# Review and Simulation of Counter-UAS Sensors for Unmanned Traffic Management
## 3 Counter-UAS Sensors Modeling
### Active Radar
#### 3.1.2 MIMO Radars  
This simulator models a MIMO radar with spatially separated antennas at high frequencies (X, Ku, K or Ka). Each radar unit will have three transmitting antennas and one receiving antenna placed with the central transmitter. Since the antennae are widely separated and will view the target from different angles, echo coherence is not expected. Therefore, incoherent integration processing is performed, since the coherent has no gain. The main characteristics are:  
* Radar cross section dependent on drone size. * Several simultaneous transmitters. * CFAR detection. * Measurement error simulation. * Simultaneous space exploration system using simultaneous antenna beams (MIMO techniques). * Minimum scan time around one second. * False alarm simulation. The basic parameters defining the model of a MIMO radar are:  
* Transmission frequency. * Position of each transmitting and receiving antenna. * Power transmitted by each transmitter. * Instrumental range. * Minimum azimuth of coverage. * Maximum azimuth of coverage. * Azimuthal width of the transmission pattern (\(\theta_{3 MBaz-T}\)). * Elevation width of the transmission pattern (\(\theta_{3 MBele-T}\)). * Number of receiver array beams (\(N_{bams}\)). * Receiver array azimuth beamwidth (\(\theta_{3 MBaz-R}\)). * Receiver array elevation beamwidth (\(\theta_{3 MBele-R}\)). * Integration time. * Bandwidth. * Distance resolution. * False alarm probability. * Scan period. * Antenna gain. * Secondary lobe level. In this case, to obtain the target echo power at the receiver, it has been added the echo powers of each of the three receivers. It is assumed that, in MIMO radar, before the incoherent integration of the signals from all transmitters, the possible clutter is coherently eliminated, but in this model, clutter is not considered. Therefore, the power received from a target will be obtained as the sum of the power received from each transmitter. \[P_{target}=P_{Tx1}+P_{Tx2}+P_{Tx3} \tag{16}\]  
\[P_{Txi}=P_{av,i}\frac{G_{Txi}G_{R}\ T\lambda^{2}\ \sigma}{\left(4\pi\right)^{3}{R_{ 1}}^{2}{R_{2}}^{2}\ L_{p}}\cdot F_{P_{Txi},Rx} \tag{17}\]  
where:  
* \(P_{av,j}\) average power of transmitter i
* \(G_{Txi}\) power gain of transmitting antenna i
* \(G_{R}\) receiver antenna power gain
* \(\lambda\) wavelength
* \(\sigma\) cross section
* \(T\) integration time
* \(R_{i}\) distance in each path
* \(L_{p}\) power losses
* \(F_{P_{-}Txi},Rx\) propagation factor in the propagation path ith-transmitter-target-receiver  
The average signal-to-noise ratio per echo for each transmitter is calculated by adding the target powers from each transmitter and dividing by the number of transmitters and dividing by the noise power. \[\left(\frac{S}{N}\right)=\frac{P_{target}/N}{P_{N\_{dec}}} \tag{18}\]  
Transmitting antennas are assumed to be uniformly patterned in coverage in the horizontal plane and to distribute their power to uniformly illuminate the scanned area from their respective positions. Transmitter gains are specified as a factor depending on the azimuth width of the scanned area. The gains are as follows:  
\[G_{T}\cong\left(\frac{4\pi}{\theta_{3 Bele-R}\theta_{3 Bazi-T}}\right) \tag{19}\]  
\[G_{R}\cong\left(\frac{4\pi}{\theta_{3 Bele-R}\theta_{3 Bazi-R}}\right) \tag{20}\]  
The propagation factor, being free space, is assumed to be 1. The system losses are assumed to be 2 dB due to Doppler filtering envelopes and CFAR detection losses. Assuming that the images of the three transmitters are integrated incoherently, it can be assumed that it is integrated a CFAR reference of three times the reference of a single system. If it is assumed for each radar a 10-cell reference, it will have 30 reference cells. For a PFA of 1e-4, this means a loss of less than 1 dB [57]. The cross section for these radars depends on the target size, so it is specified by an equal constant for all angles (0.01-0.1 m\({}^{2}\)). These radars operate at high microwave frequencies, and consequently, the predominant noise is thermal noise.