## 2 Variation in Jamming Power Reception Due to the Attitude Change  
Figure 1 shows a UAV jammed by a ground jammer in the line of sight. The radiation pattern of the UAV antenna is shown to change according to the attitude change. When the jamming power \(P_{J}\) is applied to the jammer, the jamming power received by the UAV receiver \(P_{A}\) can be expressed by the following equation:  
\[ \begin{split} P_{A}&=P_{J}\Big{(}\frac{\lambda}{4 \pi r}\Big{)}^{2}G_{J}(\theta_{J},\ \phi_{J})G_{A}(\theta_{A},\ \phi_{A})\ \Big{(}1-|\Gamma_{I}|^{2}\Big{)}\Big{(}1-|\Gamma_{A}|^{2}\Big{)}\ \big{|}\hat{\mathbf{\rho}}_{I}(\theta_{I},\ \ \phi_{J})\cdot\hat{\mathbf{\rho}}_{A}(\theta_{A},\ \phi_{A})\big{|}^{2}\\ &=P_{J}\Big{(}\frac{\lambda}{2\eta_{0}r}\Big{)}^{2}\big{|}\overline {E}^{0}_{I}(\theta_{J},\ \phi_{J})\cdot\overline{E}^{0}_{A}(\theta_{A},\ \ \phi_{A})\big{|}^{2}\ \Big{(}1-|\Gamma_{J}|^{2}\Big{)}\Big{(}1-| \Gamma_{A}|^{2}\Big{)},\end{split}  \tag{1}\]  
where \(\lambda\) is the wavelength, \(r\) is the distance between the UAV and the jammer; \(G(\theta,\ \phi)\) and \(\hat{\mathbf{\rho}}(\theta,\ \phi)\) are, respectively, the antenna gain and polarization in a given direction; \(\theta\) and \(\phi\) are, respectively, the elevation and azimuth representing the angular location of the other antenna in reference to the local coordinate system of the antenna; \(\Gamma\) is the reflection coefficient due to the impedance mismatching between the feed line and the antenna; \(\eta_{0}\) is the characteristic impedance in free space; \(\overline{E}^{0}(\theta,\ \phi)\) is the electric field component of the antenna in a given direction in a far field; and the subscripts \(J\) and \(A\), respectively, represent the quantities associated with the jammer antenna and the UAV antenna. The antenna gain \(G(\theta,\ \phi)\) can be determined by the following equation:  
\[G(\theta,\ \phi)=\frac{\frac{r^{2}}{2\eta_{0}}\big{|}\overline{E}(r,\ \theta,\ \phi)\big{|}^{2}}{P_{accepted}/4\pi}=4\pi \frac{\Big{|}\overline{E}^{0}(\theta,\ \phi)\Big{|}^{2}}{2\eta_{0}}, \tag{2}\]  
where \(P_{accepted}\) is the power accepted by the antenna \(\Big{(}=P\Big{(}1-|\Gamma|^{2}\Big{)}\Big{)}\), \(P\) is the applied power to the feed line connected to the antenna, and \(\overline{E}(r,\ \theta,\ \phi)\) is the electric field intensity of the antenna in the far-field. The electric field intensity \(\overline{E}(r,\ \theta,\ \phi)\) can be defined by the following equation:  
\[\overline{E}(r,\ \theta,\ \phi)=\sqrt{P\Big{(}1-|\Gamma|^{2}\Big{)}}\overline{E }^{0}(\theta,\ \phi)\frac{e^{-j\kappa r}}{r}. \tag{3}\]  
The electric field component \(\overline{E}^{0}(\theta,\ \phi)\) can be expressed by the following equation:  
\[ \begin{array}{rl}\overline{E}^{0}(\theta,\ \phi)&=E^{0}_{\theta}(\theta,\ \phi)\hat{\mathbf{ \theta}}+E^{0}_{\theta}(\theta,\ \phi)\hat{\mathbf{\phi}}\\ &=E^{0}_{x}(\theta,\ \phi)\hat{\mathbf{x}}+E^{0}_{y}(\theta,\ \phi)\hat{\mathbf{y}}+E^{0}_{z}( \theta,\ \phi)\hat{\mathbf{z}},\end{array}  \tag{4}\]  
where \(E^{0}_{\theta}(\theta,\ \phi)\) and \(E^{0}_{\phi}(\theta,\ \phi)\) are, respectively, the E-field component of \(\theta\) and \(\phi\) components based on the antenna local coordinate system, whose data can be obtained through a simulation or a measurement of the electric field pattern of the antenna when the antenna operates in the Tx mode; \(\hat{\mathbf{\theta}}\) and \(\hat{\mathbf{\phi}}\) are the unit vectors expressed in the spherical coordinates in reference to the local coordinate system of the antenna; and \(E^{0}_{x}(\theta,\ \phi)\), \(E^{0}_{y}(\theta,\ \phi)\), and \(E^{0}_{z}(\theta,\ \phi)\) are, respectively, the electric field components of the antenna for the \(x,\ y\), and \(z\) components, which can be obtained through coordinate system transformation. \(\hat{\mathbf{x}},\ \hat{\mathbf{y}}\), and \(\hat{\mathbf{z}}\) are, respectively, the unit vectors along the \(x-,\ y-,\ \text{and}\ z-\) axis representingthe global coordinate system.