# An Adjusted Frequency-Domain Algorithm for Arc Array Bistatic SAR Data with One-Moving Transmitter
## 4 Resolution Analysis
### AA-BiSAR with Moving Transmitter  
#### 4.1.1 Azimuth Resolution  
The azimuth resolution of AA-BiSAR is also called the azimuth angular resolution, the unit generally is an angle, which can be derived from the echo signal of the point target. According to the analysis in the above section, the raw data of AA-BiSAR with one-moving transmitter system is given by:  
\[ \begin{array}{ll}s(t_{r},t_{a})=&A_{0}\omega_{a}\left(\frac{\phi_{a}-\theta_ {a}}{\theta_{a}}\right)\omega_{r}(t_{r}-R_{\text{tr}}(t_{a})/c)\\ &\cdot\exp\Bigl{[}-j2\pi\frac{f_{z}}{c}R_{\text{tr}}(t_{a})\Bigr{]}\cdot\exp \biggl{\{}j\pi K_{r}\Bigl{[}t_{r}-\frac{R_{\text{tr}}(t_{a})}{c}\Bigr{]}^{2} \biggr{\}}\end{array}  \tag{26}\]  
Figure 4: The imaging flowchart of AA-BiSAR with one-moving transmitter. since the arc array radius \(R_{r}\)*\(\rho\) in the practical applications, the distance from the target to the receiver can be resolved as \(R_{R}=\rho-R_{r}\cos\beta_{r}\cos(\theta_{r}-\theta_{n})\), so the bistatic slant range expression can be shown as:  
\[ \begin{array}{ll}R_{\text{tr}}(t_{a})=&R_{R}+R_{T}\\ &\approx\rho-R_{r}\cos\beta_{r}\cos(\theta_{r}-\theta_{n})\\ &+\sqrt{\left(R_{n}\cos\theta_{n}-R_{t}\cos\theta_{t}-vt_{a}\right)^{2}+\left( R_{n}\sin\theta_{n}-R_{t}\sin\theta_{t}\right)^{2}+\left(H_{n}-H_{t}\right)^{2}} \\ &\text{and the instantaneous azimuth phase is given by:}\\ \psi_{a}=-\frac{2\pi f_{c}}{c}R_{\text{tr}}(t_{a})\end{array}  \tag{28}\]  
the instantaneous azimuth frequency can be expressed as:  
\[f_{\theta}=\frac{d(\psi_{a})}{d\theta_{r}}=-\frac{2\pi R_{r}\cos\beta_{r}\sin( \theta_{r}-\theta_{n})}{\lambda}+\frac{2\pi v(R_{n}\cos\theta_{n}-R_{t}\cos \theta_{t}-vt_{a})}{\lambda\varpi_{a}R_{T}} \tag{29}\]  
where \(\varpi_{a}\) is the angular velocity. It can be seen from Equation (29) that the instantaneous azimuth frequency is composed of two parts, one part is caused by the synthetic aperture, and the other part is caused by the movement of the transmitter. Due to the value range of \((\theta_{r}-\theta_{n})\) is less than or equal to half of the azimuth beam width of the receiver, which is \(\theta_{a}/2\), and \(\theta_{a}\) is generally less than 180\({}^{\circ}\). In general, the monotonicity of the instantaneous azimuth frequency is guaranteed, so the instantaneous azimuth frequency changes monotonically. Therefore, the azimuth angular resolution of AA-BiSAR with one-moving transmitter imaging system can be calculated as:  
\[\rho_{\theta}=\frac{2\pi}{\max(f_{\theta})-\min(f_{\theta})} \tag{30}\]  
Afterward, in order to compare the influences of different transmitter velocities on the azimuth resolution, four groups of comparative experiments are carried out. The detailed simulation parameters of AA-BiSAR with one-moving transmitter system are given in Table 1. The transmitter velocities are set as 0 m/s, 50 m/s, 300 m/s, and in order to amplify the influence of transmitter movement on the azimuth resolution, this article adds a set of experiments with a transmitter velocity of 1000 m/s. The initial position of the transmitter \((X_{t},Y_{t},H_{t})\) is set as (0 m, 100 m, 1000 m), and the height of the transmitter and receiver platform are set as 1000 m and 200 m since the height of the aircraft should be much larger than the helicopters for better observe the whole scene. The arc array radius of the receiver platform is set as 0.6 m, which is much smaller than the scene size. Figure 5 shows the specific simulation results of four comparative experiments, where Figure 5a is the azimuth resolution when the transmitter is stationary at a certain point. It can be observed that the azimuth resolution of the target is only related to the radius of the arc array antenna, the synthetic aperture angle and the height of the receiver platform. In short, when the transmitter is stationary, the azimuth res