# Reinforcement Learning-Based Anti-Jamming in Networked UAV Radar Systems
## 3 System Model
### Beam Synthesis Method under Low-Bit Phase Array
The spacing of array elements is \(d\), the number of coded bits is \(M\), the operating wavelength is \(\lambda=2^{M}d\), and the number of coding repetitions is \(p\). In the real scenarios, the operating wavelength is related to the frequency. It is only set here to facilitate the interpretation of the beamforming principle. Consider a case of 10 UAVs in a swarm and each UAV carries five elements, the number of ULA elements is \(N=50\). The spacing of array elements is \(d=1\)cm and \(M=2\), Figure 3a shows the antenna pattern when \(p=2\) and \(p=3\), respectively. Figure 3b shows the antenna pattern after the two coding modes are added in Figure 3a. Since the beam pointing of a phased array antenna is:  
\[\sin\theta=\frac{\varphi\lambda}{2\pi d} \tag{2}\]  
where \(\varphi=2\pi/2^{M}\), and \(\lambda=2^{M}d\), thus:  
\[\theta=\arcsin(1/p) \tag{3}\]  
The synthesized beam is now pointed to:  
\[\theta_{0}=\arcsin(\frac{1}{2}+\frac{1}{3})=56.44^{\lx@math@degree} \tag{4}\]  
From Figure 3b, the effectiveness of (1) can be verified. Therefore, this paper applies the antenna pattern generated by repetitive coding and convolution method under 3-bit condition to simulate. According to the basic theory of phased array antennas, the beam width can be controlled by the number of elements. When a wider beam needs to be generated, the UAV swarm can choose to  
Figure 3: (**a**) Antenna pattern for \(p=2\) and \(p=3\) (**b**) Antenna pattern after the two coding modes \(p=2\) and \(p=3\) are added. Figure 2: Schematic diagram of repetitive coding and convolution methods. turn off some of the array elements to reduce the number of elements. There is a trade-off for the antenna pattern policy. For a radar UAV swarm, both the antenna pointing and the mainlobe width can be tuned. When a pattern with a wide mainlobe width is selected, its antenna gain also decreases. Therefore, the algorithm does not always choose the policy of a wide mainlobe antenna pattern. In order to make the beam synthesis method more universal, we continue to consider the low bit beamforming method when the UAVs form a uniform plane array. Suppose the UAV swarm is arranged in an equivalent UPA as shown in Figure 4.  
In Figure 4, the red box represents the UPA carried by each UAV, and they constitute the equivalent UPA as a whole. The equivalent UPA is represented by a black box. Because the element number of an equivalent UPA is much more than that of a single UAV, it has better beam performance, such as higher gain and narrower mainlobe width. The equivalent UPA can adjust the azimuth and elevation of the beam independently to realize the beam synthesis in two-dimensional space. According to [28], the low-bit beam synthesis method of ULA can be extended to UPA. We assume that elevation is \(\theta\), azimuth is \(\phi\), the equivalent UPA is of \(m\) rows and \(n\) columns and the number of coded bits is \(M\). Thus, the two-dimensional radiation pattern can be written as:  
\[\left|E_{1}(\theta,\phi)\right|\approx\frac{\sin[\frac{\eta}{2}(\frac{2\pi pd_ {1}}{\lambda}\cos\theta\sin\varphi-\frac{2\pi}{2M})]}{\sin[\frac{1}{2}(\frac{2 \pi pd_{1}}{\lambda}\cos\theta\sin\varphi-\frac{2\pi}{2M})]} \tag{5}\]  
\[\left|E_{2}(\theta)\right|\approx\frac{\sin[\frac{m}{2}(\frac{2\pi pd_{2}}{ \lambda}\sin\theta-\frac{2\pi}{2M})]}{\sin[\frac{1}{2}(\frac{2\pi pd_{2}}{ \lambda}\sin\theta-\frac{2\pi}{2M})]} \tag{6}\]  
We assume that \(\mathbf{p}=[p_{1},p_{2},\cdots,p_{a}]\) represents the column coding (control azimuth) with the repetition times of \(p_{1},p_{2},\cdots,p_{a}\) respectively, and \(\mathbf{q}=[q_{1},q_{2},\cdots,q_{b}]\) represents the row coding (control elevation) with the repetition times of \(q_{1},q_{2},\cdots,q_{b}\), respectively.