# Compact Bistatic Iterative Passive Radar Based on Terrestrial Digital Video Broadcasting Signals
## 3 Proposal Description
### Calibration Process  
Before performing the estimation, the system must be calibrated. The calibration will identify the distance to the transmitter, \(d_{0}\) and the height with respect to the receiver, \(E_{z}\). As explained earlier, this method will estimate a target's position with respect to the center of the receiver, i.e., \(O\); as a consequence, the obtained coordinates are not absolute but relative to the center of the receiver. First of all, the \(x\) axis is determined as follows. As seen in Figure 3, which is a top-down view of Figure 1 without the target, the \(x\) axis is the line between the center of the receiver and the transmitter when the four antennas that receive the signal are perfectly aligned. In order to achieve alignment, all four antennas are positioned at the same height, and the receiver is moved (rotated) until the signal is received at antennas 2 and 4 at the same time, and signal at antennas 1 and 3 are received at the same time and delayed with respect to the others. Once the axis \(x\) is determined, the receiver synchronizes with the DVB-T transmitter, after which the distance \(d_{0}\) and height \(E_{z}\) can be estimated. If the receiver used the global positioning system (GPS), the GPS coordinates of the transmitter \(d_{0}\) and \(E_{z}\) can be easily obtained; alternatively, if such coordinates are lacking, the receiver can proceed as follows. Since the receiver is synchronized with the DVB-T transmitter, it can estimate the time of arrival of the signal, and with this value, it can estimate the initial distance \(l_{0}\). Then, the receiver is moved a known distance \(\Delta\) along the \(y\) axis and the time of arrival is again calculated in the new position in order to estimate the second distance \(l_{f}\). With these two distances, and some simple similarity mathematics, the values of distance \(d_{0}\) and height, \(E_{z}\) can be obtained as follows:  
\[\left. \begin{array}{l}l_{0}^{2}=E_{z}^{2}+d_{0}^{2}\\ l_{f}^{2}=E_{z}^{2}+\left(d_{0}+\Delta\right)^{2}\end{array} \right\} \tag{12}\]  
Consequently, the solution is  
\[\left. \begin{array}{l}d_{0}=\frac{l_{f}^{2}-l_{0}^{2}-\Delta^{2}}{2\Delta}\\ E_{z}=\sqrt{l_{0}^{2}-d_{0}^{2}}\end{array} \right\} \tag{13}\]  
Figure 3: Top–down view of the scenario used to demonstrate calibration. The four antennas, named 1 to 4, are shown forming a square with a separation of \(2d\). The origin is O and the \(x\) axis is in the direction of the DVB-T transmitter at a distance of \(d_{0}\) from the center of the receiver. Although the errors in the estimation of these two parameters--\(E_{z}\) and \(d_{0}\)--in the calibration process is small, in the following, we have analyzed the impact on the final estimation error in the different coordinates, even for large errors in their estimation. Let us assume that there is some error in \(E_{z}\) and \(d_{0}\) in the form of \(E_{z}^{\prime}=E_{z}+\Delta_{z}\) and \(d_{0}^{\prime}=d_{0}+\Delta_{0}\), respectively. In Figures 4 and 5, the final estimation error of each of the three coordinates is depicted. The error \(\Delta_{0}\) or \(\Delta_{z}\) is indicated in meters. Typically, this misalignment is only several centimeters, i.e., much lower than 1 m, but we have explored the behavior with errors up to \(|10|\) meters. As it can be seen in Figure 4, when \(\Delta_{z}=0\) (no error in \(E_{z}\)), the effect over the different coordinates \(T_{x},T_{y}\) and \(T_{z}\) is proportional to the error in \(d_{0}\), \(\Delta_{0}\). It can also be observed that the effect on coordinate \(T_{y}\) is very small, almost negligible even for large values of \(\Delta_{0}\), while the impact on \(T_{z}\) is more important. The reason is that the calculations for coordinate \(T_{y}\) involving either \(E_{z}\) or/and \(d_{0}\) are much fewer compared to the other two coordinates that make use of them several times (see Equations (7)-(9)). Similar behavior can be observed when we evaluate the error is \(E_{z}\) with \(\Delta_{0}=0\), but the proportional values are slightly different.