## 3 Ambiguity Suppression Method
### Ambiguous Area Imaging  
The ambiguous area imaging method is based on a classical SAR imaging algorithm called Range-Doppler algorithm (RDA) [19]. In simple terms, RDA includes three steps which are range compression, range cell migration correction (RCMC) and azimuth compression, aiming at main area imaging. To get the image of the ambiguous area, we need to make some changes to the algorithm. The general steps are shown in the right parts of Figure 2. The following contents illustrate the specific process. * Range compression Firstly, we get the signal in range frequency domain by applying the fast Fourier transform (FFT) along range direction to the echo data as shown in Equations (1)-(4): \[ \begin{array}{ll}\mathrm{S}(f_{\tau},t,n)&=\sigma_{n}\cdot W_{a}(t,n)\cdot W _{\tau}[f_{\tau}]\cdot\exp[-j4\pi(f_{0}+f_{\tau})/c\cdot R_{n}^{\prime}(t)]\\ &\cdot\exp\left[(-1)^{|n|}j\pi f_{\tau}^{2}/k_{\tau}\right]\end{array} \] (8) where \(f_{\tau}\) is the range frequency, \(f_{0}\) is the carrier frequency. It is obvious that \(n=0\) presents the main area signals in range frequency domain and others the ambiguous area signals. A range matched filter with contrary modulation rate to the main area is required to get the pulse compression result of the ambiguous area. The filter in frequency domain is as follows [19]:  
\[H_{contrary}(f_{\tau})=\exp\left(j\pi f_{\tau}^{2}/k_{\tau}\right) \tag{9}\]  
We can get the range compression result through the following matching filtering process:  
\[\mathrm{s}_{\tau}(\tau,t,n)=F^{-1}\left[\mathrm{S}(f_{\tau},t,n)*H_{contrary}(f _{\tau})\right] \tag{10}\]  
where \(F^{-1}[\cdot]\) represents the inverse FFT(IFFT). The specific expression is as follows:  
\[\mathrm{s}_{odd}(\tau,t,n) =\sigma_{n}\cdot W_{a}(t,n)\cdot\sin c\left[\tau-2R_{n}^{\prime}(t )/c\right]\cdot\exp\left[-j4\pi R_{n}^{\prime}(t)/\lambda\right] \tag{11}\] \[ \begin{array}{ll}\mathrm{s}_{even}(\tau,t,n)&=\sigma_{n}\cdot W_{a}(t,n)\cdot W_{\tau}[(\tau-2R_{n}^{\prime}(t)/c)/2]\cdot\exp[-j4\pi R_{n}^{\prime} (t)/\lambda]\\ &\cdot\exp\left[-j\pi k_{\tau}/2\cdot(\tau-2R_{n}^{\prime}(t)/c)^{2}\right] \end{array}  \tag{12}\]  
where \(s_{odd}(\tau,t,n)\) represents the compression results of odd \(n\), \(s_{even}(\tau,t,n)\) the even \(n\) (including main area with \(n=0\)). We can see that only the ambiguous area with odd \(n\) can obtain the right focused range profiles, contributing to the acquisition of the focused image of this ambiguous area. There are two steps that need to be followed, including RCMC and azimuth compression, according to the RDA. However, the coupling of range and azimuth should remain unchanged to keep the imaging process reversible, which makes it impossible to add RCMC to the imaging process. Thus, we get the image result of ambiguous area through simple azimuth matched filter compression under the assumption of tiny range migration. Here we consider \(s_{odd}(\tau,t,n)\) only since the defocusing range compression results can no longer get the focusing image through azimuth processing. We get the range compression results in Doppler frequency domain by azimuth FFT [19]:  
\[ \begin{array}{ll}S_{rd\_odd}(\tau,f_{t},n)&=\sigma_{n}\cdot W_{a}(f_{t},n) \cdot\sin c[\tau-2R_{n}^{\prime}(f_{t})/c]\\ &\cdot\exp\left[-j4\pi R_{0}/\lambda\cdot\left(\sqrt{1-\left(\lambda f_{t}/2V \right)^{2}}\right)\right]\end{array}  \tag{13}\]  
where \(f_{t}\) is the azimuth frequency, \(R_{0}\) is the center slant range, \(V\) is the equivalent velocity between satellite and target at center time. \(R\)'\(n\)(\(f_{t}\)) is the range cell migration in the range direction, which has a complex expression that can be found in [19]. The azimuth matched filter in frequency domain is as follows [19]:  
\[H_{azimuth}(f_{t})=\exp\left[j4\pi R_{ref}/\lambda\cdot\left(\sqrt{1-\left( \lambda f_{t}/2V\right)^{2}}-1\right)\right] \tag{14}\]  
where \(R_{ref}\) represents the reference slant range.