# Compact Bistatic Iterative Passive Radar Based on Terrestrial Digital Video Broadcasting Signals
## 2 System Model and Scenario  
We first describe our scenario. As it is known, the DVB-T standard uses OFDM modulation for transmitting the signal. Typical modes are 2 K, 4 K, 8 K, 16 K, and 32 K, which correspond to the number of subcarriers, 2048, 4096, 8192, 16,384, and 32,768, respectively. However, not all the subcarriers are really sending information because some are reserved for pilots and guard frequencies. Let us consider a single DVB-T transmitter in an unknown position and a receiver equipped with four antennas, three of which are required for the algorithm whilst the fourth is used for calibration purposes. The four antennas are separated more than the wavelength to avoid electromagnetic coupling among them and thus achieve better performance. In general, the distance between antennas is \(2d\), as indicated in Figure 1. As shown in this figure, the receiver establishes a coordinate system, with the origin in the middle of the four antennas. The primary reason for the four antennas is calibration. By using the signal received at all four antennas, the receiver can align with the transmitter and determine the distance \(d_{0}\). During the calibration process, all four antennas are placed at the same height (\(z\) coordinate), but once the initial calibration is achieved, each of the four antennas is displaced into the \(z\) coordinate so that each one has a different value. This step is important in preventing the appearance of a rank-deficient matrix during operation. As can be observed, the DVB-T transmitter is located at a distance \(d_{0}\) according to the new receiver-based coordinate system, and a height of \(E_{z}\) with respect to the receiver. When a target appears, it will have the unknown coordinates \(T_{x}\), \(T_{y}\), and \(T_{z}\). Thus, the distance between the DVB-T transmitter and the target is denoted \(R_{t}\), while the distance between the target and each of the receiver antennas is \(d_{i}\) (with \(i\in{1,2,3,4}\)). In the figure, only \(d_{1}\) is depicted for clarity. We seek to find the unknown coordinates of the target. In order to do this, we estimated the total distance \(dt_{i}\) between the DVB-T transmitter and each of the antennas using the following formula [6]\[R_{t}=\sqrt{\left(T_{x}-d_{0}\right)^{2}+T_{y}^{2}+\left(Tz-E_{z}\right)^{2}} \tag{1}\]  
\[d_{i}=\sqrt{\left(Rx_{i}-T_{x}\right)^{2}+\left(Ry_{i}-Ty\right)^{2}+\left(Rz_{i }-Tz\right)^{2}} \tag{2}\]  
where \(Rx_{i}\), \(Ry_{i}\), and \(Rz_{i}\) are the coordinates \(x\), \(y\), and \(z\), respectively, with respect to the receiver's coordinate system of \(i\)-th antenna. Thus, the total distance to antenna \(i\)-th can be calculated as follows:  
\[dt_{i}=R_{t}+d_{i}=\\ \sqrt{\left(T_{x}-d_{0}\right)^{2}+T_{y}^{2}+\left(Tz-E_{z}\right) ^{2}}+\sqrt{\left(Rx_{i}-T_{x}\right)^{2}+\left([PERSON]_{i}-[PERSON]\right)^{2}+\left([PERSON] _{i}-Tz\right)^{2}} \tag{3}\]  
The total distance \(dt_{i}\) can be estimated, as will be explained in the following section. Assuming we have estimated those distances, after rearranging the equation, we obtain  
\[dt_{i}-\sqrt{\left(T_{x}-d_{0}\right)^{2}+T_{y}^{2}+\left(Tz-E_{z}\right)^{2} }=\sqrt{\left(Rx_{i}-T_{x}\right)^{2}+\left([PERSON]_{i}-T_{y}\right)^{2}+\left(Rz _{i}-Tz\right)^{2}} \tag{4}\]  
By squaring both terms and making some straightforward calculations, we can obtain  
\[(Rx_{i}-d_{0})T_{x}+Ry_{i}T_{y}+T_{z}(Rz_{i}-E_{z})-dt_{i}\sqrt{ \left(T_{x}-d_{0}\right)^{2}+T_{y}^{2}+\left(T_{z}-E_{z}\right)^{2}}=\\ \frac{1}{2}\Big{(}Rx_{i}^{2}+Ry_{i}^{2}+Rz_{i}^{2}-dt_{i}^{2}-d_{ 0}^{2}-E_{z}^{2}\Big{)} \tag{5}\]  
Using matrix notation, the above equation can be expressed as follows:  
\[\mathbf{S}\mathbf{x}=\mathbf{z}+\mathbf{r}R_{t} \tag{6}\]  
Figure 1: Schematic for the scenario where the four antennas (1, 2, 3, 4) of the compact receiver, where one transmitter (in green) and the target are depicted. The main variables are also shown in the figure.