# Gyre Precoding and T-Transformation-Based GFDM System for UAV-Aided mMTC Network
## 2 System Model
### Signal Model
#### 2.2.1 Ground Segment
Considering very high LOS probability and a low excessive path loss for transmission between the main UAV and the cooperative UAVs, the path losses \(L_{01}\) and \(L_{02}\) for the MIMO flat-fading channels \(H_{01}\) and \(H_{01}\) existed between the main UAV and first UAV, and the main UAV and second UAV can be represented as:  
\[L_{01} =\eta_{1}\left(\frac{4\pi f_{c}d_{\overline{1}m}}{c}\right)^{\alpha} \tag{14a}\] \[L_{02} =\eta_{1}\left(\frac{4\pi f_{c}d_{\overline{2}m}}{c}\right)^{\alpha} \tag{14b}\]  
where \(d_{\overline{1},m}\) and \(d_{\overline{2},m}\) are the corresponding 3D distances between the main UAV and cooperative UAVs. Considering the applicability of the signal models represented in Equations (11)-(14b), the MIMO flat-fading channels from the main UAV to eight mMTC users (H1, H2, H3, H4, H5, H6, H7, H8), from the first cooperative UAV to eight mMTC users (H11, H12, H13, H14, H15, H16, H17, H18), from the second cooperative UAV to eight mMTC users (H21, H22, H23, H24, H25, H26, H27, H28), H0, \(H_{01}\), and \(H_{02}\) are estimated. The total MIMO flat-fading channel matrix \(H\) of size \(N_{T}\times N_{T}\) for all the mMTC users can be formulated as:  
\[H=[H_{1}^{T}\ H_{2}^{T}\ H_{3}^{T}\ H_{4}^{T}\ H_{5}^{T}\ H_{6}^{T}\ H_{7}^{T}\ H_{8}^{T}]^{T} \tag{15}\]  
where \((\cdot)^{T}\) indicates the non-conjugate transpose operator. Considering the GP, a novel linear multi-user multiple-input multiple-output (MU-MIMO) precoding scheme cited at [21], the ZF precoding of matrix size \(N_{T}\times N_{T}\) can be represented as:  
\[G^{(ZF)}=H^{H}(HH^{H})^{-1} \tag{16}\]  
where \((\cdot)^{H}\) indicates the conjugate transpose operator. In data matrix D, its first and second rowed data are belonging to first mMTC user, third and fourth rowed data are belonging to second mMTC user, and so on. To estimate GP, \(W^{(GP)}\), two additional components \(\Theta\) and \(\gamma^{GP}\) are in need of computation. The rotation angle matrix \(\Theta\) can be represented as:  
\[\Theta= \begin{bmatrix}e^{i\theta_{11}}&0&0&\ldots&\ldots&\ldots&\ldots&0\\ 0&e^{i\theta_{11}}&\ldots&0&\ldots&&0\\ &e^{i\theta_{22}}&&&&\\ 0&&0&e^{i\theta_{2}}&&&0\\ &&&\vdots&&&\\ &&&0&&&e^{i\theta_{16}}\\ &&&0&&&e^{i\theta_{16}}\\ \end{bmatrix}  \tag{17}\]  
where \(\theta_{\overline{k}\overline{k}}e[0,2\pi]\) and  
\[\gamma^{GP}=\frac{1}{NL}||G^{(ZF)}\Theta D||_{F}^{2} \tag{18}\]  
where \(||\cdot||_{F}\) is the Frobenius norm. The GP matrix \(W^{(GP)}\) is of size \(N_{T}\times N_{T}\) and can be represented as:  
\[W^{(GP)}=\frac{G^{(ZF)}\Theta}{\sqrt{\gamma^{GP}}} \tag{19}\]  
The precoding weights W1 through of W8, each is of \(N_{T}N_{R}\) in size for the eight mMTC users and can be represented as: \(W_{1}=W^{(GP)}(;1:2);W_{2}=W^{(GP)}(;3:4);W_{3}=W^{(GP)}(;5:6);W_{4}=W^{(GP)}(;7 :8);W_{5}=W^{(GP)}(;9:10);W_{6}=W^{(GP)}(;11:12);W_{7}=W^{(GP)}(;13:14);\) and \(W_{8}=W^{(GP)}(;15:16)\), where \(W^{(GP)}(;1:2)\) is indicative of first and second columned data of GP matrix \(W^{(GP)}\) and so on. Precoded signal \(X_{0}\) transmitted from ground BS can be represented as:  
\[X_{0}=W^{(GP)}D=W_{\overline{k}}\vec{x}_{\overline{k}}+\sum_{j=1,j\
eq k}^{ j=8}W_{j}^{\overline{j}}\vec{x}_{\overline{j}} \tag{20}\]