# A Novel Spaceborne Sliding Spotlight Range Sweep Synthetic Aperture Radar: System and Imaging
## 4 System Parameter Design  
### Range Sample Number and Pulse Width  
For a traditional SAR system, the sampling rate should be higher than the bandwidth of the transmitted LFM signal to avoid aliasing. For the SSS-RSSAR, however, as the range LFM characteristic is removed by the 2D dechirp processing, the range non-aliasing sampling restriction is different from the traditional SAR systems.  
For an observation task with an expected ground range resolution \(\rho_{\mathrm{r}}\), the bandwidth of the LFM signal, denoted by \(B_{\mathrm{r}}\), can be set as  
\[B_{\mathrm{r}}=\frac{c}{2\rho_{\mathrm{r}}\sin\beta_{\mathrm{c}}} \tag{32}\]  
However, after the 2D dechirp, there will be a unique mapping between the target position and its frequency [3] and thus, the signal will have a new bandwidth \(B_{\mathrm{m}}\) based on the geometry in Figure 2 as  
\[B_{\mathrm{m}}=\frac{2\gamma W_{\mathrm{r}}\sin\beta_{\mathrm{c}}}{c} \tag{33}\]  
The non-aliasing sampling rate \(F_{\mathrm{s}}\) should be slightly higher than the new bandwidth \(B_{\mathrm{m}}\) as  
\[F_{\mathrm{s}}=\sigma B_{\mathrm{m}} \tag{34}\]  
where \(\sigma\) denotes an oversampling factor ranging from 1.1-1.3. The chirp rate \(\gamma\) in Equation (33) can be rewritten based on the characteristic of the LFM signal as  
\[\gamma=\frac{B_{\mathrm{r}}}{T_{\mathrm{P}}} \tag{35}\]  
By substituting Equations (32), (33) and (35) into Equation (34), the sampling rate can be rewritten as  
\[F_{\mathrm{s}}=\frac{\sigma W_{\mathrm{r}}}{T_{\mathrm{P}}\rho_{\mathrm{r}}} \tag{36}\]  
On the other aspect, the echo length \(\tau_{\mathrm{w}}\) can be estimated by  
\[\tau_{\mathrm{w}}=\frac{2W_{\mathrm{r}}\sin\beta_{\mathrm{c}}}{c}+T_{\mathrm{ P}} \tag{37}\]  
The least required range sample number \(N_{\mathrm{rm}}\) can be achieved by multiplying \(F_{\mathrm{s}}\) with \(\tau_{\mathrm{w}}\) as  
\[N_{\mathrm{rm}}=\frac{\sigma W_{\mathrm{r}}}{\rho_{\mathrm{r}}}+\frac{2\sigma W _{\mathrm{r}}^{2}\sin\beta_{\mathrm{c}}}{cT_{\mathrm{P}}\rho_{\mathrm{r}}} \tag{38}\]  
On the other hand, \(N_{\mathrm{r}}\) also has an upper limit \(N_{\mathrm{rm}}\) corresponding to the case that all the space of the minimum PI is used for the data receiving as  
\[N_{\mathrm{rm}}=F_{\mathrm{s}}\left(\min_{m=-\frac{N_{\mathrm{r}}}{2},..., \frac{N_{\mathrm{s}}}{2}}PI(m)-T_{\mathrm{P}}\right) \tag{39}\]  
The final range sample number \(N_{\mathrm{r}}\) can then be set slightly higher than \(N_{\mathrm{rm}}\) but lower than \(N_{\mathrm{rm}}\). Based on Equation (38), it is clear that the longer the \(T_{\mathrm{P}}\), the smaller the required range sample number. Thus, the pulse width should be elongated as much as possible to decrease the least required totaldata amount. However, \(T_{\text{P}}\) cannot be elongated limitlessly. The upper limit for \(T_{\text{P}}\), denoted as \(T_{\text{pmax}}\), corresponds to the case that the echo length \(\tau_{\text{w}}\) equals to the minimum \(PI\) as  
\[T_{\text{pmax}}=\frac{1}{2}\Bigg{(}\min_{m=-\frac{N_{\text{a}}}{2}\ldots\frac{N _{\text{a}}}{2}}PI(m)-\frac{2W_{\text{r}}\sin\beta_{\text{c}}}{c}\Bigg{)} \tag{40}\]  
For a real SSS-RSSSAR system, \(T_{\text{P}}\) should be slightly smaller than the theoretical upper limit \(T_{\text{pmax}}\) to leave enough space for the antenna to switch between the transmitting and receiving.