# Review and Simulation of Counter-UAS Sensors for Unmanned Traffic Management
## 3 Counter-UAS Sensors Modeling
### Active Radar
#### 3.1.1 Quasi-Monostatic Radars  
This radar will be simulated using a power model from the radar equation. It will be assumed that the separation between transmitter and receiver is small compared to the distance to the target. In this first approximation, it is assumed that the radar can eliminate the clutter by doppler filtering. It is also assumed that the predominant noise is thermal. Its calculation now depends on the surrounding conditions and not only on the bandwidth. The main characteristics are:  
* Radar cross section dependent on target size [55]. * CFAR detection. * Radar parameters adapted to drone detection (integration times of the order of tens of milliseconds and range resolutions in order of meters). * Exploration times around a second. * Measurement error simulation. * Minimum scan time below a second. * False alarm simulation. The basic parameters defining the model of a quasi-monostatic radar are:  
* Instrumental range (\(R_{max}\)). * Minimum and maximum azimuth of coverage. * Distance resolution in meters (\(\Delta_{dis}\)). * Bandwidth. * Transmitting array position. * Transmitted power in W.
* Azimuthal width of the transmission pattern (\(\theta_{3 dBae-T}\)). * Elevation width of the transmission pattern (\(\theta_{3 dBele-T}\)). * Position of the receiving array. * Number of receiver array beams (\(N_{beams}\)). * Receiver array azimuth beamwidth (\(\theta_{3 dBae-R}\)). * Receiver array elevation beamwidth (\(\theta_{3 dBele-R}\)). * Dwell time. * Minimum time between scans. * Minimum and maximum frequency. * False alarm probability. The typical expression for the radar equation of a quasi-monostatic radar is simpler than that of a microwave radar since the free-space propagation losses are included within the ground-wave propagation losses. The radar equation is:  
\[\frac{S}{N}=P_{av}\frac{G_{T}G_{R}\sigma\lambda^{2}\ T_{int}}{\left(4\pi\right) ^{3}R_{T-I}^{2}R_{I-R}^{2}\ L_{s}\ N_{0}}\  
\(T_{int}\) integration time
* \(R_{T-t}\) transmitter-target distance
* \(R_{t-R}\) target-receiver distance
* \(L_{s}\) power losses of the radar system
* \(N_{0}\) system noise  
To obtain the elevation gain, the elevation width is considered, and to obtain the antenna gain, the azimuth shaping is considered. In this case and considering that the beamforming is conducted only in azimuth, the array gain is estimated approximately as 360\({}^{\circ}\) divided by the beamwidth.  
\[G_{T}\cong\left(\frac{4\pi}{\theta_{3 MBele-T}\theta_{3 MBazi-T}}\right) \tag{2}\]  
\[G_{R}\cong\left(\frac{4\pi}{\theta_{3 MBele-R}\theta_{3 MBazi-R}}\right) \tag{3}\]  
\[\theta_{3 dB-R}=\frac{\theta_{max}-\theta_{min}}{N_{beams}} \tag{4}\]  
The system losses depend on many factors such as the antenna feed or the construction of the processing. A loss factor of around 4 dB has been given as a typical value. The cross section for these frequency bands depends on the target size, and it is modeled as constant for all angles (0.01-0.1 m\({}^{2}\)). The reception noise is predominantly thermal noise due to the frequencies being used. \[N_{0}=k\ T_{0}\ 10^{F_{\alpha}/10} \tag{5}\]  
where \(k\) is the Boltzmann constant, \(T_{0}\) is the Earth temperature (typically 290\({}^{\circ}\) K) and \(F_{\alpha}\) a noise factor with a typical value of 5 dB. Once the SNR has been calculated, the detection, false alarms and measurement position must be generated. The detector is assumed to be a CA-CFAR so it is assumed that the target behaves as a Swerling I between scans and the noise residual has a Gaussian distribution [56]. The threshold of the CFAR is obtained with the following expression:  
\[\alpha=(PFA)^{-1/N}-1 \tag{6}\]  
where \(\alpha\) is the CFAR threshold factor, \(N\) is the number of CFAR cells and \(PFA\) is the false alarm probability.