## 2 System Model
### Signal, Channel and Noise Model  
The number of antennas of each UAV is \(N_{T}\); thus, the transmit signals could be denoted by \(N_{T}\)-dimension vectors. We denote the beamforming vector of the transmit signal UAV\({}_{i}\) sent to UE\({}_{j}\) as  
\[\mathbf{w}_{ij}=(w_{ij1},w_{ij2},\ldots,w_{ijn_{t}},\ldots,w_{ijN_{T}})^{T},i \in\mathcal{M},j\in\mathcal{N}, \tag{3}\]  
where \(\mathbf{w}_{ij}\) is a complex vector.  
The channel coefficient between UE\({}_{j}\) and the \(n_{t}\)th antenna of UAV\({}_{i}\) is denoted by \(h_{ijn_{t}}\), which consists of two parts: path loss and small-scale fading. We denote that \(h_{ijn_{t}}(\theta_{ijn_{t}})=\theta_{ijn_{t}}\tilde{h}_{ijn_{t}}\), where \(\theta_{ijn_{t}}\) reflects the pass loss, while \(\tilde{h}_{ijn_{t}}\) represents the small-scale fading. \(\theta_{ijn_{t}}\) is a function of \(\mathbf{u}_{jk}\) and \(\mathbf{v}_{in_{t}}\), described as  
\[\theta_{ijn_{t}}=[(x^{\mathrm{v}}_{in_{t}}-x^{\mathrm{u}}_{jk})^{2}+(y^{ \mathrm{v}}_{in_{t}}-y^{\mathrm{u}}_{jk})^{2}+H^{2}]^{-0.5}. \tag{4}\]  
\(\tilde{h}_{ijn_{t}}\) is considered to follow a Rician distribution. We adopt a Rician channel in our system model because a line of sight (LoS) signal usually exists in a UAV-assisted network. A Rician channel can be decomposed into a LoS propagation channel and a scattering channel, as shown in Equation (5).  
\[\tilde{h}_{ijn_{t}}=\sqrt{\frac{K_{R}}{1+K_{R}}}\tilde{h}_{ijn_{t}}+\sqrt{\frac {1}{1+K_{R}}}\tilde{h}_{ijn_{t}}, \tag{5}\]  
where \(\tilde{h}_{ijn_{t}}\) and \(\tilde{h}_{ijn_{t}}\) represent the LoS component and NLoS component, respectively, and \(\tilde{h}_{ijn_{t}}\) usually follows a Rayleigh distribution. \(K_{R}\) is the parameter of the Rician distribution, which determines the proportion of LoS and NLoS in the channel. These parameters are considered to be constant during the interested time slot. Let  
\[\mathbf{\tilde{h}}_{ij}=(\tilde{h}_{ij1},\tilde{h}_{ij2},\ldots,\tilde{h}_{ijN _{T}})^{\mathsf{T}}, \tag{6}\]  
and  
\[\theta_{ij}=(\theta_{ij1},\theta_{ij2},\ldots,\theta_{ijN_{T}})^{\mathsf{T}}. \tag{7}\]  
Then, we can denote the channel between UAV\({}_{i}\) and UE\({}_{j}\) as \(\mathbf{h}_{ij}=\theta_{ij}\circ\mathbf{\tilde{h}}_{ij}\), where \(\circ\) is the operator of the Hadamard product. Here, \(\mathbf{\tilde{h}}_{ij}\) is a constant complex vector, and \(\theta_{ij}\) is a complex vector variable. \(\mathbf{h}_{ij}\) is a function of variable \(\theta_{ij}\), so it is also denoted as \(\mathbf{h}_{ij}(\theta_{ij})\).  
It is assumed that MBS uses different frequency resources from UAVs, so there would be no interference between MBS and UAVs. Additive white gaussian noise (AWGN) and the interferences among UAVs are both considered. The signal to interference plus noise ratio (SINR) of the downlink transmission from UAV\({}_{i}\) to UE\({}_{j}\) could be denoted as  
\[SINR_{ij}=\frac{|\mathbf{h}_{ij}^{H}\mathbf{w}_{ij}|^{2}}{\sum\limits_{m\
eq i }\sum\limits_{n\in A_{m},n\
eq j}|\mathbf{h}_{mj}^{H}\mathbf{w}_{mn}|^{2}+ \sigma^{2}},\]  
where \(\sigma^{2}\) represents the power of AWGN. The transfer data rate from UAV\({}_{i}\) to UE\({}_{j}\) could be calculated using  
\[R_{ij}=\log_{2}(1+SINR_{ij})=\log_{2}\left(1+\frac{|\mathbf{h}_{ij}^{H} \mathbf{w}_{ij}|^{2}}{\sum\limits_{m\
eq i}\sum\limits_{n\in A_{m},n\
eq j}| \mathbf{h}_{mj}^{H}\mathbf{w}_{mn}|^{2}+\sigma^{2}}\right),i\in\mathcal{M},j \in\mathcal{N}. \tag{8}\]  
Specially, we denote the downlink data rate between MBS and UE\({}_{j}\) as \(R_{0j}\). To simplify the discussion, \(R_{0j}\) is set as constant \(R_{0}\) for \(\forall j\in\mathcal{N}\), where the link details of MBS are not considered. Generally, we set \(R_{0}\) as a lower value compared with \(R_{ij}\) because UAVs could usually achieve higher data rate by getting closer to users.