# Theory and Design of Blink Jamming
## 3 Modelling Formation Jamming
### Monopulse Receiver  
At the highest level in the simulation, shown in Figure 9, the monopulse receiver calculates the antenna gains for each monopulse beam, the RF path-loss, propagation delay, and then creates the sum and difference signals to calculate the angular error signal. Figure 8: The Simulink model for the target movement using constant velocity targets. The target kinematic information is then converted to spherical coordinates and compared to the pointing location of the tracking antenna to provide range, azimuth, and elevation information for modelling the received signal power and delay. Figure 7: The system model for the formation jamming system, replicated for each of the two jammers. A set of user operated switches, embedded in the jammer signal subsystem, allowed for selection of the three possible jamming techniques. In this monopulse receiver, sinc antenna patterns were chosen to model the tracking radars [3]. These beam patterns are described by:  
\[f(\theta)=\frac{\sin\left(\frac{\pi\eta}{\lambda}\sin\theta\right)}{\frac{\pi\eta }{\lambda}\sin\theta} \tag{17}\]  
where \(\theta\) is the azimuth, \(w\) is the aperture width, and \(\lambda\) is the wavelength. A similar equation was applied for the elevation pattern, replacing the width with height. A typical radiation pattern is shown in Figure 10 for the left and right beams in a single plane, squinted by 0.468 beamwidths. The resulting sum and difference patterns of these two antennas are shown in Figure 11.  
Figure 10: The left and right beam patterns for an amplitude comparison monopulse receiver squinted for angle measurement, units are in dBi. Figure 9: The Simulink model for the monopulse receiver, with the analog and testbed components implemented in Simulink and the digital systems on an FPGA. The antenna gain values were stored in a 2-D look-up table for each beam, shown in Figure 12. After receiving the azimuth and elevation associated with a received signal the gain in that direction was calculated for use in determining the received power. Figure 11: The sum and difference beam patterns for an amplitude comparison monopulse receiver, units are in dBi. Figure 12: The antenna gain subsystem is implemented four times to replicate the four beams of the monopulse tracking system. In each case, as shown here, it takes the received azimuth and elevation for a signal and calculates the gain in that direction. The received signal power was calculated using effective radiated power of the transmitter, receiver antenna gain, wavelength of propagation, and target range. This model used Friss transmission equation to determine the received power:  
\[P_{r}=\frac{P_{t}G_{t}G_{r}\lambda^{2}}{\left(4\pi R\right)^{2}} \tag{18}\]  
where \(P_{t}\) is the transmitted power, \(G_{t}\) is the transmitter gain, \(G_{r}\) is the receiver gain, and \(R\) is the range [3]. The model implements (18) in Figure 13, along with the phase delay. The received signal power from each of the four antennas was processed through an amplitude comparison monopulse receiver to produce the sum and delta signals, convert those to error signals, and finally pointing error in degrees for azimuth and elevation, shown in Figure 7:0 The FPGA implementation was chosen to be done in floating point math, instead of fixed point. Floating point was chosen to effectively deal with the potential high dynamic ranges involved with specific techniques. The FPGA implementation itself was done on a Kintex-7 FPGA development board, operating at the sample frequency of the complete model. The total resource utilization of the FPGA was less than 10 % due to the relatively limited signal processing required (no finite impulse response filters, or similarly computationally intensive systems). Figure 7:1 The monopulse receiver shown at the system level, calculates the error in the measured angle relative to the pointing angle of the antenna.