# Gyre Precoding and T-Transformation-Based GFDM System for UAV-Aided mMTC Network
## 2 System Model
### Signal Model
#### 2.2.2 UAV Segment
The signal received at the ground mMTC user \(\bar{k}\) from cooperative UAVs can be represented as:\[\gamma_{k1}^{2} =H_{1\bar{k}}G1Y_{01}^{1}+n_{k1}^{2}\] \[=H_{1\bar{k}}G1\left[H_{01}\sqrt{P_{1}P_{2}}\left\{W_{\bar{k}}\dddot _{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\dddot_{\bar{j}}\right\} +n_{01}^{1}\right]+n_{\bar{k}1}^{2}\] \[X_{0}=(H_{0}^{H}H_{0})^{-1}H_{0}^{H}Y_{0}\cong\sqrt{P_{1}}\left\{W_{\bar{k}} \vec{x}_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\vec{x}_{j}\right\} \tag{22}\]0 \[X_{0}=(H_{0}^{H}H_{0})^{-1}H_{0}^{H}Y_{0}\cong\sqrt{P_{1}}\left\{W_{\bar{k}} \vec{x}_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\vec{x}_{j}\right\} \tag{22}\]1 \[X_{0}=(H_{0}^{H}H_{0})^{-1}H_{0}^{H}Y_{0}\cong\sqrt{P_{1}}\left\{W_{\bar{k}} \vec{x}_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\vec{x}_{j}\right\} \tag{22}\]2 \[X_{0}=(H_{0}^{H}H_{0})^{-1}H_{0}^{H}Y_{0}\cong\sqrt{P_{1}}\left\{W_{\bar{k}} \vec{x}_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\vec{x}_{j}\right\} \tag{22}\]3  
where \(n_{\bar{k}1g}^{2}\sim\mathbb{C}\mathbb{N}(0_{N_{R}},\sigma_{k1g}^{2}I_{N_{R}})\) and \(n_{\bar{k}2g}^{2}\sim\mathbb{C}\mathbb{N}(0_{N_{R}},\sigma_{2g}^{2}I_{N_{R}})\) are the effective AWGN in transmission from the first and second cooperative UAV, respectively. In the second phase, the total signal received by the ground mMTC user \(\bar{k}\) can be represented as:  
\[X_{0}=(H_{0}^{H}H_{0})^{-1}H_{0}^{H}Y_{0}\cong\sqrt{P_{1}}\left\{W_{\bar{k}} \vec{x}_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\vec{x}_{j}\right\} \tag{22}\]4 \[X_{0}=(H_{0}^{H}H_{0})^{-1}H_{0}^{H}Y_{0}\cong\sqrt{P_{1}}\left\{W_{\bar{k}} \vec{x}_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\vec{x}_{j}\right\} \tag{22}\]5  
where the superscript 2 on different variables indicates the second time phase. By combining the signals received at the ground mMTC user \(\bar{k}\) in both phases [33], the following can be written:  
\[\gamma_{\bar{d}}^{\bar{k}}\triangleq \begin{bmatrix}Y_{\bar{k}}^{ 1}\\ Y_{\bar{k}}^{2}\end{bmatrix}  = \begin{bmatrix}H_{\bar{k}}\\ (H_{1\bar{k}}G1H_{01}+H_{2\bar{k}}G2H_{02})\end{bmatrix} \sqrt{P_{1}P_{2}} \left\{W_{\bar{k}}\dddot_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{ j}}\dddot_{\bar{j}}\right\} \begin{bmatrix}n_{\bar{k}}^{1}\\ n_{\bar{k}}^{1}\end{bmatrix} \] \[=\bar{H}_{\bar{k}}\sqrt{P_{1}P_{2}}\left\{W_{\bar{k}}\dddot_{\bar {k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\dddot_{\bar{j}}\right\}+n_ {\bar{k}} \tag{29}\]  
where \(\bar{H}_{\bar{k}}= \begin{bmatrix}H_{\bar{k}}\\ (H_{1\bar{k}}G1H_{01}+H_{2\bar{k}}G2H_{02})\end{bmatrix} \) is the effective channel between the UAVs and ground mMTC user \(\bar{k}\) and is of \(2N_{R}\times N_{T}\) matrix in size, while the combined signal \(\gamma_{\bar{d}}^{\bar{k}}\) and AWGN noise \(n_{\bar{k}}\sim\mathbb{C}\mathbb{N}(0_{2N_{R}},\sigma_{\bar{k}}^{2}I_{N_{R}})\) are of \(2N_{R}\times N_{L}\) matrix in size. On further simplification, Equation (29) can be re-written as:  
\[\gamma_{\bar{d}}^{\bar{k}}=\widehat{\bar{H}}_{\bar{k}}\left\{W_{\bar{k}}\dddot _{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{\bar{j}=8}W_{\bar{j}}\dddot_{\bar{j}}\right\} +n_{\bar{k}} \tag{30}\]  
where \(\widehat{\bar{H}}_{\bar{k}}=\widehat{H}_{\bar{k}}\sqrt{P_{1}P_{2}}\) is the effective channel for the signal model presented in Equation (30) and it is also of \(2N_{R}\times N_{T}\) matrix in size. Generally, such non-symmetric complex channel matrix is found to be sparse in nature and due to sparsity of matrix, its pseudo inverse operation for achieving exact solution from signal model presented in Equation (30) cannot be obtained.