## 4 Signal Attenuation Model for Inter-Satellite Terahertz Communication
### Signal Attenuation Model of Inter-Satellite Terahertz Communication under Pointing Error
\right] \tag{13}\]  
\[\left[ \begin{array}{c}\Delta x_{0}\\ \Delta y_{0}\\ \Delta z_{0}\end{array} \right]=A\left[ \begin{array}{c}\Delta x_{1}\\ \Delta y_{1}\\ \Delta z_{1}\end{array} \right]\]  
The pointing error angle is decomposed into the azimuth pointing error angle \(\Delta\theta_{xy}\) and the pitch pointing error angle \(\Delta\theta_{xz}\), and the pointing error angle can be expressed as Equation (14). \[\theta=\sqrt{\left(\cos\theta_{xz}\Delta\theta_{xy}+\sin\theta_{xy}\sin\theta_ {xz}\Delta\theta_{xz}\right)^{2}+\cos^{2}\theta_{xy}\Delta\theta_{xz}} \tag{14}\]  
At the receiving end, the terahertz beam offset \(R_{T}\) is expressed as \(R_{T}=L\theta\). Assuming that the receiving end is a circular surface, affected by the pointing error, there will be a certain misalignment between the beam center at the receiving end and the center of the  
Figure 6: Schematic diagram of the transmitting signal of the terahertz wave transmitter: (**a**) the change of the signal beam emission angle in the antenna coordinate system; (**b**) the change of the signal beam emission direction. circular surface. The beam trajectory change of the receiving end caused by the pointing error is shown in Figure 6.  
As shown in Figure 7, a plane rectangular coordinate system is established with the center of the receiver receiving plane as the origin, and under the condition of pointing error, the terahertz beam distribution at the receiving end is expressed by Equation (15). \[T(r-R_{T},L)=T_{0}\frac{{R_{0}}^{2}}{{R_{e}}^{2}}\exp\left(-\frac{2(r-R_{T})^{2 }}{{R_{e}}^{2}}\right) \tag{15}\]  
\(h_{err}(\theta)\) is the signal power attenuation considering the pointing error, expressed by Equation (16), which is a function of the pointing error angle. The larger the value of \(h_{err}(\theta)\), the smaller the influence of the pointing error; the smaller the value of \(h_{err}(\theta)\), the greater the influence of the pointing error. \[h_{err}(\theta)=\frac{P_{R}(\theta)}{P_{T}}=\frac{\int_{x_{1}}^{x_{2}}\int_{ \sqrt{{D_{Rx}}^{2}-(x-L\Delta\theta_{xy})^{2}}+L\Delta\theta_{xx}}^{\sqrt{{D _{Rx}}^{2}-\kappa^{2}}}T(x,y)dxdy}{\int_{-{D_{Rx}}}^{\infty}\int_{-\sqrt{{D_{ Rx}}^{2}-\kappa^{2}}}^{\sqrt{{D_{Rx}}^{2}-\kappa^{2}}}T(x,y)dxdy} \tag{16}\]  
In the Equation (16), \(0\leq h_{err}(\theta)\leq 1\), when \(h_{err}(\theta)\) is equal to 1, it means that there is no pointing error, and when \(h_{err}(\theta)\) is equal to 0, it means that the receiving end cannot receive any signal due to the extremely large pointing error. \(x_{1}\) and \(x_{2}\) can be solved by Equation (17). \[\left\{ \begin{aligned} \kappa^{2}+y^{2}={R_{e}}^{2}\\ (\kappa-L\Delta\theta_{xy})^{2}+(y-L\Delta\theta_{xz})^{2}={r_{e}}^{2} \end{aligned} \right. \tag{17}\]  
Since it is difficult to solve the integral solution of signal power attenuation in the plane rectangular coordinate system, considering that the actual receiving area of the receiving end is a circle, the integral calculation is carried out in the polar coordinate system, and Equation (18) can be obtained, that is, the analytic solution of signal power attenuation. \[h_{err}(\theta)=\frac{P_{R}(\theta)}{P_{T}}=\left\{ \begin{aligned} \frac{\exp\left(-\frac{2\kappa^{2}}{R_{T}^{2}}\right)-\exp \left(-\frac{sO_{Rx}^{2}}{{R_{e}}^{2}-R_{T}}\right)+\frac{\sqrt{2}\pi}{R_{T}} \left[\sigma r\left(\frac{\sqrt{2}R_{r}}{R_{t}}\right)+\sigma r\left(\left| \frac{2\sqrt{2}{D_{Rx}}}{\left[-R_{T}\right]R_{t}}\right|\right)\right]}{1- \exp\left(-\frac{2\mathcal{D}_{T}^{2}}{{R_{t}}^{2}}\right)},{D_{Rx}}>R_{T}\\ \frac{\exp\left(-\frac{2\kappa^{2}}{R_{T}^{2}}\right)-\exp\left(- \frac{sO_{Rx}^{2}}{{R_{e}}^{2}-R_{T}}\right)+\frac{\sqrt{2}\pi}{R_{T}}\pi}{1- \exp\left(-\frac{2\mathcal{D}_{Rx}^{2}}{{R_{t}}^{2}}\right)},{D_{Rx}}\leq R_{T }\end{aligned} \right. \tag{18}\]  
Figure 7: Schematic diagram of beam trajectories caused by pointing errors in the receiver plane.