# An Efficient Ground Moving Target Imaging Method for Airborne Circular Stripmap SAR
## 3 Target's 2-D Spectrum Based on the Proposed Range Model  
After demodulating to the baseband, the target's signal in the raw data domain can be expressed as  
\[s(t_{r},t_{a})=w_{r}\bigg{(}t_{r}-\frac{2R_{p}(t_{a})}{c}\bigg{)}\exp\bigg{\{}j \pi k_{r}\bigg{[}t_{r}-\frac{2R_{p}(t_{a})}{c}\bigg{]}^{2}\bigg{\}}\times w_{a }(t_{a})\exp\bigg{\{}-j\pi\frac{4R_{p}(t_{a})}{\lambda}\bigg{\}} \tag{17}\]  
where \(w_{r}\) (\(\cdot\)) and \(w_{a}\) (\(\cdot\)) denote the range and azimuth envelopes, respectively, \(t_{r}\) is the range time, \(k_{r}\) is the chirp rate of the transmitted signal, \(c\) is the speed of light, and \(f_{c}\) is the carrier frequency.  
Applying a Fourier transform in range to (17), one can obtain the target's signal in the range frequency domain as follows  
\[S(f_{r},t_{a})=W_{r}(f_{r})\exp\bigg{\{}-j\pi\frac{f_{r}^{2}}{k_{r}}\bigg{\}} \times w_{a}(t_{a})\exp\bigg{\{}-j\pi\frac{4(f_{c}+f_{r})R_{p}(t_{a})}{c} \bigg{\}} \tag{18}\]  
where \(f_{r}\) denotes the range frequency and \(W_{r}\) (\(\cdot\)) denotes the range frequency envelope.  
To obtain the target's 2-D spectrum, a Fourier transform in azimuth is applied to (18):  
\[S(f_{r},f_{a})=\int_{-\infty}^{+\infty}S(f_{r},t_{a})\exp\{-j2\pi f_{a}t_{a} \}dt_{a} \tag{19}\]  
Figure 3: Phase errors under different accelerations. (**a**) Proposed range equation, (**b**) second-order [PERSON] approximated range equation, and (**c**) third-order [PERSON] approximated range equation. The parameters of the target are: \(R_{c}\) = 20 km, \(v_{x}\) = 5 m/s, and \(v_{y}\) = 13 m/s.  
where \(f_{a}\) denotes the azimuth frequency. To obtain the analytical expression for the target's 2-D spectrum, the principle of stationary phase [16] is utilized. Based on (10), (18), and (19), the stationary phase point is given by  
\[t_{a}^{*}=-\frac{\sqrt{4v_{e}^{2}R_{c}^{2}-\alpha^{2}}}{2v_{e}^{2}}\Bigg{[} \frac{\Big{(}\frac{cf_{a}}{2v_{e}(f_{c}+f_{r})}+\frac{\beta}{v_{e}}\Big{)}}{ \sqrt{1-\Big{(}\frac{cf_{a}}{2v_{e}(f_{c}+f_{r})}+\frac{\beta}{v_{e}}\Big{)}^{2 }}}\Bigg{]}-\frac{\alpha}{2v_{e}^{2}} \tag{20}\]  
According to the principle of stationary phase, the target's 2-D spectrum can be obtained as follows  
\[S(f_{r},f_{a}) =S(f_{r},t_{a}^{*})\exp\{-j2\pi f_{a}t_{a}^{*}\} \tag{21}\] \[=W_{r}(f_{r})W_{a}(f_{a})\exp\{j\theta(f_{r},f_{a})\}\]  
With  
\[\begin{array}{ll}\theta(f_{r},f_{a})&=-\frac{2\pi\sqrt{4v_{e}^{2}R_{c}^{2}- \alpha^{2}}}{\lambda v_{e}}\sqrt{1-\Big{(}\frac{\lambda f_{a}}{2v_{e}}+\frac{ \beta}{v_{e}}\Big{)}^{2}+\Big{(}1-\frac{\beta}{v_{e}}\Big{(}\frac{\lambda f_{ a}}{2v_{e}}+\frac{\beta}{v_{e}}\Big{)}\Big{)}\Big{(}\frac{2f_{r}}{f_{c}} \Big{)}+\Big{(}1-\Big{(}\frac{\beta}{v_{e}}\Big{)}^{2}\Big{)}\Big{(}\frac{f_{ r}}{f_{c}}\Big{)}^{2}}\\ &+\frac{2\pi\alpha\beta(f_{e}+f_{r})}{cv_{e}^{2}}+\frac{\pi\alpha f_{a}}{v_{e} ^{2}}-\frac{\pi f_{r}^{2}}{k_{r}}\\ &\text{where }W_{a}\left(\cdot\right)\text{ denotes the azimuth frequency envelope.}\\ \end{array} \tag{22}\]  
From the above derivations it can be seen that the analytical expression of the target's 2-D spectrum can be easily derived without any approximations being made. Therefore, the proposed range model enables the derivation of an accurate analytical expression for the target's 2-D spectrum.