# A High-Resolution SAR Focusing Experiment Based on GF-3 Staring Data
## 2 Focusing Experiment Based on GF-3 Staring Data
### Two-Step Processing Algorithms  
In ST mode, the Doppler bandwidth of a target located in the imaged scene can be expressed as \(2X_{I}/(D_{az}X)\)[2], where \(X_{I}\) is a synthetic aperture, \(D_{az}\) is the antenna aperture in azimuth, and \(X\) is the length of the antenna footprint. Under the assumption that scene width is \(X\) in azimuth, the Doppler bandwidth of the whole scene is  
\[B_{a}=\frac{2X_{I}}{D_{az}X}+\frac{2}{D_{az}}=\frac{2X_{I}}{D_{az}X}+B_{as} \tag{17}\]  
where \(B_{as}\) is the Doppler bandwidth of stripmap. The \(B_{a}\) in Equation (17) is usually much larger than the PRF. If we deal with the ST data in the way of the traditional stripmap SAR, then there would be a Doppler aliasing phenomenon in azimuth, which would introduce false targets in the focused image.  
GF-3 uses the two-step processing approach--the de-ramped chirp scaling algorithm (DCSA)--to process the raw echo data. This algorithm involves two steps: the first step is prefiltering in azimuth, and the second step is dealing with the filtered data with the imaging approach of the standard stripmap SAR.  
The first step implements an azimuth convolution between the raw data and a reference chirp signal \(s_{ref}(t)\):  
\[\hat{s}_{a}(t)=s_{a}(t)*_{t}s_{ref}(t)=s_{a}(t)*_{t}\exp\left(j\pi f_{Rref}t^{ 2}+i2\pi f_{Dref}t\right) \tag{18}\]  
where \(t\) is azimuth time, \(s_{a}(t)\) is azimuth raw data, \(*_{t}\) is a convolution about \(t\), \(\hat{s}_{a}(t)\) is azimuth data after convolution, \(f_{Rref}\) is the Doppler frequency modulation rate of the reference point (in the center of imaged scene), and \(f_{Dref}\) is the Doppler centroid frequency of the reference point, and the value of \(f_{Rref}\) and \(f_{Dref}\) can be found in Table 2.  
In a discrete domain, the above convolution can be converted to a fast Fourier transform (FFT) and complex multiplication shown in the below equation [4]  
\[\hat{s}_{a}(m\cdot\Delta t^{\prime})=\exp\left\{j\pi f_{Rref}(m\cdot\Delta t^{ \prime\prime})^{2}\right\}\cdot FFT\left\{s_{a}(i\cdot\Delta t^{\prime})\cdot \exp\left[j2\pi f_{Dref}i\cdot\Delta t^{\prime}+j\pi f_{Rref}(i\cdot\Delta t^{ \prime})^{2}\right]\right\} \tag{19}\]  
where \(i=-\frac{N_{a}}{2},\cdots,\frac{N_{a}}{2}\) and \(N_{a}\) is the number of samples of raw data in azimuth, \(m=-\frac{p}{2},\cdots,\frac{p}{2}\) and \(P\) is the number of samples of prefiltered data in azimuth, \(\Delta t^{\prime}\) is the sampling interval of raw data in azimuth, and \(\Delta t^{\prime\prime}\) is the sampling interval of prefiltered data in azimuth.  
In fact, the azimuth prefiltering is equivalent to reducing the signal duration while keeping the Doppler bandwidth unchanged. The result is that the azimuth sampling rate becomes higher than the Doppler bandwidth. Then, the problem of the Doppler aliasing phenomenon in azimuth would be resolved.  
The second step is to use the classical CSA to deal with the prefiltered data. But in the process of azimuth filter and phase residual, the second-order term shown in Equation (20) should be added to eliminate the effects of the prefiltering.  
\[\Phi_{\rm cor}=\exp\left\{j\pi\frac{f_{a}^{2}}{f_{Rref}}\right\} \tag{20}\]Then, the azimuth filter and phase residual can be realized by multiplying the azimuth signal with  
\[\Phi_{3}(\tau,\,f)=\exp\left\{-j\frac{2\pi}{\lambda}c\tau\left[1-\left(\frac{ \lambda f_{a}}{2V_{(r-\frac{\tau}{2})}}\right)^{2}\right]^{\frac{1}{2}}+j\theta_ {\Delta}(f_{a},r)\right\}\times\Phi_{\rm cor} \tag{21}\]  
and the above parameters can be found in [10].