# An Adjusted Frequency-Domain Algorithm for Arc Array Bistatic SAR Data with One-Moving Transmitter
## 4 Resolution Analysis
### The Spatial Resolution of AA-BiSAR with Moving Receiver  
Assuming that the transmitter is fixed directly above the scene, and the receiver moves with speed \(v\) along the Y axis, the coordinates of the transmitter, the receiver and the point target are as follows:  
\[\left\{ \begin{array}{l}P_{t}=\left(0,0,H_{t}\right)\\ P_{r}=\left(\theta_{r},R_{arc},H_{r}\right)\\ P_{n}=\left(\theta_{n},R_{n},H_{n}\right)\end{array} \right. \tag{41}\]  
After \(t_{a}\) time, the instantaneous bistatic slant range is obtained as follows:  
\[\left\{ \begin{array}{l}R_{pt}=\sqrt{\left(R_{n}\cos\theta_{n}\right)^{2}+ \left(R_{n}\sin\theta_{n}\right)^{2}+\left(H_{n}-H_{t}\right)^{2}}\\ R_{pr}=\sqrt{\left(R_{n}\cos\theta_{n}-R_{arc}\cos\theta_{r}-vt_{a}\right)^{2 }+\left(R_{n}\sin\theta_{n}-R_{arc}\sin\theta_{r}\right)^{2}+\left(H_{n}-H_{r }\right)^{2}}\end{array} \right. \tag{42}\]  
Figure 6: The ground-range resolution of AA-BiSAR with one-moving transmitter: (**a**) the transmitter position is set as (0 m, 0 m, 1000 m); (**b**) the transmitter position is set as (0 m, 100 m, 1000 m); (**c**) the transmitter position is set as (100 m, 100 m, 1000 m); (**d**) the transmitter position is set as (0 m, 700 m, 1000 m). where \(R_{pt}\) is the distance from the transmitter to the target, \(R_{pr}\) is the distance from the target to the receiver. As in the above section to analyze AA-BiSAR with the moving transmitter, this section uses the same method to analyze the spatial resolution when the receiver is moving. The unit of azimuth resolution is angle, and the azimuth instantaneous phase can be calculated by the following formula:  
\[\psi_{a}=\frac{2\pi f_{c}}{c}R_{\text{tr}}(t_{a}) \tag{43}\]  
where \(R_{\text{tr}}(t_{a})\) is the bistatic slant range, and the specific expression is as follows:  
\[ \begin{array}{ll}R_{\text{tr}}(t_{a})&=\sqrt{\left(R_{n}\cos\theta_{n}-R_{arc }\cos\theta_{r}-vt_{a}\right)^{2}+\left(R_{n}\sin\theta_{n}-R_{arc}\sin\theta _{r}\right)^{2}+\left(H_{n}-H_{r}\right)^{2}}\\ &+\sqrt{\left(R_{n}\cos\theta_{n}\right)^{2}+\left(R_{n}\sin\theta_{n}\right) ^{2}+\left(H_{n}-H_{t}\right)^{2}}\end{array}  \tag{44}\]  
The azimuth instantaneous frequency can be obtained by approximate derivation of Equation (43) as follows:  
\[f_{\theta}=\frac{d(\psi_{a})}{d\theta_{r}}=\frac{2\pi R_{r}\cos\beta_{r}\sin( \theta_{r}-\theta_{n})}{\lambda}+\frac{2\pi\upsilon(vt_{a}-R_{n}\cos\theta_{ n})}{\lambda\omega_{a}\rho} \tag{45}\]  
where \(\rho\) is the distance from the point target to the center point of the arc array, the specific value is \(\rho=\sqrt{{H_{r}}^{2}+{R_{n}}^{2}}\), and \(\cos\beta_{r}=\frac{R_{n}}{\rho}\). The range of the azimuth angle variable \((\theta_{r}-\theta_{n})\) is 0\({}^{\circ}\)-90\({}^{\circ}\), which ensures the monotonicity of the azimuth instantaneous frequency. Therefore, the azimuth resolution of AA-BiSAR with the moving receiver is as follows:  
\[\rho_{\theta}=\frac{2\pi}{\max(f_{\theta})-\min(f_{\theta})}\approx\frac{ \lambda}{2R_{r}\cos\beta_{r}\sin(\theta_{a}/2)}+\frac{\lambda{\omega_{a}}^{2} \rho}{v^{2}\theta_{a}} \tag{46}\]  
Convert the coordinates of the transmitter, the receiver and the point target to polar coordinates as follows:  
\[\left\{ \begin{array}{l}X_{t}=0,\ Y_{t}=0,\ Z_{t}=H_{t}\\ X_{r}=R_{r}\sin\theta_{r},\ Y_{r}=R_{r}\cos\theta_{r},\ Z_{r}=H_{r}\\ X_{n}=R_{n}\sin\theta_{n},\ Y_{n}=R_{n}\cos\theta_{n},\ Z_{n}=H_{n}\end{array} \right. \tag{47}\]  
The bistatic slant range at time \(t_{a}\) can be expressed as:  
\[ \begin{array}{ll}R_{b}=&\sqrt{\left(X_{n}-X_{r}\right)^{2}+\left(Y_{n}-Y_{r }-vt_{a}\right)^{2}+\left(Z_{n}-Z_{r}\right)^{2}}\\ &\hskip 28.452756 pt+\sqrt{{X_{n}}^{2}+{Y_{n}}^{2}+\left(Z_{n}-Z_{t}\right)^{2}} \end{array}