# Gyre Precoding and T-Transformation-Based GFDM System for UAV-Aided mMTC Network
## 2 System Model
### Signal Model
#### 2.2.2 UAV Segment
Using the regularized ZF equalization technique [34] to invert the effect of the channel \(\widehat{\bar{H}}_{\bar{k}}\), the detected signal in receiver \(\hat{S}_{\bar{d}}^{\bar{k}}\) of \(N_{T}\times N_{L}\) matrix in size can be represented as:\[\bar{S}_{d}^{\bar{k}} =(\widehat{\bar{H}}_{\bar{k}}^{H}\widehat{\bar{H}}_{\bar{k}}+\mathfrak{ H}I_{N_{T}})^{-1}\widehat{\bar{H}}_{\bar{k}}^{H}Y_{d}^{\bar{k}}\] \[=(\widehat{\bar{H}}_{\bar{k}}^{H}\widehat{\bar{H}}_{\bar{k}}+ \mathfrak{H}I_{N_{T}})^{-1}\widehat{\bar{H}}_{\bar{k}}^{H}\widehat{\bar{H}}_{ \bar{k}}\Bigg{\{}W_{\bar{k}}\bar{x}_{\bar{k}}+\sum_{j=1,j\
eq\bar{k}}^{j=8}W_{ \bar{j}}\bar{x}_{j}\Bigg{\}}+(\widehat{\bar{H}}_{\bar{k}}^{H}\widehat{\bar{H}} _{\bar{k}}+\mathfrak{H}I_{N_{T}})^{-1}\widehat{\bar{H}}_{\bar{k}}^{H}n_{\bar{k}}\] \[=\Bigg{\{}W_{\bar{k}}\bar{x}_{\bar{k}}+\sum_{j=1,j\
eq k}^{j=8}W_ {\bar{j}}\bar{x}_{j}\Bigg{\}}+\hbar_{\bar{k}} \tag{31}\]  
where \(\mathfrak{H}=1.0\times 10^{-19}\) is the regularization parameter, and \(\hbar_{\bar{k}}\) is the AWGN noise. From Equation (31), it is quite obvious that the received signal in ground mMTC user \(\bar{k}\) contains its own and interference signals from other mMTC users. Multiplying Equation (31) by \(H_{\bar{k}}\), a modified form of the received signal \(\bar{S}_{d}^{\bar{k}}\) of \(N_{R}\times N_{L}\) matrix in size is obtained, and it can be represented as:  
\[\bar{S}_{d}^{\bar{k}} =H_{\bar{k}}\bar{S}_{d}^{\bar{k}}\] \[=H_{\bar{k}}W_{\bar{k}}\bar{x}_{\bar{k}}+H_{\bar{k}}\sum_{j=1,j \
eq\bar{k}}^{j=8}W_{\bar{j}}\bar{x}_{j}+\widehat{\bar{n}}_{\bar{k}} \tag{32}\]  
In Equation (32), all the components are of \(N_{R}\times N_{L}\) matrix in size, and the noise component \(\widehat{\bar{n}}_{\bar{k}}\sim\mathbb{C}\mathbb{N}(0_{N_{R}},\widehat{\bar{ \sigma}}_{\bar{k}}^{2}I_{N_{R}})\) represents the AWGN noise. Again, \(||H_{\bar{k}}W_{\bar{k}}||^{2}\) represents the instantaneous total signal power in two consecutive time phases for the mMTC user \(\bar{k}\), the total instantaneous interference power of the signal received by mMTC user \(\bar{k}\) for the \(\bar{j}\)-th mMTC user is represented by \(||H_{\bar{k}}W_{\bar{k}}||^{2}\), and \(N_{R}\bar{\bar{\sigma}}_{\bar{k}}^{2}\) denotes the AWGN noise power. The received signal-to-interference-plus-noise ratio (SINR) for the mMTC user \(\bar{k}(SINR_{\bar{k}})\) after gyre decoding can be represented as:  
\[SINR_{\bar{k}}=\frac{||\bar{P}_{\bar{k}}W_{\bar{k}}||^{2}}{\sum_{j=1,j\
eq \bar{k}}^{j=8}||\bar{P}_{\bar{k}}W_{\bar{k}}||^{2}+N_{R}\bar{\bar{\sigma}}_{ \bar{k}}^{2}} \tag{33}\]  
The logarithmic function of SINR represents the achievable ergodic rate \(R_{\bar{k}}\) for mMTC user \(\bar{k}\), and it can be written as after gyre decoding [35]:  
\[R_{\bar{k}} =\mathbb{E}\{\log_{2}(1+SINR_{\bar{k}})\}\] \[=\mathbb{E}\Bigg{\{}\log_{2}\Bigg{(}1+\frac{||\bar{P}_{\bar{k}}W_ {\bar{k}}||^{2}}{\sum_{j=1,j\
eq\bar{k}}^{j=8}||\bar{P}_{\bar{k}}W_{\bar{k}}||^ {2}+N_{R}\bar{\bar{\sigma}}_{\bar{k}}^{2}}\Bigg{)}\Bigg{\}} \tag{34}\]  
However, from the signal model presented in Equation (32), it is quite obvious that the reduction in the interfering signal power is very much significant as the GP removes MUI by designing GP beamforming weight \(W_{\bar{k}}\) assigned for mMTC user \(\bar{k}\) to fall in the null space of the total MIMO flat-fading channel matrix \(H\) and produces \(\bar{H}_{\bar{k}}W_{\bar{j}}=0\).