k=1}^{N_{j}-1}\frac{\left(k-1\right)!\lambda_{k}^{k}\tilde{\gamma}_{je}^{k} \!\left(-1\right)^{N_{j}-1-k}e^{-\frac{M_{1}}{\lambda_{a}\tilde{\gamma}_{ae} }}}{\left(M_{1}\lambda_{j}\tilde{\gamma}_{je}+\lambda_{a}\tilde{\gamma}_{ae} \left(N_{j}-1\right)\right)^{k}}\right.\] \[\times\left(\frac{\varpi_{1}}{\lambda_{a}\tilde{\gamma}_{ab}}+ \frac{1}{\lambda_{a}\tilde{\gamma}_{ae}}\right)^{N_{j}-1-k}e^{-\frac{M_{1} \varpi_{1}}{\lambda_{a}\tilde{\gamma}_{ab}}}-\left(-\frac{\varpi_{1}}{\lambda _{a}\tilde{\gamma}_{ab}}-\frac{1}{\lambda_{a}\tilde{\gamma}_{ae}}\right)^{N_{ j}-1}e^{\frac{N_{j}-1}{\lambda_{j}\tilde{\gamma}_{je}}}\] \[\times e^{\frac{\varpi_{1}\gamma_{ae}\left(N_{j}-1\right)}{ \lambda_{a}\tilde{\gamma}_{be}}}\ Ei\!\left(-\!\left(\frac{\varpi_{1}}{\lambda_{a} \tilde{\gamma}_{ab}}+\frac{1}{\lambda_{a}\tilde{\gamma}_{ae}}\right)\!\left(M_{ 1}+\frac{\lambda_{a}\tilde{\gamma}_{ae}\left(N_{j}-1\right)}{\lambda_{j} \tilde{\gamma}_{je}}\right)\right)\right).\]  
Proof.: Since \(\left\|\mathbf{h}_{je}^{\dagger}\mathbf{W}_{je}\right\|_{F}^{2}\) is a chi-squared random variable with \(2\big{(}N_{j}-1\big{)}\) degrees of freedom [33], by following similar steps as the ZFB scheme, we can obtain the desired expression in (12) after some mathematical manipulations.