# Cross-Pol Transponder with Frequency Shifter for
## 2 Materials and Methods  
### Working Principle of Monostatic/Bistatic GBSAR  
The working principle of a GBSAR that is able to acquire monostatic and bistatic images using a transponder is shown in Figure 1. A linear GBSAR acquires a (monostatic) image of the targets in itsown field of view. Using a third antenna, it acquires a second image of the same targets exploiting the bouncing of the signal through a transponder. The simplest transponder consists of a couple of antennas and an amplifier.  
By applying the radar equation [7] in monostatic and bistatic configurations, we obtain the following equations, respectively:  
\[P_{rx}^{(mono)}=P_{tx}G_{1}\frac{1}{4\pi R_{1,k}^{2}}\sigma^{(mono)}\frac{1}{4 \pi R_{1,k}^{2}}G_{2}\frac{\lambda^{2}}{4\pi} \tag{1}\]  
\[P_{rx}^{(bist)}=P_{tx}G_{1}\frac{1}{4\pi R_{1,k}^{2}}\sigma^{(bist)}\frac{1}{4 \pi R_{2}^{2}}G_{3}\frac{\lambda^{2}}{4\pi}AG_{4}\frac{1}{4\pi R_{3,k}^{2}}G_{5 }\frac{\lambda^{2}}{4\pi} \tag{2}\]  
where \(P_{tx}\) is the transmitted power; \(G_{1}\), \(G_{2}\), \(G_{3}\), \(G_{4}\), \(G_{5}\) are the gains of the five antennas; \(R_{1,k}\) is the distance between the radar head at the \(k\)-position along the mechanical guide and the target; \(R_{1}\) is the distance between the center of the mechanical guide and the target; \(R_{2}\) is the distance between the target and the transponder; \(A\) is the gain of the amplifier of the transponder; \(R_{3,k}\) is the distance between the radar head at the \(k\)-position along the mechanical guide and the transponder; \(R_{3}\) is the distance between the center of the mechanical guide and the transponder; \(\sigma^{(mono)}\) is the radar cross section in monostatic configuration; \(\sigma^{(bist)}\) is the radar cross section in bistatic configuration; and \(\lambda\) is the wavelength.  
In practical deployments, the following assumptions are reasonable: (1) All the antennas have the same gain, so \(G_{1}=G_{2}=G_{3}=G_{4}=G_{5}=G\). (2) The length (\(L_{0}\)) of mechanical guide is much smaller than the radar-target and target-transponder, so \(R_{1,k}=R_{1}\) and \(R_{3,k}=R_{3}\). (3) Although \(R_{1}\) and \(R_{3}\) are not equal, they can be reasonably similar, so we assume \(R_{1}=R_{3}\). (4) We assume \(\sigma^{(mono)}=\sigma^{(bist)}\). The last assumption is rather hard, but it can be done when the aim of the calculation is just a rough estimate of the received power. With these assumptions, we can write the following:  
\[\frac{P_{rx}^{(bist)}}{P_{rx}^{(mono)}}=A\left(\frac{G\lambda}{4\pi R_{3}} \right)^{2} \tag{3}\]  
Figure 1: Monostatic/bistatic ground-based synthetic aperture radar (GBSAR) with transponder.  
Hence, in order to compensate the loss due to the path transponder-radar, the gain of the transponder should be as follows:  
\[A=\left(\frac{4\pi R_{3}}{G\lambda}\right)^{2} \tag{4}\]  
This gain value compensates the loss; however, with this value of gain, the noise figure of the transponder is added to the noise figure of the first amplifier of the radar head. In order to improve the noise figure, the gain has to be increased. Indeed, as it is well known, the noise figure of the whole system is equal to the only noise figure of the transponder amplifier (i.e., the first stage of the receiver chain) if the gain of this is much greater than the loss of the path. In practice, a reasonable rule of thumb for the gain of the transponder could be the following:  
\[A\approx 10\left(\frac{4\pi R_{3}}{G\lambda}\right)^{2} \tag{5}\]  
For obtaining good displacement measurements in both directions with the bistatic configuration, \(L\) should be in the order of magnitude of the distance between radar and target [6].  
The operative distance of a GBSAR can reach up to 5 km. However, a value between 50 m and 500 m is reasonable in most applications. Therefore, by considering an antenna gain (\(G\)) of 15 dB and a wavelength of 3 cm (10 GHz), we obtain that the transponder gain (\(A\)) could range between 66 dB and 86 dB, with L ranging between 50 m and 500 m.