# Robust Space Time Adaptive Processing Methods for Synthetic Aperture Radar
## 2 SAR Simulation Models  
### SAR Signal Model  
Consider a multi-channel SAR system of a linear antenna array with \(N\) equi-spaced receiver channels along the azimuth direction. Figure 1 is the geometric diagram of the multi-channel SAR. Suppose all the channels and targets are located in the three-dimensional (3D) Descartes coordinate system. The radar platform travels along the \(X\)-axis at \(v\) m/s. Figure 1a shows the 3D geometric model, where \(H_{c}\) represents the height of the platform, the \(\varphi\) represents the azimuth angle, the \(\phi\) represents the elevation angle, and the \(\theta\) is the squint angle. Figure 1b shows the diagram of the geometry relationship in the slant plane, where the blue lines show the beam irradiation range when the radar is at point \(A\), and point \(B\) is the position of the beam center. When the point target is located at the center of the beam, \(R_{0}\) is the distance between the target and the receiver channel, and \(\theta_{0}\) represents the cone angle corresponding to the beam center. When the radar moves to the point \(A^{\prime}\), the point \(B^{\prime}\) is the center of the beam.  
The radar transmits a linear frequency modulated (LFM) signal and the received echo of the \(n\)th channel is base-banded and sampled, which can be written as:  
\[s(n,\hat{t},t_{m})=\exp\left[j2\pi\frac{(n-1)d\sin\theta_{0}}{\lambda}\right] \cdot A\text{rect}\left(\hat{t}-\frac{2R(t_{m})}{c}\right)\exp\left[j\pi \gamma\left(\hat{t}-\frac{2R(t_{m})}{c}\right)^{2}\right]\exp\left[-j\frac{4 \pi}{\lambda}R(t_{m})\right] \tag{1}\]  
where \(A\) is the amplitude of the target echo, \(c\) is the velocity of light, \(\lambda\) is the wavelength, \(\hat{t}\) is fast-time corresponding to the sample in the range domain of a single pulse, \(t_{m}\) is slow-time used to mark different pulses during the pulse string process with the interval of the pulse repetition interval (PRI), \(T_{p}\) is the pulse width, \(\gamma\) is the frequency rate of the LFM signal, \(B=\gamma T_{p}\) is the signal bandwidth, and \(n\) refers to the \(n\)th channel of the multi-channel receiver. The first exponential term represents the phase difference of the echo between the receiver channel \(N\) and channel \(I\). According to the geometric relation in Figure 2, the instantaneous slant distance between the carrier and the point target \(P\) can be obtained:  
\[R(t_{m})=\sqrt{(vt_{m}-X_{n})^{2}+R_{b}^{2}-2R_{b}(vt_{m}-X_{n})\sin\theta}~{} X_{n}-\frac{L}{2v}\leq t_{m}\leq X_{n}+\frac{L}{2v} \tag{2}\]  
where \(v\) is the velocity of the SAR platform, \(X_{n}\) is the azimuth coordinates of a scattering point on the ground, \(L\) is the synthetic aperture length, \(R_{b}\) represents the shortest distance between the target and the trajectory of the radar.  
After the pulse compression processing in the range domain, and the envelope difference of the echo is neglected, so the echo signal can be expressed as:  
\[s(u,\hat{t},t_{m})=\exp\left[j2\pi\frac{(n-1)d\sin\theta_{0}}{\lambda}\right] \cdot\sin\left[B(\hat{t}-\frac{2R(t_{m})}{c})\right]\exp\left[-j\frac{4\pi}{ \lambda}R(t_{m})\right] \tag{3}\]