## 4 Contribution
### Construction of Low-Altitude SULA Model of Urban Environment
#### 4.2.1 Construction of Beamforming in Low Altitude Environment  
In the process of signal transmission, the different path signals generated by the multipath effect reach the receiving end at different angles. The different paths of the signal are distinguished according to the arrival angle of the wave. Referring to [PERSON] model, it is assumed that the path generated by the multipath channel can be completely separated. Signals arriving from different paths have their own characteristics. The baseband time-varying multipath channel is:  
\[h(n,\eta)=\sum_{u=1}^{u}h_{u}(n)\sigma(\eta-\eta_{u}) \tag{53}\]  
where \(\eta_{u}\) is the sampling delay, \(u\) is the path of signal transmission, \(h_{u}(n)\) is a different sub-path. The signals transmitted by different paths will experience different attenuation due to the distribution of obstructions. When they reach the receiving end from different angles, the signals on each path will produce different Doppler shifts:  
\[h_{u}(n)=\overline{h}_{u}\exp(j2\pi nTf_{D}\cos\theta_{u}+j\psi_{u}) \tag{54}\]where \(f_{D}\) is the maximum Doppler frequency, \(\overline{h}_{u}\) is the attenuation of the signal, \(\theta_{u}\) is the angle of arrival of the signal, \(\psi_{u}\) is the phase shift of the \(u\)th path, subject to uniform distribution, and \(T\) is the sampling interval. The omnidirectional antenna has better performance in high mobility scenarios. Therefore, Uniform circular array (UCA) is adopted to achieve beamforming in order to cover the received signal in all directions. As shown in Figure 4. The spacing of each element is \(l\), and the first element is taken as a reference. The received signal in the time domain at the \(i\)th element is:  
\[s_{i}(n)=\sum_{u=1}^{u}\overline{h}_{u}x(n-\eta_{u})\exp\left(j2\pi\left( \frac{R}{\lambda}\cos(\theta_{u}-\alpha_{i})+nTf_{D}\cos\theta_{u}\right) \right)+z_{i}(n) \tag{55}\]  
where \(R=ll/2\pi\) is the radius of the uniform circular array, \(\lambda\) is wavelength of the carrier, \(\alpha_{i}=(i-1)\frac{2\pi}{T}\) is the polar angle of the \(i\)th element, and according to theoretical calculations \(R\) is about 30 cm. \(Z_{i}=\left[Z_{i}(-N_{g}),\cdots,z_{i}(N-1)\right]^{T}\) is the additive white Gaussian noise. When the UAV transmits data, the signal on each path will produce a different Doppler frequency shift. The Doppler shift can be estimated if it is assumed that there is only one angle of arrival. Firstly, the uniform circular array beamforming technology is used to separate the Doppler frequency shift caused by the multipath effect. A beamforming network with \(u\) different paths contains \(u\) beamforming vectors such as \(\gamma_{i}=[\gamma^{(1)},\cdots,\gamma^{(u)}]\). The \(\gamma^{(u)}\) of each \(u\) is an \(I\times 1\) vector. The received signal after beamforming is:  
\[s(u)=\gamma^{(u)H}s,1\leq u\leq U \tag{56}\]  
where \(s\) is the receiving signal stack on the antenna, which meets the condition:  
\[s_{i}=[s_{i}(-N_{g}),\cdots,s_{i}(N-1)]^{T} \tag{57}\]  
Assuming that the received signal meets the far-field condition. Equation (58) is used to represent the steering vector in \(\theta\) direction of a narrowband signal with carrier frequency \(f_{c}\):  
\[V(\theta)=\left[e^{j2\pi R\cos(\theta-\alpha_{1})/\lambda},\cdots,e^{j2\pi R \cos(\theta-\alpha_{1})/\lambda}\right]^{T} \tag{58}\]  
The least square method is used to calculate the weight \(\gamma^{(u)}\). Equation (59) can be obtained by the radiation pattern constraints on the path direction vector:  
\[R=\left[V(\theta_{1}),V(\theta_{2}),\cdots,V(\theta_{u})\right] \tag{59}\]  
Figure 4: Uniform circular array. The expected output under the radiation pattern constraints is represented by \(G_{R}=R^{H}\overrightarrow{\gamma}\). At the same time, the least squares constraint is set in \(Q\) directions, and then:  
\[Z=\left[V(\theta_{z_{1}}),V(\theta_{z_{2}}),\cdots,V\left(\theta_{z_{Q}}\right)\right] \tag{60}\]  
The expected output at this time is \(G_{\mathrm{Z}}=Z^{H}\overrightarrow{\gamma}\).