# Multi-Antenna Jammer-Assisted Secure Short Packet Communications in IoT Networks
## 3 Secrecy Performance Analysis
### Secrecy Throughput
#### 3.1.1 ZFB Scheme  
The closed-form expression for the secrecy throughput of the IoT downlink short packet communications system with the ZFB 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):  
\[T^{ZFB} \approx\frac{M_{1}B}{2N}\sum_{m=1}^{M_{2}}\Bigg{(}\frac{\pi}{M_{2 }}\sqrt{1-t_{m}^{2}}e^{-\frac{\omega_{1}-\omega_{2}}{4\tau_{2a}}}\Bigg{(} \frac{e^{-\frac{M_{1}(\mu_{m}+1)}{2\lambda_{a}\tau_{2a}}}}{\lambda_{a}\bar{ \gamma}_{ae}\bar{\gamma}_{je}^{N_{j}-1}}\Bigg{(}\frac{M_{1}\lambda_{j}(t_{m}+1 )}{2\lambda_{a}\bar{\gamma}_{ae}}+\frac{1}{\bar{\gamma}_{je}}\Bigg{)}^{1-N_{j}}\] \[\quad+\frac{\lambda_{j}(N_{j}-1)}{\lambda_{a}\bar{\gamma}_{ae} \bar{\gamma}_{je}^{N_{j}-1}}e^{-\frac{M_{1}(\mu_{m}+1)}{2\lambda_{a}\bar{ \gamma}_{ae}}}\Bigg{(}\frac{M_{1}\lambda_{j}(t_{m}+1)}{2\lambda_{a}\bar{\gamma }_{ae}}+\frac{1}{\bar{\gamma}_{je}}\Bigg{)}^{-N_{j}}\Bigg{)}\Bigg{)}\] \[\quad+\frac{B\lambda_{a}^{N_{j}-2}\bar{\gamma}_{ae}^{N_{j}-2}e^{ -\frac{\omega_{1}-1}{4\tau_{2a}}}}{N\lambda_{j}^{N_{j}-1}\bar{\gamma}_{je}^{N_ {j}-1}}\Bigg{(}\sum_{k=1}^{N_{j}-2}\frac{(k-1)\lambda_{j}^{k}\bar{\gamma}_{je}^ {k}(-1)^{N_{j}-2-k}e^{-\frac{M_{1}\omega_{1}}{4\tau_{2a}}}-\frac{M_{1}}{4\tau_ {2a}}}{\big{(}N_{j}-2\}!\big{(}N_{j}-2\big{)}!\big{(}M_{1}\lambda_{j}\bar{ \gamma}_{je}+\lambda_{a}\bar{\gamma}_{ae}\big{)}^{k}\] \[\quad\times\bigg{(}\frac{\omega_{1}}{A_{a}\bar{\gamma}_{ab}}+ \frac{1}{A_{a}\bar{\gamma}_{ae}}\bigg{)}^{N_{j}-2-k}-\frac{(-1)^{N_{j}-2}}{(N_ {j}-2)!}\bigg{(}\frac{\omega_{1}}{\lambda_{a}\bar{\gamma}_{ab}}+\frac{1}{ \lambda_{a}\bar{\gamma}_{ae}}\bigg{)}^{N_{j}-2}\] \[\quad\times e^{\frac{\omega_{1}\bar{\gamma}_{ae}}{4\bar{\gamma}_{ j}\bar{\gamma}_{ae}}+\frac{1}{\bar{\gamma}_{j}\bar{\gamma}_{je}}}Ei\bigg{(}-\bigg{(} \frac{\omega_{1}}{\lambda_{a}\bar{\gamma}_{ab}}+\frac{1}{\lambda_{a}\bar{\gamma }_{ae}}\bigg{)}\bigg{(}M_{1}+\frac{\lambda_{a}\bar{\gamma}_{ae}}{\lambda_{j} \bar{\gamma}_{je}}\bigg{)}\bigg{)}\bigg{)}+e^{-\frac{\omega_{1}-1}{4\tau_{2a}}}\] \[\quad\times\frac{B\lambda_{a}^{N_{j}-1}\bar{\gamma}_{ae}^{N_{j}-1 }(N_{j}-1)e^{-\frac{\omega_{1}-1}{4\tau_{2a}}}}{N\lambda_{j}^{N_{j}-1}\gamma_{je }^{N_{j}-1}(N_{j}-1)!}\Bigg{(}\sum_{k=1}^{N_{j}-1}\frac{(k-1)\lambda_{j}^{k} \bar{\gamma}_{je}^{k}\bar{\gamma}_{je}^{k}-\frac{M_{1}\omega_{1}}{4\tau_{2a}}}{ \big{(}M_{1}\lambda_{j}\bar{\gamma}_{je}+\lambda_{a}\bar{\gamma}_{ae}\big{)}^{k}}\] \[\quad\times\bigg{(}-\frac{\omega_{1}}{\lambda_{a}\bar{\gamma}_{ab}} -\frac{1}{\lambda_{a}\bar{\gamma}_{ae}}\bigg{)}^{N_{j}-1-k}-\bigg{(}-\frac{ \omega_{1}}{\lambda_{a}\bar{\gamma}_{ab}}-\frac{1}{\lambda_{a}\bar{\gamma}_{ ae}}\bigg{)}^{N_{j}-1}\] \[\quad\times e^{\frac{\omega_{1}\bar{\gamma}_{ae}}{4\bar{\gamma}_{ j}\bar{\gamma}_{ab}}+\frac{1}{\bar{\gamma}_{j}\bar{\gamma}_{je}}}Ei\bigg{(}-\bigg{(} \frac{\omega_{1}}{\lambda_{a}\bar{\gamma}_{ab}}+\frac{1}{\lambda_{a}\bar{ \gamma}_{ae}}\bigg{)}\bigg{(}M_{1}+\frac{\lambda_{a}\bar{\gamma}_{ae}}{\lambda_{ j}\bar{\gamma}_{je}}\bigg{)}\bigg{)}\bigg{)}, \tag{11}\]  
where \(Ei(\cdot)\) is the exponential integral function and \(M_{1}\), \(M_{2}\), \(\omega_{1}\), \(\omega_{m}\), and \(t_{m}\) are defined in Appendix A.  
Proof.: See Appendix A.