# Gyre Precoding and T-Transformation-Based GFDM System for UAV-Aided mMTC Network
## 2 System Model
### Signal Model
#### 2.2.1 Ground Segment  
To improve both transmission and the performance in terms of security, the 3D controlled Lorenz mapping system addressed in [10] can be represented as:  
\[\left. \begin{array}{ll}\dot{x}&=\rho(y-x)+\zeta xmod(0.001,\mu)\\ \dot{y}&=rx-y-xz+\zeta ymod(0.001,\mu)\\ \dot{z}&=xy-\beta z+\zeta zmod(0.001,\mu)\end{array} \right\} \tag{1}\]  
where the initial values of and \(x,y,z\) variables and key parameter values \((x,y,z,\rho,\zeta,r,\mu,\beta)\) have been assigned to \((1.99,2.2,2.0,9,2,35,25,8/3)\). The solution component \(\{z(t)\}_{t=1}^{768}\) can be used to produce \(768\times 1\) matrix-sized primary key \(K_{0}\) as:  
\[K_{0}=\lceil\left\{z(t)\right\}_{t=1}^{768}\rceil \tag{2}\]  
where \(\lceil\cdot\rceil\) is indicative of rounding operation to the nearest integer. The primary key \(K_{0}\) is disseminated into eight encrypted keys with each key containing 96 elements. The eight keys for the eight mMTC users can be considered as:  
\[\left. \begin{array}{ll}K_{1}&=\lceil 0.5K_{0}(1:96,1)\rceil\\ K_{2}&=\lceil 1.0K_{0}(97:192,1)\rceil\\ K_{3}&=\lceil 1.5K_{0}(193:288,1)\rceil\\ K_{4}&=\lceil 2.0K_{0}(289:384,1)\rceil\\ K_{5}&=\lceil 2.5K_{0}(385:480,1)\rceil\\ K_{6}&=\lceil 3.0K_{0}(481:576,1)\rceil\\ K_{7}&=\lceil 3.5K_{0}(577:672,1)\rceil\\ K_{8}&=\lceil 4.0K_{0}(673:768,1)\rceil\end{array} \right\} \tag{3}\]where the first to 96 th elements of \(K_{0}\) is designated by \(K_{0}(1:96,1)\), 97 th to 192 th elements of \(K_{0}\) is designated by \(K_{0}(97:192,1)\), and so on. The elemental values of all the keys \(K_{1}\) through \(K_{8}\) fall within the range, and each elemental value provides eight binary bits. In binary form, each of the generated keys \(K_{1}\) through \(K_{8}\) for eight mMTC users is of \(1\times 768\) matrix in size. By using the repmat function in MATLAB, the length of the binary data of each of the generated keys \(K_{1}\) through \(K_{8}\) is made identical to the data length of the concatenated channel encoded data. The encrypted concatenated channel encoded binary data vector \(b_{\tilde{k}}\) of data length \(\tilde{N}\) for mMTC user \(\tilde{k}\) can be represented as:  
\[\tilde{b}_{\tilde{k}}=b_{\tilde{k}}\oplus\tilde{K}_{\tilde{k}} \tag{4}\]  
where the symbol \(\oplus\) is indicative of the XOR operation, and \(\tilde{K}_{\tilde{k}}\) is the ultimately generated key for the mMTC user \(\tilde{k}\). With the data length \(\tilde{N}\), the conversion of binary data vector \(\tilde{b}_{\tilde{k}}\) as a complex symbol vector yields \(c_{\tilde{k}}\) through digital modulation. The T-transformed \(8\times 8\) sized matrix based on combining the WHT and DFT can be represented as:  
\[T=\frac{1}{\sqrt{8}} \begin{bmatrix}1&1&1&1&1&1&1\\ 1&\omega^{-1}&\omega^{-2}&\omega^{-3}&-1&\omega^{-1}&\omega^{-2}&\omega^{-3}\\ 1&\omega^{-2}&-1&-\omega^{-2}&1&\omega^{-2}&-1&-\omega^{-2}\\ 1&\omega^{-3}&-\omega^{-2}&\omega^{-1}&-1&-\omega^{-3}&\omega^{-2}&-\omega^{- 1}\\ 1&-1&1&-1&1&-1&1&-1\\ 1&-\omega^{-1}&\omega^{-2}&-\omega^{-3}&-1&\omega^{-1}&-\omega^{-2}&\omega^{- 3}\\ 1&-\omega^{-2}&-1&\omega^{-2}&1&-\omega^{-2}&-1&\omega^{-2}\\ 1&-\omega^{-3}&-\omega^{-2}&-\omega^{-1}&-1&-\omega^{-3}&\omega^{-2}&\omega^{- 1}\end{bmatrix}  \tag{5}\]  
where \(\omega=e^{2\pi/8}\). The signal model presented in Equation (5) is applicable row wise for different mMTC users \(\tilde{k}\). The first row is applicable to mMTC user 1, the second row is applicable to mMTC user 2, and so on. The application of the T-transformation spreading technique on the modulated complex symbol vector \(c_{\tilde{k}}\) for mMTC user \(\tilde{k}\) yields the complex symbol vector \(\tilde{c}_{\tilde{k}}\) of data length \(8\tilde{N}\). In GFDM structured block generation, the data vector \(\tilde{c}_{\tilde{k}}\) is rearranged into a \(K\times M\) matrix-sized GFDM block, where \(M\) represents the number of subsymbols and \(K\) denotes the number of subcarriers.