# Review and Simulation of Counter-UAS Sensors for Unmanned Traffic Management
## 3 Counter-UAS Sensors Modeling
### Passive Radar  
Passive radars will be simulated using a power model from the radar equation, which includes a multipath propagation model. These radars are of multistatic type. The predominant noise will be the direct transmitter-receiver signal interference, which will be considered in the model. The main characteristics are:  
* Radar cross section dependent on drone size. * Possibility of several simultaneous opportunity transmitters. * Opportunity transmitters self-interference simulation. * Simulation of electrical noise dependent on the frequency of the transmitter (atmospheric, human and white noise). * CFAR detection. * Measurement error simulation. * Simultaneous space exploration system using multilateration position determination techniques (bistatic or multistatic radars). * Minimum scan time in the order of seconds (>1 s). * False alarm simulation. The basic parameters defining the model of a passive radar are:
* Passive radar position. * Instrumental range in kilometers. * Minimum and maximum azimuth of coverage in degrees. * Number of receiver antenna beams. * Number of opportunity transmitters. * Position of each opportunity transmitter. * Carrier frequency of each opportunity transmitter. * Transmitted power of each opportunity transmitter. * Antenna gain. * Gain of the receiving antenna of the direct signal. * Bandwidth of each opportunity transmitter. * Integration time in seconds. * Number of receiver array beams (\(N_{beams}\)). * Receiver array azimuth beamwidth (\(\theta_{3 dBaz-R}\)). * Receiver array elevation beamwidth (\(\theta_{3 dBele-R}\)). * Receiving array sidelobe level. * Direct signal cancellation level
* False alarm probability. * Scan period in seconds. The radar equation for a passive radar is implemented in several steps. The first is to calculate the received signal power of the target echo before the correlator for an opportunity transmitter. \[P_{target}=P_{av}\frac{G_{T}G_{R}\;\lambda^{2}\;\sigma_{final}}{{(4\pi)}^{3}{R_ {1}}^{2}{R_{2}}^{2}\;L_{p}}\cdot F_{p} \tag{24}\]  
where:  
* \(P_{av}\) average power of transmitter
* \(G_{T}\) transmitting antenna power gain
* \(G_{R}\) receiver antenna power gain
* \(\lambda\) wavelength
* \(\sigma\) cross section
* \(R_{i}\) distance in each path
* \(L_{p}\) power losses
* \(F_{p}\) propagation factor  
The ratio (\(\frac{S}{N}\)) is calculated at the correlator output, where the signal will have a gain equal to the square of the product bandwidth and propagation time. The interference powers (noise and correlator side lobes of different signals, including the target) have a gain equal to the product of bandwidth and integration time. The \(\frac{S}{N}\) ratio to be used in the CFAR is computed with the following expression:  
\[\left(\frac{S}{N}\right)=\frac{P_{target}\cdot{(B\;T)}^{2}}{P_{target}\cdot{(B \;T)}+P_{N\_{c}utter}\cdot{(B\;T)}+P_{N\_{s}ignal}\cdot{(B\;T)}+P_{N\_{c}lec} \cdot{(B\;T)}} \tag{25}\]  
where \(P_{N\_{c}utter}\), \(P_{N\_{s}ignal}\) and \(P_{N\_{c}lec}\) are the clutter, direct signal and electrical noise powers at the correlator input, respectively (these are calculated in the corresponding sections); \(T\) is the integration interval and \(B\) is the bandwidth (these parameters are specified for each opportunity signal). Transmitting antennas are assumed to have uniform pattern in coverage in the horizontal plane. The transmit gain is specified as a parameter of the transmitter. This gain, if omnidirectional broadcasting is assumed, will be around that of a half-wave dipole (2.15 dBi). As the transmitter power is usually given in apparent radiated power, which considers the gain of the transmitting antenna over the half-wave dipole, 2.15 dBi is the gain over the isotropic that we will assume of the transmitting antenna. For the receiving antenna, a circular array is assumed that generates a number N of beams covering 360\({}^{\circ}\).