# An Adjusted Frequency-Domain Algorithm for Arc Array Bistatic SAR Data with One-Moving Transmitter
## 4 Resolution Analysis
### AA-BiSAR with Moving Transmitter
#### 4.1.2 Ground-Range Resolution  
The range resolution reflects the ability of radar to distinguish two close targets, which is mainly determined by the bandwidth of the transmitted signal. AA-BiSAR with one-moving transmitter, the transmitter and receiver are separated. Different from traditional monostatic SAR, the instantaneous slant range of AA-BiSAR comes from two platforms, the range of a certain platform can not be expressed as the bistatic range alone. Therefore, in this article, the ground-range resolution of AA-BiSAR with one-moving transmitter is analyzed via the gradient method. Figure 5: The azimuth angular resolution of AA-BiSAR with one-moving transmitter: (**a**) the transmitter velocity is set as 0 m/s; (**b**) the transmitter velocity is set as 50 m/s; (**c**) the transmitter velocity is set as 300 m/s; (**d**) the transmitter velocity is set as 1000 m/s. Assuming that the initial coordinates of the transmitter, receiver, and the point target are: \(P_{t}=(X_{t},Y_{t},Z_{t})\), \(P_{r}=(X_{r},Y_{r},Z_{r})\), \(P_{n}=(X_{n},Y_{n},Z_{n})\), converted to the cylindrical coordinates as follows:  
\[\left\{ \begin{array}{l}X_{t}=R_{t}\sin\theta_{t},\ Y_{t}=R_{t}\cos\theta_{t}, \ Z_{t}=H_{t}\\ X_{r}=R_{r}\sin\theta_{r},\ Y_{r}=R_{r}\cos\theta_{r},\ Z_{r}=H_{r}\\ X_{n}=R_{n}\sin\theta_{n},\ Y_{n}=R_{n}\cos\theta_{n},\ Z_{n}=H_{n}\end{array} \right. \tag{31}\]  
The bistatic slant range of point target at azimuth time \(t_{a}\) is expressed as:  
\[ \begin{array}{l}R_{b}=\\ +\sqrt{(X_{n}-X_{t})^{2}+(Y_{n}-Y_{t}-vt_{a})^{2}+(Z_{n}-Z_{t})^{2}}\end{array}  \tag{32}\]  
The gradient of any point target \((X_{n},Y_{n})\) on the isometric line is given by:  
\[\
abla R_{b}=\bigg{(}\frac{X_{n}-X_{t}}{R_{T}}+\frac{X_{n}-X_{r}}{R_{R}} \bigg{)}i+\bigg{(}\frac{Y_{n}-Y_{t}-vt_{a}}{R_{T}}+\frac{Y_{n}-Y_{r}}{R_{R}} \bigg{)}j_{y} \tag{33}\]  
where \(R_{T}\) is the instantaneous slant range from the transmitter to the target position, and \(R_{R}\) is the instantaneous slant range from the target position to the receiver. Let \(\xi_{t}\) and \(\xi_{r}\) denote the squint angles of the transmitter and receiver. Define \(\phi_{t}\) and \(\phi_{r}\) as the side-looking angles of the transmitter and receiver, they are expressed as follows:  
\[\xi_{t}=\arcsin\Bigg{[}\frac{Y_{n}-Y_{t}-vt_{a}}{\sqrt{\left(X_{n}-X_{t} \right)^{2}+\left(Y_{n}-Y_{t}-vt_{a}\right)^{2}+\left(Z_{n}-Z_{t}\right)^{2} }}\Bigg{]} \tag{34}\]  
\[\xi_{r}=\arcsin\Bigg{[}\frac{Y_{n}-Y_{r}}{\sqrt{\left(X_{n}-X_{r}\right)^{2}+ \left(Y_{n}-Y_{r}\right)^{2}+\left(Z_{n}-Z_{r}\right)^{2}}}\Bigg{]} \tag{35}\]  
\[\phi_{t}=\arcsin\Bigg{[}\frac{X_{n}-X_{t}}{\sqrt{\left(X_{n}-X_{t}\right)^{2} +\left(Z_{n}-Z_{t}\right)^{2}}}\Bigg{]} \tag{36}\]  
\[\phi_{r}=\arcsin\Bigg{[}\frac{X_{n}-X_{r}}{\sqrt{\left(X_{n}-X_{r}\right)^{2} +\left(Z_{n}-Z_{r}\right)^{2}}}\Bigg{]} \tag{37}\]  
Let the gradient be expressed by the squint angle and the side-looking angle, the gradient equation and gradient size of the point target can be obtained as follows:  
\[\
abla R_{b}=\left(\sin\phi_{t}\cos\xi_{t}+\sin\phi_{r}\cos\xi_{r}\right) \overset{\rightarrow}{i}_{x}+\left(\sin\xi_{t}+\sin\xi_{r}\right)\overset{ \rightarrow}{i}_{y} \tag{38}\]  
\[\left|\
abla R_{b}\right|=\sqrt{\left(\sin\phi_{t}\cos\xi_{t}+\sin\phi_{r}\cos \xi_{r}\right)^{2}+\left(\sin\xi_{t}+\sin\xi_{r}\right)^{2}} \tag{39}\]  
According to Equations (38) and (39), it can be concluded that the ground-range resolution of AA-BiSAR with one-moving transmitter system is expressed as follow:  
\[\rho_{r}=\frac{c/B_{r}}{\left|\
abla R\right|}=\frac{c/B_{r}}{\sqrt{\left(\sin \phi_{t}\cos\xi_{t}+\sin\phi_{r}\cos\xi_{r}\right)^{2}+\left(\sin\xi_{t}+\sin \xi_{r}\right)^{2}}} \tag{40}\]  
where \(B_{r}\) is the signal bandwidth, \(c\) is the speed of light.