# Gyre Precoding and T-Transformation-Based GFDM System for UAV-Aided mMTC Network
## 2 System Model
### Signal Model
#### 2.2.1 Ground Segment
For every GFDM block with N(=KM) samples, its first and last subsymbol contain null subcarriers to avoid collision in between two consecutive GFDM blocks, eliminate the effect of the fading channel, and additionally provide significant OOB emissions reduction for neighboring channel interference reduction. Considering data symbol \(d_{k,m_{l}}\) in each GFDM block for mMTC user \(\tilde{k}\), the \(l\)th GFDM block for mMTC user \(\tilde{k}\) can be seen as a result of circular convolution with BTRC, \(g[n]\), followed by frequency up conversion and can be written as [26]:  
\[x_{\tilde{k},l}[n]=\sum_{k=0}^{K-1}\left[g(n)\otimes\sum_{m=0}^{M-1}d_{k,m_{l }}\delta[n-mK]\right]e^{j2\pi\frac{1}{2}n} \tag{6}\]  
where \(\otimes\) denotes circular convolution, \(k=1,2,\ldots,8\) is the mMTC user identification number, \(l=1,2,\ldots,L\) is the GFDM block identification number, \(\delta[n-mK]\) is the Dirac function with \(n=0,1,2,\ldots,N-1\), and \(m=0,1,2,\ldots,M-1\). The BTRC pulse shaping filter addressed in [21] and used in signal model of Equation (5) can be written in the time domain as:  
\[g(t)=\frac{1}{T}\sin c\frac{t}{T}\times\frac{4\beta t\sin(\frac{\pi nt}{T})+2 \beta^{2}\cos(\frac{\pi nt}{T})-\beta^{2}}{\beta^{2}+(2\pi t)^{2}} \tag{7}\]  
where \(\beta=\frac{ln2}{\alpha B}\), \(T\big{(}=M\Delta T=\frac{N}{F_{s}}\big{)}\) is the GFDM symbol period, \(\alpha\) is the filter roll-off factor of the BTRC pulse shaping filter, \(\Delta T\big{(}=\frac{K}{F_{s}}\big{)}\) is the period of each subsymbol, \(F_{s}\) is the sampling frequency, \(\Delta f\big{(}=\frac{1}{\Delta T}\big{)}\) represents the subcarrier spacing, and \(B\big{(}=K\Delta f=\frac{K}{M}\big{)}\) represents the GFDM signal bandwidth. The \(l\)th discrete time domain GFDM signal (Equation (6)) for mMTC user \(\tilde{k}\) is rescaled such that \(E|x_{\tilde{k},l}|^{2}=1\), and the power rescaled signal vector of unity power is represented by \(\tilde{x}_{\tilde{k},l}[n]\). Concatenation of all the signal vectors of \(\tilde{x}_{\tilde{k},l}[n]\) yields the desired signals for the mMTC user \(\tilde{k}\) and are denoted by a data matrix \(D_{\tilde{k}}\) of size \(N\times L\). All the elements of \(D_{\tilde{k}}\) are stacked within a single column vector and further arranged into a matrix \(\bar{D}_{\bar{k}}\) of \(N_{R}\times N_{L}(=N\times L)/N_{R}\) in size, and the \(\bar{D}_{\bar{k}}\) can be represented as:  
\[\bar{D}_{\bar{k}}=\overset{\cdot}{x_{\bar{k}}} \tag{8}\]  
From Equation (8), all the data for eight mMTC users can be confined into a single data matrix D of size \(N_{T}\times N_{L}\). Prior to the application of proper transmit beamforming weights to the matrixed data, D, it is desirable to design transmit precoders utilizing the 3D geometrical MIMO flat-fading channels, which are based on the probabilistic path loss model, and in such estimated fading channels, the 3D mobility model has been incorporated to consider 3D positional uncertainty of the main UAV and cooperative UAV's positions relative to their fixed positions due to Brownian motion. The 3D mobility model can be used to present positioning errors using homogeneous stochastic differential equations in the 3D spatial (\(x,y,z\)) coordinate system as:  
\[ \begin{array}{l}dx_{t,\beta}=-\alpha_{1}\bar{x}_{t,\beta}dt+\sigma_{1}dw_{1t }\\ dy_{t,\beta}=-\alpha_{2}\bar{y}_{t,\beta}dt+\sigma_{2}dw_{2t}\\ dz_{t,\beta}=-\alpha_{3}\bar{z}_{t,\beta}dt+\sigma_{3}dw_{3t}\end{array}  \tag{9}\]  
where \(\alpha_{1},\alpha_{2},\alpha_{3},\sigma_{1},\sigma_{2},\sigma_{3}\) are the six constants (\(\alpha_{1}=\alpha_{2}=\alpha_{3}=1\) and \(\sigma_{1}=1.3,\sigma_{2}=1.0,\sigma_{3}=0.7\)); \(dw_{1t},dw_{2t},dw_{3t}\) are the differential form of the Brownian motion in three mutually perpendicular \(x\), \(y\), and \(z\) directions and are equally applicable for all UAVs. In Brownian motion, it can be assumed that \(\{dw_{1t1}\}^{2}=[dw_{2t}]^{2}=[dw_{3t}]^{2}=dt\).