# Spaceborne Sparse SAR Imaging Mode Design: From Theory to Implementation
## 2 Traditional SAR Imaging Mode
### Signal Characteristics  
The echo data \(y(t,\tau)\) of different SAR imaging modes at time \((t,\tau)\) can be written as  
\[ \begin{split} y(t,\tau)&=\iint\limits_{(p,q)}\sigma(p,q )\ \omega_{a}^{2}(\theta_{a})\ \omega_{r}^{2}(\theta_{r})\ \ \text{rect}\bigg{(}\frac{\tau-2R/c}{T_{\text{P}}}\bigg{)}\\ &\cdot\ \exp\bigg{\{}-j\frac{4\pi}{\lambda}R\bigg{\}}\cdot\exp\bigg{\{}-j \pi K_{\text{r}}\bigg{(}\tau-\frac{2R}{c}\bigg{)}^{2}\bigg{\}}dpdq\end{split}  \tag{3}\]  
where \(t\) and \(\tau\) are the azimuth and range time, \(p\) and \(q\) are the target azimuth and range positions, respectively, \(\sigma(p,q)\) is the backscattering coefficient, \(c\) is the speed of light, \(\omega_{a}(\theta_{a})\) and \(\omega_{r}(\theta_{r})\) are the azimuth and range antenna pattern weighting, \(\theta_{a}\) and \(\theta_{r}\) are the azimuth angle and range angle between the direction to the target and the antenna's electrical axis, respectively, \(T_{\text{P}}\) is the pulse duration, \(\lambda\) is the wavelength, \(K_{\text{r}}\) is the linear frequency modulated rate, \(R=\sqrt{r_{0}^{2}+(v_{r}t-x)^{2}}\) is the slant range of the target located at \((x,r_{0})\) with the azimuth position \(x\) and the closest slant range \(r_{0}\), and \(v_{r}\) is the effective radar velocity. It should be noted that due to the different modes of beam scanning, the azimuth antenna pattern is also expressed differently. The Doppler center frequency of each mode can be expressed as  
\[f_{ac}(t) = \frac{2v_{r}}{\lambda}\sin(\theta_{ac}(t)) \tag{4}\]  
Figure 1: The acquisition geometry of all kinds of typical SAR imaging modes. Figure 2: Azimuth resolution of each imaging mode. where \(\theta_{ac}(t)\) is the effective squint angle between the direction to the target and the antenna's electrical axis. In Stripmap and Scan modes, considering a subswath, \(\theta_{ac}(t)\) is a fixed value because the beam is not scanned in the azimuth direction. In Spotlight, Slide Spotlight, and TOPS modes, the Doppler center frequency varies with azimuth time when scanning the beam along the azimuth direction. The change rate of the Doppler center frequency can be written as  
\[K_{c}\ =\ \frac{df_{ac}(t)}{dt}\approx\frac{2v_{s}}{\lambda}k_{g} \tag{5}\]  
where \(v_{s}\) is the SAR velocity. It is seen that the Doppler center varies linearly when the beam is scanned at a constant speed. In the following, we conduct an analysis of the azimuth ambiguity-to-signal ratio (AASR) and range ambiguity-to-signal ratio (RASR) for all imaging modes. The occurrence of AASR arises from the finite PRF and the periodic repetition of the Doppler spectrum of the discrete signal after sampling. In Stripmap, the AASR is  
\[\text{AASR}_{\text{St}}\ =\ \frac{\sum_{m=-\infty}^{m=+\infty}\int_{-B_{D}/2}^{B_ {D}/2}G\left(f_{a}+m\cdot\text{PRF}\right)^{2}df_{a}}{\int_{-B_{D}/2}^{B_{D}/2 }G\left(f_{a}\right)^{2}df_{a}} \tag{6}\]  
where \(B_{D}\) is the Doppler bandwidth, \(G(\cdot)\) is the AAP, and \(f_{a}\) is the azimuth frequency. In Stripmap, it is seen that the AASR is associated with parameters such as PRF and Doppler bandwidth, and it is independent of the azimuth position. For Spotlight, the AASR is expressed as  
\[\text{AASR}_{\text{Sp}}\ =\ \frac{\sum_{m=-\infty}^{m=+\infty}\int_{-B_{D}/2 -K_{d}t_{0}}^{B_{D}/2-K_{d}t_{0}}G\left(m\cdot\text{PRF}-K_{d}t_{0}\right)^{2} df_{a}}{\int_{-B_{D}/2-K_{d}t_{0}}^{B_{D}/2-K_{d}t_{0}}G(-K_{a}t_{0})^{2}df_{a}} \tag{7}\]  
where \(t_{0}\) is the zero-Doppler moment of the target, and \(K_{a}\) is the Doppler-modulated frequency. It can be observed that in Spotlight mode, the AASR is dependent on the azimuth position of the target. For Scan mode, the varying weights in the AAP received at different azimuth positions contribute to distinct AASR.