# Microwave Staring Correlated Imaging Based on Unsteady Aerostat Platform
## 4 Simulation
### Effect of Different Translational Components on Imaging Performance  
This section is to study the effect of independent translational component on imaging performance. In simulations, all independent translational components use the same motion trajectory as shown in Figure 5a. Figure 12 shows the imaging results reconstructed by UPIM when only one translational component exists. Figure 11: NMSE of the imaging results by UPIM, DPDIM, and SPIM at different amplitudes of rotation. Figure 10: NMSE of the imaging results by UPIM, DPDIM, and SPIM at different amplitudes of translation. The imaging quality for each translational component under different translation amplitudes is presented in Figure 13. As shown in Figures 12 and 13, the translation along the \(X_{t}\) has the minimal influence on imaging performance, while both translation along the \(Y_{t}\) and \(Z_{t}\) has almost the same influence on imaging performance. Therefore, for improving the image performance, the position of the antenna array along the \(Y_{t}\) and the \(Z_{t}\) should be estimated more accurate. Next, we investigate the reason for the different imaging results in three-dimensional translations. Although the proposed method compensates in part of the non-cooperative motion, due to the limited number of POS data, the estimated translation errors cannot be totally eliminated. Figure 14 shows the estimation error in three-dimensional translations. The accurate calculation of the radiation field is directly related to the round-trip propagation delay of the electromagnetic wave between antenna and target. The propagation time delay error caused by estimation error of array position will lead to the calculated radiation field error. Because the translational estimation errors in three dimensions have different effect on the propagation time delay, the influence of different translational components on imaging is not the same. Figure 12: The imaging results of UPIM when only one translation component exists. (**a**) Only translational component along the \(X_{t}\) exists, the NMSE is 0.21; (**b**) Only translational component along the \(Y_{t}\) exists, the NMSE is 0.58; (**c**) Only translational component along the \(Z_{t}\) exists, the NMSE is 0.59. Figure 13: NMSE of the imaging results when only one translational component exists at different amplitudes. Assuming at \(t\) time, the coordinate of the \(i\)-th antenna is \((x_{a},y_{a},z_{a})\) and the coordinate of any point \(m\) in the imaging region is \((x,y,z)\). After \(\Delta t\), the coordinate of the antenna become \((x_{a}+\Delta x_{a},y_{a}+\Delta y_{a},z_{a}+\Delta z_{a})\). The distance from the \(i\)-th antenna to the point \(m\) in imaging region is  
\[S_{im} = \sqrt{\left(x-x_{a}\right)^{2}+\left(y-y_{a}\right)^{2}+\left(z-z_{ a}\right)^{2}}. \tag{23}\]  
The partial differential of the propagation path along three coordinate dimensions is  
\[\frac{\partial S_{im}}{\partial x_{a}} = \frac{x_{a}-x}{\sqrt{\left(x-x_{a}\right)^{2}+\left(y-y_{a}\right) ^{2}+\left(z-z_{a}\right)^{2}}}, \tag{24}\] \[\frac{\partial S_{im}}{\partial y_{a}} = \frac{y_{a}-y}{\sqrt{\left(x-x_{a}\right)^{2}+\left(y-y_{a}\right) ^{2}+\left(z-z_{a}\right)^{2}}},\] (25) \[\frac{\partial S_{im}}{\partial z_{a}} = \frac{z_{a}-z}{\sqrt{\left(x-x_{a}\right)^{2}+\left(y-y_{a}\right) ^{2}+\left(z-z_{a}\right)^{2}}}. \tag{26}\]  
In the simulation scenario, the height of the antenna array is 350 m and the antenna is squint observation with the slanting angle \(45^{\circ}\). For any point in the imaging area, its coordinate satisfies \(291.5\leq y\leq 408.5\), \(-58.5\leq x\leq 58.5\) and \(z=0\). Because the size of antenna array is much smaller than the size of imaging region, therefore for most points in the imaging region, it is satisfied that \(|x|\gg|x_{a}|\) and \(|y|\gg|y_{a}|\), and the value of partial differential function satisfy that \(|\partial S_{im}/\partial y_{a}\ |>|\partial S_{im}/\partial x_{a}\ |\).