## 2 THz Imaging Character of Parabolic Antennas
\begin{bmatrix}\cos\theta_{\omega}&-\sin\theta_{\omega}&0\\ \sin\theta_{\omega}&\cos\theta_{\omega}&0\\ 0&0&1\end{bmatrix} . \tag{11}\]  
In this scenario, the projection relationship is  
Figure 2: The observational geometry for space-based terahertz radar in detecting space targets. \[N=P\cdot R_{W}\cdot R_{V}\cdot R_{U}\cdot M=P^{{}^{\prime}}\cdot M \tag{12}\]  
where \(P^{\prime}=P\cdot R_{W}\cdot R_{V}\cdot R_{U}\). Moreover, in the geometric model of THz-ISAR imaging, the motion of the target relative to the radar can be independently decomposed into translational and rotational components. Taking into account the initial slant range and the instantaneous change in slant range due to translational motion, the instantaneous slant range \(R_{i}(t_{m})\) of scatter point \(i\) can be expressed as:  
\[R_{i}(t_{m})=r_{0}+r(t_{m})+x_{i}\sin\omega t_{m}+y_{i}\cos\omega t_{m} \tag{13}\]  
where \(t_{m}\) denotes the slow time, \(r_{0}\) represents the initial slant range, \(r_{m}\) signifies the slant range of the target's rotation center as it varies with the slow time, \(x_{i}\) and \(y_{i}\) are the coordinates of the \(i\)-th scatter point, and \(\omega\) is the rotation speed of the target. It is assumed that the radar emits a linear frequency-modulated (LFM) signal:  
\[s(\hat{t};t_{m})=\text{rect}\Big{[}\frac{\hat{t}}{T_{p}}\Big{]}\cdot\exp\Big{(} j2\pi f_{c}t+j\pi\gamma\hat{t}^{2}\Big{)}. \tag{14}\]  
Following the delay time  
\[\tau=\frac{2R_{i}(t_{m})}{c}, \tag{15}\]  
the echo signal that has been acquired is  
\[s_{i}(\hat{t};t_{m})=\text{rect}\Big{[}\frac{\hat{t}-2R_{i}(t_{m})/c}{T_{p}} \Big{]}\cdot\exp\Bigg{\{}j2\pi f_{c}\Big{[}\hat{t}-\frac{2R_{i}(t_{m})}{c} \Big{]}+j\pi\gamma_{0}\Big{[}\hat{t}-\frac{2R_{i}(t_{m})}{c}\Big{]}^{2}\Bigg{\}} \tag{16}\]  
where \(T_{p}\) denotes the pulse duration, \(f_{c}\) stands for the carrier frequency, and \(\gamma_{0}\) indicates the frequency deviation of the linear frequency-modulated (LFM) signal. We proceed to perform declirping and pulse compression on the received echo signal:  
\[S(f_{r};t_{m})=\sum_{i=0}^{Q-1}\bigg{\{}\sigma_{i}\cdot\text{sinc}\bigg{\{}T_ {p}\bigg{[}f_{r}+\frac{2R_{i}(t_{m})}{c}\bigg{]}\bigg{\}}\cdot\exp\bigg{[}-j \frac{4\pi f_{c}}{c}R_{i}(t_{m})\bigg{]}\bigg{\}} \tag{17}\]  
where \(Q\) denotes the number of scatter points and \(\sigma_{i}\) denotes the radar cross section of \(i\)-th scatter point. Based on the imaging geometry above, the imaging characteristic of one ideal parabolic antenna is first analyzed. The geometry of the paraboloid within the Cartesian coordinate system is depicted in Figure 3, and its equation could be presented in this manner:  
\[x=\frac{y^{2}+z^{2}}{4K}, \tag{18}\]  
where \(K\) represents the focal length. Given the rotational symmetry of the paraboloid, this study focuses on its projection onto the x-y plane, which forms a parabola. In Figure 3, the parabola is depicted as an orange curve. The diameter is represented by D, and \(\varphi\) denotes the angle between the electromagnetic wave transmitted by the radar and the y-axis. It is assumed that the scenario satisfies the far field condition, so the electromagnetic wave emitted by radar can be regarded as a plane wave. The focal point is denoted by G, and P is the specular reflected point on the parabola as it relates to the incident angle. \(\rho\) is the distance between G and P, and \(\beta\) is the angle between line segment GP and the y-axis.