# An Adjusted Frequency-Domain Algorithm for Arc Array Bistatic SAR Data with One-Moving Transmitter
## 3 Imaging Processing of AA-BiSAR with One-Moving Transmitter  
### Signal Model  
Without loss of generality, assuming that the receiver switches arc array unit at an angular velocity of \(\alpha_{a}\), and \(t_{a}\) is the azimuth time variable. \(R_{\mathrm{tr}}(t_{a})\) denotes the instantaneous slant range of AA-BiSAR system, it can be described as:  
\[\left\{ \begin{array}{l}R_{\mathrm{tr}}(t_{a})=R_{T}+R_{R}\\ R_{T}=\sqrt{\left(R_{n}\cos\theta_{n}-R_{t}\cos\theta_{t}-vt_{a}\right)^{2}+ \left(R_{n}\sin\theta_{n}-R_{t}\sin\theta_{t}\right)^{2}+\left(H_{n}-H_{t} \right)^{2}}\\ R_{R}=\sqrt{\left(R_{n}\cos\theta_{n}-R_{r}\cos\theta_{r}\right)^{2}+\left(R_ {n}\sin\theta_{n}-R_{r}\sin\theta_{r}\right)^{2}+\left(H_{n}-H_{r}\right)^{2} }\\ \theta_{r}=\Delta\theta_{\mathrm{interval}}\times n,\;n=1,2,3,\ldots,N_{r} \end{array} \right. \tag{1}\]  
The above Equation (1) is the precise expression of the bistatic slant range. \(R_{T}\) and \(R_{R}\) are the instantaneous slant ranges from the point target to the transmitter and to the equivalent sampling point, respectively. \(N_{r}\) is the total number of the equivalent sampling points in the azimuth direction, and the angular velocity \(\varpi_{a}\) is denoted as follows:  
\[\varpi_{a}=\frac{\theta_{r}}{t_{a}} \tag{2}\]  
Figure 3: The imaging geometry of AA-BiSAR with one-moving transmitter. Since the instantaneous bistatic slant range contains cosine terms and velocity-related terms, it is difficult to derive an accurate two-dimensional spectrum directly using the POSP, so it is necessary to do some approximates process to resolve the equation. Firstly, expand \(R_{T}\) according to the Taylor series, and ignore the higher-order phases above the third order. For ease of expression, let:  
\[R_{0}=\sqrt{R_{t}^{2}+R_{n}^{2}+H_{t}^{2}-2R_{t}R_{n}} \tag{3}\]  
After being resolved, \(R_{T}\) can be expressed as:  
\[ \begin{array}{ll}R_{T}&\approx\sqrt{{R_{0}}^{2}+2R_{t}R_{n}[1-\cos(\theta_{n }-\theta_{t})]}-\Big{(}\frac{R_{n}\cos\theta_{n}-R_{t}\cos\theta_{t}}{R_{0}} \Big{)}vt_{a}\\ &+\bigg{[}\frac{1}{2R_{0}}-\frac{(R_{n}\cos\theta_{n}-R_{t}\cos\theta_{t})^{2} }{2R_{0}}\bigg{]}(vt_{a})^{2}+\Big{(}\frac{R_{n}\cos\theta_{n}-R_{t}\cos\theta _{t}}{2R_{0}}\Big{)}(vt_{a})^{3}\\ &-\frac{1}{8R_{0}^{3}}(vt_{a})^{4}\end{array}  \tag{4}\]  
For the point target \(P_{n}\), the distance from \(P_{n}\) to the center of arc array \(O^{\prime}\) can be written as:  
\[\rho=\sqrt{{H_{r}}^{2}+{R_{n}}^{2}} \tag{5}\]  
since the arc array radius \(R_{r}\)*\(\rho\) in the practical applications, \(R_{R}\) can be resolved as:  
\[R_{R}\approx\rho-R_{r}\cos\beta_{r}\cos(\theta_{r}-\theta_{n}) \tag{6}\]  
After being approximated and resolved, the instantaneous bistatic slant range can be rewritten as:  
\[ \begin{array}{ll}\mathrm{R_{tr}}(t_{a})&\approx\rho-R_{r}\cos\beta_{r}\cos( \theta_{r}-\theta_{n})+\sqrt{{R_{0}}^{2}+2R_{t}R_{n}[1-\cos(\theta_{n}-\theta_ {t})]}\\ &-\Big{(}\frac{R_{n}\cos\theta_{n}-R_{t}\cos\theta_{t}}{R_{0}}\Big{)}vt_{a}+ \Big{[}\frac{1}{2R_{0}}-\frac{(R_{n}\cos\theta_{n}-R_{t}\cos\theta_{t})^{2}}{ 2R_{0}}\Big{]}(vt_{a})^{2}\\ &+\Big{(}\frac{R_{n}\cos\theta_{n}-R_{t}\cos\theta_{t}}{2R_{0}}\Big{)}(vt_{a} )^{3}-\frac{1}{8R_{0}}^{3}(vt_{a})^{4}\end{array}  \tag{7}\]  
According to Equation (7), the approximated slant range is composed of two parts, where the front part is independent of the velocity of the transmitter, and the latter part is related to the velocity of the transmitter. In order to facilitate the subsequent calculation, let the approximated slant range be expressed as follows:  
\[