## 2 Syntchitic Aperture Radar
### SAR Raw Signal Modeling  
In Figure 2, the equation of the chirp signal transmitted by the SAR system through its antenna at point \(P_{1}\) is represented by Equation (8) [9; 16], and the backscattered signal from the surface of interest is received at point \(P_{2}\), and is described by Equation (9) [16; 19; 20].  
\[s_{t}(t)=rect\left(\frac{t}{t_{p}}\right)exp\left(j\pi k_{r}t^{2}\right) \tag{8}\]  
where the transmitted chirp signal is \(s_{t}\), chirp modulation rate is \(k_{r}\), pulse duration is \(t_{p}\), and chirp signal received by the SAR antenna is \(s_{r}\) which has a time delay \(\Delta t\).  
\[s_{r}(t)=rect\left(\frac{t-\Delta t}{t_{p}}\right)exp\left(j\pi k_{r}(t-\Delta t )^{2}\right) \tag{9}\]  
Digital signal processing of synthetic aperture radar is achieved in phase and the amplitude can be neglected, as seen below, by expressing it with cosines.  
\[s_{r}(t)\cong cos\left(2\pi f_{0}(t-\Delta t)+\pi k_{r}(t-\Delta t)^{2}\right) \tag{10}\]  
Then, Equation (10) can be expressed in another way, as shown in Equation (11), where \(\varnothing(t)=-2\pi f_{0}\Delta t+\pi k_{r}(t-\Delta t)^{2}\) is the backscattered signal phase and \(f_{0}\) is the center frequency of the chirp signal.  
\[s_{r}(t)=cos(2\pi f_{0}t+\varnothing(t)) \tag{11}\]  
Figure 2 shows the received signal modeling by the SAR antenna based on [16; 19; 20; 21]. Where the received signal is multiplied by \(s_{r1}(t)\) and \(s_{r2}(t)\), respectively, passing the resulting signal through the low-pass filter, eliminating the carrier signal, and then these signals are digitized obtaining the SAR raw signal \(s_{a}(t)\), which is determined mathematically in the following way for both the real and imaginary channels.  
\[s_{r1}(t)=s_{r}(t)cos(2\pi f_{0}t),\ \ s_{r1}(t)=\frac{1}{2}cos(\varnothing(t))+ \frac{1}{2}cos(4\pi f_{0}t+\varnothing(t)) \tag{12}\]  
\[s_{r2}(t)=-s_{r}(t)sin(2\pi f_{0}t),\ \ s_{r2}(t)=\frac{1}{2}sin(\varnothing(t))- \frac{1}{2}sin(4\pi f_{0}t+\varnothing(t)) \tag{13}\]  
Unwanted signals must be removed; therefore, the signal \(\frac{1}{2}sin(4\pi f_{0}t+\varnothing(t))\) is removed using a low-pass filter whose frequency spectrum is shown in Figure 3. Then, the filtered signals or resulting signals are shown in Equations (14) and (15) for both the real and imaginary channels, respectively.  
Figure 2: Quadrature demodulation of the SAR system [19].  
\[s_{rr}(t)=\frac{1}{2}cos(\varnothing(t)) \tag{14}\]  
\[s_{ri}(t)=\frac{1}{2}sin(\varnothing(t)) \tag{15}\]  
\[s_{a}(t)=\frac{1}{2}(cos(\varnothing(t))+1 jsin(\varnothing(t))) \tag{16}\]  
Equation (16) can be expressed exponentially using [PERSON]'s relation as follows: \(\frac{1}{2}exp(1j\varnothing(t))\). Then, returning the rectangular function to Equation (16), the SAR raw signal is obtained as shown in Equation (17), similar to that in [16; 19; 20]. Where the phase \(exp(1j\varnothing(t))\) is used for generating SAR images through digital signal processing.  
\[s_{a}(t)\cong rect\bigg{(}\frac{t-\Delta t}{t_{p}}\bigg{)}exp(1j\varnothing(t)) \tag{17}\]  
Figure 1 shows that the chirp signal is transmitted at point \(P_{1}\), and said chirp signal with a time delay is received by the antenna at point \(P_{2}\). In pulsed SAR systems, an antenna is used for the transmission and reception of chirp signals, as shown in Figure 4 where the pulse repetition interval \((PRI)\) is the signal period in Figure 4, and in a system pulse, \(n\) is transmitted and \(m\) is received, then pulse \(n+1\) is transmitted and receives \(m+1\), and so on.