## 3 Numerical Simulations
### Observation Capabilities  
This section investigates the observational capabilities of BIRALET sensor. The analysis is carried out considering the NORAD LEO population as target objects. The population and related Two-Line Elements were retrieved from SPACE-TRACK website on July 25, 2019 and consist of 3015 objects. Of these objects, only those with known radar cross section (RCS) are considered, giving a total number of objects of 2301. A dedicated software for simulating the passages of the objects was developed. For each catalogued object, the software propagates its TLE by means of the SGP4 dynamical model and computes possible passages that could be detected by BIRALET. A passage is considered detectable if the constraints on the RX and TX minimum elevation are satisfied, and the detected SNR is larger than an imposed threshold (typically 6 dB). For each detectable passage, the software provides a list of time epochs and required pointing directions, i.e., RX and TX azimuth and elevation, along with the measured SNR value and Doppler shift measurements. Only time epochs for which the SNR limit and the RX and TX rotational speeds constraints are satisfied are considered. For the considered simulations, the TLE propagated with the SGP4 dynamical model represents the real dynamics of the system. In order to model possible discrepancies between the real object dynamics and the available model, a uniform random error in the time variable is assumed, i.e., for each passage, a time shift \(\delta\) is guessed by sampling from a uniform distribution \(\Delta~{}\sim~{}\mathcal{U}(-0.2,~{}0.2)\)s, and the RX and TX pointing direction are computed considering a time epoch.  
\[t_{shift}=t_{real}+\delta \tag{1}\]  
where \(t_{real}\) represents the time epoch predicted by the TLE (the real world), whereas \(t_{shift}\) represents the time epoch used to compute the sensor pointing directions. This model essentially assumes that the error is mainly in the along-track direction, which is what typically happens when an estimate is propagated in time. The uncertainty interval for the time variable was selected according to data obtained from some observation campaigns performed with the sensor.  
The analysis considers an observation window of one week, from 26 July 2019, 00:00:00 UTC to 2 August 2019, 00:00:00 UTC. Figure 5a shows the results of the simulation in terms of objects orbital inclination distribution as a function of the perigee height. Objects that are detected during the seven-days observation campaign are represented with grey dots, whereas non-detected objects are plotted with black triangles. Starting from a catalogue of 2301 objects, 2262 objects are detected at least once. That it, the sensor potentially covers more than 98% of the target population. As can be seen, the sensor cannot observe only objects with a low orbital inclination and perigee height. Figure 5b, instead, shows the distribution of the number of passages per object. The vast majority of the objects has a number of passages between 10 and 20. The results previously described represent the upper bound of the performance of BIRALET. The sensor, indeed, is assumed to be capable of observing multiple objects at the same time, and no time separation between consecutive observations is imposed. These constraints are introduced when orbit determination is performed, as described in the following paragraphs.