# Multi-Antenna Jammer-Assisted Secure Short Packet Communications in IoT Networks
## 2 System Model  
We consider an IoT downlink short packet communications system, as shown in Figure 1, which consists of an AP ([PERSON]), a legitimate actuator (Bob), a cooperative jammer, and an eavesdropper (Eve). Similar to [29; 30], we assume that all nodes are equipped with a single antenna, except that the jammer has \(N_{j}\) antennas. We also assume that all the involved channels follow quasi-static Rayleigh block fading, such that the channel coefficients keep constant within each transmission block and change independently between different blocks. To exploit the advantages of both multiple antenna and cooperative jamming techniques, we consider three different secure transmission schemes, i.e., the ZFB scheme, NAN scheme, and TAS scheme. For the ZFB scheme, the jammer aims to maximize the interference at Eve, while avoiding interference to Bob. Thus, the beamforming vector \(\mathbf{w_{ZF}}\) is the solution of the following optimization problem:  
\[ \begin{array}{l}\max_{\mathbf{w_{ZF}}}\left|\mathbf{h}_{je}^{\dagger}\mathbf{ w_{ZF}}\right|\\ s.t.\,\,\,\left|\mathbf{h}_{jb}^{\dagger}\mathbf{w_{ZF}}\right|=0\&\epsilon \left\|\mathbf{w_{ZF}}\right\|_{F}=1,\end{array}  \tag{1}\]  
where \(\dagger\) is the conjugate transpose operator, \(\left\|\cdot\right\|_{F}\) is the Frobenius norm, \(\mathbf{h}_{je}\) is the \(N_{j}\times 1\) channel vector for the jammer to Eve link with entries following identical and independently distributed (i.i.d.) Rayleigh fading with parameter \(\tilde{\gamma}_{je}\), and \(\mathbf{h}_{jb}\) is the \(N_{j}\times 1\) channel vector for the jammer to Bob link with entries following i.i.d. Rayleigh fading with parameter \(\tilde{\gamma}_{jb}\). By using projection matrix theory [31], the beamforming vector \(\mathbf{w_{ZF}}\) can be expressed as:  
\[\mathbf{w_{ZF}}=\frac{\mathbb{N}^{\perp}\mathbf{h}_{je}}{\left\|\mathbb{N}^{ \perp}\mathbf{h}_{je}\right\|_{F}}, \tag{2}\]  
where \(\mathbb{N}^{\perp}=\mathbf{I}-\mathbf{h}_{jb}\left(\mathbf{h}_{jb}^{\dagger} \mathbf{h}_{jb}\right)^{-1}\mathbf{h}_{jb}^{\dagger}\) is the projection idempotent matrix with rank \(N_{j}-1\). Accordingly, the received signal-to-noise ratios (SNRs) at Bob and Eve of the ZFB scheme, respectively, can be expressed as:  
Figure 1: A multi-antenna jammer-assisted secure short packet communications in the presence of an eavesdropper. \[\gamma_{b}^{ZBF}=\frac{P_{a}}{\sigma^{2}}|h_{ab}|^{2} \tag{3}\]  
and  
\[\gamma_{e}^{ZBF}=\frac{P_{a}|h_{ae}|^{2}}{P_{j}|\mathbf{h}_{je}^{\dagger}\mathbf{ w}_{\mathbf{ZF}}|^{2}+\sigma^{2}}, \tag{4}\]  
where \(P_{a}\) is the transmit power of Alice, \(P_{j}\) is the transmit power of the jammer, \(h_{ab}\) is the Rayleigh channel coefficient for [PERSON] to Bob link with parameter \(\bar{\gamma}_{ab}\), \(h_{ae}\) is the Rayleigh channel coefficient for [PERSON] to [PERSON] link with parameter \(\bar{\gamma}_{ae}\), and \(\sigma^{2}\) is the noise variance at each receiver. For the NAN scheme, the jammer sends artificial noise in the null space of the legitimate channel to guarantee secure communication. Thus, the beamforming matrix at the jammer is designed as \(\mathbf{W}=\left[\mathbf{w}_{jb},\mathbf{W}_{je}\right]\), where \(\mathbf{w}_{jb}\in\mathbb{C}^{N_{j}\times 1}\) is a weighted vector used for the jammer to Bob link, and \(\mathbf{W}_{je}\in\mathbb{C}^{N_{j}\times N_{j}-1}\) is a weighted matrix for the null space of \(\mathbf{h}_{jb}\), i.e., \(\mathbf{h}_{jb}\mathbf{W}_{je}=0\). As in [29; 32], we assume that the transmit power \(P_{j}\) of the jammer is uniformly allocated in the \(N_{j}-1\) directions. Accordingly, the received SNRs at Bob and Eve of the NAN scheme, respectively, can be expressed as:  
\[\gamma_{b}^{NAN}=\frac{P_{a}}{\sigma^{2}}|h_{ab}|^{2} \tag{5}\]  
and  
\[\gamma_{e}^{NAN}=\frac{P_{a}|h_{ae}|^{2}}{\frac{P_{j}}{N_{j}-1}\left\|\mathbf{ h}_{je}^{\dagger}\mathbf{W}_{je}\right\|_{F}^{2}+\sigma^{2}}. \tag{6}\]  
In addition to transmit beamforming, TAS is regarded as another effective method to improve the physical layer security performance.