# Motion Planning in UAV-Aided Data Collection with Dynamic Jamming
## 2 Problem Statement and Formulation
### Channel Model  
The communication channel of the UAV has a crucial influence on data collection. Thus, we present the channel model first. The distance between the UAV and jammer \(j\) in the \(n\)-th slot is as follows:  
\[d_{j}[n]=\big{\|}p[n]-p_{j}[n]\big{\|},\forall n\in\mathcal{N}, \tag{1}\]  
The distance between the UAV and SN \(k\) in the \(n\)-th slot is as follows:  
\[d_{k}[n]=\|p[n]-p_{k}\|,\forall n\in\mathcal{N},k\in\mathcal{K}, \tag{2}\]  
We assume that the G2A channel comprises large- and small-scale fading. Small-scale fading is regarded as an identically distributed and independent Rician channel, and the Rice factor is \(\mathcal{A}\); the channel coefficient \(h_{i}[n]\) can be expressed as follows:  
\[h_{i}[n]=\hat{h}_{i}\cdot g_{i}[n], \tag{3}\]  
where \(\hat{h}_{i}\) and \(g_{i}[n]\) are the path-loss coefficients and small-scale fading, respectively. We can express the path-loss coefficient as follows:  
\[g_{i}[n]=\sqrt{\mu_{0}\cdot(d_{i}[n])^{-\alpha}}, \tag{4}\]  
where \(\mu_{0}\) is the mean channel power gain over the reference distance \(d_{0}=1\) m, and \(\alpha\) is the path-loss exponent, which typically exceeds 2 for a Rician fading channel. Small-scale fading \(\hat{h}_{i}\) consists of the LoS component \(\bar{h}_{i}\), where \(\big{|}\bar{h}_{i}\big{|}=1\), and a random NLoS component \(\bar{h}_{i}\), where \(\bar{h}_{i}\sim\mathcal{CN}(0,1)\). Small-scale fading \(\hat{h}_{i}\) is determined as follows:  
\[\hat{h}_{i}=\sqrt{\frac{\mathcal{A}}{\mathcal{A}+1}}\cdot\bar{h}_{i}+\sqrt{ \frac{1}{\mathcal{A}+1}}\cdot\bar{h}_{i}, \tag{5}\]  
Figure 1: UAV-aided Data Collection with Dynamic Jamming.  
Assuming that each jammer is transmitting at a constant power \(P_{j}\), the instantaneous signal to interference plus noise ratio (SINR) of the SN k is as follows:  
\[\xi_{k}[n]=\frac{P_{k}x_{k}u_{0}d_{k}[n]^{-\alpha}}{\sum\limits_{j=0}^{J-1}x_{j}u _{0}d_{j}[n]^{-\alpha}+\sigma^{2}}, \tag{6}\]  
where \(x_{i}=\left|\hat{h}_{i}\right|^{2}\), which is a random variable that obeys a non-central \(\chi^{2}\) distribution. To simplify the optimization problem presented below, we further perform the homogeneous approximation of the SINR function, which considers the expected value of the SINR and yields the following result:  
\[\mathbb{E}[\xi_{k}[n]]=\frac{P_{k}\cdot\beta_{k}\cdot d_{k}[n]^{-\alpha}}{ \sum\limits_{j=0}^{J-1}P_{j}\cdot\beta_{j}\cdot d_{j}[n]^{-\alpha}+\sigma^{2}}, \tag{7}\]  
where  
\[\beta_{k}=\int_{0}^{\infty}x_{k}\cdot f(x_{k})\cdot\mu_{0}\cdot dx_{k}, \tag{8}\]  
\[\beta_{j}=\int_{\begin{subarray}{c}0\\ x_{0},\cdots,x_{J-1}\end{subarray}}^{\infty}\cdots\int_{0}^{\infty}x_{j}\mu_{0 }\prod\limits_{j}^{J-1}f(x_{j})dx_{0}\cdots dx_{J-1}, \tag{9}\]  
\[f(x)=\frac{\mathcal{A}+1}{\mathcal{P}}\exp(-\mathcal{A}-\frac{(\mathcal{A}+1 )x}{\mathcal{P}})l_{0}(2\sqrt{\frac{\mathcal{A}(\mathcal{A}+1)x}{\mathcal{P}} }), \tag{10}\]  
where \(\mathcal{P}\) is the signal model parameter, which, in general, can be set to 1. Moreover, \(I_{0}(\cdot)\) is a Bessel function of order zero. \(\beta_{k}\) and \(\beta_{j}\) are constants in each independent simulation that can be generated according to the Rician channel model. When the time slot is sufficiently small or when the flight speed is low, we consider the position of the UAV a fixed point in each time slot. The expected information rate (bits/s) between the UAV and SN \(k\) in the \(n\)-th slot is represented as follows:  
\[r_{k}[n]=\frac{B}{K}\cdot\log_{2}(1+\frac{P_{k}\cdot\beta_{k}\cdot d_{k}[n]^{- \alpha}}{\sum\limits_{j=0}^{J-1}P_{j}\cdot\beta_{j}\cdot d_{j}[n]^{-\alpha}+ \sigma^{2}}). \tag{11}\]  
where \(B\) is the total channel bandwidth in Hz, and \(\sigma^{2}\) represents the noise power spectral density.