# An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems
## 2 Signal Modelling
### Derivation of the Target's 2-D Spectrum  
Supposing that a chirp signal is transmitted, then the signal after demodulation and range compression can be expressed as [1]  
\[s_{n}(t_{r},t_{a})=p_{r}\bigg{(}t_{r}-\frac{2R_{n}(t_{a})}{c}\bigg{)}w_{a}(t_{ a})\exp\bigg{\{}-j\frac{4\pi}{\lambda}R_{n}(t_{a})\bigg{\}}, \tag{19}\]  
where \(p_{r}(\cdot)\) defines the function of range impulse response,\(w_{a}(\cdot)\) indicates the azimuth window function, \(t_{r}\) is the range time, \(c\) represents the velocity of light, and \(\lambda\) defines the wavelength. In the paper, the channels are assumed to be well calibrated. Applying a FT to (19) in the range direction, the target signal in the range frequency domain can be obtained:  
\[S_{n}(f_{r},t_{a})=W_{r}(f_{r})w_{a}(t_{a})\exp\bigg{\{}-j4\pi(f_{c}+f_{r}) \frac{R_{n}(t_{a})}{\lambda}\bigg{\}}, \tag{20}\]  
where \(f_{r}\) is the range frequency, \(W_{r}(\cdot)\) refers to the envelope of range spectrum, \(f_{c}\) defines the carrier frequency. The target's 2-D spectrum can be obtained via taking a FT to (20) with respect to \(t_{a}\). Since there are quadratic and cubic terms of \(t_{a}\), the series reversion and the principle of stationary phase are used to solve the Fourier integral. The obtained expression for the target signal in the 2-D frequency domain is given by  
\[S_{n}(f_{r},f_{a})=W_{r}(f_{r})W_{a}(f_{a})\exp\{j\theta_{n}(f_{r},f_{a})\}, \tag{21}\]  
with  
\[ \begin{array}{rl}\theta_{n}(f_{r},f_{a})&=-4\pi\frac{(f_{c}+f_{r})}{c}\bigg{(} R_{0}-\frac{(l_{1}+(n-1)da)^{2}}{4l_{2}}-\frac{(l_{1}+(n-1)da)^{3}l_{3}}{8l_{2} ^{3}}\bigg{)}\\ &+2\pi(f_{a}+M\cdot PRF)\bigg{(}\frac{l_{1}+(n-1)da}{2l_{2}}+\frac{3(l_{1}+(n- 1)da)^{2}l_{3}}{8l_{2}^{3}}\bigg{)}\\ &+\frac{\pi c}{f_{c}+f_{r}}(f_{a}+M\cdot PRF)^{2}\bigg{(}\frac{3(l_{1}+(n-1)da) l_{3}}{8l_{2}^{3}}+\frac{1}{4l_{2}}\bigg{)}-\frac{\pi c^{2}}{2(f_{c}+f_{r})^{2}} (f_{a}+M\cdot PRF)^{3}(-\frac{l_{3}}{8l_{2}^{3}}),\end{array}  \tag{22}\]  
where \(W_{a}(\cdot)\) indicates the envelope of azimuth spectrum, \(f_{a}\) defines the Doppler frequency, \(M\) represents the Doppler ambiguity number, \(PRF\) indicates the pulse repetition frequency. To facilitate the design of the imaging method, \(\theta_{n}\) (\(f_{r},f_{a}\)) is decomposed via using power series expansions of \((1+f_{r}/f_{c})^{-1}\) and \((1+f_{r}/f_{c})^{-2}\) (keeping terms up to the third order), and also is rearranged according to the physical meaning. The decomposed and rearranged phase is given by  
\[ \begin{array}{ll}\theta_{n}(f_{r},f_{a})&=-\frac{4\pi R_{a}}{\lambda}+\frac{ \pi(l_{1}+(n-1)da)^{2}}{\lambda l_{2}}+\frac{\pi(l_{1}+(n-1)da)^{3}l_{3}}{2 \lambda l_{2}}+\frac{\pi(l_{1}+(n-1)da)}{l_{2}}(f_{a}+M\cdot PRF)\\ &+\frac{3\pi(l_{1}+(n-1)da)^{2}l_{3}}{4l_{2}^{2}}(f_{a}+M\cdot PRF)+\frac{3 \pi l_{2}}{4l_{2}^{2}}(f_{a}+M\cdot PRF)^{2}+\frac{3\pi\lambda(l_{1}+(n-1)da)l_ {3}}{8l_{2}^{2}}(f_{a}+M\cdot PRF)^{2}\\ &+\frac{\pi l_{3}\lambda^{2}}{16l_{2}^{2}}(f_{a}+M\cdot PRF)^{3}-2\pi f_{r} \tau_{n}(f_{a})+\frac{\pi f_{r}^{2}}{F_{n}(f_{a})}+\frac{\pi f_{r}^{3}}{M_{n}( f_{a})},\end{array}  \tag{23}\]  
where \(\tau(f_{a})\) represents echo delay, \(F_{n}(f_{a})\) and \(M_{n}(f_{a})\) indicate the residual quadratic and cubic range frequency modulation rates, respectively. Their expressions are as follows  
\[ \begin{array}{ll}\tau_{n}(f_{a})&=\frac{2R_{a}}{c}-\frac{(l_{1}+(n-1)da)^{2} }{2c_{2}}-\frac{(l_{1}+(n-1)da)^{3}l_{3}}{4c_{2}^{2}}\\ &+\frac{3c(l_{1}+(n-1)da)}{16f_{c}^{2}l_{2}^{2}}(f_{a}+M\cdot PRF)^{2}+\frac{c }{8f_{c}^{2}l_{2}}(f_{a}+M\cdot PRF)^{2}+\frac{c^{2}l_{3}}{16f_{c}^{3}l_{2}^{2 }}(f_{a}+M\cdot PRF)^{3}\end{array} , \tag{24}\]  
\[ \begin{array}{ll}\frac{1}{F_{n}(f_{a})}&=\frac{3\lambda(l_{1}+(n-1)da)l_{3}} {8f_{c}2l_{2}^{3}}(f_{a}+M\cdot PRF)^{2}+\frac{\lambda}{4f_{c}2l_{2}^{2}}(f_{ a}+M\cdot PRF)^{2}+\frac{3\lambda^{2}l_{3}}{16f_{c}2l_{2}^{3}}(f_{a}+M\cdot PRF)^{3}, \end{array}  \tag{25}\]  
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