# Spaceborne Sparse SAR Imaging Mode Design: From Theory to Implementation
## 3 Design and Performance Analysis of Sparse SAR Imaging Mode
### System Performance Analysis of Sliding Spotlight Mode  
Firstly, the system performances of both traditional and designed sparse modes are calculated. Then, the relationship between AASR and PRF at different azimuth steering angles is analyzed. This section analyzes the effect of azimuth steering angle on AASR based on the system parameters of both conventional and sparse Sliding Spotlight modes. Finally, we analyzed the change in RASR with PRF and compared the RASR values using the system parameters of these two modes. **System Parameter Calculation**. According to Figure 3c, the basic system parameters of the sparse and traditional Sliding Spotlight modes are calculated in Table 1. The swath width of the traditional mode is 45.03 km, and PRF \(=\) 2300 Hz, while in the design of sparse Sliding Spotlight, PRF is reduced to 1150 Hz to obtain the 90.16 km swath width. **Grating Lobes Analysis**. In order to achieve antenna beam sweeping, the Sliding Spotlight mode usually adopts the phased array antenna. The AAP of total antenna is  
\[G(\theta,t)\ =\ G_{\mathrm{e}}(\theta)\cdot G_{\mathrm{T}}(\theta,t)\cdot G_{ \mathrm{R}}(\theta,t) \tag{12}\]  
where \(\theta\) is the azimuth illumination angle, and \(G_{\mathrm{e}}(\theta)\) is the single-element AAP, which is written as  
\[G_{\mathrm{e}}(\theta)\ =\ \sin\!\mathrm{c}^{2}\bigg{(}\frac{L_{\mathrm{e}}}{ \lambda}\sin\theta\bigg{)}. \tag{13}\]  
\(G_{\mathrm{T}}(\theta,t)\) and \(G_{\mathrm{R}}(\theta,t)\) are the transmit and receive AAP, i.e.,  
\[G_{\mathrm{T}}(\theta,t)\ =\ \Big{|}\frac{1}{M}\sum_{k=0}^{M-1}C_{ \mathrm{T}}\exp\bigg{(}j\frac{2\pi k}{\lambda}L_{\mathrm{e}}\sin\theta\bigg{)}\Big{|} \tag{14a}\] \[G_{\mathrm{R}}(\theta,t)\ =\ \Big{|}\frac{1}{M}\sum_{k=0}^{M-1}C_{ \mathrm{R}}\exp\bigg{(}j\frac{2\pi k}{\lambda}L_{\mathrm{e}}\sin\theta\bigg{)}\Big{|} \tag{14b}\]  
where \(M\) is the number of transmit and receive modules, \(L_{\mathrm{e}}=L_{\mathrm{a}}/M\) is the single-element antenna size, \(L_{\mathrm{a}}\) is the total antenna size, \(C_{\mathrm{T}}\) and \(C_{\mathrm{R}}\) are the transmit and receive excitation coefficients, i.e.,  
\[C_{\mathrm{T}}\ =\ a_{\mathrm{T}}\cdot\exp\bigg{(}j\frac{2\pi k}{ \lambda}L_{\mathrm{e}}\sin\phi(t)\bigg{)} \tag{15a}\] \[C_{\mathrm{R}}\ =\ a_{\mathrm{R}}\cdot\exp\bigg{(}j\frac{2\pi k}{ \lambda}L_{\mathrm{e}}\sin\phi(t+\Delta t)\bigg{)} \tag{15b}\] \begin{table}
\begin{tabular}{l c c} \hline  & **Sparse** & **Traditional** \\ \hline PRF [Hz] & 1150 & 2300 \\ Pulse duration [ms] & 0.02 & 0.02 \\ Incident angle (near) [deg] & 32.32 & 36.87 \\ Incident angle (far) [deg] & 38.15 & 39.94 \\ Middle look angle [deg] & 35.24 & 38.41 \\ Azimuth resolution [m] & 0.5 & 0.5 \\ Azimuth beamwidth [deg] & 0.22 & 0.22 \\ Platform velocity [m/s] & 7600 & 7600 \\ Maximum steering angle [deg] & \(\pm\)3.5 & \(\pm\)3.5 \\ Azimuth swath width [km] & 45 & 45 \\ Range swath width [km] & 90.16 & 45.03 \\ \hline \end{tabular}
\end{table} 
Table 1: The parameters of designed sparse and traditional Sliding Spotlight modes. where \(\Delta t\) is the delay time, and \(a_{\text{T}}\) and \(a_{\text{R}}\) are the amplitude of transmit and receive excitation coefficients, respectively. We can analyze the grating lobes of the Sliding Spotlight mode by two-way AAP. Based on the equations above, it is found that the AAP is affected by the steering angle in Sliding Spotlight. The total and single-element AAPs are shown in Figure 4. The sidelobes are proportional to the steering angle, which means that the azimuth beam scanning will affect the system's performance. In practice, we expect the system's performance to be unaffected by azimuth beam scanning. **Azimuth Ambiguity Analysis**. Figure 5 shows the AASR versus PRF under different steering angles. The results demonstrate that AASR is inversely related to PRF, and the steering angle has an effect on AASR. Moreover, the change in AASR becomes more obvious with the PRF increasing.