## 2 The Waveform Superposition Algorithm
### Waveform Superposition  
Each node emits a directional jamming beam aimed at the target, as shown in Figure 4. These beams overlap at various positions in space, resulting in varying degrees of constructive and destructive interference. If these beams arrive at the target point simultaneously and are perfectly aligned in phase and frequency, the strongest interference effect can be achieved at the target location, maximizing the interference across the entire space. The superposed waveform can be expressed as  
\[x(t)=\sum_{k=1}^{N}x_{k}(t) \tag{4}\]  
where \(x_{k}(t)\) is the waveform of node \(k\):  
\[x_{k}(t)=A_{tk}\cos(2\pi f_{c}t+\varphi_{k0}) \tag{5}\]  
where \(A_{tk}\) is the amplitude of the waveform, \(f_{c}\) is the carrier frequency, and \(\varphi_{k0}\) is the initial phase of the waveform, which is set according to the distance of the node from the target location \(L_{k0}\):  
\[\varphi_{k0}=-KL_{k0} \tag{6}\]  
where \(\lambda\) is the carrier wavelength. The amplitude relationship between the receiver and transmitter is  
\[\frac{A_{rk}(\theta,\phi)}{A_{tk}}=\sqrt{\frac{P_{rk}(\theta,\phi)}{P_{tk}}} \tag{7}\]  
where \(P_{tk}\) and \(P_{rk}(\theta,\phi)\) are the transmitted and received power of the node, respectively. Figure 4: The diagram of wave superposition. Figure 3: The gain of URA. According to the Friis formula, the received power is given by  
\[P_{rk}(\theta,\phi)=P_{tk}\cdot g(\theta^{\prime}_{k^{\prime}},\phi^{\prime}_{k} )\cdot G_{r}(\theta,\phi)\cdot\left(\frac{\lambda}{4\pi L_{k}(\theta,\phi)} \right)^{2} \tag{8}\]  
where \(G_{r}(\theta,\phi)\) is the gain of the target antenna, and \(L_{k}(\theta,\phi)\) is the distance between the node and the point on the sphere. The \(\theta^{\prime}_{k},\phi^{\prime}_{k}\) represents the azimuth and elevation angle of the point on the sphere concerning node \(k\), respectively. In order to calculate \(\theta^{\prime}_{k},\phi^{\prime}_{k}\), node \(k\) is regarded as the origin of the coordinates, and the \(z\) axis is the target's direction. As shown in Figure 5, the coordinate is transformed. Assume \(P(x,y,z)\) is a point in the coordinate system. First, move the origin point to node \(k\), whose coordinates are \((x_{k},y_{k},z_{k})\); secondly, rotate \(\phi\) counterclockwise with \(z\) as the axis, and then rotate \(\theta\) counterclockwise with \(y\) as the axis. The coordinate of \(P\) in the new coordinate system is given by  
\[ \begin{bmatrix}x^{\prime\prime}\\ y^{\prime\prime}\\ z^{\prime\prime}\end{bmatrix} = \begin{bmatrix}\cos\theta\cos\phi&\cos\theta \sin\phi&\sin\theta\\ -\sin\phi&\cos\phi&0\\ -\sin\theta\cos\phi&-\sin\theta\sin\phi&\cos\theta\end{bmatrix}\begin{bmatrix} x\\ y\\ z\end{bmatrix}  \tag{9}\]  
Based on the new coordinate, the azimuth and elevation angle of the point on the sphere concerning node \(k\) are given by  
\[\theta^{\prime}_{k}=\arctan\left(\frac{z}{\sqrt{x^{\prime\prime 2}+y^{\prime \prime 2}}}\right) \tag{10}\] \[\phi^{\prime}_{k}=\arctan 2(y^{\prime\prime},x^{\prime\prime})\]  
Bring (8) to (7), and \(A_{rk}(\theta,\phi)\) is written as  
\[A_{rk}(\theta,\phi)=\frac{A_{rk}\lambda}{4\pi L_{k}(\theta,\phi)}\cdot\sqrt{g (\theta^{\prime}_{k^{\prime}},\phi^{\prime}_{k})\cdot G_{r}(\theta,\phi)} \tag{11}\]  
Figure 5: The diagram of coordinate system transformation: the coordinates change from \((x,y,z)\) to \((x^{\prime\prime},y^{\prime\prime},z^{\prime\prime})\). where \(N\) is the number of nodes of the sparse array, and \(d_{k}(\theta,\phi)\) is the distance between node \(k\) and the point \((L_{1},\theta,\phi)\) on the far-field sphere:  
\[d_{k}^{2}(\theta,\phi)=L_{1}^{2}+r_{k}^{2}-2L_{1}r_{k}\cdot\big{[}\sin\theta \sin\theta_{k}\cos(\phi-\phi_{k})+\cos\theta\cos\theta_{k}\big{]} \tag{19}\]  
The 16 UAV nodes are distributed in the \(xoy\) plane, the distributed radius \(R\) is 5 m, the distance \(L_{1}\) is 1000 m, and the carrier frequency is 400 MHz.