# Analysis of a Bistatic Ground-Based Synthetic Aperture Radar System and Indoor Experiments
## Materials and Methods
### Bistatic GB-SAR Focusing Algorithms
#### 2.2.3 Range-Doppler Algorithm  
BPA is accurate in geometry but time-consuming due to the computational burden of the time-domain convolution. Therefore, frequency-based focusing algorithms have been developed to reduce the processing time at the cost of image quality. The range-Doppler algorithm (RDA) is one of the widely used algorithms for many satellite and airborne SAR missions.  
Formulation of RDA for the bistatic GB-SAR begins with a Taylor series expansion of the phase at \(x=x_{c}\) so that  
\[\phi(x)=\phi(x_{c})+\phi^{\prime}(x_{c})(x-x_{c})+\frac{\phi^{\prime\prime}(x _{c})}{2!}(x-x_{c})^{2}+\cdots\]  
\[=\phi(x_{c})+2\pi u_{Dc}(x-x_{c})+\pi u_{r}(x-x_{c})^{2}+\cdots, \tag{9}\]  
where \(u_{Dc}\) and \(u_{r}\) are Doppler centroid and Doppler rate, respectively, defined by \(u_{Dc}:=\frac{\phi^{\prime}(x_{c})}{2\pi}\) and \(u_{r}:=\frac{\phi^{\prime\prime}(x_{c})}{2\pi}\).  
Azimuth compression for RDA begins with the Fourier transform of the received signal to the Doppler (wavenumber) \(u\) domain by:  
\[S(u)=\int_{-\infty}^{\infty}s(x)e^{-j2\pi ux}dx\]  
\[=e^{j\phi(x_{c})}\int_{-\infty}^{\infty}e^{2\pi j[u_{Dc}(x-x_{c})+u_{r}(x-x_{c })^{2}/2]}e^{-j2\pi ux}dx. \tag{10}\]  
Stationary phase occurs when the derivative of the phase of the integrand in (10) is zero, so that  
\[x-x_{c}=(u-u_{Dc})/u_{r}. \tag{11}\]  
This is the locking relationship between the rail position in space and the wavenumber Doppler, enabling range migration in wavenumber domain. Note that (11) was acquired by cancelling the terms higher than the quadrature.  
After range migration, azimuth compression is performed as follows.  
\[G(u)=H^{-1}(u)S(u), \tag{12}\]  
after which the focused signal \(g(x)\) is obtained by the inverse Fourier transformation of the above signal.  
\[g(x)=\int_{-\infty}^{\infty}G(u)e^{j2\pi ux}du. \tag{13}\]  
We need to evaluate the Doppler centroid and Doppler rate for the bistatic GB-SAR. Doppler is a wavenumber defined by the change of phase with respect to \(x\).  
\[u_{D}:=\frac{\phi^{\prime}}{2\pi}=-\frac{1}{\lambda}\left(\frac{dr_{1}}{dx}+ \frac{dr_{2}}{dx}\right)\]  
\[=-\frac{1}{\lambda}\left\{\frac{x-x_{c}-\frac{B}{2}}{\sqrt{\left(x-x_{c}- \frac{B}{2}\right)^{2}+y_{c}^{2}}}+\frac{x-x_{c}+\frac{B}{2}}{\sqrt{\left(x-x _{c}+\frac{B}{2}\right)^{2}+y_{c}^{2}}}\right\} \tag{14}\]  
Doppler centroid is then \(u_{Dc}:=u_{D}|_{x=x_{c}}=0\), i.e., the Doppler centroid of the bistatic GB-SAR is zero, the same with the monostatic case in [10].  
The derivative of Doppler is as follows.  
\[u^{\prime}_{D}:=\frac{\phi^{\prime\prime}}{2\pi}\]  
\[=-\frac{1}{\lambda}\left\{\frac{y_{c}^{2}}{\left[\left(x-x_{c}-\frac{B}{2} \right)^{2}+y_{c}^{2}\right]^{3}}+\frac{y_{c}^{2}}{\left[\left(x-x_{c}+\frac{B }{2}\right)^{2}+y_{c}^{2}\right]^{\frac{3}{2}}}\right\} \tag{15}\]  
Doppler rate is then defined by  
\[u_{r}:=\left.u^{\prime}_{D}\right|_{x=x_{c}}=-\frac{2y_{c}^{2}}{\lambda\left[ \left(\frac{B}{2}\right)^{2}+y_{c}^{2}\right]^{\frac{3}{2}}} \tag{16}\]  
If \(B=0\), \(u_{r}=-\frac{2}{\lambda y_{c}}\), which is the same with the monostatic case in [10]. Figure 3 shows Doppler rate versus range with various \(B\) values. The value deviates from the monostatic case (\(B=0\)) especially at near range, which means the cancellation of the higher term in (11) would be inappropriate and causes image degradation at the near range for RDA. However, the bistatic Doppler rate approaches 97% of the monostatic case at \(r=4\) m and the image degradation becomes negligible at far range. This effect can be seen in the experimental results in the next section.