# Gyre Precoding and T-Transformation-Based GFDM System for UAV-Aided mMTC Network
## 2 System Model
### Signal Model
#### 2.2.1 Ground Segment
The \((x_{t,\beta},y_{t,\beta},z_{t,\beta})\) are the 3D positional coordinates of the \(s\)-th UAV (\(s=0\) for main UAV, \(s=1\) for cooperative UAV #1 and \(s=2\) for cooperative UAV #2). The \((\bar{x}_{t,\beta},\bar{y}_{t,\beta}z_{t,\beta})\) are the 3D directional coordinates of the \(s\)-th UAV due to Brownian motion [30; 31]. With the consideration of Brownian motion, the \(x\), \(y\), and \(z\) coordinates of the main UAV and the cooperative first and second UAVs are \((\bar{x}_{t,0},\bar{y}_{t,0}z_{t,0})\), \((\bar{x}_{t,1},\bar{y}_{t,1}\bar{z}_{t,1})\), and \((\bar{x}_{t,2},\bar{y}_{t,2}\bar{z}_{t,2})\), respectively. The 3D distance \(d_{\overline{k_{\overline{j}}}}\) between main/cooperative UAV \(\bar{j}\) and the ground mMTC user \(\bar{k}\) with its 3D coordinates \((x_{\bar{k}},y_{\bar{k}},z_{\bar{k}})\) can be represented as: \(d_{\overline{k_{\overline{j}}}}=\sqrt{(\bar{x}_{t,\beta}-x_{\bar{k}})^{2}+( \bar{y}_{t,\beta}-y_{\bar{k}})^{2}+(\bar{z}_{t,\beta}-h_{\bar{k}})^{2}}\), and the elevation angle \(\theta_{\overline{k_{\overline{j}}}}\) can be represented as: \(\theta_{\overline{k_{\overline{j}}}}=\frac{180}{\pi}\times\sin^{-1}\frac{ \bar{z}_{t,\beta}-h_{\bar{k}}}{d_{\overline{k_{\overline{j}}}}}\). The LOS probability from UAVs to ground mMTC communications can be represented as [32]:  
\[P_{LOS}^{\overline{k_{\overline{j}}}}=\frac{1}{1+pexp[-\beta_{0}(\theta_{ \overline{k_{\overline{j}}}}-\psi)]} \tag{10}\]  
where \(s=0,1,2\), and \(\psi\) (=11.95) and \(\beta_{0}\) (=0.14) are the functions of the carrier frequency and environment. The path loss between the main/cooperative UAV \(\bar{j}\) and the ground mMTC user \(\bar{k}\) can be represented as:  
\[L_{\overline{k_{\overline{j}}}}=\left\{ \begin{array}{ll}\eta_{1}\bigg{(} \frac{4\pi f_{c}d_{\overline{k_{\overline{j}}}}}{c}\bigg{)}^{\alpha},&\text{ LOS Link}\\ \eta_{2}\bigg{(}\frac{4\pi f_{c}d_{\overline{k_{\overline{j}}}}}{c}\bigg{)}^{ \alpha},&\text{NLOS Link}\end{array} \right\} \tag{11}\]  
where \(\alpha\) is the path loss exponent, \(f_{c}\) (=28 GHz) is the carrier frequency, \(\eta_{1}\) (=10\({}^{3/10}\)) represents the excessive path loss coefficient for LOS link, \(\eta_{2}\) (=10\({}^{23/10}\)) denotes the excessive path loss coefficient for NLOS link, and \(c\) (=3\(\times\) 10\({}^{8}\)) m/s represents the speed of light. The simple form of NLOS probability is \(P_{NLOS}^{\overline{k_{\overline{j}}}}=1-P_{LOS}^{\overline{k_{\overline{j}}}}\). Between the main/cooperative UAVs \(\bar{j}\) and the ground mMTC user \(\bar{k}\), the average path loss equation can be represented as:  
\[L_{\overline{k_{\overline{j}}}}=P_{LOS}^{\overline{k_{\overline{j}}}}\eta_{1} \bigg{(}\frac{4\pi f_{c}d_{\overline{k_{\overline{j}}}}}{c}\bigg{)}^{\alpha}+P _{NLOS}^{\bar{k_{\overline{j}}}}\eta_{2}\bigg{(}\frac{4\pi f_{c}d_{\overline{k_ {\overline{j}}}}}{c}\bigg{)}^{\alpha} \tag{12}\]  
With its 3D coordinates \([x_{0},y_{\sigma},z_{0}]\) to the main UAV, the path loss \(L_{0}\) of the MIMO flat-fading channel \(H_{0}\) for the case of uplink transmission from ground BS can be represented as:  
\[L_{0}=P_{LOS}^{\overline{k_{\overline{m}}}}\eta_{1}\bigg{(}\frac{4\pi f_{c}d_{ \overline{k_{\overline{j}}}}}{c}\bigg{)}^{\alpha} \tag{13}\]  
where \(d_{\overline{k_{\overline{m}}}}=\sqrt{(\bar{x}_{t,0}-x_{0})^{2}+(\bar{y}_{t,0}- y_{0})^{2}+(\bar{z}_{t,0}-z_{0})^{2}}\) and \(P_{LOS}^{\overline{k_{\overline{m}}}}=\frac{1}{1+\psi exp}\bigg{[}-\bar{\psi}_{t }(\frac{180}{\pi}\times\sin^{-1}\frac{\bar{z}_{t,0}-z_{0}}{2\psi}-\psi)\bigg{]}\).