## 2 The Waveform Superposition Algorithm  
### The Gain of Nodes  
The diagram of collaborative jamming is shown in Figure 1. The jammers are formed by a uniform rectangular array. Each node aims at the target and sends a directional jamming beam, which is \(d_{k}\) far from the target, and the steering vector of the beam is \(e^{2\pi d_{k}/\lambda}\). When the beams overlap, they form a high-power field within a specific range, jamming the communication receiver effectively. The jamming array is regarded as a whole, and the beam emitted by each node would form a directional combined beam towards the target under the far-field condition. In addition, redundant UAVs at the rear support the damaged UAVs to ensure a continuous jamming effect.  
Figure 1: The diagram of collaborative jamming. The jammers are distributed in the _xoy_ plane, the coordinate of jammer \(k\) is (\(r_{k},\phi_{k}\)), and the target is located at \((L,\theta_{0},\phi_{0})\).  
The normalized field pattern of node \(k\) is written as  
\[f(\theta,\phi)=\sum_{i=1}^{n}e^{-jK(\mathbf{u}-\mathbf{u_{0}})\cdot\mathbf{x_{i}}} \tag{1}\]  
where \(K=2\pi/\lambda\), \(\mathbf{u}\) is the unit vector from the node's element origin to the point \((L,\theta,\phi)\) on the sphere, and \(\mathbf{x_{i}}\) is the vector of the node's element pointing to the origin of the target. Thus, the gain of the node is expressed as  
\[g(\theta,\phi)=4\pi\eta_{e}\frac{|f(\theta,\phi)|^{2}}{p_{total}} \tag{2}\]  
where \(\eta_{e}\) is the efficiency, and \(p_{total}\) is the total radiated power of the node in the sphere, which is provided by  
\[p_{total}=\frac{2\pi^{2}}{N_{e}N_{a}}\sum_{i=1}^{N_{e}-1}\sum_{j=1}^{N_{e}-1} \lvert f(\theta_{i},\phi_{j})\rvert^{2}\sin\theta_{i} \tag{3}\]  
where \(N_{e}\) is the number of elevation grid points and \(N_{a}\) is the number of azimuth grid points. The gain of the uniform linear array (ULA) is shown in Figure 2, which is composed of 16 isotropic antennas, assuming \(\phi\) and \(\theta\) of the target are 0 and \(\pi/2\), respectively, and the space between the nodes is 0.5\(\lambda\). In the horizontal direction, the main lobe is concentrated at 0\({}^{\circ}\) and \(\pm\)180\({}^{\circ}\). The symmetry in the gain pattern around these angles suggests that the array has a balanced radiation profile in the horizontal plane. The gain pattern exhibits uniform amplitude distribution in the vertical direction, implying that the ULA maintains consistent radiation characteristics across different elevation angles. The side lobes, visible at other angles, have significantly lower gain, indicating effective suppression of unwanted radiation in non-primary directions.  
The gain of the uniform rectangular array (URA) is shown in Figure 3. Assume \(\phi\) and \(\theta\) of the target are 0 and \(\pi/2\), respectively, and there are 4 main lobes of the array. This contrasts with the ULA, which typically exhibits fewer main lobes. The URA's gain pattern shows that the energy is more concentrated in the main lobes, with significantly reduced side lobes compared to the ULA. This characteristic allows for more efficient focusing of jamming power towards the intended directions, thereby enhancing the effectiveness of the jamming signal. However, the presence of multiple main lobes means that only the lobe directed toward the target is useful for jamming. The power transmitted through the other main lobes is wasted and does not contribute to the jamming effort.  
Figure 2: The gain of ULA.