# Multi-Antenna Jammer-Assisted Secure Short Packet Communications in IoT Networks
## 3 Secrecy Performance Analysis
### Secrecy Throughput
#### 3.1.3. TAS Scheme  
The closed-form expression for the secrecy throughput of the IoT downlink short packet communications system with the TAS scheme can be approximated as (here, we assume that \(c_{1}\
eq\frac{\lambda_{a}\tilde{\gamma}_{ae}}{\lambda_{j}\tilde{\gamma}_{je}}\), when \(c_{1}=\frac{\lambda_{a}\tilde{\gamma}_{ae}}{\lambda_{j}\tilde{\gamma}_{je}}\), the result can be obtained by similar analysis):  
\[T^{TAS} \approx \frac{M_{1}B}{2N}\sum_{m=1}^{M_{2}}\Bigg{(}\frac{\pi N_{j} \lambda_{a}\tilde{\gamma}_{ab}\sqrt{1-t_{m}^{2}}e^{-\frac{\varpi_{m}}{\lambda_{ a}\tilde{\gamma}_{ae}}}}{M_{2}\varpi_{a}\lambda_{j}\tilde{\gamma}_{j}+M_{2}\lambda_{a} \tilde{\gamma}_{ab}N_{j}}\Bigg{(}\frac{2e^{-\frac{M_{1}\left(\varpi_{1}+1\right)} {2\lambda_{a}\tilde{\gamma}_{ae}}}}{\lambda_{j}\tilde{\gamma}_{je}M_{1}\! \left(t_{m+1}+2\lambda_{a}\tilde{\gamma}_{ae}\right.} \tag{13}\] \[+\frac{4\lambda_{a}\tilde{\gamma}_{ae}\lambda_{j}\tilde{\gamma}_{je} e^{-\frac{M_{1}\left(\varpi_{1}\right)}{2\lambda_{a}\tilde{\gamma}_{ae}}}}{\left( \lambda_{j}\tilde{\gamma}_{je}M_{1}\!\left(t_{m+1}+2\lambda_{a}\tilde{\gamma}_{ ae}\right)^{2}\right)}\Bigg{)}-\frac{BN_{j}\lambda_{a}^{2}\tilde{\gamma}_{ab}\tilde{\gamma}_{ ae}\tilde{\gamma}_{je}^{-\frac{M_{1}\alpha_{1}}{\lambda_{a}\tilde{\gamma}_{je}}}-\frac{M_{1} \alpha_{1}}{\lambda_{a}\tilde{\gamma}_{je}}}{N\lambda_{0}\gamma_{1}\gamma_{j} \left(M_{1}\lambda_{j}\tilde{\gamma}_{je}+\lambda_{a}\tilde{\gamma}_{ae} \right)}\] \[\times\frac{1}{\lambda_{a}\tilde{\gamma}_{ae}-c_{1}\lambda_{j} \tilde{\gamma}_{je}}+\left(\frac{\lambda_{a}\lambda_{j}\tilde{\gamma}_{je}}{ \left(\lambda_{a}\tilde{\gamma}_{ae}-c_{1}\lambda_{j}\tilde{\gamma}_{je}\right)^{2} }-\frac{\varpi_{1}}{\left(\lambda_{a}\tilde{\gamma}_{ae}-c_{1}\lambda_{j} \tilde{\gamma}_{je}\right)^{2}}-\frac{\varpi_{1}}{\left(\lambda_{a}\tilde{\gamma}_ {ae}-c_{1}\lambda_{j}\tilde{\gamma}_{je}\right)^{2}}\right)\] \[\times\frac{BN_{j}\lambda_{a}\tilde{\gamma}_{ab}\tilde{\gamma}_{ae} e^{\frac{\varpi_{1}\gamma_{ae}}{\lambda_{a}\tilde{\gamma}_{je}}+\frac{1}{\lambda_{a}\tilde{\gamma}_{ ae}}}Ei\!\left(-\frac{\varpi_{1}\left(M_{1}+c_{1}\right)}{\lambda_{a}\tilde{\gamma}_{ ae}}-\frac{M_{1}+c_{1}}{\lambda_{a}\tilde{\gamma}_{ae}}\right)\]where \(c_{1}=\frac{(\alpha_{1}-1)\lambda_{j}\gamma_{jb}+\lambda_{a}\tau_{ab}N_{j}}{ \alpha_{1}\lambda_{j}\gamma_{jb}}\).  
Proof.: Since \(\left|h_{j_{i}\epsilon}\right|^{2}\) is an exponentially distributed random variable and the probability density function (PDF) of \(Z_{1}=\min_{1\leq i\leq N_{j}}\left|h_{jb}\right|^{2}\) is given by \(f_{Z_{1}}(z)=\frac{N_{j}}{\gamma_{jb}}e^{-\frac{N_{j}^{2}}{\gamma_{jb}}}\)[29], by following a similar procedure as the ZFB scheme, we can derive the desired result in (13) after some mathematical manipulations.  
The derived analytical results in (11)-(13) provide an efficient means to evaluate the secrecy throughput of the IoT downlink short packet communications system with the three proposed transmission schemes. Moreover, the approximations for the secrecy throughput of finite blocklength will be verified via simulations.