# On IRS-Assisted Covert Communication with a Friendly UAV
## 2 System Model
### Channel Model  
In a practical system, due to the high altitude of the UAV the UAV-to-ground communications can be treated as line-of-sight (LOS) links. Therefore, we model the UAV-to-ground communication channel as a free-space path loss model, where the signal can propagate along a straight line without obstacles between the transmitter and the receiver. In this scenario, the gain of the channel between the UAV and Bob can be described as  
\[h_{UB}[t]=\sqrt{\beta_{0}d_{UB}^{-2}[t]}=\sqrt{\frac{\beta_{0}}{H^{2}+\|c[t]- \omega_{B}\|}}. \tag{2}\]  
Similarly, the channel gain between the UAV and Willie is  
\[h_{UW}[t]=\sqrt{\beta_{0}d_{UW}^{-2}[t]}=\sqrt{\frac{\beta_{0}}{H^{2}+\|c[t]- \omega_{W}\|}}, \tag{3}\]  
where \(\beta_{0}\) is the channel power gain over the channel at distance \(D_{0}=1\) m. The channel between [PERSON], [PERSON], and [PERSON] is considered a non-line-of-sight (NLOS) channel due to the blockage caused by obstacles. Consider the IRS as a uniform linear array (ULA) antenna, i.e., a system of antennas arranged in a plane according to a linear law, called an antenna array. Then, the gain of the channel from Alice and IRS is  
\[h_{AI}[t]=\sqrt{\beta_{0}d_{AI}^{-a}[t]}\Big{[}1,e^{-i\frac{2\pi}{\lambda}d \phi_{AI}[t]},\ldots,e^{-i\frac{2\pi}{\lambda}d(M-1)\phi_{AI}[t]}\Big{]}^{T}, \tag{4}\]  
Figure 1: IRS-assisted communication system with a friendly UAV. where \(\alpha\) denotes the path loss, \(d_{AI}[t]=\sqrt{\left\|w_{A}-w_{I}\right\|^{2}}\) is the distance between Alice and IRS, \(\phi_{AI}[n]\) is the cosine of the AoA of the signal from the UAV to the IRS in the \(t\)-th time slot, \(d\) is the antenna distance, and \(\lambda\) is the carrier wavelength. Similarly, the channels from the IRS to Bob and Willie are considered LOS channels; thus, the fading of the channels between IRS and Bob and Willie can be respectively expressed as  
\[h_{IB}[t]=\sqrt{\beta_{0}d_{IB}^{-\alpha}[t]}\Big{[}1,e^{-j\frac{2\pi}{\lambda} d\phi_{IB}[t]},\ldots,e^{-j\frac{2\pi}{\lambda}d(M-1)\phi_{IB}[t]}\Big{]}^{T}, \tag{5}\]  
and  
\[h_{IW}[t]=\sqrt{\beta_{0}d_{IW}^{-\alpha}[t]}\Big{[}1,e^{-j\frac{2\pi}{\lambda} d\phi_{IW}[t]},\ldots,e^{-j\frac{2\pi}{\lambda}d(M-1)\phi_{IW}[t]}\Big{]}^{T}, \tag{6}\]  
where \(d_{IB}[t]=\sqrt{\left\|w_{I}-w_{B}\right\|^{2}}\) is the distance between the IRS and [PERSON], \(d_{IW}[t]=\sqrt{\left\|w_{I}-w_{W}\right\|^{2}}\) is the distance between the IRS and [PERSON], and \(\phi_{IB}[t]\) and \(\phi_{IW}[t]\) are the cosines of the angles of departure (AoD) of the signals that [PERSON] and [PERSON] receive from the IRS, respectively. Assuming that the channel utilization tends to be maximized in each time slot according to (3)-(5), the signal received by [PERSON] is  
\[y_{B}[t]=\sqrt{P_{a}[t]}\Big{(}h_{IB}^{H}[t]\Theta[t]h_{AI}[t]\Big{)}x_{a}[t] +\sqrt{P_{a}[t]}h_{UB}[t]x_{u}[t]+n_{B}[t], \tag{7}\]  
where \(P_{a}[t]\) is the transmitting power of [PERSON] and \(x_{a}[t]\) is the signal sent by [PERSON] in the \(t\)-th time slot, which follows a complex Gaussian distribution with mean 0 and variance 1, i.e., \(x_{a}[t]\sim CN(0,1)\). Moreover, \(x_{u}[t]\) is the AN signal transmitted by the UAV, which satisfies \(\mathbb{E}\Big{[}\left|x_{u}[t]\right|^{2}\Big{]}=1\); \(n_{B}[t]\) is the AWGN, with mean 0 and variance \(\sigma_{b}^{2}\) at Bob, i.e., \(n_{B}[t]\sim CN(0,\sigma_{b}^{2})\); and \(P_{u}[t]\) is the AN transmit power of the UAV, which follows a uniform distribution over the interval \(\big{[}0,\tilde{P}_{u}[t]\big{]}\), where \(\tilde{P}_{u}[t]\) is the maximum AN transmit power of the UAV and takes values in the range \(\big{[}0,\tilde{P}_{u,\max}[t]\big{]}\). Accordingly, the Probability Density Function (PDF) is expressed as  
\[f_{P_{u}[t]}(x)=\left\{ \begin{array}{ll}\frac{1}{P_{u}[t]},&0\leq x\leq \hat{P}_{u}[t]\\ 0,&otherwise.\end{array} \right.